topology.separation ⟷ Mathlib.Topology.Separation

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -727,7 +727,7 @@ theorem continuousOn_update_iff [T1Space α] [DecidableEq α] [TopologicalSpace
     ContinuousOn (Function.update f x y) s ↔
       ContinuousOn f (s \ {x}) ∧ (x ∈ s → Tendsto f (𝓝[s \ {x}] x) (𝓝 y)) :=
   by
-  rw [ContinuousOn, ← and_forall_ne x, and_comm']
+  rw [ContinuousOn, ← and_forall_ne x, and_comm]
   refine' and_congr ⟨fun H z hz => _, fun H z hzx hzs => _⟩ (forall_congr' fun hxs => _)
   · specialize H z hz.2 hz.1
     rw [continuousWithinAt_update_of_ne hz.2] at H
@@ -1127,7 +1127,7 @@ theorem t2Space_iff_disjoint_nhds : T2Space α ↔ ∀ x y : α, x ≠ y → Dis
   by
   refine' (t2Space_iff α).trans (forall₃_congr fun x y hne => _)
   simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y), exists_prop, ← exists_and_left,
-    and_assoc, and_comm', and_left_comm]
+    and_assoc, and_comm, and_left_comm]
 #align t2_space_iff_disjoint_nhds t2Space_iff_disjoint_nhds
 -/
 
@@ -1209,7 +1209,7 @@ theorem t2_separation_nhds [T2Space α] {x y : α} (h : x ≠ y) :
 #print t2_separation_compact_nhds /-
 theorem t2_separation_compact_nhds [LocallyCompactSpace α] [T2Space α] {x y : α} (h : x ≠ y) :
     ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v := by
-  simpa only [exists_prop, ← exists_and_left, and_comm', and_assoc, and_left_comm] using
+  simpa only [exists_prop, ← exists_and_left, and_comm, and_assoc, and_left_comm] using
     ((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h)
 #align t2_separation_compact_nhds t2_separation_compact_nhds
 -/
@@ -1893,7 +1893,7 @@ theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
   by
   tfae_have 1 ↔ 5
   · rw [regularSpace_iff, (@compl_surjective (Set X) _).forall, forall_swap]
-    simp only [isClosed_compl_iff, mem_compl_iff, Classical.not_not, @and_comm' (_ ∈ _),
+    simp only [isClosed_compl_iff, mem_compl_iff, Classical.not_not, @and_comm (_ ∈ _),
       (nhds_basis_opens _).lift'_closure.le_basis_iffₓ (nhds_basis_opens _), and_imp,
       (nhds_basis_opens _).disjoint_iff_rightₓ, exists_prop, ← subset_interior_iff_mem_nhdsSet,
       interior_compl, compl_subset_compl]
@@ -1994,7 +1994,7 @@ theorem hasBasis_opens_closure (a : α) : (𝓝 a).HasBasis (fun s => a ∈ s 
 theorem TopologicalSpace.IsTopologicalBasis.nhds_basis_closure {B : Set (Set α)}
     (hB : TopologicalSpace.IsTopologicalBasis B) (a : α) :
     (𝓝 a).HasBasis (fun s : Set α => a ∈ s ∧ s ∈ B) closure := by
-  simpa only [and_comm'] using hB.nhds_has_basis.nhds_closure
+  simpa only [and_comm] using hB.nhds_has_basis.nhds_closure
 #align topological_space.is_topological_basis.nhds_basis_closure TopologicalSpace.IsTopologicalBasis.nhds_basis_closure
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
 import Topology.SubsetProperties
-import Topology.Connected
+import Topology.Connected.Basic
 import Topology.NhdsSet
 import Topology.Inseparable
 
@@ -306,7 +306,7 @@ instance : T0Space (SeparationQuotient α) :=
     Quotient.inductionOn₂' x' y' fun x y h =>
       SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.inducing_mk.inseparable_iff.1 h⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_subsingleton /-
 theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : s.Subsingleton :=
@@ -321,7 +321,7 @@ theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsC
 #align minimal_nonempty_closed_subsingleton minimal_nonempty_closed_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_eq_singleton /-
 theorem minimal_nonempty_closed_eq_singleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} :=
@@ -343,7 +343,7 @@ theorem IsClosed.exists_closed_singleton {α : Type _} [TopologicalSpace α] [T0
 #align is_closed.exists_closed_singleton IsClosed.exists_closed_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_subsingleton /-
 theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : s.Subsingleton :=
@@ -358,7 +358,7 @@ theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_subsingleton minimal_nonempty_open_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_eq_singleton /-
 theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} :=
@@ -366,7 +366,7 @@ theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_eq_singleton minimal_nonempty_open_eq_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊂ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊂ » s) -/
 #print exists_isOpen_singleton_of_isOpen_finite /-
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
 theorem exists_isOpen_singleton_of_isOpen_finite [T0Space α] {s : Set α} (hfin : s.Finite)
@@ -1882,7 +1882,7 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
 #align regular_space RegularSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
 #print regularSpace_TFAE /-
 theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     TFAE
@@ -2145,8 +2145,8 @@ instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [
     T3Space (∀ i, π i) :=
   ⟨⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
 #print disjoint_nested_nhds /-
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
 with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1601,8 +1601,8 @@ theorem Set.EqOn.of_subset_closure [T2Space α] {s t : Set β} {f g : β → α}
 #align set.eq_on.of_subset_closure Set.EqOn.of_subset_closure
 -/
 
-#print Function.LeftInverse.closed_range /-
-theorem Function.LeftInverse.closed_range [T2Space α] {f : α → β} {g : β → α}
+#print Function.LeftInverse.isClosed_range /-
+theorem Function.LeftInverse.isClosed_range [T2Space α] {f : α → β} {g : β → α}
     (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsClosed (range g) :=
   have : EqOn (g ∘ f) id (closure <| range g) :=
     h.rightInvOn_range.EqOn.closure (hg.comp hf) continuous_id
@@ -1610,13 +1610,13 @@ theorem Function.LeftInverse.closed_range [T2Space α] {f : α → β} {g : β 
     calc
       x = g (f x) := (this hx).symm
       _ ∈ _ := mem_range_self _
-#align function.left_inverse.closed_range Function.LeftInverse.closed_range
+#align function.left_inverse.closed_range Function.LeftInverse.isClosed_range
 -/
 
 #print Function.LeftInverse.closedEmbedding /-
 theorem Function.LeftInverse.closedEmbedding [T2Space α] {f : α → β} {g : β → α}
     (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : ClosedEmbedding g :=
-  ⟨h.Embedding hf hg, h.closed_range hf hg⟩
+  ⟨h.Embedding hf hg, h.isClosed_range hf hg⟩
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -434,7 +434,7 @@ theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set α, x ∈
   refine' ⟨fun x y hxy => _⟩
   rcases h x y hxy with ⟨s, hxs, hys, hs⟩; skip
   lift x to s using hxs; lift y to s using hys
-  rw [← subtype_inseparable_iff] at hxy 
+  rw [← subtype_inseparable_iff] at hxy
   exact congr_arg coe hxy.eq
 #align t0_space.of_cover T0Space.of_cover
 -/
@@ -506,7 +506,7 @@ theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
   filter_upwards [eventually_nhds_nhds_within.mpr ha] with b hb
   by_cases a = b
   · subst h; exact hb
-  · rw [(Ne.symm h).nhdsWithin_compl_singleton] at hb 
+  · rw [(Ne.symm h).nhdsWithin_compl_singleton] at hb
     exact hb.filter_mono nhdsWithin_le_nhds
 #align is_open_set_of_eventually_nhds_within isOpen_setOf_eventually_nhdsWithin
 -/
@@ -558,7 +558,7 @@ theorem Bornology.relativelyCompact.isBounded_iff [T1Space α] {s : Set α} :
   rw [Filter.mem_coclosedCompact]
   constructor
   · rintro ⟨t, ht₁, ht₂, hst⟩
-    rw [compl_subset_compl] at hst 
+    rw [compl_subset_compl] at hst
     exact IsCompact.of_isClosed_subset ht₂ isClosed_closure (closure_minimal hst ht₁)
   · intro h
     exact ⟨closure s, isClosed_closure, h, compl_subset_compl.mpr subset_closure⟩
@@ -730,7 +730,7 @@ theorem continuousOn_update_iff [T1Space α] [DecidableEq α] [TopologicalSpace
   rw [ContinuousOn, ← and_forall_ne x, and_comm']
   refine' and_congr ⟨fun H z hz => _, fun H z hzx hzs => _⟩ (forall_congr' fun hxs => _)
   · specialize H z hz.2 hz.1
-    rw [continuousWithinAt_update_of_ne hz.2] at H 
+    rw [continuousWithinAt_update_of_ne hz.2] at H
     exact H.mono (diff_subset _ _)
   · rw [continuousWithinAt_update_of_ne hzx]
     refine' (H z ⟨hzs, hzx⟩).mono_of_mem (inter_mem_nhdsWithin _ _)
@@ -867,7 +867,7 @@ theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s
   refine' ⟨_, fun h => monotone_nhdsSet h⟩
   simp_rw [Filter.le_def]; intro h x hx
   specialize h ({x}ᶜ)
-  simp_rw [compl_singleton_mem_nhdsSet_iff] at h 
+  simp_rw [compl_singleton_mem_nhdsSet_iff] at h
   by_contra hxt
   exact h hxt hx
 #align nhds_set_le_iff nhdsSet_le_iff
@@ -982,7 +982,7 @@ theorem isOpen_singleton_of_finite_mem_nhds {α : Type _} [TopologicalSpace α]
   have B : IsClosed (s \ {x}) := (hsf.subset (diff_subset _ _)).IsClosed
   have C : (s \ {x})ᶜ ∈ 𝓝 x := B.is_open_compl.mem_nhds fun h => h.2 rfl
   have D : {x} ∈ 𝓝 x := by simpa only [← diff_eq, diff_diff_cancel_left A] using inter_mem hs C
-  rwa [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, subset_interior_iff_isOpen] at D 
+  rwa [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, subset_interior_iff_isOpen] at D
 #align is_open_singleton_of_finite_mem_nhds isOpen_singleton_of_finite_mem_nhds
 -/
 
@@ -1013,7 +1013,7 @@ theorem PreconnectedSpace.trivial_of_discrete [PreconnectedSpace α] [DiscreteTo
     Subsingleton α := by
   rw [← not_nontrivial_iff_subsingleton]
   rintro ⟨x, y, hxy⟩
-  rw [Ne.def, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy 
+  rw [Ne.def, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
   exact hxy (mem_univ x)
 #align preconnected_space.trivial_of_discrete PreconnectedSpace.trivial_of_discrete
 -/
@@ -1373,7 +1373,7 @@ theorem isOpen_iff_ultrafilter' [CompactSpace α] (U : Set α) :
   rw [isOpen_iff_ultrafilter]
   refine' ⟨fun h F hF => h F.lim hF F F.le_nhds_lim, _⟩
   intro cond x hx f h
-  rw [← Ultrafilter.lim_eq_iff_le_nhds.2 h] at hx 
+  rw [← Ultrafilter.lim_eq_iff_le_nhds.2 h] at hx
   exact cond _ hx
 #align is_open_iff_ultrafilter' isOpen_iff_ultrafilter'
 -/
@@ -1582,8 +1582,8 @@ theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ
 theorem eqOn_closure₂ [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ → α}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (uncurry f))
     (hg : Continuous (uncurry g)) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
-  eqOn_closure₂' h (fun x => continuous_uncurry_left x hf) (fun x => continuous_uncurry_right x hf)
-    (fun y => continuous_uncurry_left y hg) fun y => continuous_uncurry_right y hg
+  eqOn_closure₂' h (fun x => Continuous.uncurry_left x hf) (fun x => Continuous.uncurry_right x hf)
+    (fun y => Continuous.uncurry_left y hg) fun y => Continuous.uncurry_right y hg
 #align eq_on_closure₂ eqOn_closure₂
 -/
 
@@ -1753,8 +1753,8 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
   induction' t using Finset.induction with x t hx ih generalizing U hU s hs hsC
   · refine' ⟨fun _ => ∅, fun i => isCompact_empty, fun i => empty_subset _, _⟩
     simpa only [subset_empty_iff, Union_false, Union_empty] using hsC
-  simp only [Finset.set_biUnion_insert] at hsC 
-  simp only [Finset.mem_insert] at hU 
+  simp only [Finset.set_biUnion_insert] at hsC
+  simp only [Finset.mem_insert] at hU
   have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU i (Or.inr hi)
   rcases hs.binary_compact_cover (hU x (Or.inl rfl)) (isOpen_biUnion hU') hsC with
     ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
@@ -1762,10 +1762,10 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
   refine' ⟨update K x K₁, _, _, _⟩
   · intro i; by_cases hi : i = x
     · simp only [update_same, hi, h1K₁]
-    · rw [← Ne.def] at hi ; simp only [update_noteq hi, h1K]
+    · rw [← Ne.def] at hi; simp only [update_noteq hi, h1K]
   · intro i; by_cases hi : i = x
     · simp only [update_same, hi, h2K₁]
-    · rw [← Ne.def] at hi ; simp only [update_noteq hi, h2K]
+    · rw [← Ne.def] at hi; simp only [update_noteq hi, h2K]
   · simp only [set_bUnion_insert_update _ hx, hK, h3K]
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 -/
@@ -1900,7 +1900,7 @@ theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
   tfae_have 5 → 6; exact fun h a => (h a).antisymm (𝓝 _).le_lift'_closure
   tfae_have 6 → 4
   · intro H a s hs
-    rw [← H] at hs 
+    rw [← H] at hs
     rcases(𝓝 a).basis_sets.lift'_closure.mem_iff.mp hs with ⟨U, hU, hUs⟩
     exact ⟨closure U, mem_of_superset hU subset_closure, isClosed_closure, hUs⟩
   tfae_have 4 → 2
@@ -2078,7 +2078,7 @@ theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @
 #print regularSpace_iInf /-
 theorem regularSpace_iInf {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @RegularSpace X (t i)) :
     @RegularSpace X (iInf t) :=
-  regularSpace_sInf <| forall_range_iff.mpr h
+  regularSpace_sInf <| forall_mem_range.mpr h
 #align regular_space_infi regularSpace_iInf
 -/
 
@@ -2120,7 +2120,7 @@ instance (priority := 100) T3Space.t25Space [T3Space α] : T25Space α :=
     (t0Space_iff_or_not_mem_closure α).mp inferInstance x y hne
   wlog H : x ∉ closure ({y} : Set α)
   · refine' (this y x hne.symm aux.symm (aux.resolve_left H)).symm
-  · rwa [← disjoint_nhds_nhdsSet, nhdsSet_singleton] at H 
+  · rwa [← disjoint_nhds_nhdsSet, nhdsSet_singleton] at H
 #align t3_space.t2_5_space T3Space.t25Space
 -/
 
@@ -2277,7 +2277,7 @@ theorem NormalSpace.of_regularSpace_secondCountableTopology [SecondCountableTopo
                 ∀ n : ℕ, IsClosed (⋃ (u ∈ U) (h : Encodable.encode u ≤ n), closure (u : Set α)) :=
     by
     intro s t hc hd
-    rw [disjoint_left] at hd 
+    rw [disjoint_left] at hd
     have : ∀ x ∈ s, ∃ U ∈ countable_basis α, x ∈ U ∧ Disjoint (closure U) t :=
       by
       intro x hx
@@ -2413,7 +2413,7 @@ theorem connectedComponent_eq_iInter_isClopen [T2Space α] [CompactSpace α] (x
   -- in whichever of u or v x lies in and hence will be a subset of either a or b.
   rsuffices ⟨Z, H⟩ : ∃ Z : Set α, IsClopen Z ∧ x ∈ Z ∧ Z ⊆ u ∪ v
   · have H1 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hu hv huv
-    rw [union_comm] at H 
+    rw [union_comm] at H
     have H2 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hv hu huv.symm
     by_cases x ∈ u
     -- The x ∈ u case.
@@ -2452,7 +2452,7 @@ theorem connectedComponent_eq_iInter_isClopen [T2Space α] [CompactSpace α] (x
   have H1 :=
     (hu.union hv).isClosed_compl.IsCompact.inter_iInter_nonempty
       (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z) fun Z => Z.2.1.2
-  rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, Classical.not_forall] at H1 
+  rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, Classical.not_forall] at H1
   cases' H1 (disjoint_compl_left_iff_subset.2 <| hab.trans <| union_subset_union hau hbv) with Zi H2
   refine' ⟨⋂ U ∈ Zi, Subtype.val U, _, _, _⟩
   · exact isClopen_biInter_finset fun Z hZ => Z.2.1
@@ -2509,7 +2509,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
     constructor
     · have : connectedComponent x = {x} :=
         totally_disconnected_space_iff_connected_component_singleton.mp ‹_› x
-      rw [connectedComponent_eq_iInter_isClopen] at this 
+      rw [connectedComponent_eq_iInter_isClopen] at this
       intro hU
       let N := { Z // IsClopen Z ∧ x ∈ Z }
       rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ Z : N, Z.val ⊆ U
@@ -2523,7 +2523,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
       have h_nhd : ∀ y ∈ ⋂ Z : N, Z.val, U ∈ 𝓝 y :=
         by
         intro y y_in
-        erw [this, mem_singleton_iff] at y_in 
+        erw [this, mem_singleton_iff] at y_in
         rwa [y_in]
       exact exists_subset_nhds_of_compactSpace hdir hNcl h_nhd
     · rintro ⟨V, ⟨hxV, V_op, -⟩, hUV : V ⊆ U⟩
@@ -2623,10 +2623,10 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
   -- Proof follows that of: https://stacks.math.columbia.edu/tag/0900
   -- Fix 2 distinct connected components, with points a and b
   refine' ⟨connected_components.surjective_coe.forall₂.2 fun a b ne => _⟩
-  rw [ConnectedComponents.coe_ne_coe] at ne 
+  rw [ConnectedComponents.coe_ne_coe] at ne
   have h := connectedComponent_disjoint Ne
   -- write ↑b as the intersection of all clopen subsets containing it
-  rw [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h 
+  rw [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h
   -- Now we show that this can be reduced to some clopen containing `↑b` being disjoint to `↑a`
   obtain ⟨U, V, hU, ha, hb, rfl⟩ :
     ∃ (U : Set α) (V : Set (ConnectedComponents α)),
@@ -2640,7 +2640,7 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
     exact
       ⟨U, coe '' U, hU, ha, subset_Inter₂ fun Z _ => Z.2.1.connectedComponent_subset Z.2.2,
         (connectedComponents_preimage_image U).symm ▸ hU.bUnion_connected_component_eq⟩
-  rw [connected_components.quotient_map_coe.is_clopen_preimage] at hU 
+  rw [connected_components.quotient_map_coe.is_clopen_preimage] at hU
   refine' ⟨Vᶜ, V, hU.compl.is_open, hU.is_open, _, hb mem_connectedComponent, disjoint_compl_left⟩
   exact fun h => flip Set.Nonempty.ne_empty ha ⟨a, mem_connectedComponent, h⟩
 #align connected_components.t2 ConnectedComponents.t2

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1917,25 +1917,25 @@ theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
 #align regular_space_tfae regularSpace_TFAE
 -/
 
-#print RegularSpace.ofLift'_closure /-
-theorem RegularSpace.ofLift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 𝓝 a) : RegularSpace α :=
+#print RegularSpace.of_lift'_closure /-
+theorem RegularSpace.of_lift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 𝓝 a) : RegularSpace α :=
   Iff.mpr ((regularSpace_TFAE α).out 0 5) h
-#align regular_space.of_lift'_closure RegularSpace.ofLift'_closure
+#align regular_space.of_lift'_closure RegularSpace.of_lift'_closure
 -/
 
-#print RegularSpace.ofBasis /-
-theorem RegularSpace.ofBasis {ι : α → Sort _} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set α}
+#print RegularSpace.of_hasBasis /-
+theorem RegularSpace.of_hasBasis {ι : α → Sort _} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set α}
     (h₁ : ∀ a, (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ a i, p a i → IsClosed (s a i)) :
     RegularSpace α :=
-  RegularSpace.ofLift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a)
-#align regular_space.of_basis RegularSpace.ofBasis
+  RegularSpace.of_lift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a)
+#align regular_space.of_basis RegularSpace.of_hasBasis
 -/
 
-#print RegularSpace.ofExistsMemNhdsIsClosedSubset /-
-theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
+#print RegularSpace.of_exists_mem_nhds_isClosed_subset /-
+theorem RegularSpace.of_exists_mem_nhds_isClosed_subset
     (h : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s) : RegularSpace α :=
   Iff.mpr ((regularSpace_TFAE α).out 0 3) h
-#align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.ofExistsMemNhdsIsClosedSubset
+#align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.of_exists_mem_nhds_isClosed_subset
 -/
 
 variable [RegularSpace α] {a : α} {s : Set α}
@@ -2043,8 +2043,9 @@ theorem isClosed_setOf_inseparable : IsClosed {p : α × α | Inseparable p.1 p.
 #print Inducing.regularSpace /-
 protected theorem Inducing.regularSpace [TopologicalSpace β] {f : β → α} (hf : Inducing f) :
     RegularSpace β :=
-  RegularSpace.ofBasis (fun b => by rw [hf.nhds_eq_comap b]; exact (closed_nhds_basis _).comap _)
-    fun b s hs => hs.2.Preimage hf.Continuous
+  RegularSpace.of_hasBasis
+    (fun b => by rw [hf.nhds_eq_comap b]; exact (closed_nhds_basis _).comap _) fun b s hs =>
+    hs.2.Preimage hf.Continuous
 #align inducing.regular_space Inducing.regularSpace
 -/
 
@@ -2069,7 +2070,7 @@ theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @
     intro a
     rw [nhds_sInf, ← iInf_subtype'']
     exact has_basis_infi fun t : T => @closed_nhds_basis X t (h t t.2) a
-  refine' RegularSpace.ofBasis this fun a If hIf => isClosed_iInter fun i => _
+  refine' RegularSpace.of_hasBasis this fun a If hIf => isClosed_iInter fun i => _
   exact (hIf.2 i).2.mono (sInf_le (i : T).2)
 #align regular_space_Inf regularSpace_sInf
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1748,7 +1748,25 @@ open Finset Function
 /-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
 theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact s) {ι} (t : Finset ι)
     (U : ι → Set α) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
-    ∃ K : ι → Set α, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by classical
+    ∃ K : ι → Set α, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
+  classical
+  induction' t using Finset.induction with x t hx ih generalizing U hU s hs hsC
+  · refine' ⟨fun _ => ∅, fun i => isCompact_empty, fun i => empty_subset _, _⟩
+    simpa only [subset_empty_iff, Union_false, Union_empty] using hsC
+  simp only [Finset.set_biUnion_insert] at hsC 
+  simp only [Finset.mem_insert] at hU 
+  have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU i (Or.inr hi)
+  rcases hs.binary_compact_cover (hU x (Or.inl rfl)) (isOpen_biUnion hU') hsC with
+    ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
+  rcases ih U hU' h1K₂ h2K₂ with ⟨K, h1K, h2K, h3K⟩
+  refine' ⟨update K x K₁, _, _, _⟩
+  · intro i; by_cases hi : i = x
+    · simp only [update_same, hi, h1K₁]
+    · rw [← Ne.def] at hi ; simp only [update_noteq hi, h1K]
+  · intro i; by_cases hi : i = x
+    · simp only [update_same, hi, h2K₁]
+    · rw [← Ne.def] at hi ; simp only [update_noteq hi, h2K]
+  · simp only [set_bUnion_insert_update _ hx, hK, h3K]
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1238,8 +1238,8 @@ section Separated
 
 open SeparatedNhds Finset
 
-#print separatedNhds_of_finset_finset /-
-theorem separatedNhds_of_finset_finset [T2Space α] :
+#print SeparatedNhds.of_finset_finset /-
+theorem SeparatedNhds.of_finset_finset [T2Space α] :
     ∀ s t : Finset α, Disjoint s t → SeparatedNhds (s : Set α) t :=
   by
   refine'
@@ -1250,14 +1250,14 @@ theorem separatedNhds_of_finset_finset [T2Space α] :
   · intro a b c ac bc d
     apply_mod_cast union_left (ac (disjoint_of_subset_left (a.subset_union_left b) d)) (bc _)
     exact disjoint_of_subset_left (a.subset_union_right b) d
-#align finset_disjoint_finset_opens_of_t2 separatedNhds_of_finset_finset
+#align finset_disjoint_finset_opens_of_t2 SeparatedNhds.of_finset_finset
 -/
 
-#print point_disjoint_finset_opens_of_t2 /-
-theorem point_disjoint_finset_opens_of_t2 [T2Space α] {x : α} {s : Finset α} (h : x ∉ s) :
+#print SeparatedNhds.of_singleton_finset /-
+theorem SeparatedNhds.of_singleton_finset [T2Space α] {x : α} {s : Finset α} (h : x ∉ s) :
     SeparatedNhds ({x} : Set α) s := by
-  exact_mod_cast separatedNhds_of_finset_finset {x} s (finset.disjoint_singleton_left.mpr h)
-#align point_disjoint_finset_opens_of_t2 point_disjoint_finset_opens_of_t2
+  exact_mod_cast SeparatedNhds.of_finset_finset {x} s (finset.disjoint_singleton_left.mpr h)
+#align point_disjoint_finset_opens_of_t2 SeparatedNhds.of_singleton_finset
 -/
 
 end Separated
@@ -1620,12 +1620,12 @@ theorem Function.LeftInverse.closedEmbedding [T2Space α] {f : α → β} {g : 
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 -/
 
-#print separatedNhds_of_isCompact_isCompact /-
-theorem separatedNhds_of_isCompact_isCompact [T2Space α] {s t : Set α} (hs : IsCompact s)
+#print SeparatedNhds.of_isCompact_isCompact /-
+theorem SeparatedNhds.of_isCompact_isCompact [T2Space α] {s t : Set α} (hs : IsCompact s)
     (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t := by
   simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst ⊢ <;>
     exact generalized_tube_lemma hs ht is_closed_diagonal.is_open_compl hst
-#align is_compact_is_compact_separated separatedNhds_of_isCompact_isCompact
+#align is_compact_is_compact_separated SeparatedNhds.of_isCompact_isCompact
 -/
 
 #print IsCompact.isClosed /-
@@ -1634,7 +1634,7 @@ theorem IsCompact.isClosed [T2Space α] {s : Set α} (hs : IsCompact s) : IsClos
   isOpen_compl_iff.1 <|
     isOpen_iff_forall_mem_open.mpr fun x hx =>
       let ⟨u, v, uo, vo, su, xv, uv⟩ :=
-        separatedNhds_of_isCompact_isCompact hs isCompact_singleton (disjoint_singleton_right.2 hx)
+        SeparatedNhds.of_isCompact_isCompact hs isCompact_singleton (disjoint_singleton_right.2 hx)
       ⟨v, (uv.mono_left <| show s ≤ u from su).subset_compl_left, vo, by simpa using xv⟩
 #align is_compact.is_closed IsCompact.isClosed
 -/
@@ -1679,20 +1679,19 @@ theorem IsCompact.inter [T2Space α] {s t : Set α} (hs : IsCompact s) (ht : IsC
 #align is_compact.inter IsCompact.inter
 -/
 
-#print isCompact_closure_of_subset_compact /-
-theorem isCompact_closure_of_subset_compact [T2Space α] {s t : Set α} (ht : IsCompact t)
-    (h : s ⊆ t) : IsCompact (closure s) :=
+#print IsCompact.closure_of_subset /-
+theorem IsCompact.closure_of_subset [T2Space α] {s t : Set α} (ht : IsCompact t) (h : s ⊆ t) :
+    IsCompact (closure s) :=
   IsCompact.of_isClosed_subset ht isClosed_closure (closure_minimal h ht.IsClosed)
-#align is_compact_closure_of_subset_compact isCompact_closure_of_subset_compact
+#align is_compact_closure_of_subset_compact IsCompact.closure_of_subset
 -/
 
-#print exists_compact_superset_iff /-
+#print exists_isCompact_superset_iff /-
 @[simp]
-theorem exists_compact_superset_iff [T2Space α] {s : Set α} :
+theorem exists_isCompact_superset_iff [T2Space α] {s : Set α} :
     (∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) :=
-  ⟨fun ⟨K, hK, hsK⟩ => isCompact_closure_of_subset_compact hK hsK, fun h =>
-    ⟨closure s, h, subset_closure⟩⟩
-#align exists_compact_superset_iff exists_compact_superset_iff
+  ⟨fun ⟨K, hK, hsK⟩ => IsCompact.closure_of_subset hK hsK, fun h => ⟨closure s, h, subset_closure⟩⟩
+#align exists_compact_superset_iff exists_isCompact_superset_iff
 -/
 
 #print image_closure_of_isCompact /-
@@ -1710,7 +1709,7 @@ theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCom
     ∃ K₁ K₂ : Set α, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ :=
   by
   obtain ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩ :=
-    separatedNhds_of_isCompact_isCompact (hK.diff hU) (hK.diff hV)
+    SeparatedNhds.of_isCompact_isCompact (hK.diff hU) (hK.diff hV)
       (by rwa [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty])
   exact
     ⟨_, _, hK.diff h1O₁, hK.diff h1O₂, by rwa [diff_subset_comm], by rwa [diff_subset_comm], by
@@ -1749,25 +1748,7 @@ open Finset Function
 /-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
 theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact s) {ι} (t : Finset ι)
     (U : ι → Set α) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
-    ∃ K : ι → Set α, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
-  classical
-  induction' t using Finset.induction with x t hx ih generalizing U hU s hs hsC
-  · refine' ⟨fun _ => ∅, fun i => isCompact_empty, fun i => empty_subset _, _⟩
-    simpa only [subset_empty_iff, Union_false, Union_empty] using hsC
-  simp only [Finset.set_biUnion_insert] at hsC 
-  simp only [Finset.mem_insert] at hU 
-  have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU i (Or.inr hi)
-  rcases hs.binary_compact_cover (hU x (Or.inl rfl)) (isOpen_biUnion hU') hsC with
-    ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
-  rcases ih U hU' h1K₂ h2K₂ with ⟨K, h1K, h2K, h3K⟩
-  refine' ⟨update K x K₁, _, _, _⟩
-  · intro i; by_cases hi : i = x
-    · simp only [update_same, hi, h1K₁]
-    · rw [← Ne.def] at hi ; simp only [update_noteq hi, h1K]
-  · intro i; by_cases hi : i = x
-    · simp only [update_same, hi, h2K₁]
-    · rw [← Ne.def] at hi ; simp only [update_noteq hi, h2K]
-  · simp only [set_bUnion_insert_update _ hx, hK, h3K]
+    ∃ K : ι → Set α, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by classical
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 -/
 
@@ -1784,7 +1765,7 @@ theorem WeaklyLocallyCompactSpace.locallyCompactSpace [T2Space α]
     -- we may find open sets V, W separating x from K \ U.
     -- Then K \ W is a compact neighborhood of x contained in U.
     let ⟨v, w, vo, wo, xv, kuw, vw⟩ :=
-      separatedNhds_of_isCompact_isCompact isCompact_singleton (kc.diffₓ uo)
+      SeparatedNhds.of_isCompact_isCompact isCompact_singleton (kc.diffₓ uo)
         (disjoint_singleton_left.2 fun h => h.2 xu)
     have wn : wᶜ ∈ 𝓝 x :=
       mem_nhds_iff.mpr ⟨v, vw.subset_compl_right, vo, singleton_subset_iff.mp xv⟩
@@ -1801,27 +1782,26 @@ instance (priority := 100) locally_compact_of_compact [T2Space α] [CompactSpace
 #align locally_compact_of_compact locally_compact_of_compact
 -/
 
-#print exists_open_with_compact_closure /-
+#print exists_isOpen_mem_isCompact_closure /-
 /-- In a locally compact T₂ space, every point has an open neighborhood with compact closure -/
-theorem exists_open_with_compact_closure [LocallyCompactSpace α] [T2Space α] (x : α) :
+theorem exists_isOpen_mem_isCompact_closure [LocallyCompactSpace α] [T2Space α] (x : α) :
     ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) :=
   by
   rcases WeaklyLocallyCompactSpace.exists_compact_mem_nhds x with ⟨K, hKc, hxK⟩
   rcases mem_nhds_iff.1 hxK with ⟨t, h1t, h2t, h3t⟩
-  exact ⟨t, h2t, h3t, isCompact_closure_of_subset_compact hKc h1t⟩
-#align exists_open_with_compact_closure exists_open_with_compact_closure
+  exact ⟨t, h2t, h3t, IsCompact.closure_of_subset hKc h1t⟩
+#align exists_open_with_compact_closure exists_isOpen_mem_isCompact_closure
 -/
 
-#print exists_open_superset_and_isCompact_closure /-
+#print exists_isOpen_superset_and_isCompact_closure /-
 /-- In a locally compact T₂ space, every compact set has an open neighborhood with compact closure.
 -/
-theorem exists_open_superset_and_isCompact_closure [LocallyCompactSpace α] [T2Space α] {K : Set α}
+theorem exists_isOpen_superset_and_isCompact_closure [LocallyCompactSpace α] [T2Space α] {K : Set α}
     (hK : IsCompact K) : ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) :=
   by
   rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
-  refine'
-    ⟨interior K', isOpen_interior, hKK', isCompact_closure_of_subset_compact hK' interior_subset⟩
-#align exists_open_superset_and_is_compact_closure exists_open_superset_and_isCompact_closure
+  refine' ⟨interior K', isOpen_interior, hKK', IsCompact.closure_of_subset hK' interior_subset⟩
+#align exists_open_superset_and_is_compact_closure exists_isOpen_superset_and_isCompact_closure
 -/
 
 #print exists_open_between_and_isCompact_closure /-
@@ -1835,7 +1815,7 @@ theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace α] [T2Sp
   rcases exists_compact_between hK hU hKU with ⟨V, hV, hKV, hVU⟩
   exact
     ⟨interior V, isOpen_interior, hKV, (closure_minimal interior_subset hV.is_closed).trans hVU,
-      isCompact_closure_of_subset_compact hV interior_subset⟩
+      IsCompact.closure_of_subset hV interior_subset⟩
 #align exists_open_between_and_is_compact_closure exists_open_between_and_isCompact_closure
 -/
 
@@ -2222,7 +2202,7 @@ instance (priority := 100) T4Space.t3Space [NormalSpace α] : T3Space α
 #print T4Space.of_compactSpace_t2Space /-
 -- We can't make this an instance because it could cause an instance loop.
 theorem T4Space.of_compactSpace_t2Space [CompactSpace α] [T2Space α] : NormalSpace α :=
-  ⟨fun s t hs ht => separatedNhds_of_isCompact_isCompact hs.IsCompact ht.IsCompact⟩
+  ⟨fun s t hs ht => SeparatedNhds.of_isCompact_isCompact hs.IsCompact ht.IsCompact⟩
 #align normal_of_compact_t2 T4Space.of_compactSpace_t2Space
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -1238,8 +1238,8 @@ section Separated
 
 open SeparatedNhds Finset
 
-#print finset_disjoint_finset_opens_of_t2 /-
-theorem finset_disjoint_finset_opens_of_t2 [T2Space α] :
+#print separatedNhds_of_finset_finset /-
+theorem separatedNhds_of_finset_finset [T2Space α] :
     ∀ s t : Finset α, Disjoint s t → SeparatedNhds (s : Set α) t :=
   by
   refine'
@@ -1250,13 +1250,13 @@ theorem finset_disjoint_finset_opens_of_t2 [T2Space α] :
   · intro a b c ac bc d
     apply_mod_cast union_left (ac (disjoint_of_subset_left (a.subset_union_left b) d)) (bc _)
     exact disjoint_of_subset_left (a.subset_union_right b) d
-#align finset_disjoint_finset_opens_of_t2 finset_disjoint_finset_opens_of_t2
+#align finset_disjoint_finset_opens_of_t2 separatedNhds_of_finset_finset
 -/
 
 #print point_disjoint_finset_opens_of_t2 /-
 theorem point_disjoint_finset_opens_of_t2 [T2Space α] {x : α} {s : Finset α} (h : x ∉ s) :
     SeparatedNhds ({x} : Set α) s := by
-  exact_mod_cast finset_disjoint_finset_opens_of_t2 {x} s (finset.disjoint_singleton_left.mpr h)
+  exact_mod_cast separatedNhds_of_finset_finset {x} s (finset.disjoint_singleton_left.mpr h)
 #align point_disjoint_finset_opens_of_t2 point_disjoint_finset_opens_of_t2
 -/
 
@@ -1620,12 +1620,12 @@ theorem Function.LeftInverse.closedEmbedding [T2Space α] {f : α → β} {g : 
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 -/
 
-#print isCompact_isCompact_separated /-
-theorem isCompact_isCompact_separated [T2Space α] {s t : Set α} (hs : IsCompact s)
+#print separatedNhds_of_isCompact_isCompact /-
+theorem separatedNhds_of_isCompact_isCompact [T2Space α] {s t : Set α} (hs : IsCompact s)
     (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t := by
   simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst ⊢ <;>
     exact generalized_tube_lemma hs ht is_closed_diagonal.is_open_compl hst
-#align is_compact_is_compact_separated isCompact_isCompact_separated
+#align is_compact_is_compact_separated separatedNhds_of_isCompact_isCompact
 -/
 
 #print IsCompact.isClosed /-
@@ -1634,7 +1634,7 @@ theorem IsCompact.isClosed [T2Space α] {s : Set α} (hs : IsCompact s) : IsClos
   isOpen_compl_iff.1 <|
     isOpen_iff_forall_mem_open.mpr fun x hx =>
       let ⟨u, v, uo, vo, su, xv, uv⟩ :=
-        isCompact_isCompact_separated hs isCompact_singleton (disjoint_singleton_right.2 hx)
+        separatedNhds_of_isCompact_isCompact hs isCompact_singleton (disjoint_singleton_right.2 hx)
       ⟨v, (uv.mono_left <| show s ≤ u from su).subset_compl_left, vo, by simpa using xv⟩
 #align is_compact.is_closed IsCompact.isClosed
 -/
@@ -1710,7 +1710,7 @@ theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCom
     ∃ K₁ K₂ : Set α, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ :=
   by
   obtain ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩ :=
-    isCompact_isCompact_separated (hK.diff hU) (hK.diff hV)
+    separatedNhds_of_isCompact_isCompact (hK.diff hU) (hK.diff hV)
       (by rwa [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty])
   exact
     ⟨_, _, hK.diff h1O₁, hK.diff h1O₂, by rwa [diff_subset_comm], by rwa [diff_subset_comm], by
@@ -1784,7 +1784,7 @@ theorem WeaklyLocallyCompactSpace.locallyCompactSpace [T2Space α]
     -- we may find open sets V, W separating x from K \ U.
     -- Then K \ W is a compact neighborhood of x contained in U.
     let ⟨v, w, vo, wo, xv, kuw, vw⟩ :=
-      isCompact_isCompact_separated isCompact_singleton (kc.diffₓ uo)
+      separatedNhds_of_isCompact_isCompact isCompact_singleton (kc.diffₓ uo)
         (disjoint_singleton_left.2 fun h => h.2 xu)
     have wn : wᶜ ∈ 𝓝 x :=
       mem_nhds_iff.mpr ⟨v, vw.subset_compl_right, vo, singleton_subset_iff.mp xv⟩
@@ -2222,7 +2222,7 @@ instance (priority := 100) T4Space.t3Space [NormalSpace α] : T3Space α
 #print T4Space.of_compactSpace_t2Space /-
 -- We can't make this an instance because it could cause an instance loop.
 theorem T4Space.of_compactSpace_t2Space [CompactSpace α] [T2Space α] : NormalSpace α :=
-  ⟨fun s t hs ht => isCompact_isCompact_separated hs.IsCompact ht.IsCompact⟩
+  ⟨fun s t hs ht => separatedNhds_of_isCompact_isCompact hs.IsCompact ht.IsCompact⟩
 #align normal_of_compact_t2 T4Space.of_compactSpace_t2Space
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -367,9 +367,9 @@ theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpe
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊂ » s) -/
-#print exists_open_singleton_of_open_finite /-
+#print exists_isOpen_singleton_of_isOpen_finite /-
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
-theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s.Finite)
+theorem exists_isOpen_singleton_of_isOpen_finite [T0Space α] {s : Set α} (hfin : s.Finite)
     (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set α) :=
   by
   lift s to Finset α using hfin
@@ -382,13 +382,14 @@ theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s
     refine' fun t hts htne hto => of_not_not fun hts' => ht _
     lift t to Finset α using s.finite_to_set.subset hts
     exact ⟨t, ssubset_iff_subset_ne.2 ⟨hts, mt Finset.coe_inj.2 hts'⟩, htne, hto⟩
-#align exists_open_singleton_of_open_finite exists_open_singleton_of_open_finite
+#align exists_open_singleton_of_open_finite exists_isOpen_singleton_of_isOpen_finite
 -/
 
 #print exists_open_singleton_of_finite /-
 theorem exists_open_singleton_of_finite [T0Space α] [Finite α] [Nonempty α] :
     ∃ x : α, IsOpen ({x} : Set α) :=
-  let ⟨x, _, h⟩ := exists_open_singleton_of_open_finite (Set.toFinite _) univ_nonempty isOpen_univ
+  let ⟨x, _, h⟩ :=
+    exists_isOpen_singleton_of_isOpen_finite (Set.toFinite _) univ_nonempty isOpen_univ
   ⟨x, h⟩
 #align exists_open_singleton_of_fintype exists_open_singleton_of_finite
 -/
@@ -902,7 +903,7 @@ theorem nhds_le_nhdsSet_iff [T1Space α] {s : Set α} {x : α} : 𝓝 x ≤ 𝓝
 /-- Removing a non-isolated point from a dense set, one still obtains a dense set. -/
 theorem Dense.diff_singleton [T1Space α] {s : Set α} (hs : Dense s) (x : α) [NeBot (𝓝[≠] x)] :
     Dense (s \ {x}) :=
-  hs.inter_of_open_right (dense_compl_singleton x) isOpen_compl_singleton
+  hs.inter_of_isOpen_right (dense_compl_singleton x) isOpen_compl_singleton
 #align dense.diff_singleton Dense.diff_singleton
 -/
 
@@ -2532,8 +2533,8 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
 #align nhds_basis_clopen nhds_basis_clopen
 -/
 
-#print isTopologicalBasis_clopen /-
-theorem isTopologicalBasis_clopen : IsTopologicalBasis {s : Set α | IsClopen s} :=
+#print isTopologicalBasis_isClopen /-
+theorem isTopologicalBasis_isClopen : IsTopologicalBasis {s : Set α | IsClopen s} :=
   by
   apply is_topological_basis_of_open_of_nhds fun U (hU : IsClopen U) => hU.1
   intro x U hxU U_op
@@ -2541,16 +2542,16 @@ theorem isTopologicalBasis_clopen : IsTopologicalBasis {s : Set α | IsClopen s}
   rcases(nhds_basis_clopen x).mem_iff.mp this with ⟨V, ⟨hxV, hV⟩, hVU : V ⊆ U⟩
   use V
   tauto
-#align is_topological_basis_clopen isTopologicalBasis_clopen
+#align is_topological_basis_clopen isTopologicalBasis_isClopen
 -/
 
-#print compact_exists_clopen_in_open /-
+#print compact_exists_isClopen_in_isOpen /-
 /-- Every member of an open set in a compact Hausdorff totally disconnected space
   is contained in a clopen set contained in the open set.  -/
-theorem compact_exists_clopen_in_open {x : α} {U : Set α} (is_open : IsOpen U) (memU : x ∈ U) :
+theorem compact_exists_isClopen_in_isOpen {x : α} {U : Set α} (is_open : IsOpen U) (memU : x ∈ U) :
     ∃ (V : Set α) (hV : IsClopen V), x ∈ V ∧ V ⊆ U :=
-  (IsTopologicalBasis.mem_nhds_iff isTopologicalBasis_clopen).1 (IsOpen.mem_nhds memU)
-#align compact_exists_clopen_in_open compact_exists_clopen_in_open
+  (IsTopologicalBasis.mem_nhds_iff isTopologicalBasis_isClopen).1 (IsOpen.mem_nhds memU)
+#align compact_exists_clopen_in_open compact_exists_isClopen_in_isOpen
 -/
 
 end Profinite
@@ -2573,7 +2574,7 @@ theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
   let X : s := ⟨x, h xt⟩
   have Xu : X ∈ u := xs
   haveI : CompactSpace s := isCompact_iff_compactSpace.1 comp
-  obtain ⟨V : Set s, clopen_in_s, Vx, V_sub⟩ := compact_exists_clopen_in_open u_open_in_s Xu
+  obtain ⟨V : Set s, clopen_in_s, Vx, V_sub⟩ := compact_exists_isClopen_in_isOpen u_open_in_s Xu
   have V_clopen : IsClopen ((coe : s → H) '' V) :=
     by
     refine' ⟨_, comp.is_closed.closed_embedding_subtype_coe.closed_iff_image_closed.1 clopen_in_s.2⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -2387,13 +2387,13 @@ instance [T5Space α] : T5Space (SeparationQuotient α)
 
 end CompletelyNormal
 
-#print connectedComponent_eq_iInter_clopen /-
+#print connectedComponent_eq_iInter_isClopen /-
 /-- In a compact t2 space, the connected component of a point equals the intersection of all
 its clopen neighbourhoods. -/
-theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x : α) :
+theorem connectedComponent_eq_iInter_isClopen [T2Space α] [CompactSpace α] (x : α) :
     connectedComponent x = ⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, Z :=
   by
-  apply eq_of_subset_of_subset connectedComponent_subset_iInter_clopen
+  apply eq_of_subset_of_subset connectedComponent_subset_iInter_isClopen
   -- Reduce to showing that the clopen intersection is connected.
   refine' IsPreconnected.subset_connectedComponent _ (mem_Inter.2 fun Z => Z.2.2)
   -- We do this by showing that any disjoint cover by two closed sets implies
@@ -2458,7 +2458,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
   · exact isClopen_biInter_finset fun Z hZ => Z.2.1
   · exact mem_Inter₂.2 fun Z hZ => Z.2.2
   · rwa [← disjoint_compl_left_iff_subset, disjoint_iff_inter_eq_empty, ← not_nonempty_iff_eq_empty]
-#align connected_component_eq_Inter_clopen connectedComponent_eq_iInter_clopen
+#align connected_component_eq_Inter_clopen connectedComponent_eq_iInter_isClopen
 -/
 
 section Profinite
@@ -2491,7 +2491,7 @@ theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace α ↔ Totall
     intro hyp
     suffices x ∈ connectedComponent y by
       simpa [totallyDisconnectedSpace_iff_connectedComponent_singleton.1 h y, mem_singleton_iff]
-    rw [connectedComponent_eq_iInter_clopen, mem_Inter]
+    rw [connectedComponent_eq_iInter_isClopen, mem_Inter]
     rintro ⟨w : Set α, hw : IsClopen w, hy : y ∈ w⟩
     by_contra hx
     exact
@@ -2509,7 +2509,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
     constructor
     · have : connectedComponent x = {x} :=
         totally_disconnected_space_iff_connected_component_singleton.mp ‹_› x
-      rw [connectedComponent_eq_iInter_clopen] at this 
+      rw [connectedComponent_eq_iInter_isClopen] at this 
       intro hU
       let N := { Z // IsClopen Z ∧ x ∈ Z }
       rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ Z : N, Z.val ⊆ U
@@ -2626,7 +2626,7 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
   rw [ConnectedComponents.coe_ne_coe] at ne 
   have h := connectedComponent_disjoint Ne
   -- write ↑b as the intersection of all clopen subsets containing it
-  rw [connectedComponent_eq_iInter_clopen b, disjoint_iff_inter_eq_empty] at h 
+  rw [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h 
   -- Now we show that this can be reduced to some clopen containing `↑b` being disjoint to `↑a`
   obtain ⟨U, V, hU, ha, hb, rfl⟩ :
     ∃ (U : Set α) (V : Set (ConnectedComponents α)),

mathlib commit https://github.com/leanprover-community/mathlib/commit/b1abe23ae96fef89ad30d9f4362c307f72a55010

@@ -282,7 +282,7 @@ theorem inseparable_eq_eq [T0Space α] : Inseparable = @Eq α :=
 #print t0Space_iff_exists_isOpen_xor'_mem /-
 theorem t0Space_iff_exists_isOpen_xor'_mem (α : Type u) [TopologicalSpace α] :
     T0Space α ↔ ∀ x y, x ≠ y → ∃ U : Set α, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by
-  simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,
+  simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, Classical.not_forall, exists_prop,
     inseparable_iff_forall_open]
 #align t0_space_iff_exists_is_open_xor_mem t0Space_iff_exists_isOpen_xor'_mem
 -/
@@ -1512,7 +1512,7 @@ instance {α : Type _} {β : Type _} [t₁ : TopologicalSpace α] [T2Space α] [
 instance Pi.t2Space {α : Type _} {β : α → Type v} [t₂ : ∀ a, TopologicalSpace (β a)]
     [∀ a, T2Space (β a)] : T2Space (∀ a, β a) :=
   ⟨fun x y h =>
-    let ⟨i, hi⟩ := not_forall.mp (mt funext h)
+    let ⟨i, hi⟩ := Classical.not_forall.mp (mt funext h)
     separated_by_continuous (continuous_apply i) hi⟩
 #align Pi.t2_space Pi.t2Space
 -/
@@ -2315,7 +2315,7 @@ theorem NormalSpace.of_regularSpace_secondCountableTopology [SecondCountableTopo
     simp only [mem_Union]
     rintro ⟨u, huU, -, hxu⟩
     exact (hUd u huU).le_bot ⟨hxu, hx⟩
-  · simp only [disjoint_left, mem_Union, mem_diff, not_exists, not_and, not_forall,
+  · simp only [disjoint_left, mem_Union, mem_diff, not_exists, not_and, Classical.not_forall,
       Classical.not_not]
     rintro a ⟨u, huU, hau, haV⟩ v hvV hav
     cases' le_total (Encodable.encode u) (Encodable.encode v) with hle hle
@@ -2452,7 +2452,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
   have H1 :=
     (hu.union hv).isClosed_compl.IsCompact.inter_iInter_nonempty
       (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z) fun Z => Z.2.1.2
-  rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, not_forall] at H1 
+  rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, Classical.not_forall] at H1 
   cases' H1 (disjoint_compl_left_iff_subset.2 <| hab.trans <| union_subset_union hau hbv) with Zi H2
   refine' ⟨⋂ U ∈ Zi, Subtype.val U, _, _, _⟩
   · exact isClopen_biInter_finset fun Z hZ => Z.2.1

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,10 +3,10 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
-import Mathbin.Topology.SubsetProperties
-import Mathbin.Topology.Connected
-import Mathbin.Topology.NhdsSet
-import Mathbin.Topology.Inseparable
+import Topology.SubsetProperties
+import Topology.Connected
+import Topology.NhdsSet
+import Topology.Inseparable
 
 #align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
 
@@ -306,7 +306,7 @@ instance : T0Space (SeparationQuotient α) :=
     Quotient.inductionOn₂' x' y' fun x y h =>
       SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.inducing_mk.inseparable_iff.1 h⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_subsingleton /-
 theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : s.Subsingleton :=
@@ -321,7 +321,7 @@ theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsC
 #align minimal_nonempty_closed_subsingleton minimal_nonempty_closed_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_eq_singleton /-
 theorem minimal_nonempty_closed_eq_singleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} :=
@@ -343,7 +343,7 @@ theorem IsClosed.exists_closed_singleton {α : Type _} [TopologicalSpace α] [T0
 #align is_closed.exists_closed_singleton IsClosed.exists_closed_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_subsingleton /-
 theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : s.Subsingleton :=
@@ -358,7 +358,7 @@ theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_subsingleton minimal_nonempty_open_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_eq_singleton /-
 theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} :=
@@ -366,7 +366,7 @@ theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_eq_singleton minimal_nonempty_open_eq_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊂ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (t «expr ⊂ » s) -/
 #print exists_open_singleton_of_open_finite /-
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
 theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s.Finite)
@@ -558,7 +558,7 @@ theorem Bornology.relativelyCompact.isBounded_iff [T1Space α] {s : Set α} :
   constructor
   · rintro ⟨t, ht₁, ht₂, hst⟩
     rw [compl_subset_compl] at hst 
-    exact isCompact_of_isClosed_subset ht₂ isClosed_closure (closure_minimal hst ht₁)
+    exact IsCompact.of_isClosed_subset ht₂ isClosed_closure (closure_minimal hst ht₁)
   · intro h
     exact ⟨closure s, isClosed_closure, h, compl_subset_compl.mpr subset_closure⟩
 #align bornology.relatively_compact.is_bounded_iff Bornology.relativelyCompact.isBounded_iff
@@ -1681,7 +1681,7 @@ theorem IsCompact.inter [T2Space α] {s t : Set α} (hs : IsCompact s) (ht : IsC
 #print isCompact_closure_of_subset_compact /-
 theorem isCompact_closure_of_subset_compact [T2Space α] {s t : Set α} (ht : IsCompact t)
     (h : s ⊆ t) : IsCompact (closure s) :=
-  isCompact_of_isClosed_subset ht isClosed_closure (closure_minimal h ht.IsClosed)
+  IsCompact.of_isClosed_subset ht isClosed_closure (closure_minimal h ht.IsClosed)
 #align is_compact_closure_of_subset_compact isCompact_closure_of_subset_compact
 -/
 
@@ -1883,7 +1883,7 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
 #align regular_space RegularSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
 #print regularSpace_TFAE /-
 theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     TFAE
@@ -2145,8 +2145,8 @@ instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [
     T3Space (∀ i, π i) :=
   ⟨⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
 #print disjoint_nested_nhds /-
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
 with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/001ffdc42920050657fd45bd2b8bfbec8eaaeb29

@@ -2208,25 +2208,25 @@ theorem normal_exists_closure_subset [NormalSpace α] {s t : Set α} (hs : IsClo
 #align normal_exists_closure_subset normal_exists_closure_subset
 -/
 
-#print NormalSpace.t3Space /-
+#print T4Space.t3Space /-
 -- see Note [lower instance priority]
-instance (priority := 100) NormalSpace.t3Space [NormalSpace α] : T3Space α
+instance (priority := 100) T4Space.t3Space [NormalSpace α] : T3Space α
     where regular s x hs hxs :=
     let ⟨u, v, hu, hv, hsu, hxv, huv⟩ :=
       normal_separation hs isClosed_singleton (disjoint_singleton_right.mpr hxs)
     disjoint_of_disjoint_of_mem huv (hu.mem_nhdsSet.2 hsu) (hv.mem_nhds <| hxv rfl)
-#align normal_space.t3_space NormalSpace.t3Space
+#align normal_space.t3_space T4Space.t3Space
 -/
 
-#print normalOfCompactT2 /-
+#print T4Space.of_compactSpace_t2Space /-
 -- We can't make this an instance because it could cause an instance loop.
-theorem normalOfCompactT2 [CompactSpace α] [T2Space α] : NormalSpace α :=
+theorem T4Space.of_compactSpace_t2Space [CompactSpace α] [T2Space α] : NormalSpace α :=
   ⟨fun s t hs ht => isCompact_isCompact_separated hs.IsCompact ht.IsCompact⟩
-#align normal_of_compact_t2 normalOfCompactT2
+#align normal_of_compact_t2 T4Space.of_compactSpace_t2Space
 -/
 
-#print ClosedEmbedding.normalSpace /-
-protected theorem ClosedEmbedding.normalSpace [TopologicalSpace β] [NormalSpace β] {f : α → β}
+#print ClosedEmbedding.t4Space /-
+protected theorem ClosedEmbedding.t4Space [TopologicalSpace β] [NormalSpace β] {f : α → β}
     (hf : ClosedEmbedding f) : NormalSpace α :=
   { to_t1Space := hf.toEmbedding.T1Space
     normal := by
@@ -2236,7 +2236,7 @@ protected theorem ClosedEmbedding.normalSpace [TopologicalSpace β] [NormalSpace
           (disjoint_image_of_injective hf.inj hst)
       exact
         (H.preimage hf.continuous).mono (subset_preimage_image _ _) (subset_preimage_image _ _) }
-#align closed_embedding.normal_space ClosedEmbedding.normalSpace
+#align closed_embedding.normal_space ClosedEmbedding.t4Space
 -/
 
 namespace SeparationQuotient
@@ -2261,10 +2261,11 @@ end SeparationQuotient
 
 variable (α)
 
-#print normalSpaceOfT3SecondCountable /-
+#print NormalSpace.of_regularSpace_secondCountableTopology /-
 /-- A T₃ topological space with second countable topology is a normal space.
 This lemma is not an instance to avoid a loop. -/
-theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α] : NormalSpace α :=
+theorem NormalSpace.of_regularSpace_secondCountableTopology [SecondCountableTopology α]
+    [T3Space α] : NormalSpace α :=
   by
   have key :
     ∀ {s t : Set α},
@@ -2319,7 +2320,7 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
     rintro a ⟨u, huU, hau, haV⟩ v hvV hav
     cases' le_total (Encodable.encode u) (Encodable.encode v) with hle hle
     exacts [⟨u, huU, hle, subset_closure hau⟩, (haV _ hvV hle <| subset_closure hav).elim]
-#align normal_space_of_t3_second_countable normalSpaceOfT3SecondCountable
+#align normal_space_of_t3_second_countable NormalSpace.of_regularSpace_secondCountableTopology
 -/
 
 end Normality
@@ -2359,14 +2360,14 @@ theorem Embedding.t5Space [TopologicalSpace β] [T5Space β] {e : α → β} (he
 instance [T5Space α] {p : α → Prop} : T5Space { x // p x } :=
   embedding_subtype_val.T5Space
 
-#print T5Space.toNormalSpace /-
+#print T5Space.toT4Space /-
 -- see Note [lower instance priority]
 /-- A `T₅` space is a `T₄` space. -/
-instance (priority := 100) T5Space.toNormalSpace [T5Space α] : NormalSpace α :=
+instance (priority := 100) T5Space.toT4Space [T5Space α] : NormalSpace α :=
   ⟨fun s t hs ht hd =>
     separatedNhds_iff_disjoint.2 <|
       completely_normal (by rwa [hs.closure_eq]) (by rwa [ht.closure_eq])⟩
-#align t5_space.to_normal_space T5Space.toNormalSpace
+#align t5_space.to_normal_space T5Space.toT4Space
 -/
 
 open SeparationQuotient
@@ -2402,7 +2403,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
     isClosed_iInter fun Z => Z.2.1.2
   rw [isPreconnected_iff_subset_of_fully_disjoint_closed hs]
   intro a b ha hb hab ab_disj
-  haveI := @normalOfCompactT2 α _ _ _
+  haveI := @T4Space.of_compactSpace_t2Space α _ _ _
   -- Since our space is normal, we get two larger disjoint open sets containing the disjoint
   -- closed sets. If we can show that our intersection is a subset of any of these we can then
   -- "descend" this to show that it is a subset of either a or b.

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -514,7 +514,7 @@ theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
 protected theorem Set.Finite.isClosed [T1Space α] {s : Set α} (hs : Set.Finite s) : IsClosed s :=
   by
   rw [← bUnion_of_singleton s]
-  exact isClosed_biUnion hs fun i hi => isClosed_singleton
+  exact Set.Finite.isClosed_biUnion hs fun i hi => isClosed_singleton
 #align set.finite.is_closed Set.Finite.isClosed
 -/
 
@@ -2291,7 +2291,7 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
       exact fun x hx => mem_Union.2 ⟨⟨x, hx⟩, hxu x hx⟩
     · simp only [← supr_eq_Union, iSup_and']
       exact
-        isClosed_biUnion
+        Set.Finite.isClosed_biUnion
           (((finite_le_nat n).preimage_embedding (Encodable.encode' _)).Subset <|
             inter_subset_right _ _)
           fun u hu => isClosed_closure

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -1772,9 +1772,9 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
 
 end
 
-#print locally_compact_of_compact_nhds /-
-theorem locally_compact_of_compact_nhds [T2Space α] (h : ∀ x : α, ∃ s, s ∈ 𝓝 x ∧ IsCompact s) :
-    LocallyCompactSpace α :=
+#print WeaklyLocallyCompactSpace.locallyCompactSpace /-
+theorem WeaklyLocallyCompactSpace.locallyCompactSpace [T2Space α]
+    (h : ∀ x : α, ∃ s, s ∈ 𝓝 x ∧ IsCompact s) : LocallyCompactSpace α :=
   ⟨fun x n hn =>
     let ⟨u, un, uo, xu⟩ := mem_nhds_iff.mp hn
     let ⟨k, kx, kc⟩ := h x
@@ -1788,14 +1788,15 @@ theorem locally_compact_of_compact_nhds [T2Space α] (h : ∀ x : α, ∃ s, s 
     have wn : wᶜ ∈ 𝓝 x :=
       mem_nhds_iff.mpr ⟨v, vw.subset_compl_right, vo, singleton_subset_iff.mp xv⟩
     ⟨k \ w, Filter.inter_mem kx wn, Subset.trans (diff_subset_comm.mp kuw) un, kc.diffₓ wo⟩⟩
-#align locally_compact_of_compact_nhds locally_compact_of_compact_nhds
+#align locally_compact_of_compact_nhds WeaklyLocallyCompactSpace.locallyCompactSpace
 -/
 
 #print locally_compact_of_compact /-
 -- see Note [lower instance priority]
 instance (priority := 100) locally_compact_of_compact [T2Space α] [CompactSpace α] :
     LocallyCompactSpace α :=
-  locally_compact_of_compact_nhds fun x => ⟨univ, isOpen_univ.mem_nhds trivial, isCompact_univ⟩
+  WeaklyLocallyCompactSpace.locallyCompactSpace fun x =>
+    ⟨univ, isOpen_univ.mem_nhds trivial, isCompact_univ⟩
 #align locally_compact_of_compact locally_compact_of_compact
 -/
 
@@ -1804,7 +1805,7 @@ instance (priority := 100) locally_compact_of_compact [T2Space α] [CompactSpace
 theorem exists_open_with_compact_closure [LocallyCompactSpace α] [T2Space α] (x : α) :
     ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) :=
   by
-  rcases exists_compact_mem_nhds x with ⟨K, hKc, hxK⟩
+  rcases WeaklyLocallyCompactSpace.exists_compact_mem_nhds x with ⟨K, hKc, hxK⟩
   rcases mem_nhds_iff.1 hxK with ⟨t, h1t, h2t, h3t⟩
   exact ⟨t, h2t, h3t, isCompact_closure_of_subset_compact hKc h1t⟩
 #align exists_open_with_compact_closure exists_open_with_compact_closure

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -1848,7 +1848,7 @@ theorem isPreirreducible_iff_subsingleton [T2Space α] {S : Set α} :
 #align is_preirreducible_iff_subsingleton isPreirreducible_iff_subsingleton
 -/
 
-alias isPreirreducible_iff_subsingleton ↔ IsPreirreducible.subsingleton _
+alias ⟨IsPreirreducible.subsingleton, _⟩ := isPreirreducible_iff_subsingleton
 #align is_preirreducible.subsingleton IsPreirreducible.subsingleton
 
 attribute [protected] IsPreirreducible.subsingleton
@@ -2018,7 +2018,7 @@ theorem specializes_comm {a b : α} : a ⤳ b ↔ b ⤳ a := by
 #align specializes_comm specializes_comm
 -/
 
-alias specializes_comm ↔ Specializes.symm _
+alias ⟨Specializes.symm, _⟩ := specializes_comm
 #align specializes.symm Specializes.symm
 
 #print specializes_iff_inseparable /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,17 +2,14 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
-
-! This file was ported from Lean 3 source module topology.separation
-! leanprover-community/mathlib commit d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.SubsetProperties
 import Mathbin.Topology.Connected
 import Mathbin.Topology.NhdsSet
 import Mathbin.Topology.Inseparable
 
+#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
+
 /-!
 # Separation properties of topological spaces.
 
@@ -309,7 +306,7 @@ instance : T0Space (SeparationQuotient α) :=
     Quotient.inductionOn₂' x' y' fun x y h =>
       SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.inducing_mk.inseparable_iff.1 h⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_subsingleton /-
 theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : s.Subsingleton :=
@@ -324,7 +321,7 @@ theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsC
 #align minimal_nonempty_closed_subsingleton minimal_nonempty_closed_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_eq_singleton /-
 theorem minimal_nonempty_closed_eq_singleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} :=
@@ -346,7 +343,7 @@ theorem IsClosed.exists_closed_singleton {α : Type _} [TopologicalSpace α] [T0
 #align is_closed.exists_closed_singleton IsClosed.exists_closed_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_subsingleton /-
 theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : s.Subsingleton :=
@@ -361,7 +358,7 @@ theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_subsingleton minimal_nonempty_open_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_eq_singleton /-
 theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} :=
@@ -369,7 +366,7 @@ theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_eq_singleton minimal_nonempty_open_eq_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊂ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊂ » s) -/
 #print exists_open_singleton_of_open_finite /-
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
 theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s.Finite)
@@ -1885,7 +1882,7 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
 #align regular_space RegularSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
 #print regularSpace_TFAE /-
 theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     TFAE
@@ -2147,8 +2144,8 @@ instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [
     T3Space (∀ i, π i) :=
   ⟨⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
 #print disjoint_nested_nhds /-
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
 with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -113,10 +113,12 @@ def SeparatedNhds : Set α → Set α → Prop := fun s t : Set α =>
 #align separated_nhds SeparatedNhds
 -/
 
+#print separatedNhds_iff_disjoint /-
 theorem separatedNhds_iff_disjoint {s t : Set α} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by
   simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, ←
     exists_and_left, and_assoc, and_comm, and_left_comm]
 #align separated_nhds_iff_disjoint separatedNhds_iff_disjoint
+-/
 
 namespace SeparatedNhds
 
@@ -144,19 +146,25 @@ theorem preimage [TopologicalSpace β] {f : α → β} {s t : Set β} (h : Separ
 #align separated_nhds.preimage SeparatedNhds.preimage
 -/
 
+#print SeparatedNhds.disjoint /-
 protected theorem disjoint (h : SeparatedNhds s t) : Disjoint s t :=
   let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h
   hd.mono hsU htV
 #align separated_nhds.disjoint SeparatedNhds.disjoint
+-/
 
+#print SeparatedNhds.disjoint_closure_left /-
 theorem disjoint_closure_left (h : SeparatedNhds s t) : Disjoint (closure s) t :=
   let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h
   (hd.closure_left hV).mono (closure_mono hsU) htV
 #align separated_nhds.disjoint_closure_left SeparatedNhds.disjoint_closure_left
+-/
 
+#print SeparatedNhds.disjoint_closure_right /-
 theorem disjoint_closure_right (h : SeparatedNhds s t) : Disjoint s (closure t) :=
   h.symm.disjoint_closure_left.symm
 #align separated_nhds.disjoint_closure_right SeparatedNhds.disjoint_closure_right
+-/
 
 #print SeparatedNhds.empty_right /-
 theorem empty_right (s : Set α) : SeparatedNhds s ∅ :=
@@ -177,13 +185,17 @@ theorem mono (h : SeparatedNhds s₂ t₂) (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t
 #align separated_nhds.mono SeparatedNhds.mono
 -/
 
+#print SeparatedNhds.union_left /-
 theorem union_left : SeparatedNhds s u → SeparatedNhds t u → SeparatedNhds (s ∪ t) u := by
   simpa only [separatedNhds_iff_disjoint, nhdsSet_union, disjoint_sup_left] using And.intro
 #align separated_nhds.union_left SeparatedNhds.union_left
+-/
 
+#print SeparatedNhds.union_right /-
 theorem union_right (ht : SeparatedNhds s t) (hu : SeparatedNhds s u) : SeparatedNhds s (t ∪ u) :=
   (ht.symm.union_left hu.symm).symm
 #align separated_nhds.union_right SeparatedNhds.union_right
+-/
 
 end SeparatedNhds
 
@@ -358,6 +370,7 @@ theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpe
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊂ » s) -/
+#print exists_open_singleton_of_open_finite /-
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
 theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s.Finite)
     (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set α) :=
@@ -373,6 +386,7 @@ theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s
     lift t to Finset α using s.finite_to_set.subset hts
     exact ⟨t, ssubset_iff_subset_ne.2 ⟨hts, mt Finset.coe_inj.2 hts'⟩, htne, hto⟩
 #align exists_open_singleton_of_open_finite exists_open_singleton_of_open_finite
+-/
 
 #print exists_open_singleton_of_finite /-
 theorem exists_open_singleton_of_finite [T0Space α] [Finite α] [Nonempty α] :
@@ -450,9 +464,11 @@ theorem isClosed_singleton [T1Space α] {x : α} : IsClosed ({x} : Set α) :=
 #align is_closed_singleton isClosed_singleton
 -/
 
+#print isOpen_compl_singleton /-
 theorem isOpen_compl_singleton [T1Space α] {x : α} : IsOpen ({x}ᶜ : Set α) :=
   isClosed_singleton.isOpen_compl
 #align is_open_compl_singleton isOpen_compl_singleton
+-/
 
 #print isOpen_ne /-
 theorem isOpen_ne [T1Space α] {x : α} : IsOpen {y | y ≠ x} :=
@@ -469,17 +485,22 @@ theorem Continuous.isOpen_mulSupport [T1Space α] [One α] [TopologicalSpace β]
 #align continuous.is_open_support Continuous.isOpen_support
 -/
 
+#print Ne.nhdsWithin_compl_singleton /-
 theorem Ne.nhdsWithin_compl_singleton [T1Space α] {x y : α} (h : x ≠ y) : 𝓝[{y}ᶜ] x = 𝓝 x :=
   isOpen_ne.nhdsWithin_eq h
 #align ne.nhds_within_compl_singleton Ne.nhdsWithin_compl_singleton
+-/
 
+#print Ne.nhdsWithin_diff_singleton /-
 theorem Ne.nhdsWithin_diff_singleton [T1Space α] {x y : α} (h : x ≠ y) (s : Set α) :
     𝓝[s \ {y}] x = 𝓝[s] x :=
   by
   rw [diff_eq, inter_comm, nhdsWithin_inter_of_mem]
   exact mem_nhdsWithin_of_mem_nhds (is_open_ne.mem_nhds h)
 #align ne.nhds_within_diff_singleton Ne.nhdsWithin_diff_singleton
+-/
 
+#print isOpen_setOf_eventually_nhdsWithin /-
 theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
     IsOpen {x | ∀ᶠ y in 𝓝[≠] x, p y} :=
   by
@@ -490,6 +511,7 @@ theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
   · rw [(Ne.symm h).nhdsWithin_compl_singleton] at hb 
     exact hb.filter_mono nhdsWithin_le_nhds
 #align is_open_set_of_eventually_nhds_within isOpen_setOf_eventually_nhdsWithin
+-/
 
 #print Set.Finite.isClosed /-
 protected theorem Set.Finite.isClosed [T1Space α] {s : Set α} (hs : Set.Finite s) : IsClosed s :=
@@ -499,17 +521,21 @@ protected theorem Set.Finite.isClosed [T1Space α] {s : Set α} (hs : Set.Finite
 #align set.finite.is_closed Set.Finite.isClosed
 -/
 
+#print TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne /-
 theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space α] {b : Set (Set α)}
     (hb : IsTopologicalBasis b) {x y : α} (h : x ≠ y) : ∃ a ∈ b, x ∈ a ∧ y ∉ a :=
   by
   rcases hb.is_open_iff.1 isOpen_ne x h with ⟨a, ab, xa, ha⟩
   exact ⟨a, ab, xa, fun h => ha h rfl⟩
 #align topological_space.is_topological_basis.exists_mem_of_ne TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne
+-/
 
+#print Filter.coclosedCompact_le_cofinite /-
 theorem Filter.coclosedCompact_le_cofinite [T1Space α] :
     Filter.coclosedCompact α ≤ Filter.cofinite := fun s hs =>
   compl_compl s ▸ hs.IsCompact.compl_mem_coclosedCompact_of_isClosed hs.IsClosed
 #align filter.coclosed_compact_le_cofinite Filter.coclosedCompact_le_cofinite
+-/
 
 variable (α)
 
@@ -547,6 +573,7 @@ protected theorem Finset.isClosed [T1Space α] (s : Finset α) : IsClosed (s : S
 #align finset.is_closed Finset.isClosed
 -/
 
+#print t1Space_TFAE /-
 theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
     TFAE
       [T1Space α, ∀ x, IsClosed ({x} : Set α), ∀ x, IsOpen ({x}ᶜ : Set α),
@@ -582,28 +609,39 @@ theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
       mem_singleton_iff, eq_comm]
   tfae_finish
 #align t1_space_tfae t1Space_TFAE
+-/
 
+#print t1Space_iff_continuous_cofinite_of /-
 theorem t1Space_iff_continuous_cofinite_of {α : Type _} [TopologicalSpace α] :
     T1Space α ↔ Continuous (@CofiniteTopology.of α) :=
   (t1Space_TFAE α).out 0 3
 #align t1_space_iff_continuous_cofinite_of t1Space_iff_continuous_cofinite_of
+-/
 
+#print CofiniteTopology.continuous_of /-
 theorem CofiniteTopology.continuous_of [T1Space α] : Continuous (@CofiniteTopology.of α) :=
   t1Space_iff_continuous_cofinite_of.mp ‹_›
 #align cofinite_topology.continuous_of CofiniteTopology.continuous_of
+-/
 
+#print t1Space_iff_exists_open /-
 theorem t1Space_iff_exists_open :
     T1Space α ↔ ∀ x y, x ≠ y → ∃ (U : Set α) (hU : IsOpen U), x ∈ U ∧ y ∉ U :=
   (t1Space_TFAE α).out 0 6
 #align t1_space_iff_exists_open t1Space_iff_exists_open
+-/
 
+#print t1Space_iff_disjoint_pure_nhds /-
 theorem t1Space_iff_disjoint_pure_nhds : T1Space α ↔ ∀ ⦃x y : α⦄, x ≠ y → Disjoint (pure x) (𝓝 y) :=
   (t1Space_TFAE α).out 0 8
 #align t1_space_iff_disjoint_pure_nhds t1Space_iff_disjoint_pure_nhds
+-/
 
+#print t1Space_iff_disjoint_nhds_pure /-
 theorem t1Space_iff_disjoint_nhds_pure : T1Space α ↔ ∀ ⦃x y : α⦄, x ≠ y → Disjoint (𝓝 x) (pure y) :=
   (t1Space_TFAE α).out 0 7
 #align t1_space_iff_disjoint_nhds_pure t1Space_iff_disjoint_nhds_pure
+-/
 
 #print t1Space_iff_specializes_imp_eq /-
 theorem t1Space_iff_specializes_imp_eq : T1Space α ↔ ∀ ⦃x y : α⦄, x ⤳ y → x = y :=
@@ -611,13 +649,17 @@ theorem t1Space_iff_specializes_imp_eq : T1Space α ↔ ∀ ⦃x y : α⦄, x 
 #align t1_space_iff_specializes_imp_eq t1Space_iff_specializes_imp_eq
 -/
 
+#print disjoint_pure_nhds /-
 theorem disjoint_pure_nhds [T1Space α] {x y : α} (h : x ≠ y) : Disjoint (pure x) (𝓝 y) :=
   t1Space_iff_disjoint_pure_nhds.mp ‹_› h
 #align disjoint_pure_nhds disjoint_pure_nhds
+-/
 
+#print disjoint_nhds_pure /-
 theorem disjoint_nhds_pure [T1Space α] {x y : α} (h : x ≠ y) : Disjoint (𝓝 x) (pure y) :=
   t1Space_iff_disjoint_nhds_pure.mp ‹_› h
 #align disjoint_nhds_pure disjoint_nhds_pure
+-/
 
 #print Specializes.eq /-
 theorem Specializes.eq [T1Space α] {x y : α} (h : x ⤳ y) : x = y :=
@@ -638,24 +680,30 @@ theorem specializes_eq_eq [T1Space α] : (· ⤳ ·) = @Eq α :=
 #align specializes_eq_eq specializes_eq_eq
 -/
 
+#print pure_le_nhds_iff /-
 @[simp]
 theorem pure_le_nhds_iff [T1Space α] {a b : α} : pure a ≤ 𝓝 b ↔ a = b :=
   specializes_iff_pure.symm.trans specializes_iff_eq
 #align pure_le_nhds_iff pure_le_nhds_iff
+-/
 
+#print nhds_le_nhds_iff /-
 @[simp]
 theorem nhds_le_nhds_iff [T1Space α] {a b : α} : 𝓝 a ≤ 𝓝 b ↔ a = b :=
   specializes_iff_eq
 #align nhds_le_nhds_iff nhds_le_nhds_iff
+-/
 
 instance {α : Type _} : T1Space (CofiniteTopology α) :=
   t1Space_iff_continuous_cofinite_of.mpr continuous_id
 
+#print t1Space_antitone /-
 theorem t1Space_antitone {α : Type _} : Antitone (@T1Space α) :=
   by
   simp only [Antitone, t1Space_iff_continuous_cofinite_of, continuous_iff_le_induced]
   exact fun t₁ t₂ h => h.trans
 #align t1_space_antitone t1Space_antitone
+-/
 
 #print continuousWithinAt_update_of_ne /-
 theorem continuousWithinAt_update_of_ne [T1Space α] [DecidableEq α] [TopologicalSpace β] {f : α → β}
@@ -675,6 +723,7 @@ theorem continuousAt_update_of_ne [T1Space α] [DecidableEq α] [TopologicalSpac
 #align continuous_at_update_of_ne continuousAt_update_of_ne
 -/
 
+#print continuousOn_update_iff /-
 theorem continuousOn_update_iff [T1Space α] [DecidableEq α] [TopologicalSpace β] {f : α → β}
     {s : Set α} {x : α} {y : β} :
     ContinuousOn (Function.update f x y) s ↔
@@ -690,6 +739,7 @@ theorem continuousOn_update_iff [T1Space α] [DecidableEq α] [TopologicalSpace
     exact is_open_ne.mem_nhds hzx
   · exact continuousWithinAt_update_same
 #align continuous_on_update_iff continuousOn_update_iff
+-/
 
 #print t1Space_of_injective_of_continuous /-
 theorem t1Space_of_injective_of_continuous [TopologicalSpace β] {f : α → β}
@@ -726,14 +776,18 @@ instance (priority := 100) T1Space.t0Space [T1Space α] : T0Space α :=
 #align t1_space.t0_space T1Space.t0Space
 -/
 
+#print compl_singleton_mem_nhds_iff /-
 @[simp]
 theorem compl_singleton_mem_nhds_iff [T1Space α] {x y : α} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x :=
   isOpen_compl_singleton.mem_nhds_iffₓ
 #align compl_singleton_mem_nhds_iff compl_singleton_mem_nhds_iff
+-/
 
+#print compl_singleton_mem_nhds /-
 theorem compl_singleton_mem_nhds [T1Space α] {x y : α} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y :=
   compl_singleton_mem_nhds_iff.mpr h
 #align compl_singleton_mem_nhds compl_singleton_mem_nhds
+-/
 
 #print closure_singleton /-
 @[simp]
@@ -756,10 +810,12 @@ theorem subsingleton_closure [T1Space α] {s : Set α} : (closure s).Subsingleto
 #align subsingleton_closure subsingleton_closure
 -/
 
+#print isClosedMap_const /-
 theorem isClosedMap_const {α β} [TopologicalSpace α] [TopologicalSpace β] [T1Space β] {y : β} :
     IsClosedMap (Function.const α y) :=
   IsClosedMap.of_nonempty fun s hs h2s => by simp_rw [h2s.image_const, isClosed_singleton]
 #align is_closed_map_const isClosedMap_const
+-/
 
 #print nhdsWithin_insert_of_ne /-
 theorem nhdsWithin_insert_of_ne [T1Space α] {x y : α} {s : Set α} (hxy : x ≠ y) :
@@ -787,6 +843,7 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
 #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert
 -/
 
+#print biInter_basis_nhds /-
 theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι → Set α} {x : α}
     (h : (𝓝 x).HasBasis p s) : (⋂ (i) (h : p i), s i) = {x} :=
   by
@@ -796,12 +853,16 @@ theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι
   rcases h.mem_iff.1 (compl_singleton_mem_nhds hy.symm) with ⟨i, hi, hsub⟩
   exact ⟨i, hi, fun h => hsub h rfl⟩
 #align bInter_basis_nhds biInter_basis_nhds
+-/
 
+#print compl_singleton_mem_nhdsSet_iff /-
 @[simp]
 theorem compl_singleton_mem_nhdsSet_iff [T1Space α] {x : α} {s : Set α} : {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s := by
   rwa [is_open_compl_singleton.mem_nhds_set, subset_compl_singleton_iff]
 #align compl_singleton_mem_nhds_set_iff compl_singleton_mem_nhdsSet_iff
+-/
 
+#print nhdsSet_le_iff /-
 @[simp]
 theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t :=
   by
@@ -812,6 +873,7 @@ theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s
   by_contra hxt
   exact h hxt hx
 #align nhds_set_le_iff nhdsSet_le_iff
+-/
 
 #print nhdsSet_inj_iff /-
 @[simp]
@@ -826,21 +888,28 @@ theorem injective_nhdsSet [T1Space α] : Function.Injective (𝓝ˢ : Set α →
 #align injective_nhds_set injective_nhdsSet
 -/
 
+#print strictMono_nhdsSet /-
 theorem strictMono_nhdsSet [T1Space α] : StrictMono (𝓝ˢ : Set α → Filter α) :=
   monotone_nhdsSet.strictMono_of_injective injective_nhdsSet
 #align strict_mono_nhds_set strictMono_nhdsSet
+-/
 
+#print nhds_le_nhdsSet_iff /-
 @[simp]
 theorem nhds_le_nhdsSet_iff [T1Space α] {s : Set α} {x : α} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s := by
   rw [← nhdsSet_singleton, nhdsSet_le_iff, singleton_subset_iff]
 #align nhds_le_nhds_set_iff nhds_le_nhdsSet_iff
+-/
 
+#print Dense.diff_singleton /-
 /-- Removing a non-isolated point from a dense set, one still obtains a dense set. -/
 theorem Dense.diff_singleton [T1Space α] {s : Set α} (hs : Dense s) (x : α) [NeBot (𝓝[≠] x)] :
     Dense (s \ {x}) :=
   hs.inter_of_open_right (dense_compl_singleton x) isOpen_compl_singleton
 #align dense.diff_singleton Dense.diff_singleton
+-/
 
+#print Dense.diff_finset /-
 /-- Removing a finset from a dense set in a space without isolated points, one still
 obtains a dense set. -/
 theorem Dense.diff_finset [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Set α} (hs : Dense s)
@@ -851,7 +920,9 @@ theorem Dense.diff_finset [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Se
   · rw [Finset.coe_insert, ← union_singleton, ← diff_diff]
     exact ih.diff_singleton _
 #align dense.diff_finset Dense.diff_finset
+-/
 
+#print Dense.diff_finite /-
 /-- Removing a finite set from a dense set in a space without isolated points, one still
 obtains a dense set. -/
 theorem Dense.diff_finite [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Set α} (hs : Dense s)
@@ -860,6 +931,7 @@ theorem Dense.diff_finite [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Se
   convert hs.diff_finset ht.to_finset
   exact (finite.coe_to_finset _).symm
 #align dense.diff_finite Dense.diff_finite
+-/
 
 #print eq_of_tendsto_nhds /-
 /-- If a function to a `t1_space` tends to some limit `b` at some point `a`, then necessarily
@@ -873,10 +945,12 @@ theorem eq_of_tendsto_nhds [TopologicalSpace β] [T1Space β] {f : α → β} {a
 #align eq_of_tendsto_nhds eq_of_tendsto_nhds
 -/
 
+#print Filter.Tendsto.eventually_ne /-
 theorem Filter.Tendsto.eventually_ne [TopologicalSpace β] [T1Space β] {α : Type _} {g : α → β}
     {l : Filter α} {b₁ b₂ : β} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ :=
   hg.Eventually (isOpen_compl_singleton.eventually_mem hb)
 #align filter.tendsto.eventually_ne Filter.Tendsto.eventually_ne
+-/
 
 #print ContinuousAt.eventually_ne /-
 theorem ContinuousAt.eventually_ne [TopologicalSpace β] [T1Space β] {g : α → β} {a : α} {b : β}
@@ -914,6 +988,7 @@ theorem isOpen_singleton_of_finite_mem_nhds {α : Type _} [TopologicalSpace α]
 #align is_open_singleton_of_finite_mem_nhds isOpen_singleton_of_finite_mem_nhds
 -/
 
+#print infinite_of_mem_nhds /-
 /-- If the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is
 infinite. -/
 theorem infinite_of_mem_nhds {α} [TopologicalSpace α] [T1Space α] (x : α) [hx : NeBot (𝓝[≠] x)]
@@ -923,6 +998,7 @@ theorem infinite_of_mem_nhds {α} [TopologicalSpace α] [T1Space α] (x : α) [h
   rw [← isOpen_singleton_iff_punctured_nhds]
   exact isOpen_singleton_of_finite_mem_nhds x hs hsf
 #align infinite_of_mem_nhds infinite_of_mem_nhds
+-/
 
 #print discrete_of_t1_of_finite /-
 theorem discrete_of_t1_of_finite {X : Type _} [TopologicalSpace X] [T1Space X] [Finite X] :
@@ -979,6 +1055,7 @@ theorem nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α} (h
 #align nhds_within_of_mem_discrete nhdsWithin_of_mem_discrete
 -/
 
+#print Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete /-
 theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type _} {p : ι → Prop}
     {t : ι → Set α} {s : Set α} [DiscreteTopology s] {x : α} (hb : (𝓝 x).HasBasis p t)
     (hx : x ∈ s) : ∃ (i : _) (hi : p i), t i ∩ s = {x} :=
@@ -988,14 +1065,18 @@ theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type _}
   exact
     ⟨i, hi, subset.antisymm hix <| singleton_subset_iff.2 ⟨mem_of_mem_nhds <| hb.mem_of_mem hi, hx⟩⟩
 #align filter.has_basis.exists_inter_eq_singleton_of_mem_discrete Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete
+-/
 
+#print nhds_inter_eq_singleton_of_mem_discrete /-
 /-- A point `x` in a discrete subset `s` of a topological space admits a neighbourhood
 that only meets `s` at `x`.  -/
 theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α}
     (hx : x ∈ s) : ∃ U ∈ 𝓝 x, U ∩ s = {x} := by
   simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx
 #align nhds_inter_eq_singleton_of_mem_discrete nhds_inter_eq_singleton_of_mem_discrete
+-/
 
+#print disjoint_nhdsWithin_of_mem_discrete /-
 /-- For point `x` in a discrete subset `s` of a topological space, there is a set `U`
 such that
 1. `U` is a punctured neighborhood of `x` (ie. `U ∪ {x}` is a neighbourhood of `x`),
@@ -1007,6 +1088,7 @@ theorem disjoint_nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x
   ⟨{x}ᶜ ∩ V, inter_mem_nhdsWithin _ h,
     disjoint_iff_inter_eq_empty.mpr (by rw [inter_assoc, h', compl_inter_self])⟩
 #align disjoint_nhds_within_of_mem_discrete disjoint_nhdsWithin_of_mem_discrete
+-/
 
 #print TopologicalSpace.subset_trans /-
 /-- Let `X` be a topological space and let `s, t ⊆ X` be two subsets.  If there is an inclusion
@@ -1034,38 +1116,51 @@ class T2Space (α : Type u) [TopologicalSpace α] : Prop where
 #align t2_space T2Space
 -/
 
+#print t2_separation /-
 /-- Two different points can be separated by open sets. -/
 theorem t2_separation [T2Space α] {x y : α} (h : x ≠ y) :
     ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
   T2Space.t2 x y h
 #align t2_separation t2_separation
+-/
 
+#print t2Space_iff_disjoint_nhds /-
 theorem t2Space_iff_disjoint_nhds : T2Space α ↔ ∀ x y : α, x ≠ y → Disjoint (𝓝 x) (𝓝 y) :=
   by
   refine' (t2Space_iff α).trans (forall₃_congr fun x y hne => _)
   simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y), exists_prop, ← exists_and_left,
     and_assoc, and_comm', and_left_comm]
 #align t2_space_iff_disjoint_nhds t2Space_iff_disjoint_nhds
+-/
 
+#print disjoint_nhds_nhds /-
 @[simp]
 theorem disjoint_nhds_nhds [T2Space α] {x y : α} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y :=
   ⟨fun hd he => by simpa [he, nhds_ne_bot.ne] using hd, t2Space_iff_disjoint_nhds.mp ‹_› x y⟩
 #align disjoint_nhds_nhds disjoint_nhds_nhds
+-/
 
+#print pairwise_disjoint_nhds /-
 theorem pairwise_disjoint_nhds [T2Space α] : Pairwise (Disjoint on (𝓝 : α → Filter α)) := fun x y =>
   disjoint_nhds_nhds.2
 #align pairwise_disjoint_nhds pairwise_disjoint_nhds
+-/
 
+#print Set.pairwiseDisjoint_nhds /-
 protected theorem Set.pairwiseDisjoint_nhds [T2Space α] (s : Set α) : s.PairwiseDisjoint 𝓝 :=
   pairwise_disjoint_nhds.set_pairwise s
 #align set.pairwise_disjoint_nhds Set.pairwiseDisjoint_nhds
+-/
 
+#print Set.Finite.t2_separation /-
 /-- Points of a finite set can be separated by open sets from each other. -/
 theorem Set.Finite.t2_separation [T2Space α] {s : Set α} (hs : s.Finite) :
     ∃ U : α → Set α, (∀ x, x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U :=
   s.pairwise_disjoint_nhds.exists_mem_filter_basisₓ hs nhds_basis_opens
 #align set.finite.t2_separation Set.Finite.t2_separation
+-/
 
+#print isOpen_setOf_disjoint_nhds_nhds /-
 theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen {p : α × α | Disjoint (𝓝 p.1) (𝓝 p.2)} :=
   by
   simp only [isOpen_iff_mem_nhds, Prod.forall, mem_set_of_eq]
@@ -1076,6 +1171,7 @@ theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen {p : α × α | Disjoint (𝓝
       ⟨U, hU.2.mem_nhds hU.1, V, hV.2.mem_nhds hV.1, fun ⟨x', y'⟩ ⟨hx', hy'⟩ =>
         disjoint_of_disjoint_of_mem hd (hU.2.mem_nhds hx') (hV.2.mem_nhds hy')⟩
 #align is_open_set_of_disjoint_nhds_nhds isOpen_setOf_disjoint_nhds_nhds
+-/
 
 #print T2Space.t1Space /-
 -- see Note [lower instance priority]
@@ -1085,49 +1181,66 @@ instance (priority := 100) T2Space.t1Space [T2Space α] : T1Space α :=
 #align t2_space.t1_space T2Space.t1Space
 -/
 
+#print t2_iff_nhds /-
 /-- A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. -/
 theorem t2_iff_nhds : T2Space α ↔ ∀ {x y : α}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
   simp only [t2Space_iff_disjoint_nhds, disjoint_iff, ne_bot_iff, Ne.def, not_imp_comm]
 #align t2_iff_nhds t2_iff_nhds
+-/
 
+#print eq_of_nhds_neBot /-
 theorem eq_of_nhds_neBot [T2Space α] {x y : α} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y :=
   t2_iff_nhds.mp ‹_› h
 #align eq_of_nhds_ne_bot eq_of_nhds_neBot
+-/
 
+#print t2Space_iff_nhds /-
 theorem t2Space_iff_nhds : T2Space α ↔ ∀ {x y : α}, x ≠ y → ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V := by
   simp only [t2Space_iff_disjoint_nhds, Filter.disjoint_iff]
 #align t2_space_iff_nhds t2Space_iff_nhds
+-/
 
+#print t2_separation_nhds /-
 theorem t2_separation_nhds [T2Space α] {x y : α} (h : x ≠ y) :
     ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ Disjoint u v :=
   let ⟨u, v, open_u, open_v, x_in, y_in, huv⟩ := t2_separation h
   ⟨u, v, open_u.mem_nhds x_in, open_v.mem_nhds y_in, huv⟩
 #align t2_separation_nhds t2_separation_nhds
+-/
 
+#print t2_separation_compact_nhds /-
 theorem t2_separation_compact_nhds [LocallyCompactSpace α] [T2Space α] {x y : α} (h : x ≠ y) :
     ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v := by
   simpa only [exists_prop, ← exists_and_left, and_comm', and_assoc, and_left_comm] using
     ((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h)
 #align t2_separation_compact_nhds t2_separation_compact_nhds
+-/
 
+#print t2_iff_ultrafilter /-
 theorem t2_iff_ultrafilter :
     T2Space α ↔ ∀ {x y : α} (f : Ultrafilter α), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y :=
   t2_iff_nhds.trans <| by simp only [← exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp]
 #align t2_iff_ultrafilter t2_iff_ultrafilter
+-/
 
+#print t2_iff_isClosed_diagonal /-
 theorem t2_iff_isClosed_diagonal : T2Space α ↔ IsClosed (diagonal α) := by
   simp only [t2Space_iff_disjoint_nhds, ← isOpen_compl_iff, isOpen_iff_mem_nhds, Prod.forall,
     nhds_prod_eq, compl_diagonal_mem_prod, mem_compl_iff, mem_diagonal_iff]
 #align t2_iff_is_closed_diagonal t2_iff_isClosed_diagonal
+-/
 
+#print isClosed_diagonal /-
 theorem isClosed_diagonal [T2Space α] : IsClosed (diagonal α) :=
   t2_iff_isClosed_diagonal.mp ‹_›
 #align is_closed_diagonal isClosed_diagonal
+-/
 
 section Separated
 
 open SeparatedNhds Finset
 
+#print finset_disjoint_finset_opens_of_t2 /-
 theorem finset_disjoint_finset_opens_of_t2 [T2Space α] :
     ∀ s t : Finset α, Disjoint s t → SeparatedNhds (s : Set α) t :=
   by
@@ -1140,6 +1253,7 @@ theorem finset_disjoint_finset_opens_of_t2 [T2Space α] :
     apply_mod_cast union_left (ac (disjoint_of_subset_left (a.subset_union_left b) d)) (bc _)
     exact disjoint_of_subset_left (a.subset_union_right b) d
 #align finset_disjoint_finset_opens_of_t2 finset_disjoint_finset_opens_of_t2
+-/
 
 #print point_disjoint_finset_opens_of_t2 /-
 theorem point_disjoint_finset_opens_of_t2 [T2Space α] {x : α} {s : Finset α} (h : x ∉ s) :
@@ -1188,11 +1302,13 @@ class T25Space (α : Type u) [TopologicalSpace α] : Prop where
 #align t2_5_space T25Space
 -/
 
+#print disjoint_lift'_closure_nhds /-
 @[simp]
 theorem disjoint_lift'_closure_nhds [T25Space α] {x y : α} :
     Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure) ↔ x ≠ y :=
   ⟨fun h hxy => by simpa [hxy, nhds_ne_bot.ne] using h, fun h => T25Space.t2_5 h⟩
 #align disjoint_lift'_closure_nhds disjoint_lift'_closure_nhds
+-/
 
 #print T25Space.t2Space /-
 -- see Note [lower instance priority]
@@ -1202,12 +1318,15 @@ instance (priority := 100) T25Space.t2Space [T25Space α] : T2Space α :=
 #align t2_5_space.t2_space T25Space.t2Space
 -/
 
+#print exists_nhds_disjoint_closure /-
 theorem exists_nhds_disjoint_closure [T25Space α] {x y : α} (h : x ≠ y) :
     ∃ s ∈ 𝓝 x, ∃ t ∈ 𝓝 y, Disjoint (closure s) (closure t) :=
   ((𝓝 x).basis_sets.lift'_closure.disjoint_iff (𝓝 y).basis_sets.lift'_closure).1 <|
     disjoint_lift'_closure_nhds.2 h
 #align exists_nhds_disjoint_closure exists_nhds_disjoint_closure
+-/
 
+#print exists_open_nhds_disjoint_closure /-
 theorem exists_open_nhds_disjoint_closure [T25Space α] {x y : α} (h : x ≠ y) :
     ∃ u : Set α,
       x ∈ u ∧ IsOpen u ∧ ∃ v : Set α, y ∈ v ∧ IsOpen v ∧ Disjoint (closure u) (closure v) :=
@@ -1216,6 +1335,7 @@ theorem exists_open_nhds_disjoint_closure [T25Space α] {x y : α} (h : x ≠ y)
     ((nhds_basis_opens x).lift'_closure.disjoint_iff (nhds_basis_opens y).lift'_closure).1
       (disjoint_lift'_closure_nhds.2 h)
 #align exists_open_nhds_disjoint_closure exists_open_nhds_disjoint_closure
+-/
 
 section limUnder
 
@@ -1229,18 +1349,24 @@ are useful without a `nonempty α` instance.
 -/
 
 
+#print lim_eq /-
 theorem lim_eq {a : α} [NeBot f] (h : f ≤ 𝓝 a) : @lim _ _ ⟨a⟩ f = a :=
   tendsto_nhds_unique (le_nhds_lim ⟨a, h⟩) h
 #align Lim_eq lim_eq
+-/
 
+#print lim_eq_iff /-
 theorem lim_eq_iff [NeBot f] (h : ∃ a : α, f ≤ nhds a) {a} : @lim _ _ ⟨a⟩ f = a ↔ f ≤ 𝓝 a :=
   ⟨fun c => c ▸ le_nhds_lim h, lim_eq⟩
 #align Lim_eq_iff lim_eq_iff
+-/
 
+#print Ultrafilter.lim_eq_iff_le_nhds /-
 theorem Ultrafilter.lim_eq_iff_le_nhds [CompactSpace α] {x : α} {F : Ultrafilter α} :
     F.lim = x ↔ ↑F ≤ 𝓝 x :=
   ⟨fun h => h ▸ F.le_nhds_lim, lim_eq⟩
 #align ultrafilter.Lim_eq_iff_le_nhds Ultrafilter.lim_eq_iff_le_nhds
+-/
 
 #print isOpen_iff_ultrafilter' /-
 theorem isOpen_iff_ultrafilter' [CompactSpace α] (U : Set α) :
@@ -1332,13 +1458,16 @@ instance (priority := 100) DiscreteTopology.toT2Space {α : Type _} [Topological
 #align discrete_topology.to_t2_space DiscreteTopology.toT2Space
 -/
 
+#print separated_by_continuous /-
 theorem separated_by_continuous {α : Type _} {β : Type _} [TopologicalSpace α] [TopologicalSpace β]
     [T2Space β] {f : α → β} (hf : Continuous f) {x y : α} (h : f x ≠ f y) :
     ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
   let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
   ⟨f ⁻¹' u, f ⁻¹' v, uo.Preimage hf, vo.Preimage hf, xu, yv, uv.Preimage _⟩
 #align separated_by_continuous separated_by_continuous
+-/
 
+#print separated_by_openEmbedding /-
 theorem separated_by_openEmbedding {α β : Type _} [TopologicalSpace α] [TopologicalSpace β]
     [T2Space α] {f : α → β} (hf : OpenEmbedding f) {x y : α} (h : x ≠ y) :
     ∃ u v : Set β, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v :=
@@ -1346,6 +1475,7 @@ theorem separated_by_openEmbedding {α β : Type _} [TopologicalSpace α] [Topol
   ⟨f '' u, f '' v, hf.IsOpenMap _ uo, hf.IsOpenMap _ vo, mem_image_of_mem _ xu,
     mem_image_of_mem _ yv, disjoint_image_of_injective hf.inj uv⟩
 #align separated_by_open_embedding separated_by_openEmbedding
+-/
 
 instance {α : Type _} {p : α → Prop} [t : TopologicalSpace α] [T2Space α] : T2Space (Subtype p) :=
   ⟨fun x y h => separated_by_continuous continuous_subtype_val (mt Subtype.eq h)⟩
@@ -1390,6 +1520,7 @@ instance Pi.t2Space {α : Type _} {β : α → Type v} [t₂ : ∀ a, Topologica
 #align Pi.t2_space Pi.t2Space
 -/
 
+#print Sigma.t2Space /-
 instance Sigma.t2Space {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)]
     [∀ a, T2Space (α a)] : T2Space (Σ i, α i) :=
   by
@@ -1403,6 +1534,7 @@ instance Sigma.t2Space {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpa
       ⟨_, _, isOpen_range_sigmaMk, isOpen_range_sigmaMk, ⟨x, rfl⟩, ⟨y, rfl⟩,
         set.disjoint_left.mpr <| by tidy⟩
 #align sigma.t2_space Sigma.t2Space
+-/
 
 variable {γ : Type _} [TopologicalSpace β] [TopologicalSpace γ]
 
@@ -1437,6 +1569,7 @@ theorem Continuous.ext_on [T2Space α] {s : Set β} (hs : Dense s) {f g : β →
 #align continuous.ext_on Continuous.ext_on
 -/
 
+#print eqOn_closure₂' /-
 theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ → α}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf₁ : ∀ x, Continuous (f x))
     (hf₂ : ∀ y, Continuous fun x => f x y) (hg₁ : ∀ x, Continuous (g x))
@@ -1445,13 +1578,16 @@ theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ
   (closure_minimal fun x hx => mem_iInter₂.2 <| Set.EqOn.closure (h x hx) (hf₁ _) (hg₁ _)) <|
     isClosed_biInter fun y hy => isClosed_eq (hf₂ _) (hg₂ _)
 #align eq_on_closure₂' eqOn_closure₂'
+-/
 
+#print eqOn_closure₂ /-
 theorem eqOn_closure₂ [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ → α}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (uncurry f))
     (hg : Continuous (uncurry g)) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
   eqOn_closure₂' h (fun x => continuous_uncurry_left x hf) (fun x => continuous_uncurry_right x hf)
     (fun y => continuous_uncurry_left y hg) fun y => continuous_uncurry_right y hg
 #align eq_on_closure₂ eqOn_closure₂
+-/
 
 #print Set.EqOn.of_subset_closure /-
 /-- If `f x = g x` for all `x ∈ s` and `f`, `g` are continuous on `t`, `s ⊆ t ⊆ closure s`, then
@@ -1486,11 +1622,13 @@ theorem Function.LeftInverse.closedEmbedding [T2Space α] {f : α → β} {g : 
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 -/
 
+#print isCompact_isCompact_separated /-
 theorem isCompact_isCompact_separated [T2Space α] {s t : Set α} (hs : IsCompact s)
     (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t := by
   simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst ⊢ <;>
     exact generalized_tube_lemma hs ht is_closed_diagonal.is_open_compl hst
 #align is_compact_is_compact_separated isCompact_isCompact_separated
+-/
 
 #print IsCompact.isClosed /-
 /-- In a `t2_space`, every compact set is closed. -/
@@ -1518,6 +1656,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 #align bornology.relatively_compact_eq_in_compact Bornology.relativelyCompact_eq_inCompact
 -/
 
+#print exists_subset_nhds_of_isCompact /-
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we
 don't need to assume each `V i` closed because it follows from compactness since `α` is
@@ -1527,6 +1666,7 @@ theorem exists_subset_nhds_of_isCompact [T2Space α] {ι : Type _} [Nonempty ι]
     (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U :=
   exists_subset_nhds_of_isCompact' hV hV_cpct (fun i => (hV_cpct i).IsClosed) hU
 #align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompact
+-/
 
 #print CompactExhaustion.isClosed /-
 theorem CompactExhaustion.isClosed [T2Space α] (K : CompactExhaustion α) (n : ℕ) : IsClosed (K n) :=
@@ -1534,10 +1674,12 @@ theorem CompactExhaustion.isClosed [T2Space α] (K : CompactExhaustion α) (n :
 #align compact_exhaustion.is_closed CompactExhaustion.isClosed
 -/
 
+#print IsCompact.inter /-
 theorem IsCompact.inter [T2Space α] {s t : Set α} (hs : IsCompact s) (ht : IsCompact t) :
     IsCompact (s ∩ t) :=
   hs.inter_right <| ht.IsClosed
 #align is_compact.inter IsCompact.inter
+-/
 
 #print isCompact_closure_of_subset_compact /-
 theorem isCompact_closure_of_subset_compact [T2Space α] {s t : Set α} (ht : IsCompact t)
@@ -1563,6 +1705,7 @@ theorem image_closure_of_isCompact [T2Space β] {s : Set α} (hs : IsCompact (cl
 #align image_closure_of_is_compact image_closure_of_isCompact
 -/
 
+#print IsCompact.binary_compact_cover /-
 /-- If a compact set is covered by two open sets, then we can cover it by two compact subsets. -/
 theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCompact K)
     (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
@@ -1575,6 +1718,7 @@ theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCom
     ⟨_, _, hK.diff h1O₁, hK.diff h1O₂, by rwa [diff_subset_comm], by rwa [diff_subset_comm], by
       rw [← diff_inter, hO.inter_eq, diff_empty]⟩
 #align is_compact.binary_compact_cover IsCompact.binary_compact_cover
+-/
 
 #print Continuous.isClosedMap /-
 /-- A continuous map from a compact space to a Hausdorff space is a closed map. -/
@@ -1603,6 +1747,7 @@ section
 
 open Finset Function
 
+#print IsCompact.finite_compact_cover /-
 /-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
 theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact s) {ι} (t : Finset ι)
     (U : ι → Set α) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
@@ -1626,6 +1771,7 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
     · rw [← Ne.def] at hi ; simp only [update_noteq hi, h2K]
   · simp only [set_bUnion_insert_update _ hx, hK, h3K]
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
+-/
 
 end
 
@@ -1740,6 +1886,7 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
+#print regularSpace_TFAE /-
 theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     TFAE
       [RegularSpace X, ∀ (s : Set X) (a) (_ : a ∉ closure s), Disjoint (𝓝ˢ s) (𝓝 a),
@@ -1771,6 +1918,7 @@ theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
   tfae_have 3 → 1; exact fun H => ⟨fun s a hs ha => (H _ _).mpr <| hs.closure_eq.symm ▸ ha⟩
   tfae_finish
 #align regular_space_tfae regularSpace_TFAE
+-/
 
 #print RegularSpace.ofLift'_closure /-
 theorem RegularSpace.ofLift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 𝓝 a) : RegularSpace α :=
@@ -1778,31 +1926,41 @@ theorem RegularSpace.ofLift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 
 #align regular_space.of_lift'_closure RegularSpace.ofLift'_closure
 -/
 
+#print RegularSpace.ofBasis /-
 theorem RegularSpace.ofBasis {ι : α → Sort _} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set α}
     (h₁ : ∀ a, (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ a i, p a i → IsClosed (s a i)) :
     RegularSpace α :=
   RegularSpace.ofLift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a)
 #align regular_space.of_basis RegularSpace.ofBasis
+-/
 
+#print RegularSpace.ofExistsMemNhdsIsClosedSubset /-
 theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
     (h : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s) : RegularSpace α :=
   Iff.mpr ((regularSpace_TFAE α).out 0 3) h
 #align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.ofExistsMemNhdsIsClosedSubset
+-/
 
 variable [RegularSpace α] {a : α} {s : Set α}
 
+#print disjoint_nhdsSet_nhds /-
 theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 a) ↔ a ∉ closure s :=
   Iff.mp ((regularSpace_TFAE α).out 0 2) ‹_› _ _
 #align disjoint_nhds_set_nhds disjoint_nhdsSet_nhds
+-/
 
+#print disjoint_nhds_nhdsSet /-
 theorem disjoint_nhds_nhdsSet : Disjoint (𝓝 a) (𝓝ˢ s) ↔ a ∉ closure s :=
   disjoint_comm.trans disjoint_nhdsSet_nhds
 #align disjoint_nhds_nhds_set disjoint_nhds_nhdsSet
+-/
 
+#print exists_mem_nhds_isClosed_subset /-
 theorem exists_mem_nhds_isClosed_subset {a : α} {s : Set α} (h : s ∈ 𝓝 a) :
     ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s :=
   Iff.mp ((regularSpace_TFAE α).out 0 3) ‹_› _ _ h
 #align exists_mem_nhds_is_closed_subset exists_mem_nhds_isClosed_subset
+-/
 
 #print closed_nhds_basis /-
 theorem closed_nhds_basis (a : α) : (𝓝 a).HasBasis (fun s : Set α => s ∈ 𝓝 a ∧ IsClosed s) id :=
@@ -1816,10 +1974,12 @@ theorem lift'_nhds_closure (a : α) : (𝓝 a).lift' closure = 𝓝 a :=
 #align lift'_nhds_closure lift'_nhds_closure
 -/
 
+#print Filter.HasBasis.nhds_closure /-
 theorem Filter.HasBasis.nhds_closure {ι : Sort _} {a : α} {p : ι → Prop} {s : ι → Set α}
     (h : (𝓝 a).HasBasis p s) : (𝓝 a).HasBasis p fun i => closure (s i) :=
   lift'_nhds_closure a ▸ h.lift'_closure
 #align filter.has_basis.nhds_closure Filter.HasBasis.nhds_closure
+-/
 
 #print hasBasis_nhds_closure /-
 theorem hasBasis_nhds_closure (a : α) : (𝓝 a).HasBasis (fun s => s ∈ 𝓝 a) closure :=
@@ -1841,15 +2001,19 @@ theorem TopologicalSpace.IsTopologicalBasis.nhds_basis_closure {B : Set (Set α)
 #align topological_space.is_topological_basis.nhds_basis_closure TopologicalSpace.IsTopologicalBasis.nhds_basis_closure
 -/
 
+#print TopologicalSpace.IsTopologicalBasis.exists_closure_subset /-
 theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set α)}
     (hB : TopologicalSpace.IsTopologicalBasis B) {a : α} {s : Set α} (h : s ∈ 𝓝 a) :
     ∃ t ∈ B, a ∈ t ∧ closure t ⊆ s := by
   simpa only [exists_prop, and_assoc] using hB.nhds_has_basis.nhds_closure.mem_iff.mp h
 #align topological_space.is_topological_basis.exists_closure_subset TopologicalSpace.IsTopologicalBasis.exists_closure_subset
+-/
 
+#print disjoint_nhds_nhds_iff_not_specializes /-
 theorem disjoint_nhds_nhds_iff_not_specializes {a b : α} : Disjoint (𝓝 a) (𝓝 b) ↔ ¬a ⤳ b := by
   rw [← nhdsSet_singleton, disjoint_nhdsSet_nhds, specializes_iff_mem_closure]
 #align disjoint_nhds_nhds_iff_not_specializes disjoint_nhds_nhds_iff_not_specializes
+-/
 
 #print specializes_comm /-
 theorem specializes_comm {a b : α} : a ⤳ b ↔ b ⤳ a := by
@@ -1866,14 +2030,18 @@ theorem specializes_iff_inseparable {a b : α} : a ⤳ b ↔ Inseparable a b :=
 #align specializes_iff_inseparable specializes_iff_inseparable
 -/
 
+#print isClosed_setOf_specializes /-
 theorem isClosed_setOf_specializes : IsClosed {p : α × α | p.1 ⤳ p.2} := by
   simp only [← isOpen_compl_iff, compl_set_of, ← disjoint_nhds_nhds_iff_not_specializes,
     isOpen_setOf_disjoint_nhds_nhds]
 #align is_closed_set_of_specializes isClosed_setOf_specializes
+-/
 
+#print isClosed_setOf_inseparable /-
 theorem isClosed_setOf_inseparable : IsClosed {p : α × α | Inseparable p.1 p.2} := by
   simp only [← specializes_iff_inseparable, isClosed_setOf_specializes]
 #align is_closed_set_of_inseparable isClosed_setOf_inseparable
+-/
 
 #print Inducing.regularSpace /-
 protected theorem Inducing.regularSpace [TopologicalSpace β] {f : β → α} (hf : Inducing f) :
@@ -1890,6 +2058,7 @@ theorem regularSpace_induced (f : β → α) : @RegularSpace β (induced f ‹_
 #align regular_space_induced regularSpace_induced
 -/
 
+#print regularSpace_sInf /-
 theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @RegularSpace X t) :
     @RegularSpace X (sInf T) := by
   letI := Inf T
@@ -1906,16 +2075,21 @@ theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @
   refine' RegularSpace.ofBasis this fun a If hIf => isClosed_iInter fun i => _
   exact (hIf.2 i).2.mono (sInf_le (i : T).2)
 #align regular_space_Inf regularSpace_sInf
+-/
 
+#print regularSpace_iInf /-
 theorem regularSpace_iInf {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @RegularSpace X (t i)) :
     @RegularSpace X (iInf t) :=
   regularSpace_sInf <| forall_range_iff.mpr h
 #align regular_space_infi regularSpace_iInf
+-/
 
+#print RegularSpace.inf /-
 theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSpace X t₁)
     (h₂ : @RegularSpace X t₂) : @RegularSpace X (t₁ ⊓ t₂) := by rw [inf_eq_iInf];
   exact regularSpace_iInf (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align regular_space.inf RegularSpace.inf
+-/
 
 instance {p : α → Prop} : RegularSpace (Subtype p) :=
   embedding_subtype_val.to_inducing.RegularSpace
@@ -1975,6 +2149,7 @@ instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
+#print disjoint_nested_nhds /-
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
 with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. -/
 theorem disjoint_nested_nhds [T3Space α] {x y : α} (h : x ≠ y) :
@@ -1989,6 +2164,7 @@ theorem disjoint_nested_nhds [T3Space α] {x y : α} (h : x ≠ y) :
     ⟨U₁, mem_of_superset V₁_in h₁, V₁, V₁_in, U₂, mem_of_superset V₂_in h₂, V₂, V₂_in, V₁_closed,
       V₂_closed, U₁_op, U₂_op, h₁, h₂, H⟩
 #align disjoint_nested_nhds disjoint_nested_nhds
+-/
 
 open SeparationQuotient
 
@@ -2013,10 +2189,12 @@ class NormalSpace (α : Type u) [TopologicalSpace α] extends T1Space α : Prop
 #align normal_space NormalSpace
 -/
 
+#print normal_separation /-
 theorem normal_separation [NormalSpace α] {s t : Set α} (H1 : IsClosed s) (H2 : IsClosed t)
     (H3 : Disjoint s t) : SeparatedNhds s t :=
   NormalSpace.normal s t H1 H2 H3
 #align normal_separation normal_separation
+-/
 
 #print normal_exists_closure_subset /-
 theorem normal_exists_closure_subset [NormalSpace α] {s t : Set α} (hs : IsClosed s) (ht : IsOpen t)
@@ -2367,12 +2545,14 @@ theorem isTopologicalBasis_clopen : IsTopologicalBasis {s : Set α | IsClopen s}
 #align is_topological_basis_clopen isTopologicalBasis_clopen
 -/
 
+#print compact_exists_clopen_in_open /-
 /-- Every member of an open set in a compact Hausdorff totally disconnected space
   is contained in a clopen set contained in the open set.  -/
 theorem compact_exists_clopen_in_open {x : α} {U : Set α} (is_open : IsOpen U) (memU : x ∈ U) :
     ∃ (V : Set α) (hV : IsClopen V), x ∈ V ∧ V ⊆ U :=
   (IsTopologicalBasis.mem_nhds_iff isTopologicalBasis_clopen).1 (IsOpen.mem_nhds memU)
 #align compact_exists_clopen_in_open compact_exists_clopen_in_open
+-/
 
 end Profinite
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -1476,7 +1476,6 @@ theorem Function.LeftInverse.closed_range [T2Space α] {f : α → β} {g : β 
     calc
       x = g (f x) := (this hx).symm
       _ ∈ _ := mem_range_self _
-      
 #align function.left_inverse.closed_range Function.LeftInverse.closed_range
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -297,7 +297,7 @@ instance : T0Space (SeparationQuotient α) :=
     Quotient.inductionOn₂' x' y' fun x y h =>
       SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.inducing_mk.inseparable_iff.1 h⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_subsingleton /-
 theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : s.Subsingleton :=
@@ -312,7 +312,7 @@ theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsC
 #align minimal_nonempty_closed_subsingleton minimal_nonempty_closed_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_eq_singleton /-
 theorem minimal_nonempty_closed_eq_singleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} :=
@@ -334,7 +334,7 @@ theorem IsClosed.exists_closed_singleton {α : Type _} [TopologicalSpace α] [T0
 #align is_closed.exists_closed_singleton IsClosed.exists_closed_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_subsingleton /-
 theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : s.Subsingleton :=
@@ -349,7 +349,7 @@ theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_subsingleton minimal_nonempty_open_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_eq_singleton /-
 theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} :=
@@ -357,7 +357,7 @@ theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_eq_singleton minimal_nonempty_open_eq_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊂ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊂ » s) -/
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
 theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s.Finite)
     (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set α) :=
@@ -1740,7 +1740,7 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
 #align regular_space RegularSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
 theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     TFAE
       [RegularSpace X, ∀ (s : Set X) (a) (_ : a ∉ closure s), Disjoint (𝓝ˢ s) (𝓝 a),
@@ -1974,8 +1974,8 @@ instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [
     T3Space (∀ i, π i) :=
   ⟨⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
 with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. -/
 theorem disjoint_nested_nhds [T3Space α] {x y : α} (h : x ≠ y) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -455,7 +455,7 @@ theorem isOpen_compl_singleton [T1Space α] {x : α} : IsOpen ({x}ᶜ : Set α)
 #align is_open_compl_singleton isOpen_compl_singleton
 
 #print isOpen_ne /-
-theorem isOpen_ne [T1Space α] {x : α} : IsOpen { y | y ≠ x } :=
+theorem isOpen_ne [T1Space α] {x : α} : IsOpen {y | y ≠ x} :=
   isOpen_compl_singleton
 #align is_open_ne isOpen_ne
 -/
@@ -481,10 +481,10 @@ theorem Ne.nhdsWithin_diff_singleton [T1Space α] {x y : α} (h : x ≠ y) (s :
 #align ne.nhds_within_diff_singleton Ne.nhdsWithin_diff_singleton
 
 theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
-    IsOpen { x | ∀ᶠ y in 𝓝[≠] x, p y } :=
+    IsOpen {x | ∀ᶠ y in 𝓝[≠] x, p y} :=
   by
   refine' is_open_iff_mem_nhds.mpr fun a ha => _
-  filter_upwards [eventually_nhds_nhds_within.mpr ha]with b hb
+  filter_upwards [eventually_nhds_nhds_within.mpr ha] with b hb
   by_cases a = b
   · subst h; exact hb
   · rw [(Ne.symm h).nhdsWithin_compl_singleton] at hb 
@@ -1066,7 +1066,7 @@ theorem Set.Finite.t2_separation [T2Space α] {s : Set α} (hs : s.Finite) :
   s.pairwise_disjoint_nhds.exists_mem_filter_basisₓ hs nhds_basis_opens
 #align set.finite.t2_separation Set.Finite.t2_separation
 
-theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : α × α | Disjoint (𝓝 p.1) (𝓝 p.2) } :=
+theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen {p : α × α | Disjoint (𝓝 p.1) (𝓝 p.2)} :=
   by
   simp only [isOpen_iff_mem_nhds, Prod.forall, mem_set_of_eq]
   intro x y h
@@ -1408,14 +1408,14 @@ variable {γ : Type _} [TopologicalSpace β] [TopologicalSpace γ]
 
 #print isClosed_eq /-
 theorem isClosed_eq [T2Space α] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
-    IsClosed { x : β | f x = g x } :=
+    IsClosed {x : β | f x = g x} :=
   continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_diagonal
 #align is_closed_eq isClosed_eq
 -/
 
 #print isOpen_ne_fun /-
 theorem isOpen_ne_fun [T2Space α] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
-    IsOpen { x : β | f x ≠ g x } :=
+    IsOpen {x : β | f x ≠ g x} :=
   isOpen_compl_iff.mpr <| isClosed_eq hf hg
 #align is_open_ne_fun isOpen_ne_fun
 -/
@@ -1441,7 +1441,7 @@ theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf₁ : ∀ x, Continuous (f x))
     (hf₂ : ∀ y, Continuous fun x => f x y) (hg₁ : ∀ x, Continuous (g x))
     (hg₂ : ∀ y, Continuous fun x => g x y) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
-  suffices closure s ⊆ ⋂ y ∈ closure t, { x | f x y = g x y } by simpa only [subset_def, mem_Inter]
+  suffices closure s ⊆ ⋂ y ∈ closure t, {x | f x y = g x y} by simpa only [subset_def, mem_Inter]
   (closure_minimal fun x hx => mem_iInter₂.2 <| Set.EqOn.closure (h x hx) (hf₁ _) (hg₁ _)) <|
     isClosed_biInter fun y hy => isClosed_eq (hf₂ _) (hg₂ _)
 #align eq_on_closure₂' eqOn_closure₂'
@@ -1609,23 +1609,23 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
     (U : ι → Set α) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
     ∃ K : ι → Set α, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
   classical
-    induction' t using Finset.induction with x t hx ih generalizing U hU s hs hsC
-    · refine' ⟨fun _ => ∅, fun i => isCompact_empty, fun i => empty_subset _, _⟩
-      simpa only [subset_empty_iff, Union_false, Union_empty] using hsC
-    simp only [Finset.set_biUnion_insert] at hsC 
-    simp only [Finset.mem_insert] at hU 
-    have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU i (Or.inr hi)
-    rcases hs.binary_compact_cover (hU x (Or.inl rfl)) (isOpen_biUnion hU') hsC with
-      ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
-    rcases ih U hU' h1K₂ h2K₂ with ⟨K, h1K, h2K, h3K⟩
-    refine' ⟨update K x K₁, _, _, _⟩
-    · intro i; by_cases hi : i = x
-      · simp only [update_same, hi, h1K₁]
-      · rw [← Ne.def] at hi ; simp only [update_noteq hi, h1K]
-    · intro i; by_cases hi : i = x
-      · simp only [update_same, hi, h2K₁]
-      · rw [← Ne.def] at hi ; simp only [update_noteq hi, h2K]
-    · simp only [set_bUnion_insert_update _ hx, hK, h3K]
+  induction' t using Finset.induction with x t hx ih generalizing U hU s hs hsC
+  · refine' ⟨fun _ => ∅, fun i => isCompact_empty, fun i => empty_subset _, _⟩
+    simpa only [subset_empty_iff, Union_false, Union_empty] using hsC
+  simp only [Finset.set_biUnion_insert] at hsC 
+  simp only [Finset.mem_insert] at hU 
+  have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU i (Or.inr hi)
+  rcases hs.binary_compact_cover (hU x (Or.inl rfl)) (isOpen_biUnion hU') hsC with
+    ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
+  rcases ih U hU' h1K₂ h2K₂ with ⟨K, h1K, h2K, h3K⟩
+  refine' ⟨update K x K₁, _, _, _⟩
+  · intro i; by_cases hi : i = x
+    · simp only [update_same, hi, h1K₁]
+    · rw [← Ne.def] at hi ; simp only [update_noteq hi, h1K]
+  · intro i; by_cases hi : i = x
+    · simp only [update_same, hi, h2K₁]
+    · rw [← Ne.def] at hi ; simp only [update_noteq hi, h2K]
+  · simp only [set_bUnion_insert_update _ hx, hK, h3K]
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 
 end
@@ -1867,12 +1867,12 @@ theorem specializes_iff_inseparable {a b : α} : a ⤳ b ↔ Inseparable a b :=
 #align specializes_iff_inseparable specializes_iff_inseparable
 -/
 
-theorem isClosed_setOf_specializes : IsClosed { p : α × α | p.1 ⤳ p.2 } := by
+theorem isClosed_setOf_specializes : IsClosed {p : α × α | p.1 ⤳ p.2} := by
   simp only [← isOpen_compl_iff, compl_set_of, ← disjoint_nhds_nhds_iff_not_specializes,
     isOpen_setOf_disjoint_nhds_nhds]
 #align is_closed_set_of_specializes isClosed_setOf_specializes
 
-theorem isClosed_setOf_inseparable : IsClosed { p : α × α | Inseparable p.1 p.2 } := by
+theorem isClosed_setOf_inseparable : IsClosed {p : α × α | Inseparable p.1 p.2} := by
   simp only [← specializes_iff_inseparable, isClosed_setOf_specializes]
 #align is_closed_set_of_inseparable isClosed_setOf_inseparable
 
@@ -2290,7 +2290,7 @@ section Profinite
 #print totallySeparatedSpace_of_t1_of_basis_clopen /-
 /-- A T1 space with a clopen basis is totally separated. -/
 theorem totallySeparatedSpace_of_t1_of_basis_clopen [T1Space α]
-    (h : IsTopologicalBasis { s : Set α | IsClopen s }) : TotallySeparatedSpace α :=
+    (h : IsTopologicalBasis {s : Set α | IsClopen s}) : TotallySeparatedSpace α :=
   by
   constructor
   rintro x - y - hxy
@@ -2357,7 +2357,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
 -/
 
 #print isTopologicalBasis_clopen /-
-theorem isTopologicalBasis_clopen : IsTopologicalBasis { s : Set α | IsClopen s } :=
+theorem isTopologicalBasis_clopen : IsTopologicalBasis {s : Set α | IsClopen s} :=
   by
   apply is_topological_basis_of_open_of_nhds fun U (hU : IsClopen U) => hU.1
   intro x U hxU U_op
@@ -2384,7 +2384,7 @@ variable {H : Type _} [TopologicalSpace H] [LocallyCompactSpace H] [T2Space H]
 #print loc_compact_Haus_tot_disc_of_zero_dim /-
 /-- A locally compact Hausdorff totally disconnected space has a basis with clopen elements. -/
 theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
-    IsTopologicalBasis { s : Set H | IsClopen s } :=
+    IsTopologicalBasis {s : Set H | IsClopen s} :=
   by
   refine' is_topological_basis_of_open_of_nhds (fun u hu => hu.1) _
   rintro x U memU hU

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -364,7 +364,7 @@ theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s
   by
   lift s to Finset α using hfin
   induction' s using Finset.strongInductionOn with s ihs
-  rcases em (∃ (t : _)(_ : t ⊂ s), t.Nonempty ∧ IsOpen (t : Set α)) with (⟨t, hts, htne, hto⟩ | ht)
+  rcases em (∃ (t : _) (_ : t ⊂ s), t.Nonempty ∧ IsOpen (t : Set α)) with (⟨t, hts, htne, hto⟩ | ht)
   · rcases ihs t hts htne hto with ⟨x, hxt, hxo⟩
     exact ⟨x, hts.1 hxt, hxo⟩
   · rcases minimal_nonempty_open_eq_singleton ho hne _ with ⟨x, hx⟩
@@ -422,7 +422,7 @@ theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set α, x ∈
   refine' ⟨fun x y hxy => _⟩
   rcases h x y hxy with ⟨s, hxs, hys, hs⟩; skip
   lift x to s using hxs; lift y to s using hys
-  rw [← subtype_inseparable_iff] at hxy
+  rw [← subtype_inseparable_iff] at hxy 
   exact congr_arg coe hxy.eq
 #align t0_space.of_cover T0Space.of_cover
 -/
@@ -487,7 +487,7 @@ theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
   filter_upwards [eventually_nhds_nhds_within.mpr ha]with b hb
   by_cases a = b
   · subst h; exact hb
-  · rw [(Ne.symm h).nhdsWithin_compl_singleton] at hb
+  · rw [(Ne.symm h).nhdsWithin_compl_singleton] at hb 
     exact hb.filter_mono nhdsWithin_le_nhds
 #align is_open_set_of_eventually_nhds_within isOpen_setOf_eventually_nhdsWithin
 
@@ -534,7 +534,7 @@ theorem Bornology.relativelyCompact.isBounded_iff [T1Space α] {s : Set α} :
   rw [Filter.mem_coclosedCompact]
   constructor
   · rintro ⟨t, ht₁, ht₂, hst⟩
-    rw [compl_subset_compl] at hst
+    rw [compl_subset_compl] at hst 
     exact isCompact_of_isClosed_subset ht₂ isClosed_closure (closure_minimal hst ht₁)
   · intro h
     exact ⟨closure s, isClosed_closure, h, compl_subset_compl.mpr subset_closure⟩
@@ -552,7 +552,7 @@ theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
       [T1Space α, ∀ x, IsClosed ({x} : Set α), ∀ x, IsOpen ({x}ᶜ : Set α),
         Continuous (@CofiniteTopology.of α), ∀ ⦃x y : α⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
         ∀ ⦃x y : α⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
-        ∀ ⦃x y : α⦄, x ≠ y → ∃ (U : Set α)(hU : IsOpen U), x ∈ U ∧ y ∉ U,
+        ∀ ⦃x y : α⦄, x ≠ y → ∃ (U : Set α) (hU : IsOpen U), x ∈ U ∧ y ∉ U,
         ∀ ⦃x y : α⦄, x ≠ y → Disjoint (𝓝 x) (pure y), ∀ ⦃x y : α⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
         ∀ ⦃x y : α⦄, x ⤳ y → x = y] :=
   by
@@ -573,7 +573,7 @@ theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
   tfae_have 1 → 4
   · simp only [continuous_def, CofiniteTopology.isOpen_iff']
     rintro H s (rfl | hs)
-    exacts[isOpen_empty, compl_compl s ▸ (@Set.Finite.isClosed _ _ H _ hs).isOpen_compl]
+    exacts [isOpen_empty, compl_compl s ▸ (@Set.Finite.isClosed _ _ H _ hs).isOpen_compl]
   tfae_have 4 → 2
   exact fun h x => (CofiniteTopology.isClosed_iff.2 <| Or.inr (finite_singleton _)).Preimage h
   tfae_have 2 ↔ 10
@@ -593,7 +593,7 @@ theorem CofiniteTopology.continuous_of [T1Space α] : Continuous (@CofiniteTopol
 #align cofinite_topology.continuous_of CofiniteTopology.continuous_of
 
 theorem t1Space_iff_exists_open :
-    T1Space α ↔ ∀ x y, x ≠ y → ∃ (U : Set α)(hU : IsOpen U), x ∈ U ∧ y ∉ U :=
+    T1Space α ↔ ∀ x y, x ≠ y → ∃ (U : Set α) (hU : IsOpen U), x ∈ U ∧ y ∉ U :=
   (t1Space_TFAE α).out 0 6
 #align t1_space_iff_exists_open t1Space_iff_exists_open
 
@@ -683,7 +683,7 @@ theorem continuousOn_update_iff [T1Space α] [DecidableEq α] [TopologicalSpace
   rw [ContinuousOn, ← and_forall_ne x, and_comm']
   refine' and_congr ⟨fun H z hz => _, fun H z hzx hzs => _⟩ (forall_congr' fun hxs => _)
   · specialize H z hz.2 hz.1
-    rw [continuousWithinAt_update_of_ne hz.2] at H
+    rw [continuousWithinAt_update_of_ne hz.2] at H 
     exact H.mono (diff_subset _ _)
   · rw [continuousWithinAt_update_of_ne hzx]
     refine' (H z ⟨hzs, hzx⟩).mono_of_mem (inter_mem_nhdsWithin _ _)
@@ -808,7 +808,7 @@ theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s
   refine' ⟨_, fun h => monotone_nhdsSet h⟩
   simp_rw [Filter.le_def]; intro h x hx
   specialize h ({x}ᶜ)
-  simp_rw [compl_singleton_mem_nhdsSet_iff] at h
+  simp_rw [compl_singleton_mem_nhdsSet_iff] at h 
   by_contra hxt
   exact h hxt hx
 #align nhds_set_le_iff nhdsSet_le_iff
@@ -910,7 +910,7 @@ theorem isOpen_singleton_of_finite_mem_nhds {α : Type _} [TopologicalSpace α]
   have B : IsClosed (s \ {x}) := (hsf.subset (diff_subset _ _)).IsClosed
   have C : (s \ {x})ᶜ ∈ 𝓝 x := B.is_open_compl.mem_nhds fun h => h.2 rfl
   have D : {x} ∈ 𝓝 x := by simpa only [← diff_eq, diff_diff_cancel_left A] using inter_mem hs C
-  rwa [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, subset_interior_iff_isOpen] at D
+  rwa [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, subset_interior_iff_isOpen] at D 
 #align is_open_singleton_of_finite_mem_nhds isOpen_singleton_of_finite_mem_nhds
 -/
 
@@ -939,7 +939,7 @@ theorem PreconnectedSpace.trivial_of_discrete [PreconnectedSpace α] [DiscreteTo
     Subsingleton α := by
   rw [← not_nontrivial_iff_subsingleton]
   rintro ⟨x, y, hxy⟩
-  rw [Ne.def, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
+  rw [Ne.def, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy 
   exact hxy (mem_univ x)
 #align preconnected_space.trivial_of_discrete PreconnectedSpace.trivial_of_discrete
 -/
@@ -981,7 +981,7 @@ theorem nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α} (h
 
 theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type _} {p : ι → Prop}
     {t : ι → Set α} {s : Set α} [DiscreteTopology s] {x : α} (hb : (𝓝 x).HasBasis p t)
-    (hx : x ∈ s) : ∃ (i : _)(hi : p i), t i ∩ s = {x} :=
+    (hx : x ∈ s) : ∃ (i : _) (hi : p i), t i ∩ s = {x} :=
   by
   rcases(nhdsWithin_hasBasis hb s).mem_iff.1 (singleton_mem_nhdsWithin_of_mem_discrete hx) with
     ⟨i, hi, hix⟩
@@ -1249,7 +1249,7 @@ theorem isOpen_iff_ultrafilter' [CompactSpace α] (U : Set α) :
   rw [isOpen_iff_ultrafilter]
   refine' ⟨fun h F hF => h F.lim hF F F.le_nhds_lim, _⟩
   intro cond x hx f h
-  rw [← Ultrafilter.lim_eq_iff_le_nhds.2 h] at hx
+  rw [← Ultrafilter.lim_eq_iff_le_nhds.2 h] at hx 
   exact cond _ hx
 #align is_open_iff_ultrafilter' isOpen_iff_ultrafilter'
 -/
@@ -1391,7 +1391,7 @@ instance Pi.t2Space {α : Type _} {β : α → Type v} [t₂ : ∀ a, Topologica
 -/
 
 instance Sigma.t2Space {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)]
-    [∀ a, T2Space (α a)] : T2Space (Σi, α i) :=
+    [∀ a, T2Space (α a)] : T2Space (Σ i, α i) :=
   by
   constructor
   rintro ⟨i, x⟩ ⟨j, y⟩ neq
@@ -1489,7 +1489,7 @@ theorem Function.LeftInverse.closedEmbedding [T2Space α] {f : α → β} {g : 
 
 theorem isCompact_isCompact_separated [T2Space α] {s t : Set α} (hs : IsCompact s)
     (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t := by
-  simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst⊢ <;>
+  simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst ⊢ <;>
     exact generalized_tube_lemma hs ht is_closed_diagonal.is_open_compl hst
 #align is_compact_is_compact_separated isCompact_isCompact_separated
 
@@ -1612,8 +1612,8 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
     induction' t using Finset.induction with x t hx ih generalizing U hU s hs hsC
     · refine' ⟨fun _ => ∅, fun i => isCompact_empty, fun i => empty_subset _, _⟩
       simpa only [subset_empty_iff, Union_false, Union_empty] using hsC
-    simp only [Finset.set_biUnion_insert] at hsC
-    simp only [Finset.mem_insert] at hU
+    simp only [Finset.set_biUnion_insert] at hsC 
+    simp only [Finset.mem_insert] at hU 
     have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU i (Or.inr hi)
     rcases hs.binary_compact_cover (hU x (Or.inl rfl)) (isOpen_biUnion hU') hsC with
       ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
@@ -1621,10 +1621,10 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
     refine' ⟨update K x K₁, _, _, _⟩
     · intro i; by_cases hi : i = x
       · simp only [update_same, hi, h1K₁]
-      · rw [← Ne.def] at hi; simp only [update_noteq hi, h1K]
+      · rw [← Ne.def] at hi ; simp only [update_noteq hi, h1K]
     · intro i; by_cases hi : i = x
       · simp only [update_same, hi, h2K₁]
-      · rw [← Ne.def] at hi; simp only [update_noteq hi, h2K]
+      · rw [← Ne.def] at hi ; simp only [update_noteq hi, h2K]
     · simp only [set_bUnion_insert_update _ hx, hK, h3K]
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 
@@ -1757,7 +1757,7 @@ theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
   tfae_have 5 → 6; exact fun h a => (h a).antisymm (𝓝 _).le_lift'_closure
   tfae_have 6 → 4
   · intro H a s hs
-    rw [← H] at hs
+    rw [← H] at hs 
     rcases(𝓝 a).basis_sets.lift'_closure.mem_iff.mp hs with ⟨U, hU, hUs⟩
     exact ⟨closure U, mem_of_superset hU subset_closure, isClosed_closure, hUs⟩
   tfae_have 4 → 2
@@ -1897,7 +1897,7 @@ theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @
   have :
     ∀ a,
       (𝓝 a).HasBasis
-        (fun If : ΣI : Set T, I → Set X =>
+        (fun If : Σ I : Set T, I → Set X =>
           If.1.Finite ∧ ∀ i : If.1, If.2 i ∈ @nhds X i a ∧ is_closed[↑i] (If.2 i))
         fun If => ⋂ i : If.1, If.snd i :=
     by
@@ -1949,7 +1949,7 @@ instance (priority := 100) T3Space.t25Space [T3Space α] : T25Space α :=
     (t0Space_iff_or_not_mem_closure α).mp inferInstance x y hne
   wlog H : x ∉ closure ({y} : Set α)
   · refine' (this y x hne.symm aux.symm (aux.resolve_left H)).symm
-  · rwa [← disjoint_nhds_nhdsSet, nhdsSet_singleton] at H
+  · rwa [← disjoint_nhds_nhdsSet, nhdsSet_singleton] at H 
 #align t3_space.t2_5_space T3Space.t25Space
 -/
 
@@ -1979,7 +1979,8 @@ instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
 with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. -/
 theorem disjoint_nested_nhds [T3Space α] {x y : α} (h : x ≠ y) :
-    ∃ (U₁ : _)(_ : U₁ ∈ 𝓝 x)(V₁ : _)(_ : V₁ ∈ 𝓝 x)(U₂ : _)(_ : U₂ ∈ 𝓝 y)(V₂ : _)(_ : V₂ ∈ 𝓝 y),
+    ∃ (U₁ : _) (_ : U₁ ∈ 𝓝 x) (V₁ : _) (_ : V₁ ∈ 𝓝 x) (U₂ : _) (_ : U₂ ∈ 𝓝 y) (V₂ : _) (_ :
+      V₂ ∈ 𝓝 y),
       IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂ :=
   by
   rcases t2_separation h with ⟨U₁, U₂, U₁_op, U₂_op, x_in, y_in, H⟩
@@ -2100,7 +2101,7 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
                 ∀ n : ℕ, IsClosed (⋃ (u ∈ U) (h : Encodable.encode u ≤ n), closure (u : Set α)) :=
     by
     intro s t hc hd
-    rw [disjoint_left] at hd
+    rw [disjoint_left] at hd 
     have : ∀ x ∈ s, ∃ U ∈ countable_basis α, x ∈ U ∧ Disjoint (closure U) t :=
       by
       intro x hx
@@ -2142,7 +2143,7 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
       Classical.not_not]
     rintro a ⟨u, huU, hau, haV⟩ v hvV hav
     cases' le_total (Encodable.encode u) (Encodable.encode v) with hle hle
-    exacts[⟨u, huU, hle, subset_closure hau⟩, (haV _ hvV hle <| subset_closure hav).elim]
+    exacts [⟨u, huU, hle, subset_closure hau⟩, (haV _ hvV hle <| subset_closure hav).elim]
 #align normal_space_of_t3_second_countable normalSpaceOfT3SecondCountable
 -/
 
@@ -2206,7 +2207,7 @@ instance [T5Space α] : T5Space (SeparationQuotient α)
     by
     rw [← disjoint_comap_iff surjective_mk, comap_mk_nhds_set, comap_mk_nhds_set]
     apply T5Space.completely_normal <;> rw [← preimage_mk_closure]
-    exacts[hd₁.preimage mk, hd₂.preimage mk]
+    exacts [hd₁.preimage mk, hd₂.preimage mk]
 
 end CompletelyNormal
 
@@ -2236,7 +2237,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
   -- in whichever of u or v x lies in and hence will be a subset of either a or b.
   rsuffices ⟨Z, H⟩ : ∃ Z : Set α, IsClopen Z ∧ x ∈ Z ∧ Z ⊆ u ∪ v
   · have H1 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hu hv huv
-    rw [union_comm] at H
+    rw [union_comm] at H 
     have H2 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hv hu huv.symm
     by_cases x ∈ u
     -- The x ∈ u case.
@@ -2275,7 +2276,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
   have H1 :=
     (hu.union hv).isClosed_compl.IsCompact.inter_iInter_nonempty
       (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z) fun Z => Z.2.1.2
-  rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, not_forall] at H1
+  rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, not_forall] at H1 
   cases' H1 (disjoint_compl_left_iff_subset.2 <| hab.trans <| union_subset_union hau hbv) with Zi H2
   refine' ⟨⋂ U ∈ Zi, Subtype.val U, _, _, _⟩
   · exact isClopen_biInter_finset fun Z hZ => Z.2.1
@@ -2332,7 +2333,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
     constructor
     · have : connectedComponent x = {x} :=
         totally_disconnected_space_iff_connected_component_singleton.mp ‹_› x
-      rw [connectedComponent_eq_iInter_clopen] at this
+      rw [connectedComponent_eq_iInter_clopen] at this 
       intro hU
       let N := { Z // IsClopen Z ∧ x ∈ Z }
       rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ Z : N, Z.val ⊆ U
@@ -2346,7 +2347,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
       have h_nhd : ∀ y ∈ ⋂ Z : N, Z.val, U ∈ 𝓝 y :=
         by
         intro y y_in
-        erw [this, mem_singleton_iff] at y_in
+        erw [this, mem_singleton_iff] at y_in 
         rwa [y_in]
       exact exists_subset_nhds_of_compactSpace hdir hNcl h_nhd
     · rintro ⟨V, ⟨hxV, V_op, -⟩, hUV : V ⊆ U⟩
@@ -2370,7 +2371,7 @@ theorem isTopologicalBasis_clopen : IsTopologicalBasis { s : Set α | IsClopen s
 /-- Every member of an open set in a compact Hausdorff totally disconnected space
   is contained in a clopen set contained in the open set.  -/
 theorem compact_exists_clopen_in_open {x : α} {U : Set α} (is_open : IsOpen U) (memU : x ∈ U) :
-    ∃ (V : Set α)(hV : IsClopen V), x ∈ V ∧ V ⊆ U :=
+    ∃ (V : Set α) (hV : IsClopen V), x ∈ V ∧ V ⊆ U :=
   (IsTopologicalBasis.mem_nhds_iff isTopologicalBasis_clopen).1 (IsOpen.mem_nhds memU)
 #align compact_exists_clopen_in_open compact_exists_clopen_in_open
 
@@ -2444,13 +2445,13 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
   -- Proof follows that of: https://stacks.math.columbia.edu/tag/0900
   -- Fix 2 distinct connected components, with points a and b
   refine' ⟨connected_components.surjective_coe.forall₂.2 fun a b ne => _⟩
-  rw [ConnectedComponents.coe_ne_coe] at ne
+  rw [ConnectedComponents.coe_ne_coe] at ne 
   have h := connectedComponent_disjoint Ne
   -- write ↑b as the intersection of all clopen subsets containing it
-  rw [connectedComponent_eq_iInter_clopen b, disjoint_iff_inter_eq_empty] at h
+  rw [connectedComponent_eq_iInter_clopen b, disjoint_iff_inter_eq_empty] at h 
   -- Now we show that this can be reduced to some clopen containing `↑b` being disjoint to `↑a`
   obtain ⟨U, V, hU, ha, hb, rfl⟩ :
-    ∃ (U : Set α)(V : Set (ConnectedComponents α)),
+    ∃ (U : Set α) (V : Set (ConnectedComponents α)),
       IsClopen U ∧ connectedComponent a ∩ U = ∅ ∧ connectedComponent b ⊆ U ∧ coe ⁻¹' V = U :=
     by
     cases' is_closed_connected_component.is_compact.elim_finite_subfamily_closed _ _ h with fin_a ha
@@ -2461,7 +2462,7 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
     exact
       ⟨U, coe '' U, hU, ha, subset_Inter₂ fun Z _ => Z.2.1.connectedComponent_subset Z.2.2,
         (connectedComponents_preimage_image U).symm ▸ hU.bUnion_connected_component_eq⟩
-  rw [connected_components.quotient_map_coe.is_clopen_preimage] at hU
+  rw [connected_components.quotient_map_coe.is_clopen_preimage] at hU 
   refine' ⟨Vᶜ, V, hU.compl.is_open, hU.is_open, _, hb mem_connectedComponent, disjoint_compl_left⟩
   exact fun h => flip Set.Nonempty.ne_empty ha ⟨a, mem_connectedComponent, h⟩
 #align connected_components.t2 ConnectedComponents.t2

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -1592,11 +1592,13 @@ theorem Continuous.closedEmbedding [CompactSpace α] [T2Space β] {f : α → β
 #align continuous.closed_embedding Continuous.closedEmbedding
 -/
 
+#print QuotientMap.of_surjective_continuous /-
 /-- A surjective continuous map from a compact space to a Hausdorff space is a quotient map. -/
 theorem QuotientMap.of_surjective_continuous [CompactSpace α] [T2Space β] {f : α → β}
     (hsurj : Surjective f) (hcont : Continuous f) : QuotientMap f :=
   hcont.IsClosedMap.to_quotientMap hcont hsurj
 #align quotient_map.of_surjective_continuous QuotientMap.of_surjective_continuous
+-/
 
 section
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -95,7 +95,7 @@ https://en.wikipedia.org/wiki/Separation_axiom
 
 open Function Set Filter TopologicalSpace
 
-open Topology Filter Classical
+open scoped Topology Filter Classical
 
 universe u v
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -113,12 +113,6 @@ def SeparatedNhds : Set α → Set α → Prop := fun s t : Set α =>
 #align separated_nhds SeparatedNhds
 -/
 
-/- warning: separated_nhds_iff_disjoint -> separatedNhds_iff_disjoint is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (SeparatedNhds.{u1} α _inst_1 s t) (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhdsSet.{u1} α _inst_1 t))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (SeparatedNhds.{u1} α _inst_1 s t) (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhdsSet.{u1} α _inst_1 t))
-Case conversion may be inaccurate. Consider using '#align separated_nhds_iff_disjoint separatedNhds_iff_disjointₓ'. -/
 theorem separatedNhds_iff_disjoint {s t : Set α} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by
   simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, ←
     exists_and_left, and_assoc, and_comm, and_left_comm]
@@ -150,34 +144,16 @@ theorem preimage [TopologicalSpace β] {f : α → β} {s t : Set β} (h : Separ
 #align separated_nhds.preimage SeparatedNhds.preimage
 -/
 
-/- warning: separated_nhds.disjoint -> SeparatedNhds.disjoint is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) s t)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) s t)
-Case conversion may be inaccurate. Consider using '#align separated_nhds.disjoint SeparatedNhds.disjointₓ'. -/
 protected theorem disjoint (h : SeparatedNhds s t) : Disjoint s t :=
   let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h
   hd.mono hsU htV
 #align separated_nhds.disjoint SeparatedNhds.disjoint
 
-/- warning: separated_nhds.disjoint_closure_left -> SeparatedNhds.disjoint_closure_left is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) (closure.{u1} α _inst_1 s) t)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (closure.{u1} α _inst_1 s) t)
-Case conversion may be inaccurate. Consider using '#align separated_nhds.disjoint_closure_left SeparatedNhds.disjoint_closure_leftₓ'. -/
 theorem disjoint_closure_left (h : SeparatedNhds s t) : Disjoint (closure s) t :=
   let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h
   (hd.closure_left hV).mono (closure_mono hsU) htV
 #align separated_nhds.disjoint_closure_left SeparatedNhds.disjoint_closure_left
 
-/- warning: separated_nhds.disjoint_closure_right -> SeparatedNhds.disjoint_closure_right is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) s (closure.{u1} α _inst_1 t))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) s (closure.{u1} α _inst_1 t))
-Case conversion may be inaccurate. Consider using '#align separated_nhds.disjoint_closure_right SeparatedNhds.disjoint_closure_rightₓ'. -/
 theorem disjoint_closure_right (h : SeparatedNhds s t) : Disjoint s (closure t) :=
   h.symm.disjoint_closure_left.symm
 #align separated_nhds.disjoint_closure_right SeparatedNhds.disjoint_closure_right
@@ -201,22 +177,10 @@ theorem mono (h : SeparatedNhds s₂ t₂) (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t
 #align separated_nhds.mono SeparatedNhds.mono
 -/
 
-/- warning: separated_nhds.union_left -> SeparatedNhds.union_left is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {u : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s u) -> (SeparatedNhds.{u1} α _inst_1 t u) -> (SeparatedNhds.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t) u)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {u : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s u) -> (SeparatedNhds.{u1} α _inst_1 t u) -> (SeparatedNhds.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t) u)
-Case conversion may be inaccurate. Consider using '#align separated_nhds.union_left SeparatedNhds.union_leftₓ'. -/
 theorem union_left : SeparatedNhds s u → SeparatedNhds t u → SeparatedNhds (s ∪ t) u := by
   simpa only [separatedNhds_iff_disjoint, nhdsSet_union, disjoint_sup_left] using And.intro
 #align separated_nhds.union_left SeparatedNhds.union_left
 
-/- warning: separated_nhds.union_right -> SeparatedNhds.union_right is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {u : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s t) -> (SeparatedNhds.{u1} α _inst_1 s u) -> (SeparatedNhds.{u1} α _inst_1 s (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) t u))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α} {u : Set.{u1} α}, (SeparatedNhds.{u1} α _inst_1 s t) -> (SeparatedNhds.{u1} α _inst_1 s u) -> (SeparatedNhds.{u1} α _inst_1 s (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) t u))
-Case conversion may be inaccurate. Consider using '#align separated_nhds.union_right SeparatedNhds.union_rightₓ'. -/
 theorem union_right (ht : SeparatedNhds s t) (hu : SeparatedNhds s u) : SeparatedNhds s (t ∪ u) :=
   (ht.symm.union_left hu.symm).symm
 #align separated_nhds.union_right SeparatedNhds.union_right
@@ -393,12 +357,6 @@ theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_eq_singleton minimal_nonempty_open_eq_singleton
 -/
 
-/- warning: exists_open_singleton_of_open_finite -> exists_open_singleton_of_open_finite is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T0Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Finite.{u1} α s) -> (Set.Nonempty.{u1} α s) -> (IsOpen.{u1} α _inst_1 s) -> (Exists.{succ u1} α (fun (x : α) => Exists.{0} (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) => IsOpen.{u1} α _inst_1 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T0Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Finite.{u1} α s) -> (Set.Nonempty.{u1} α s) -> (IsOpen.{u1} α _inst_1 s) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (IsOpen.{u1} α _inst_1 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))))
-Case conversion may be inaccurate. Consider using '#align exists_open_singleton_of_open_finite exists_open_singleton_of_open_finiteₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊂ » s) -/
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
 theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s.Finite)
@@ -492,12 +450,6 @@ theorem isClosed_singleton [T1Space α] {x : α} : IsClosed ({x} : Set α) :=
 #align is_closed_singleton isClosed_singleton
 -/
 
-/- warning: is_open_compl_singleton -> isOpen_compl_singleton is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α}, IsOpen.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α}, IsOpen.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))
-Case conversion may be inaccurate. Consider using '#align is_open_compl_singleton isOpen_compl_singletonₓ'. -/
 theorem isOpen_compl_singleton [T1Space α] {x : α} : IsOpen ({x}ᶜ : Set α) :=
   isClosed_singleton.isOpen_compl
 #align is_open_compl_singleton isOpen_compl_singleton
@@ -517,22 +469,10 @@ theorem Continuous.isOpen_mulSupport [T1Space α] [One α] [TopologicalSpace β]
 #align continuous.is_open_support Continuous.isOpen_support
 -/
 
-/- warning: ne.nhds_within_compl_singleton -> Ne.nhdsWithin_compl_singleton is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) y))) (nhds.{u1} α _inst_1 x))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) y))) (nhds.{u1} α _inst_1 x))
-Case conversion may be inaccurate. Consider using '#align ne.nhds_within_compl_singleton Ne.nhdsWithin_compl_singletonₓ'. -/
 theorem Ne.nhdsWithin_compl_singleton [T1Space α] {x y : α} (h : x ≠ y) : 𝓝[{y}ᶜ] x = 𝓝 x :=
   isOpen_ne.nhdsWithin_eq h
 #align ne.nhds_within_compl_singleton Ne.nhdsWithin_compl_singleton
 
-/- warning: ne.nhds_within_diff_singleton -> Ne.nhdsWithin_diff_singleton is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (forall (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 x (SDiff.sdiff.{u1} (Set.{u1} α) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) y))) (nhdsWithin.{u1} α _inst_1 x s))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (forall (s : Set.{u1} α), Eq.{succ u1} (Filter.{u1} α) (nhdsWithin.{u1} α _inst_1 x (SDiff.sdiff.{u1} (Set.{u1} α) (Set.instSDiffSet.{u1} α) s (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) y))) (nhdsWithin.{u1} α _inst_1 x s))
-Case conversion may be inaccurate. Consider using '#align ne.nhds_within_diff_singleton Ne.nhdsWithin_diff_singletonₓ'. -/
 theorem Ne.nhdsWithin_diff_singleton [T1Space α] {x y : α} (h : x ≠ y) (s : Set α) :
     𝓝[s \ {y}] x = 𝓝[s] x :=
   by
@@ -540,12 +480,6 @@ theorem Ne.nhdsWithin_diff_singleton [T1Space α] {x y : α} (h : x ≠ y) (s :
   exact mem_nhdsWithin_of_mem_nhds (is_open_ne.mem_nhds h)
 #align ne.nhds_within_diff_singleton Ne.nhdsWithin_diff_singleton
 
-/- warning: is_open_set_of_eventually_nhds_within -> isOpen_setOf_eventually_nhdsWithin is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {p : α -> Prop}, IsOpen.{u1} α _inst_1 (setOf.{u1} α (fun (x : α) => Filter.Eventually.{u1} α (fun (y : α) => p y) (nhdsWithin.{u1} α _inst_1 x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {p : α -> Prop}, IsOpen.{u1} α _inst_1 (setOf.{u1} α (fun (x : α) => Filter.Eventually.{u1} α (fun (y : α) => p y) (nhdsWithin.{u1} α _inst_1 x (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)))))
-Case conversion may be inaccurate. Consider using '#align is_open_set_of_eventually_nhds_within isOpen_setOf_eventually_nhdsWithinₓ'. -/
 theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
     IsOpen { x | ∀ᶠ y in 𝓝[≠] x, p y } :=
   by
@@ -565,12 +499,6 @@ protected theorem Set.Finite.isClosed [T1Space α] {s : Set α} (hs : Set.Finite
 #align set.finite.is_closed Set.Finite.isClosed
 -/
 
-/- warning: topological_space.is_topological_basis.exists_mem_of_ne -> TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne is a dubious translation:
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α _inst_1 b) -> (forall {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (a : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) a b) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x a) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y a))))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.exists_mem_of_ne TopologicalSpace.IsTopologicalBasis.exists_mem_of_neₓ'. -/
 theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space α] {b : Set (Set α)}
     (hb : IsTopologicalBasis b) {x y : α} (h : x ≠ y) : ∃ a ∈ b, x ∈ a ∧ y ∉ a :=
   by
@@ -578,12 +506,6 @@ theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space α] {b : S
   exact ⟨a, ab, xa, fun h => ha h rfl⟩
 #align topological_space.is_topological_basis.exists_mem_of_ne TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne
 
-/- warning: filter.coclosed_compact_le_cofinite -> Filter.coclosedCompact_le_cofinite is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.coclosedCompact.{u1} α _inst_1) (Filter.cofinite.{u1} α)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align filter.coclosed_compact_le_cofinite Filter.coclosedCompact_le_cofiniteₓ'. -/
 theorem Filter.coclosedCompact_le_cofinite [T1Space α] :
     Filter.coclosedCompact α ≤ Filter.cofinite := fun s hs =>
   compl_compl s ▸ hs.IsCompact.compl_mem_coclosedCompact_of_isClosed hs.IsClosed
@@ -625,12 +547,6 @@ protected theorem Finset.isClosed [T1Space α] (s : Finset α) : IsClosed (s : S
 #align finset.is_closed Finset.isClosed
 -/
 
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-Case conversion may be inaccurate. Consider using '#align t1_space_tfae t1Space_TFAEₓ'. -/
 theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
     TFAE
       [T1Space α, ∀ x, IsClosed ({x} : Set α), ∀ x, IsOpen ({x}ᶜ : Set α),
@@ -667,54 +583,24 @@ theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
   tfae_finish
 #align t1_space_tfae t1Space_TFAE
 
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 theorem t1Space_iff_continuous_cofinite_of {α : Type _} [TopologicalSpace α] :
     T1Space α ↔ Continuous (@CofiniteTopology.of α) :=
   (t1Space_TFAE α).out 0 3
 #align t1_space_iff_continuous_cofinite_of t1Space_iff_continuous_cofinite_of
 
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 theorem CofiniteTopology.continuous_of [T1Space α] : Continuous (@CofiniteTopology.of α) :=
   t1Space_iff_continuous_cofinite_of.mp ‹_›
 #align cofinite_topology.continuous_of CofiniteTopology.continuous_of
 
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 theorem t1Space_iff_exists_open :
     T1Space α ↔ ∀ x y, x ≠ y → ∃ (U : Set α)(hU : IsOpen U), x ∈ U ∧ y ∉ U :=
   (t1Space_TFAE α).out 0 6
 #align t1_space_iff_exists_open t1Space_iff_exists_open
 
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 theorem t1Space_iff_disjoint_pure_nhds : T1Space α ↔ ∀ ⦃x y : α⦄, x ≠ y → Disjoint (pure x) (𝓝 y) :=
   (t1Space_TFAE α).out 0 8
 #align t1_space_iff_disjoint_pure_nhds t1Space_iff_disjoint_pure_nhds
 
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 theorem t1Space_iff_disjoint_nhds_pure : T1Space α ↔ ∀ ⦃x y : α⦄, x ≠ y → Disjoint (𝓝 x) (pure y) :=
   (t1Space_TFAE α).out 0 7
 #align t1_space_iff_disjoint_nhds_pure t1Space_iff_disjoint_nhds_pure
@@ -725,22 +611,10 @@ theorem t1Space_iff_specializes_imp_eq : T1Space α ↔ ∀ ⦃x y : α⦄, x 
 #align t1_space_iff_specializes_imp_eq t1Space_iff_specializes_imp_eq
 -/
 
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 theorem disjoint_pure_nhds [T1Space α] {x y : α} (h : x ≠ y) : Disjoint (pure x) (𝓝 y) :=
   t1Space_iff_disjoint_pure_nhds.mp ‹_› h
 #align disjoint_pure_nhds disjoint_pure_nhds
 
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 theorem disjoint_nhds_pure [T1Space α] {x y : α} (h : x ≠ y) : Disjoint (𝓝 x) (pure y) :=
   t1Space_iff_disjoint_nhds_pure.mp ‹_› h
 #align disjoint_nhds_pure disjoint_nhds_pure
@@ -764,23 +638,11 @@ theorem specializes_eq_eq [T1Space α] : (· ⤳ ·) = @Eq α :=
 #align specializes_eq_eq specializes_eq_eq
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align pure_le_nhds_iff pure_le_nhds_iffₓ'. -/
 @[simp]
 theorem pure_le_nhds_iff [T1Space α] {a b : α} : pure a ≤ 𝓝 b ↔ a = b :=
   specializes_iff_pure.symm.trans specializes_iff_eq
 #align pure_le_nhds_iff pure_le_nhds_iff
 
-/- warning: nhds_le_nhds_iff -> nhds_le_nhds_iff is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align nhds_le_nhds_iff nhds_le_nhds_iffₓ'. -/
 @[simp]
 theorem nhds_le_nhds_iff [T1Space α] {a b : α} : 𝓝 a ≤ 𝓝 b ↔ a = b :=
   specializes_iff_eq
@@ -789,12 +651,6 @@ theorem nhds_le_nhds_iff [T1Space α] {a b : α} : 𝓝 a ≤ 𝓝 b ↔ a = b :
 instance {α : Type _} : T1Space (CofiniteTopology α) :=
   t1Space_iff_continuous_cofinite_of.mpr continuous_id
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align t1_space_antitone t1Space_antitoneₓ'. -/
 theorem t1Space_antitone {α : Type _} : Antitone (@T1Space α) :=
   by
   simp only [Antitone, t1Space_iff_continuous_cofinite_of, continuous_iff_le_induced]
@@ -819,12 +675,6 @@ theorem continuousAt_update_of_ne [T1Space α] [DecidableEq α] [TopologicalSpac
 #align continuous_at_update_of_ne continuousAt_update_of_ne
 -/
 
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-Case conversion may be inaccurate. Consider using '#align continuous_on_update_iff continuousOn_update_iffₓ'. -/
 theorem continuousOn_update_iff [T1Space α] [DecidableEq α] [TopologicalSpace β] {f : α → β}
     {s : Set α} {x : α} {y : β} :
     ContinuousOn (Function.update f x y) s ↔
@@ -876,23 +726,11 @@ instance (priority := 100) T1Space.t0Space [T1Space α] : T0Space α :=
 #align t1_space.t0_space T1Space.t0Space
 -/
 
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-but is expected to have type
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 @[simp]
 theorem compl_singleton_mem_nhds_iff [T1Space α] {x y : α} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x :=
   isOpen_compl_singleton.mem_nhds_iffₓ
 #align compl_singleton_mem_nhds_iff compl_singleton_mem_nhds_iff
 
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-Case conversion may be inaccurate. Consider using '#align compl_singleton_mem_nhds compl_singleton_mem_nhdsₓ'. -/
 theorem compl_singleton_mem_nhds [T1Space α] {x y : α} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y :=
   compl_singleton_mem_nhds_iff.mpr h
 #align compl_singleton_mem_nhds compl_singleton_mem_nhds
@@ -918,12 +756,6 @@ theorem subsingleton_closure [T1Space α] {s : Set α} : (closure s).Subsingleto
 #align subsingleton_closure subsingleton_closure
 -/
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_closed_map_const isClosedMap_constₓ'. -/
 theorem isClosedMap_const {α β} [TopologicalSpace α] [TopologicalSpace β] [T1Space β] {y : β} :
     IsClosedMap (Function.const α y) :=
   IsClosedMap.of_nonempty fun s hs h2s => by simp_rw [h2s.image_const, isClosed_singleton]
@@ -955,12 +787,6 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
 #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align bInter_basis_nhds biInter_basis_nhdsₓ'. -/
 theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι → Set α} {x : α}
     (h : (𝓝 x).HasBasis p s) : (⋂ (i) (h : p i), s i) = {x} :=
   by
@@ -971,23 +797,11 @@ theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι
   exact ⟨i, hi, fun h => hsub h rfl⟩
 #align bInter_basis_nhds biInter_basis_nhds
 
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-Case conversion may be inaccurate. Consider using '#align compl_singleton_mem_nhds_set_iff compl_singleton_mem_nhdsSet_iffₓ'. -/
 @[simp]
 theorem compl_singleton_mem_nhdsSet_iff [T1Space α] {x : α} {s : Set α} : {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s := by
   rwa [is_open_compl_singleton.mem_nhds_set, subset_compl_singleton_iff]
 #align compl_singleton_mem_nhds_set_iff compl_singleton_mem_nhdsSet_iff
 
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-Case conversion may be inaccurate. Consider using '#align nhds_set_le_iff nhdsSet_le_iffₓ'. -/
 @[simp]
 theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t :=
   by
@@ -1012,45 +826,21 @@ theorem injective_nhdsSet [T1Space α] : Function.Injective (𝓝ˢ : Set α →
 #align injective_nhds_set injective_nhdsSet
 -/
 
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-Case conversion may be inaccurate. Consider using '#align strict_mono_nhds_set strictMono_nhdsSetₓ'. -/
 theorem strictMono_nhdsSet [T1Space α] : StrictMono (𝓝ˢ : Set α → Filter α) :=
   monotone_nhdsSet.strictMono_of_injective injective_nhdsSet
 #align strict_mono_nhds_set strictMono_nhdsSet
 
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 @[simp]
 theorem nhds_le_nhdsSet_iff [T1Space α] {s : Set α} {x : α} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s := by
   rw [← nhdsSet_singleton, nhdsSet_le_iff, singleton_subset_iff]
 #align nhds_le_nhds_set_iff nhds_le_nhdsSet_iff
 
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 /-- Removing a non-isolated point from a dense set, one still obtains a dense set. -/
 theorem Dense.diff_singleton [T1Space α] {s : Set α} (hs : Dense s) (x : α) [NeBot (𝓝[≠] x)] :
     Dense (s \ {x}) :=
   hs.inter_of_open_right (dense_compl_singleton x) isOpen_compl_singleton
 #align dense.diff_singleton Dense.diff_singleton
 
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-Case conversion may be inaccurate. Consider using '#align dense.diff_finset Dense.diff_finsetₓ'. -/
 /-- Removing a finset from a dense set in a space without isolated points, one still
 obtains a dense set. -/
 theorem Dense.diff_finset [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Set α} (hs : Dense s)
@@ -1062,12 +852,6 @@ theorem Dense.diff_finset [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Se
     exact ih.diff_singleton _
 #align dense.diff_finset Dense.diff_finset
 
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-Case conversion may be inaccurate. Consider using '#align dense.diff_finite Dense.diff_finiteₓ'. -/
 /-- Removing a finite set from a dense set in a space without isolated points, one still
 obtains a dense set. -/
 theorem Dense.diff_finite [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Set α} (hs : Dense s)
@@ -1089,12 +873,6 @@ theorem eq_of_tendsto_nhds [TopologicalSpace β] [T1Space β] {f : α → β} {a
 #align eq_of_tendsto_nhds eq_of_tendsto_nhds
 -/
 
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 theorem Filter.Tendsto.eventually_ne [TopologicalSpace β] [T1Space β] {α : Type _} {g : α → β}
     {l : Filter α} {b₁ b₂ : β} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ :=
   hg.Eventually (isOpen_compl_singleton.eventually_mem hb)
@@ -1136,12 +914,6 @@ theorem isOpen_singleton_of_finite_mem_nhds {α : Type _} [TopologicalSpace α]
 #align is_open_singleton_of_finite_mem_nhds isOpen_singleton_of_finite_mem_nhds
 -/
 
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 /-- If the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is
 infinite. -/
 theorem infinite_of_mem_nhds {α} [TopologicalSpace α] [T1Space α] (x : α) [hx : NeBot (𝓝[≠] x)]
@@ -1207,12 +979,6 @@ theorem nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α} (h
 #align nhds_within_of_mem_discrete nhdsWithin_of_mem_discrete
 -/
 
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 theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type _} {p : ι → Prop}
     {t : ι → Set α} {s : Set α} [DiscreteTopology s] {x : α} (hb : (𝓝 x).HasBasis p t)
     (hx : x ∈ s) : ∃ (i : _)(hi : p i), t i ∩ s = {x} :=
@@ -1223,12 +989,6 @@ theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type _}
     ⟨i, hi, subset.antisymm hix <| singleton_subset_iff.2 ⟨mem_of_mem_nhds <| hb.mem_of_mem hi, hx⟩⟩
 #align filter.has_basis.exists_inter_eq_singleton_of_mem_discrete Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete
 
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 /-- A point `x` in a discrete subset `s` of a topological space admits a neighbourhood
 that only meets `s` at `x`.  -/
 theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α}
@@ -1236,12 +996,6 @@ theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set α} [DiscreteTopology s
   simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx
 #align nhds_inter_eq_singleton_of_mem_discrete nhds_inter_eq_singleton_of_mem_discrete
 
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 /-- For point `x` in a discrete subset `s` of a topological space, there is a set `U`
 such that
 1. `U` is a punctured neighborhood of `x` (ie. `U ∪ {x}` is a neighbourhood of `x`),
@@ -1280,24 +1034,12 @@ class T2Space (α : Type u) [TopologicalSpace α] : Prop where
 #align t2_space T2Space
 -/
 
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 /-- Two different points can be separated by open sets. -/
 theorem t2_separation [T2Space α] {x y : α} (h : x ≠ y) :
     ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
   T2Space.t2 x y h
 #align t2_separation t2_separation
 
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 theorem t2Space_iff_disjoint_nhds : T2Space α ↔ ∀ x y : α, x ≠ y → Disjoint (𝓝 x) (𝓝 y) :=
   by
   refine' (t2Space_iff α).trans (forall₃_congr fun x y hne => _)
@@ -1305,55 +1047,25 @@ theorem t2Space_iff_disjoint_nhds : T2Space α ↔ ∀ x y : α, x ≠ y → Dis
     and_assoc, and_comm', and_left_comm]
 #align t2_space_iff_disjoint_nhds t2Space_iff_disjoint_nhds
 
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 @[simp]
 theorem disjoint_nhds_nhds [T2Space α] {x y : α} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y :=
   ⟨fun hd he => by simpa [he, nhds_ne_bot.ne] using hd, t2Space_iff_disjoint_nhds.mp ‹_› x y⟩
 #align disjoint_nhds_nhds disjoint_nhds_nhds
 
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 theorem pairwise_disjoint_nhds [T2Space α] : Pairwise (Disjoint on (𝓝 : α → Filter α)) := fun x y =>
   disjoint_nhds_nhds.2
 #align pairwise_disjoint_nhds pairwise_disjoint_nhds
 
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 protected theorem Set.pairwiseDisjoint_nhds [T2Space α] (s : Set α) : s.PairwiseDisjoint 𝓝 :=
   pairwise_disjoint_nhds.set_pairwise s
 #align set.pairwise_disjoint_nhds Set.pairwiseDisjoint_nhds
 
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 /-- Points of a finite set can be separated by open sets from each other. -/
 theorem Set.Finite.t2_separation [T2Space α] {s : Set α} (hs : s.Finite) :
     ∃ U : α → Set α, (∀ x, x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U :=
   s.pairwise_disjoint_nhds.exists_mem_filter_basisₓ hs nhds_basis_opens
 #align set.finite.t2_separation Set.Finite.t2_separation
 
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 theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : α × α | Disjoint (𝓝 p.1) (𝓝 p.2) } :=
   by
   simp only [isOpen_iff_mem_nhds, Prod.forall, mem_set_of_eq]
@@ -1373,89 +1085,41 @@ instance (priority := 100) T2Space.t1Space [T2Space α] : T1Space α :=
 #align t2_space.t1_space T2Space.t1Space
 -/
 
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 /-- A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. -/
 theorem t2_iff_nhds : T2Space α ↔ ∀ {x y : α}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
   simp only [t2Space_iff_disjoint_nhds, disjoint_iff, ne_bot_iff, Ne.def, not_imp_comm]
 #align t2_iff_nhds t2_iff_nhds
 
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 theorem eq_of_nhds_neBot [T2Space α] {x y : α} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y :=
   t2_iff_nhds.mp ‹_› h
 #align eq_of_nhds_ne_bot eq_of_nhds_neBot
 
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 theorem t2Space_iff_nhds : T2Space α ↔ ∀ {x y : α}, x ≠ y → ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V := by
   simp only [t2Space_iff_disjoint_nhds, Filter.disjoint_iff]
 #align t2_space_iff_nhds t2Space_iff_nhds
 
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 theorem t2_separation_nhds [T2Space α] {x y : α} (h : x ≠ y) :
     ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ Disjoint u v :=
   let ⟨u, v, open_u, open_v, x_in, y_in, huv⟩ := t2_separation h
   ⟨u, v, open_u.mem_nhds x_in, open_v.mem_nhds y_in, huv⟩
 #align t2_separation_nhds t2_separation_nhds
 
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 theorem t2_separation_compact_nhds [LocallyCompactSpace α] [T2Space α] {x y : α} (h : x ≠ y) :
     ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v := by
   simpa only [exists_prop, ← exists_and_left, and_comm', and_assoc, and_left_comm] using
     ((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h)
 #align t2_separation_compact_nhds t2_separation_compact_nhds
 
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 theorem t2_iff_ultrafilter :
     T2Space α ↔ ∀ {x y : α} (f : Ultrafilter α), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y :=
   t2_iff_nhds.trans <| by simp only [← exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp]
 #align t2_iff_ultrafilter t2_iff_ultrafilter
 
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 theorem t2_iff_isClosed_diagonal : T2Space α ↔ IsClosed (diagonal α) := by
   simp only [t2Space_iff_disjoint_nhds, ← isOpen_compl_iff, isOpen_iff_mem_nhds, Prod.forall,
     nhds_prod_eq, compl_diagonal_mem_prod, mem_compl_iff, mem_diagonal_iff]
 #align t2_iff_is_closed_diagonal t2_iff_isClosed_diagonal
 
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 theorem isClosed_diagonal [T2Space α] : IsClosed (diagonal α) :=
   t2_iff_isClosed_diagonal.mp ‹_›
 #align is_closed_diagonal isClosed_diagonal
@@ -1464,12 +1128,6 @@ section Separated
 
 open SeparatedNhds Finset
 
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 theorem finset_disjoint_finset_opens_of_t2 [T2Space α] :
     ∀ s t : Finset α, Disjoint s t → SeparatedNhds (s : Set α) t :=
   by
@@ -1530,12 +1188,6 @@ class T25Space (α : Type u) [TopologicalSpace α] : Prop where
 #align t2_5_space T25Space
 -/
 
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 @[simp]
 theorem disjoint_lift'_closure_nhds [T25Space α] {x y : α} :
     Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure) ↔ x ≠ y :=
@@ -1550,24 +1202,12 @@ instance (priority := 100) T25Space.t2Space [T25Space α] : T2Space α :=
 #align t2_5_space.t2_space T25Space.t2Space
 -/
 
-/- warning: exists_nhds_disjoint_closure -> exists_nhds_disjoint_closure is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align exists_nhds_disjoint_closure exists_nhds_disjoint_closureₓ'. -/
 theorem exists_nhds_disjoint_closure [T25Space α] {x y : α} (h : x ≠ y) :
     ∃ s ∈ 𝓝 x, ∃ t ∈ 𝓝 y, Disjoint (closure s) (closure t) :=
   ((𝓝 x).basis_sets.lift'_closure.disjoint_iff (𝓝 y).basis_sets.lift'_closure).1 <|
     disjoint_lift'_closure_nhds.2 h
 #align exists_nhds_disjoint_closure exists_nhds_disjoint_closure
 
-/- warning: exists_open_nhds_disjoint_closure -> exists_open_nhds_disjoint_closure is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align exists_open_nhds_disjoint_closure exists_open_nhds_disjoint_closureₓ'. -/
 theorem exists_open_nhds_disjoint_closure [T25Space α] {x y : α} (h : x ≠ y) :
     ∃ u : Set α,
       x ∈ u ∧ IsOpen u ∧ ∃ v : Set α, y ∈ v ∧ IsOpen v ∧ Disjoint (closure u) (closure v) :=
@@ -1589,32 +1229,14 @@ are useful without a `nonempty α` instance.
 -/
 
 
-/- warning: Lim_eq -> lim_eq is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align Lim_eq lim_eqₓ'. -/
 theorem lim_eq {a : α} [NeBot f] (h : f ≤ 𝓝 a) : @lim _ _ ⟨a⟩ f = a :=
   tendsto_nhds_unique (le_nhds_lim ⟨a, h⟩) h
 #align Lim_eq lim_eq
 
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-Case conversion may be inaccurate. Consider using '#align Lim_eq_iff lim_eq_iffₓ'. -/
 theorem lim_eq_iff [NeBot f] (h : ∃ a : α, f ≤ nhds a) {a} : @lim _ _ ⟨a⟩ f = a ↔ f ≤ 𝓝 a :=
   ⟨fun c => c ▸ le_nhds_lim h, lim_eq⟩
 #align Lim_eq_iff lim_eq_iff
 
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-Case conversion may be inaccurate. Consider using '#align ultrafilter.Lim_eq_iff_le_nhds Ultrafilter.lim_eq_iff_le_nhdsₓ'. -/
 theorem Ultrafilter.lim_eq_iff_le_nhds [CompactSpace α] {x : α} {F : Ultrafilter α} :
     F.lim = x ↔ ↑F ≤ 𝓝 x :=
   ⟨fun h => h ▸ F.le_nhds_lim, lim_eq⟩
@@ -1710,9 +1332,6 @@ instance (priority := 100) DiscreteTopology.toT2Space {α : Type _} [Topological
 #align discrete_topology.to_t2_space DiscreteTopology.toT2Space
 -/
 
-/- warning: separated_by_continuous -> separated_by_continuous is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align separated_by_continuous separated_by_continuousₓ'. -/
 theorem separated_by_continuous {α : Type _} {β : Type _} [TopologicalSpace α] [TopologicalSpace β]
     [T2Space β] {f : α → β} (hf : Continuous f) {x y : α} (h : f x ≠ f y) :
     ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
@@ -1720,9 +1339,6 @@ theorem separated_by_continuous {α : Type _} {β : Type _} [TopologicalSpace α
   ⟨f ⁻¹' u, f ⁻¹' v, uo.Preimage hf, vo.Preimage hf, xu, yv, uv.Preimage _⟩
 #align separated_by_continuous separated_by_continuous
 
-/- warning: separated_by_open_embedding -> separated_by_openEmbedding is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align separated_by_open_embedding separated_by_openEmbeddingₓ'. -/
 theorem separated_by_openEmbedding {α β : Type _} [TopologicalSpace α] [TopologicalSpace β]
     [T2Space α] {f : α → β} (hf : OpenEmbedding f) {x y : α} (h : x ≠ y) :
     ∃ u v : Set β, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v :=
@@ -1774,12 +1390,6 @@ instance Pi.t2Space {α : Type _} {β : α → Type v} [t₂ : ∀ a, Topologica
 #align Pi.t2_space Pi.t2Space
 -/
 
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-Case conversion may be inaccurate. Consider using '#align sigma.t2_space Sigma.t2Spaceₓ'. -/
 instance Sigma.t2Space {ι : Type _} {α : ι → Type _} [∀ i, TopologicalSpace (α i)]
     [∀ a, T2Space (α a)] : T2Space (Σi, α i) :=
   by
@@ -1827,12 +1437,6 @@ theorem Continuous.ext_on [T2Space α] {s : Set β} (hs : Dense s) {f g : β →
 #align continuous.ext_on Continuous.ext_on
 -/
 
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-Case conversion may be inaccurate. Consider using '#align eq_on_closure₂' eqOn_closure₂'ₓ'. -/
 theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ → α}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf₁ : ∀ x, Continuous (f x))
     (hf₂ : ∀ y, Continuous fun x => f x y) (hg₁ : ∀ x, Continuous (g x))
@@ -1842,12 +1446,6 @@ theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ
     isClosed_biInter fun y hy => isClosed_eq (hf₂ _) (hg₂ _)
 #align eq_on_closure₂' eqOn_closure₂'
 
-/- warning: eq_on_closure₂ -> eqOn_closure₂ is a dubious translation:
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 theorem eqOn_closure₂ [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ → α}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (uncurry f))
     (hg : Continuous (uncurry g)) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
@@ -1889,12 +1487,6 @@ theorem Function.LeftInverse.closedEmbedding [T2Space α] {f : α → β} {g : 
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 -/
 
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 theorem isCompact_isCompact_separated [T2Space α] {s t : Set α} (hs : IsCompact s)
     (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t := by
   simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst⊢ <;>
@@ -1927,12 +1519,6 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 #align bornology.relatively_compact_eq_in_compact Bornology.relativelyCompact_eq_inCompact
 -/
 
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 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we
 don't need to assume each `V i` closed because it follows from compactness since `α` is
@@ -1949,12 +1535,6 @@ theorem CompactExhaustion.isClosed [T2Space α] (K : CompactExhaustion α) (n :
 #align compact_exhaustion.is_closed CompactExhaustion.isClosed
 -/
 
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 theorem IsCompact.inter [T2Space α] {s t : Set α} (hs : IsCompact s) (ht : IsCompact t) :
     IsCompact (s ∩ t) :=
   hs.inter_right <| ht.IsClosed
@@ -1984,12 +1564,6 @@ theorem image_closure_of_isCompact [T2Space β] {s : Set α} (hs : IsCompact (cl
 #align image_closure_of_is_compact image_closure_of_isCompact
 -/
 
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 /-- If a compact set is covered by two open sets, then we can cover it by two compact subsets. -/
 theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCompact K)
     (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
@@ -2028,12 +1602,6 @@ section
 
 open Finset Function
 
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 /-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
 theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact s) {ι} (t : Finset ι)
     (U : ι → Set α) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
@@ -2170,12 +1738,6 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
 #align regular_space RegularSpace
 -/
 
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-Case conversion may be inaccurate. Consider using '#align regular_space_tfae regularSpace_TFAEₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
 theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     TFAE
@@ -2215,24 +1777,12 @@ theorem RegularSpace.ofLift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 
 #align regular_space.of_lift'_closure RegularSpace.ofLift'_closure
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align regular_space.of_basis RegularSpace.ofBasisₓ'. -/
 theorem RegularSpace.ofBasis {ι : α → Sort _} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set α}
     (h₁ : ∀ a, (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ a i, p a i → IsClosed (s a i)) :
     RegularSpace α :=
   RegularSpace.ofLift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a)
 #align regular_space.of_basis RegularSpace.ofBasis
 
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-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], (forall (a : α) (s : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_1 a)) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t (nhds.{u1} α _inst_1 a)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) t (nhds.{u1} α _inst_1 a)) => And (IsClosed.{u1} α _inst_1 t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s))))) -> (RegularSpace.{u1} α _inst_1)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], (forall (a : α) (s : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_1 a)) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t (nhds.{u1} α _inst_1 a)) (And (IsClosed.{u1} α _inst_1 t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s))))) -> (RegularSpace.{u1} α _inst_1)
-Case conversion may be inaccurate. Consider using '#align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.ofExistsMemNhdsIsClosedSubsetₓ'. -/
 theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
     (h : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s) : RegularSpace α :=
   Iff.mpr ((regularSpace_TFAE α).out 0 3) h
@@ -2240,32 +1790,14 @@ theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
 
 variable [RegularSpace α] {a : α} {s : Set α}
 
-/- warning: disjoint_nhds_set_nhds -> disjoint_nhdsSet_nhds is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align disjoint_nhds_set_nhds disjoint_nhdsSet_nhdsₓ'. -/
 theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 a) ↔ a ∉ closure s :=
   Iff.mp ((regularSpace_TFAE α).out 0 2) ‹_› _ _
 #align disjoint_nhds_set_nhds disjoint_nhdsSet_nhds
 
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 a) (nhdsSet.{u1} α _inst_1 s)) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α _inst_1 s)))
-Case conversion may be inaccurate. Consider using '#align disjoint_nhds_nhds_set disjoint_nhds_nhdsSetₓ'. -/
 theorem disjoint_nhds_nhdsSet : Disjoint (𝓝 a) (𝓝ˢ s) ↔ a ∉ closure s :=
   disjoint_comm.trans disjoint_nhdsSet_nhds
 #align disjoint_nhds_nhds_set disjoint_nhds_nhdsSet
 
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-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_1 a)) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) t (nhds.{u1} α _inst_1 a)) (And (IsClosed.{u1} α _inst_1 t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s))))
-Case conversion may be inaccurate. Consider using '#align exists_mem_nhds_is_closed_subset exists_mem_nhds_isClosed_subsetₓ'. -/
 theorem exists_mem_nhds_isClosed_subset {a : α} {s : Set α} (h : s ∈ 𝓝 a) :
     ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s :=
   Iff.mp ((regularSpace_TFAE α).out 0 3) ‹_› _ _ h
@@ -2283,12 +1815,6 @@ theorem lift'_nhds_closure (a : α) : (𝓝 a).lift' closure = 𝓝 a :=
 #align lift'_nhds_closure lift'_nhds_closure
 -/
 
-/- warning: filter.has_basis.nhds_closure -> Filter.HasBasis.nhds_closure is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : RegularSpace.{u2} α _inst_1] {ι : Sort.{u1}} {a : α} {p : ι -> Prop} {s : ι -> (Set.{u2} α)}, (Filter.HasBasis.{u2, u1} α ι (nhds.{u2} α _inst_1 a) p s) -> (Filter.HasBasis.{u2, u1} α ι (nhds.{u2} α _inst_1 a) p (fun (i : ι) => closure.{u2} α _inst_1 (s i)))
-Case conversion may be inaccurate. Consider using '#align filter.has_basis.nhds_closure Filter.HasBasis.nhds_closureₓ'. -/
 theorem Filter.HasBasis.nhds_closure {ι : Sort _} {a : α} {p : ι → Prop} {s : ι → Set α}
     (h : (𝓝 a).HasBasis p s) : (𝓝 a).HasBasis p fun i => closure (s i) :=
   lift'_nhds_closure a ▸ h.lift'_closure
@@ -2314,24 +1840,12 @@ theorem TopologicalSpace.IsTopologicalBasis.nhds_basis_closure {B : Set (Set α)
 #align topological_space.is_topological_basis.nhds_basis_closure TopologicalSpace.IsTopologicalBasis.nhds_basis_closure
 -/
 
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-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α _inst_1 B) -> (forall {a : α} {s : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_1 a)) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t B) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t B) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (closure.{u1} α _inst_1 t) s)))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α _inst_1 B) -> (forall {a : α} {s : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_1 a)) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t B) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) (closure.{u1} α _inst_1 t) s)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.exists_closure_subset TopologicalSpace.IsTopologicalBasis.exists_closure_subsetₓ'. -/
 theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set α)}
     (hB : TopologicalSpace.IsTopologicalBasis B) {a : α} {s : Set α} (h : s ∈ 𝓝 a) :
     ∃ t ∈ B, a ∈ t ∧ closure t ⊆ s := by
   simpa only [exists_prop, and_assoc] using hB.nhds_has_basis.nhds_closure.mem_iff.mp h
 #align topological_space.is_topological_basis.exists_closure_subset TopologicalSpace.IsTopologicalBasis.exists_closure_subset
 
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-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {b : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 a) (nhds.{u1} α _inst_1 b)) (Not (Specializes.{u1} α _inst_1 a b))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {b : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 a) (nhds.{u1} α _inst_1 b)) (Not (Specializes.{u1} α _inst_1 a b))
-Case conversion may be inaccurate. Consider using '#align disjoint_nhds_nhds_iff_not_specializes disjoint_nhds_nhds_iff_not_specializesₓ'. -/
 theorem disjoint_nhds_nhds_iff_not_specializes {a b : α} : Disjoint (𝓝 a) (𝓝 b) ↔ ¬a ⤳ b := by
   rw [← nhdsSet_singleton, disjoint_nhdsSet_nhds, specializes_iff_mem_closure]
 #align disjoint_nhds_nhds_iff_not_specializes disjoint_nhds_nhds_iff_not_specializes
@@ -2351,23 +1865,11 @@ theorem specializes_iff_inseparable {a b : α} : a ⤳ b ↔ Inseparable a b :=
 #align specializes_iff_inseparable specializes_iff_inseparable
 -/
 
-/- warning: is_closed_set_of_specializes -> isClosed_setOf_specializes is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1], IsClosed.{u1} (Prod.{u1, u1} α α) (Prod.topologicalSpace.{u1, u1} α α _inst_1 _inst_1) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => Specializes.{u1} α _inst_1 (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1], IsClosed.{u1} (Prod.{u1, u1} α α) (instTopologicalSpaceProd.{u1, u1} α α _inst_1 _inst_1) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => Specializes.{u1} α _inst_1 (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)))
-Case conversion may be inaccurate. Consider using '#align is_closed_set_of_specializes isClosed_setOf_specializesₓ'. -/
 theorem isClosed_setOf_specializes : IsClosed { p : α × α | p.1 ⤳ p.2 } := by
   simp only [← isOpen_compl_iff, compl_set_of, ← disjoint_nhds_nhds_iff_not_specializes,
     isOpen_setOf_disjoint_nhds_nhds]
 #align is_closed_set_of_specializes isClosed_setOf_specializes
 
-/- warning: is_closed_set_of_inseparable -> isClosed_setOf_inseparable is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1], IsClosed.{u1} (Prod.{u1, u1} α α) (Prod.topologicalSpace.{u1, u1} α α _inst_1 _inst_1) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => Inseparable.{u1} α _inst_1 (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1], IsClosed.{u1} (Prod.{u1, u1} α α) (instTopologicalSpaceProd.{u1, u1} α α _inst_1 _inst_1) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => Inseparable.{u1} α _inst_1 (Prod.fst.{u1, u1} α α p) (Prod.snd.{u1, u1} α α p)))
-Case conversion may be inaccurate. Consider using '#align is_closed_set_of_inseparable isClosed_setOf_inseparableₓ'. -/
 theorem isClosed_setOf_inseparable : IsClosed { p : α × α | Inseparable p.1 p.2 } := by
   simp only [← specializes_iff_inseparable, isClosed_setOf_specializes]
 #align is_closed_set_of_inseparable isClosed_setOf_inseparable
@@ -2387,12 +1889,6 @@ theorem regularSpace_induced (f : β → α) : @RegularSpace β (induced f ‹_
 #align regular_space_induced regularSpace_induced
 -/
 
-/- warning: regular_space_Inf -> regularSpace_sInf is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {T : Set.{u1} (TopologicalSpace.{u1} X)}, (forall (t : TopologicalSpace.{u1} X), (Membership.Mem.{u1, u1} (TopologicalSpace.{u1} X) (Set.{u1} (TopologicalSpace.{u1} X)) (Set.hasMem.{u1} (TopologicalSpace.{u1} X)) t T) -> (RegularSpace.{u1} X t)) -> (RegularSpace.{u1} X (InfSet.sInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toHasInf.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.completeLattice.{u1} X))) T))
-but is expected to have type
-  forall {X : Type.{u1}} {T : Set.{u1} (TopologicalSpace.{u1} X)}, (forall (t : TopologicalSpace.{u1} X), (Membership.mem.{u1, u1} (TopologicalSpace.{u1} X) (Set.{u1} (TopologicalSpace.{u1} X)) (Set.instMembershipSet.{u1} (TopologicalSpace.{u1} X)) t T) -> (RegularSpace.{u1} X t)) -> (RegularSpace.{u1} X (InfSet.sInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toInfSet.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X))) T))
-Case conversion may be inaccurate. Consider using '#align regular_space_Inf regularSpace_sInfₓ'. -/
 theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @RegularSpace X t) :
     @RegularSpace X (sInf T) := by
   letI := Inf T
@@ -2410,23 +1906,11 @@ theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @
   exact (hIf.2 i).2.mono (sInf_le (i : T).2)
 #align regular_space_Inf regularSpace_sInf
 
-/- warning: regular_space_infi -> regularSpace_iInf is a dubious translation:
-lean 3 declaration is
-  forall {ι : Sort.{u1}} {X : Type.{u2}} {t : ι -> (TopologicalSpace.{u2} X)}, (forall (i : ι), RegularSpace.{u2} X (t i)) -> (RegularSpace.{u2} X (iInf.{u2, u1} (TopologicalSpace.{u2} X) (ConditionallyCompleteLattice.toHasInf.{u2} (TopologicalSpace.{u2} X) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} X) (TopologicalSpace.completeLattice.{u2} X))) ι t))
-but is expected to have type
-  forall {ι : Sort.{u2}} {X : Type.{u1}} {t : ι -> (TopologicalSpace.{u1} X)}, (forall (i : ι), RegularSpace.{u1} X (t i)) -> (RegularSpace.{u1} X (iInf.{u1, u2} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toInfSet.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X))) ι t))
-Case conversion may be inaccurate. Consider using '#align regular_space_infi regularSpace_iInfₓ'. -/
 theorem regularSpace_iInf {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @RegularSpace X (t i)) :
     @RegularSpace X (iInf t) :=
   regularSpace_sInf <| forall_range_iff.mpr h
 #align regular_space_infi regularSpace_iInf
 
-/- warning: regular_space.inf -> RegularSpace.inf is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {t₁ : TopologicalSpace.{u1} X} {t₂ : TopologicalSpace.{u1} X}, (RegularSpace.{u1} X t₁) -> (RegularSpace.{u1} X t₂) -> (RegularSpace.{u1} X (Inf.inf.{u1} (TopologicalSpace.{u1} X) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.{u1} X) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.completeLattice.{u1} X))))) t₁ t₂))
-but is expected to have type
-  forall {X : Type.{u1}} {t₁ : TopologicalSpace.{u1} X} {t₂ : TopologicalSpace.{u1} X}, (RegularSpace.{u1} X t₁) -> (RegularSpace.{u1} X t₂) -> (RegularSpace.{u1} X (Inf.inf.{u1} (TopologicalSpace.{u1} X) (Lattice.toInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X)))) t₁ t₂))
-Case conversion may be inaccurate. Consider using '#align regular_space.inf RegularSpace.infₓ'. -/
 theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSpace X t₁)
     (h₂ : @RegularSpace X t₂) : @RegularSpace X (t₁ ⊓ t₂) := by rw [inf_eq_iInf];
   exact regularSpace_iInf (Bool.forall_bool.2 ⟨h₂, h₁⟩)
@@ -2488,9 +1972,6 @@ instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [
     T3Space (∀ i, π i) :=
   ⟨⟩
 
-/- warning: disjoint_nested_nhds -> disjoint_nested_nhds is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align disjoint_nested_nhds disjoint_nested_nhdsₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
@@ -2530,12 +2011,6 @@ class NormalSpace (α : Type u) [TopologicalSpace α] extends T1Space α : Prop
 #align normal_space NormalSpace
 -/
 
-/- warning: normal_separation -> normal_separation is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : NormalSpace.{u1} α _inst_1] {s : Set.{u1} α} {t : Set.{u1} α}, (IsClosed.{u1} α _inst_1 s) -> (IsClosed.{u1} α _inst_1 t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) s t) -> (SeparatedNhds.{u1} α _inst_1 s t)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : NormalSpace.{u1} α _inst_1] {s : Set.{u1} α} {t : Set.{u1} α}, (IsClosed.{u1} α _inst_1 s) -> (IsClosed.{u1} α _inst_1 t) -> (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) s t) -> (SeparatedNhds.{u1} α _inst_1 s t)
-Case conversion may be inaccurate. Consider using '#align normal_separation normal_separationₓ'. -/
 theorem normal_separation [NormalSpace α] {s t : Set α} (H1 : IsClosed s) (H2 : IsClosed t)
     (H3 : Disjoint s t) : SeparatedNhds s t :=
   NormalSpace.normal s t H1 H2 H3
@@ -2890,12 +2365,6 @@ theorem isTopologicalBasis_clopen : IsTopologicalBasis { s : Set α | IsClopen s
 #align is_topological_basis_clopen isTopologicalBasis_clopen
 -/
 
-/- warning: compact_exists_clopen_in_open -> compact_exists_clopen_in_open is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : CompactSpace.{u1} α _inst_1] [_inst_4 : TotallyDisconnectedSpace.{u1} α _inst_1] {x : α} {U : Set.{u1} α}, (IsOpen.{u1} α _inst_1 U) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x U) -> (Exists.{succ u1} (Set.{u1} α) (fun (V : Set.{u1} α) => Exists.{0} (IsClopen.{u1} α _inst_1 V) (fun (hV : IsClopen.{u1} α _inst_1 V) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x V) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) V U))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : CompactSpace.{u1} α _inst_1] [_inst_4 : TotallyDisconnectedSpace.{u1} α _inst_1] {x : α} {U : Set.{u1} α}, (IsOpen.{u1} α _inst_1 U) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x U) -> (Exists.{succ u1} (Set.{u1} α) (fun (V : Set.{u1} α) => And (IsClopen.{u1} α _inst_1 V) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x V) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) V U))))
-Case conversion may be inaccurate. Consider using '#align compact_exists_clopen_in_open compact_exists_clopen_in_openₓ'. -/
 /-- Every member of an open set in a compact Hausdorff totally disconnected space
   is contained in a clopen set contained in the open set.  -/
 theorem compact_exists_clopen_in_open {x : α} {U : Set α} (is_open : IsOpen U) (memU : x ∈ U) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -552,8 +552,7 @@ theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
   refine' is_open_iff_mem_nhds.mpr fun a ha => _
   filter_upwards [eventually_nhds_nhds_within.mpr ha]with b hb
   by_cases a = b
-  · subst h
-    exact hb
+  · subst h; exact hb
   · rw [(Ne.symm h).nhdsWithin_compl_singleton] at hb
     exact hb.filter_mono nhdsWithin_le_nhds
 #align is_open_set_of_eventually_nhds_within isOpen_setOf_eventually_nhdsWithin
@@ -1002,10 +1001,8 @@ theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s
 
 #print nhdsSet_inj_iff /-
 @[simp]
-theorem nhdsSet_inj_iff [T1Space α] {s t : Set α} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t :=
-  by
-  simp_rw [le_antisymm_iff]
-  exact and_congr nhdsSet_le_iff nhdsSet_le_iff
+theorem nhdsSet_inj_iff [T1Space α] {s t : Set α} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t := by
+  simp_rw [le_antisymm_iff]; exact and_congr nhdsSet_le_iff nhdsSet_le_iff
 #align nhds_set_inj_iff nhdsSet_inj_iff
 -/
 
@@ -2052,16 +2049,12 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
       ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
     rcases ih U hU' h1K₂ h2K₂ with ⟨K, h1K, h2K, h3K⟩
     refine' ⟨update K x K₁, _, _, _⟩
-    · intro i
-      by_cases hi : i = x
+    · intro i; by_cases hi : i = x
       · simp only [update_same, hi, h1K₁]
-      · rw [← Ne.def] at hi
-        simp only [update_noteq hi, h1K]
-    · intro i
-      by_cases hi : i = x
+      · rw [← Ne.def] at hi; simp only [update_noteq hi, h1K]
+    · intro i; by_cases hi : i = x
       · simp only [update_same, hi, h2K₁]
-      · rw [← Ne.def] at hi
-        simp only [update_noteq hi, h2K]
+      · rw [← Ne.def] at hi; simp only [update_noteq hi, h2K]
     · simp only [set_bUnion_insert_update _ hx, hK, h3K]
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 
@@ -2382,10 +2375,7 @@ theorem isClosed_setOf_inseparable : IsClosed { p : α × α | Inseparable p.1 p
 #print Inducing.regularSpace /-
 protected theorem Inducing.regularSpace [TopologicalSpace β] {f : β → α} (hf : Inducing f) :
     RegularSpace β :=
-  RegularSpace.ofBasis
-    (fun b => by
-      rw [hf.nhds_eq_comap b]
-      exact (closed_nhds_basis _).comap _)
+  RegularSpace.ofBasis (fun b => by rw [hf.nhds_eq_comap b]; exact (closed_nhds_basis _).comap _)
     fun b s hs => hs.2.Preimage hf.Continuous
 #align inducing.regular_space Inducing.regularSpace
 -/
@@ -2438,9 +2428,7 @@ but is expected to have type
   forall {X : Type.{u1}} {t₁ : TopologicalSpace.{u1} X} {t₂ : TopologicalSpace.{u1} X}, (RegularSpace.{u1} X t₁) -> (RegularSpace.{u1} X t₂) -> (RegularSpace.{u1} X (Inf.inf.{u1} (TopologicalSpace.{u1} X) (Lattice.toInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X)))) t₁ t₂))
 Case conversion may be inaccurate. Consider using '#align regular_space.inf RegularSpace.infₓ'. -/
 theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSpace X t₁)
-    (h₂ : @RegularSpace X t₂) : @RegularSpace X (t₁ ⊓ t₂) :=
-  by
-  rw [inf_eq_iInf]
+    (h₂ : @RegularSpace X t₂) : @RegularSpace X (t₁ ⊓ t₂) := by rw [inf_eq_iInf];
   exact regularSpace_iInf (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align regular_space.inf RegularSpace.inf
 
@@ -2793,8 +2781,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
           (mem_of_subset_of_mem (subset.trans hab (union_subset_union hau hbv))
             (mem_Inter.2 fun i => i.2.2)) with
         h1 h1
-      · exfalso
-        exact h h1
+      · exfalso; exact h h1
       · exact h1
     right
     suffices (⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, ↑Z) ⊆ v
@@ -2844,8 +2831,7 @@ variable [T2Space α] [CompactSpace α]
 theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace α ↔ TotallySeparatedSpace α :=
   by
   constructor
-  · intro h
-    constructor
+  · intro h; constructor
     rintro x - y -
     contrapose!
     intro hyp
@@ -2997,8 +2983,7 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
       IsClopen U ∧ connectedComponent a ∩ U = ∅ ∧ connectedComponent b ⊆ U ∧ coe ⁻¹' V = U :=
     by
     cases' is_closed_connected_component.is_compact.elim_finite_subfamily_closed _ _ h with fin_a ha
-    swap
-    · exact fun Z => Z.2.1.2
+    swap; · exact fun Z => Z.2.1.2
     -- This clopen and its complement will separate the connected components of `a` and `b`
     set U : Set α := ⋂ (i : { Z // IsClopen Z ∧ b ∈ Z }) (H : i ∈ fin_a), i
     have hU : IsClopen U := isClopen_biInter_finset fun i j => i.2.1

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 
 ! This file was ported from Lean 3 source module topology.separation
-! leanprover-community/mathlib commit 195fcd60ff2bfe392543bceb0ec2adcdb472db4c
+! leanprover-community/mathlib commit d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1714,10 +1714,7 @@ instance (priority := 100) DiscreteTopology.toT2Space {α : Type _} [Topological
 -/
 
 /- warning: separated_by_continuous -> separated_by_continuous is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : T2Space.{u2} β _inst_3] {f : α -> β}, (Continuous.{u1, u2} α β _inst_2 _inst_3 f) -> (forall {x : α} {y : α}, (Ne.{succ u2} β (f x) (f y)) -> (Exists.{succ u1} (Set.{u1} α) (fun (u : Set.{u1} α) => Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => And (IsOpen.{u1} α _inst_2 u) (And (IsOpen.{u1} α _inst_2 v) (And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x u) (And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y v) (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) u v))))))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : T2Space.{u1} β _inst_3] {f : α -> β}, (Continuous.{u2, u1} α β _inst_2 _inst_3 f) -> (forall {x : α} {y : α}, (Ne.{succ u1} β (f x) (f y)) -> (Exists.{succ u2} (Set.{u2} α) (fun (u : Set.{u2} α) => Exists.{succ u2} (Set.{u2} α) (fun (v : Set.{u2} α) => And (IsOpen.{u2} α _inst_2 u) (And (IsOpen.{u2} α _inst_2 v) (And (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x u) (And (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) y v) (Disjoint.{u2} (Set.{u2} α) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} α) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} α) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} α) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} α) (Set.instCompleteBooleanAlgebraSet.{u2} α)))))) (BoundedOrder.toOrderBot.{u2} (Set.{u2} α) (Preorder.toLE.{u2} (Set.{u2} α) (PartialOrder.toPreorder.{u2} (Set.{u2} α) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} α) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} α) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} α) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} α) (Set.instCompleteBooleanAlgebraSet.{u2} α)))))))) (CompleteLattice.toBoundedOrder.{u2} (Set.{u2} α) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} α) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} α) (Set.instCompleteBooleanAlgebraSet.{u2} α)))))) u v))))))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align separated_by_continuous separated_by_continuousₓ'. -/
 theorem separated_by_continuous {α : Type _} {β : Type _} [TopologicalSpace α] [TopologicalSpace β]
     [T2Space β] {f : α → β} (hf : Continuous f) {x y : α} (h : f x ≠ f y) :
@@ -1727,10 +1724,7 @@ theorem separated_by_continuous {α : Type _} {β : Type _} [TopologicalSpace α
 #align separated_by_continuous separated_by_continuous
 
 /- warning: separated_by_open_embedding -> separated_by_openEmbedding is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u1} α] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : T2Space.{u1} α _inst_2] {f : α -> β}, (OpenEmbedding.{u1, u2} α β _inst_2 _inst_3 f) -> (forall {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Exists.{succ u2} (Set.{u2} β) (fun (u : Set.{u2} β) => Exists.{succ u2} (Set.{u2} β) (fun (v : Set.{u2} β) => And (IsOpen.{u2} β _inst_3 u) (And (IsOpen.{u2} β _inst_3 v) (And (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) (f x) u) (And (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) (f y) v) (Disjoint.{u2} (Set.{u2} β) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} β) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} β) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} β) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} β) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} β) (Set.completeBooleanAlgebra.{u2} β)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} β) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} β) (Set.booleanAlgebra.{u2} β))) u v))))))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u2} α] [_inst_3 : TopologicalSpace.{u1} β] [_inst_4 : T2Space.{u2} α _inst_2] {f : α -> β}, (OpenEmbedding.{u2, u1} α β _inst_2 _inst_3 f) -> (forall {x : α} {y : α}, (Ne.{succ u2} α x y) -> (Exists.{succ u1} (Set.{u1} β) (fun (u : Set.{u1} β) => Exists.{succ u1} (Set.{u1} β) (fun (v : Set.{u1} β) => And (IsOpen.{u1} β _inst_3 u) (And (IsOpen.{u1} β _inst_3 v) (And (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) (f x) u) (And (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) (f y) v) (Disjoint.{u1} (Set.{u1} β) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} β) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} β) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} β) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} β) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} β) (Set.instCompleteBooleanAlgebraSet.{u1} β)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} β) (Preorder.toLE.{u1} (Set.{u1} β) (PartialOrder.toPreorder.{u1} (Set.{u1} β) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} β) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} β) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} β) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} β) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} β) (Set.instCompleteBooleanAlgebraSet.{u1} β)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} β) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} β) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} β) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} β) (Set.instCompleteBooleanAlgebraSet.{u1} β)))))) u v))))))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align separated_by_open_embedding separated_by_openEmbeddingₓ'. -/
 theorem separated_by_openEmbedding {α β : Type _} [TopologicalSpace α] [TopologicalSpace β]
     [T2Space α] {f : α → β} (hf : OpenEmbedding f) {x y : α} (h : x ≠ y) :
@@ -1940,7 +1934,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.iInter.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13050 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13052 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13050 x._@.Mathlib.Topology.Separation._hyg.13052) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.iInter.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13055 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13057 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13055 x._@.Mathlib.Topology.Separation._hyg.13057) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.iInter.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
 Case conversion may be inaccurate. Consider using '#align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompactₓ'. -/
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we
@@ -2013,18 +2007,26 @@ theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCom
 #align is_compact.binary_compact_cover IsCompact.binary_compact_cover
 
 #print Continuous.isClosedMap /-
-theorem Continuous.isClosedMap [CompactSpace α] [T2Space β] {f : α → β} (h : Continuous f) :
-    IsClosedMap f := fun s hs => (hs.IsCompact.image h).IsClosed
+/-- A continuous map from a compact space to a Hausdorff space is a closed map. -/
+protected theorem Continuous.isClosedMap [CompactSpace α] [T2Space β] {f : α → β}
+    (h : Continuous f) : IsClosedMap f := fun s hs => (hs.IsCompact.image h).IsClosed
 #align continuous.is_closed_map Continuous.isClosedMap
 -/
 
 #print Continuous.closedEmbedding /-
+/-- An injective continuous map from a compact space to a Hausdorff space is a closed embedding. -/
 theorem Continuous.closedEmbedding [CompactSpace α] [T2Space β] {f : α → β} (h : Continuous f)
     (hf : Function.Injective f) : ClosedEmbedding f :=
   closedEmbedding_of_continuous_injective_closed h hf h.IsClosedMap
 #align continuous.closed_embedding Continuous.closedEmbedding
 -/
 
+/-- A surjective continuous map from a compact space to a Hausdorff space is a quotient map. -/
+theorem QuotientMap.of_surjective_continuous [CompactSpace α] [T2Space β] {f : α → β}
+    (hsurj : Surjective f) (hcont : Continuous f) : QuotientMap f :=
+  hcont.IsClosedMap.to_quotientMap hcont hsurj
+#align quotient_map.of_surjective_continuous QuotientMap.of_surjective_continuous
+
 section
 
 open Finset Function
@@ -2499,10 +2501,7 @@ instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [
   ⟨⟩
 
 /- warning: disjoint_nested_nhds -> disjoint_nested_nhds is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T3Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U₁ : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U₁ (nhds.{u1} α _inst_1 x)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U₁ (nhds.{u1} α _inst_1 x)) => Exists.{succ u1} (Set.{u1} α) (fun (V₁ : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) V₁ (nhds.{u1} α _inst_1 x)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) V₁ (nhds.{u1} α _inst_1 x)) => Exists.{succ u1} (Set.{u1} α) (fun (U₂ : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U₂ (nhds.{u1} α _inst_1 y)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U₂ (nhds.{u1} α _inst_1 y)) => Exists.{succ u1} (Set.{u1} α) (fun (V₂ : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) V₂ (nhds.{u1} α _inst_1 y)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) V₂ (nhds.{u1} α _inst_1 y)) => And (IsClosed.{u1} α _inst_1 V₁) (And (IsClosed.{u1} α _inst_1 V₂) (And (IsOpen.{u1} α _inst_1 U₁) (And (IsOpen.{u1} α _inst_1 U₂) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) V₁ U₁) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) V₂ U₂) (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.completeBooleanAlgebra.{u1} α)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} α) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α))) U₁ U₂)))))))))))))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T3Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U₁ : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) U₁ (nhds.{u1} α _inst_1 x)) (Exists.{succ u1} (Set.{u1} α) (fun (V₁ : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) V₁ (nhds.{u1} α _inst_1 x)) (Exists.{succ u1} (Set.{u1} α) (fun (U₂ : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) U₂ (nhds.{u1} α _inst_1 y)) (Exists.{succ u1} (Set.{u1} α) (fun (V₂ : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) V₂ (nhds.{u1} α _inst_1 y)) (And (IsClosed.{u1} α _inst_1 V₁) (And (IsClosed.{u1} α _inst_1 V₂) (And (IsOpen.{u1} α _inst_1 U₁) (And (IsOpen.{u1} α _inst_1 U₂) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) V₁ U₁) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) V₂ U₂) (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) U₁ U₂)))))))))))))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align disjoint_nested_nhds disjoint_nested_nhdsₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -630,7 +630,7 @@ protected theorem Finset.isClosed [T1Space α] (s : Finset α) : IsClosed (s : S
 lean 3 declaration is
   forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.topologicalSpace.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) => α -> (CofiniteTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_2 x)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_2 x)) => Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (IsOpen.{u1} α _inst_2 U) (fun (hU : IsOpen.{u1} α _inst_2 U) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x U) (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
 but is expected to have type
-  forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_2 x)) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (IsOpen.{u1} α _inst_2 U) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x U) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
+  forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_2 x)) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (IsOpen.{u1} α _inst_2 U) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x U) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
 Case conversion may be inaccurate. Consider using '#align t1_space_tfae t1Space_TFAEₓ'. -/
 theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
     TFAE
@@ -672,7 +672,7 @@ theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_2) (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.topologicalSpace.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) => α -> (CofiniteTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_2) (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α)))
+  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_2) (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α)))
 Case conversion may be inaccurate. Consider using '#align t1_space_iff_continuous_cofinite_of t1Space_iff_continuous_cofinite_ofₓ'. -/
 theorem t1Space_iff_continuous_cofinite_of {α : Type _} [TopologicalSpace α] :
     T1Space α ↔ Continuous (@CofiniteTopology.of α) :=
@@ -683,7 +683,7 @@ theorem t1Space_iff_continuous_cofinite_of {α : Type _} [TopologicalSpace α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_1 (CofiniteTopology.topologicalSpace.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) => α -> (CofiniteTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_1 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_1 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))
 Case conversion may be inaccurate. Consider using '#align cofinite_topology.continuous_of CofiniteTopology.continuous_ofₓ'. -/
 theorem CofiniteTopology.continuous_of [T1Space α] : Continuous (@CofiniteTopology.of α) :=
   t1Space_iff_continuous_cofinite_of.mp ‹_›

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -115,7 +115,7 @@ def SeparatedNhds : Set α → Set α → Prop := fun s t : Set α =>
 
 /- warning: separated_nhds_iff_disjoint -> separatedNhds_iff_disjoint is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (SeparatedNhds.{u1} α _inst_1 s t) (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhdsSet.{u1} α _inst_1 t))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (SeparatedNhds.{u1} α _inst_1 s t) (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhdsSet.{u1} α _inst_1 t))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (SeparatedNhds.{u1} α _inst_1 s t) (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhdsSet.{u1} α _inst_1 t))
 Case conversion may be inaccurate. Consider using '#align separated_nhds_iff_disjoint separatedNhds_iff_disjointₓ'. -/
@@ -581,7 +581,7 @@ theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space α] {b : S
 
 /- warning: filter.coclosed_compact_le_cofinite -> Filter.coclosedCompact_le_cofinite is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.coclosedCompact.{u1} α _inst_1) (Filter.cofinite.{u1} α)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.coclosedCompact.{u1} α _inst_1) (Filter.cofinite.{u1} α)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Filter.coclosedCompact.{u1} α _inst_1) (Filter.cofinite.{u1} α)
 Case conversion may be inaccurate. Consider using '#align filter.coclosed_compact_le_cofinite Filter.coclosedCompact_le_cofiniteₓ'. -/
@@ -628,7 +628,7 @@ protected theorem Finset.isClosed [T1Space α] (s : Finset α) : IsClosed (s : S
 
 /- warning: t1_space_tfae -> t1Space_TFAE is a dubious translation:
 lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.topologicalSpace.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) => α -> (CofiniteTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_2 x)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_2 x)) => Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (IsOpen.{u1} α _inst_2 U) (fun (hU : IsOpen.{u1} α _inst_2 U) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x U) (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
+  forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.topologicalSpace.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) => α -> (CofiniteTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_2 x)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_2 x)) => Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (IsOpen.{u1} α _inst_2 U) (fun (hU : IsOpen.{u1} α _inst_2 U) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x U) (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
 but is expected to have type
   forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_2 x)) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (IsOpen.{u1} α _inst_2 U) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x U) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
 Case conversion may be inaccurate. Consider using '#align t1_space_tfae t1Space_TFAEₓ'. -/
@@ -702,7 +702,7 @@ theorem t1Space_iff_exists_open :
 
 /- warning: t1_space_iff_disjoint_pure_nhds -> t1Space_iff_disjoint_pure_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_1) (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α x) (nhds.{u1} α _inst_1 y)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_1) (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α x) (nhds.{u1} α _inst_1 y)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_1) (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α x) (nhds.{u1} α _inst_1 y)))
 Case conversion may be inaccurate. Consider using '#align t1_space_iff_disjoint_pure_nhds t1Space_iff_disjoint_pure_nhdsₓ'. -/
@@ -712,7 +712,7 @@ theorem t1Space_iff_disjoint_pure_nhds : T1Space α ↔ ∀ ⦃x y : α⦄, x 
 
 /- warning: t1_space_iff_disjoint_nhds_pure -> t1Space_iff_disjoint_nhds_pure is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_1) (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α y)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_1) (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α y)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_1) (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α y)))
 Case conversion may be inaccurate. Consider using '#align t1_space_iff_disjoint_nhds_pure t1Space_iff_disjoint_nhds_pureₓ'. -/
@@ -728,7 +728,7 @@ theorem t1Space_iff_specializes_imp_eq : T1Space α ↔ ∀ ⦃x y : α⦄, x 
 
 /- warning: disjoint_pure_nhds -> disjoint_pure_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α x) (nhds.{u1} α _inst_1 y))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α x) (nhds.{u1} α _inst_1 y))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α x) (nhds.{u1} α _inst_1 y))
 Case conversion may be inaccurate. Consider using '#align disjoint_pure_nhds disjoint_pure_nhdsₓ'. -/
@@ -738,7 +738,7 @@ theorem disjoint_pure_nhds [T1Space α] {x y : α} (h : x ≠ y) : Disjoint (pur
 
 /- warning: disjoint_nhds_pure -> disjoint_nhds_pure is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α y))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α y))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α y))
 Case conversion may be inaccurate. Consider using '#align disjoint_nhds_pure disjoint_nhds_pureₓ'. -/
@@ -767,7 +767,7 @@ theorem specializes_eq_eq [T1Space α] : (· ⤳ ·) = @Eq α :=
 
 /- warning: pure_le_nhds_iff -> pure_le_nhds_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {a : α} {b : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α a) (nhds.{u1} α _inst_1 b)) (Eq.{succ u1} α a b)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {a : α} {b : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α a) (nhds.{u1} α _inst_1 b)) (Eq.{succ u1} α a b)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {a : α} {b : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α a) (nhds.{u1} α _inst_1 b)) (Eq.{succ u1} α a b)
 Case conversion may be inaccurate. Consider using '#align pure_le_nhds_iff pure_le_nhds_iffₓ'. -/
@@ -778,7 +778,7 @@ theorem pure_le_nhds_iff [T1Space α] {a b : α} : pure a ≤ 𝓝 b ↔ a = b :
 
 /- warning: nhds_le_nhds_iff -> nhds_le_nhds_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {a : α} {b : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhds.{u1} α _inst_1 a) (nhds.{u1} α _inst_1 b)) (Eq.{succ u1} α a b)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {a : α} {b : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhds.{u1} α _inst_1 a) (nhds.{u1} α _inst_1 b)) (Eq.{succ u1} α a b)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {a : α} {b : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhds.{u1} α _inst_1 a) (nhds.{u1} α _inst_1 b)) (Eq.{succ u1} α a b)
 Case conversion may be inaccurate. Consider using '#align nhds_le_nhds_iff nhds_le_nhds_iffₓ'. -/
@@ -985,7 +985,7 @@ theorem compl_singleton_mem_nhdsSet_iff [T1Space α] {x : α} {s : Set α} : {x}
 
 /- warning: nhds_set_le_iff -> nhdsSet_le_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhdsSet.{u1} α _inst_1 t)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s t)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhdsSet.{u1} α _inst_1 t)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s t)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α} {t : Set.{u1} α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhdsSet.{u1} α _inst_1 t)) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s t)
 Case conversion may be inaccurate. Consider using '#align nhds_set_le_iff nhdsSet_le_iffₓ'. -/
@@ -1027,7 +1027,7 @@ theorem strictMono_nhdsSet [T1Space α] : StrictMono (𝓝ˢ : Set α → Filter
 
 /- warning: nhds_le_nhds_set_iff -> nhds_le_nhdsSet_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α} {x : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhds.{u1} α _inst_1 x) (nhdsSet.{u1} α _inst_1 s)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α} {x : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (nhds.{u1} α _inst_1 x) (nhdsSet.{u1} α _inst_1 s)) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α} {x : α}, Iff (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (nhds.{u1} α _inst_1 x) (nhdsSet.{u1} α _inst_1 s)) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s)
 Case conversion may be inaccurate. Consider using '#align nhds_le_nhds_set_iff nhds_le_nhdsSet_iffₓ'. -/
@@ -1297,7 +1297,7 @@ theorem t2_separation [T2Space α] {x y : α} (h : x ≠ y) :
 
 /- warning: t2_space_iff_disjoint_nhds -> t2Space_iff_disjoint_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall (x : α) (y : α), (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall (x : α) (y : α), (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall (x : α) (y : α), (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y)))
 Case conversion may be inaccurate. Consider using '#align t2_space_iff_disjoint_nhds t2Space_iff_disjoint_nhdsₓ'. -/
@@ -1310,7 +1310,7 @@ theorem t2Space_iff_disjoint_nhds : T2Space α ↔ ∀ x y : α, x ≠ y → Dis
 
 /- warning: disjoint_nhds_nhds -> disjoint_nhds_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {x : α} {y : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y)) (Ne.{succ u1} α x y)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {x : α} {y : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y)) (Ne.{succ u1} α x y)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {x : α} {y : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y)) (Ne.{succ u1} α x y)
 Case conversion may be inaccurate. Consider using '#align disjoint_nhds_nhds disjoint_nhds_nhdsₓ'. -/
@@ -1321,7 +1321,7 @@ theorem disjoint_nhds_nhds [T2Space α] {x y : α} : Disjoint (𝓝 x) (𝓝 y)
 
 /- warning: pairwise_disjoint_nhds -> pairwise_disjoint_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1], Pairwise.{u1} α (Function.onFun.{succ u1, succ u1, 1} α (Filter.{u1} α) Prop (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α)))) (nhds.{u1} α _inst_1))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1], Pairwise.{u1} α (Function.onFun.{succ u1, succ u1, 1} α (Filter.{u1} α) Prop (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α)))) (nhds.{u1} α _inst_1))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1], Pairwise.{u1} α (Function.onFun.{succ u1, succ u1, 1} α (Filter.{u1} α) Prop (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α)))) (nhds.{u1} α _inst_1))
 Case conversion may be inaccurate. Consider using '#align pairwise_disjoint_nhds pairwise_disjoint_nhdsₓ'. -/
@@ -1331,7 +1331,7 @@ theorem pairwise_disjoint_nhds [T2Space α] : Pairwise (Disjoint on (𝓝 : α 
 
 /- warning: set.pairwise_disjoint_nhds -> Set.pairwiseDisjoint_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] (s : Set.{u1} α), Set.PairwiseDisjoint.{u1, u1} (Filter.{u1} α) α (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) s (nhds.{u1} α _inst_1)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] (s : Set.{u1} α), Set.PairwiseDisjoint.{u1, u1} (Filter.{u1} α) α (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) s (nhds.{u1} α _inst_1)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] (s : Set.{u1} α), Set.PairwiseDisjoint.{u1, u1} (Filter.{u1} α) α (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) s (nhds.{u1} α _inst_1)
 Case conversion may be inaccurate. Consider using '#align set.pairwise_disjoint_nhds Set.pairwiseDisjoint_nhdsₓ'. -/
@@ -1353,7 +1353,7 @@ theorem Set.Finite.t2_separation [T2Space α] {s : Set α} (hs : s.Finite) :
 
 /- warning: is_open_set_of_disjoint_nhds_nhds -> isOpen_setOf_disjoint_nhds_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], IsOpen.{u1} (Prod.{u1, u1} α α) (Prod.topologicalSpace.{u1, u1} α α _inst_1 _inst_1) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 (Prod.fst.{u1, u1} α α p)) (nhds.{u1} α _inst_1 (Prod.snd.{u1, u1} α α p))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], IsOpen.{u1} (Prod.{u1, u1} α α) (Prod.topologicalSpace.{u1, u1} α α _inst_1 _inst_1) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 (Prod.fst.{u1, u1} α α p)) (nhds.{u1} α _inst_1 (Prod.snd.{u1, u1} α α p))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], IsOpen.{u1} (Prod.{u1, u1} α α) (instTopologicalSpaceProd.{u1, u1} α α _inst_1 _inst_1) (setOf.{u1} (Prod.{u1, u1} α α) (fun (p : Prod.{u1, u1} α α) => Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 (Prod.fst.{u1, u1} α α p)) (nhds.{u1} α _inst_1 (Prod.snd.{u1, u1} α α p))))
 Case conversion may be inaccurate. Consider using '#align is_open_set_of_disjoint_nhds_nhds isOpen_setOf_disjoint_nhds_nhdsₓ'. -/
@@ -1433,7 +1433,7 @@ theorem t2_separation_compact_nhds [LocallyCompactSpace α] [T2Space α] {x y :
 
 /- warning: t2_iff_ultrafilter -> t2_iff_ultrafilter is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall {x : α} {y : α} (f : Ultrafilter.{u1} α), (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ultrafilter.{u1} α) (Filter.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (Ultrafilter.Filter.hasCoeT.{u1} α))) f) (nhds.{u1} α _inst_1 x)) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ultrafilter.{u1} α) (Filter.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (Ultrafilter.Filter.hasCoeT.{u1} α))) f) (nhds.{u1} α _inst_1 y)) -> (Eq.{succ u1} α x y))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall {x : α} {y : α} (f : Ultrafilter.{u1} α), (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ultrafilter.{u1} α) (Filter.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (Ultrafilter.Filter.hasCoeT.{u1} α))) f) (nhds.{u1} α _inst_1 x)) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ultrafilter.{u1} α) (Filter.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (Ultrafilter.Filter.hasCoeT.{u1} α))) f) (nhds.{u1} α _inst_1 y)) -> (Eq.{succ u1} α x y))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall {x : α} {y : α} (f : Ultrafilter.{u1} α), (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Ultrafilter.toFilter.{u1} α f) (nhds.{u1} α _inst_1 x)) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Ultrafilter.toFilter.{u1} α f) (nhds.{u1} α _inst_1 y)) -> (Eq.{succ u1} α x y))
 Case conversion may be inaccurate. Consider using '#align t2_iff_ultrafilter t2_iff_ultrafilterₓ'. -/
@@ -1535,7 +1535,7 @@ class T25Space (α : Type u) [TopologicalSpace α] : Prop where
 
 /- warning: disjoint_lift'_closure_nhds -> disjoint_lift'_closure_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T25Space.{u1} α _inst_1] {x : α} {y : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Filter.lift'.{u1, u1} α α (nhds.{u1} α _inst_1 x) (closure.{u1} α _inst_1)) (Filter.lift'.{u1, u1} α α (nhds.{u1} α _inst_1 y) (closure.{u1} α _inst_1))) (Ne.{succ u1} α x y)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T25Space.{u1} α _inst_1] {x : α} {y : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Filter.lift'.{u1, u1} α α (nhds.{u1} α _inst_1 x) (closure.{u1} α _inst_1)) (Filter.lift'.{u1, u1} α α (nhds.{u1} α _inst_1 y) (closure.{u1} α _inst_1))) (Ne.{succ u1} α x y)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T25Space.{u1} α _inst_1] {x : α} {y : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Filter.lift'.{u1, u1} α α (nhds.{u1} α _inst_1 x) (closure.{u1} α _inst_1)) (Filter.lift'.{u1, u1} α α (nhds.{u1} α _inst_1 y) (closure.{u1} α _inst_1))) (Ne.{succ u1} α x y)
 Case conversion may be inaccurate. Consider using '#align disjoint_lift'_closure_nhds disjoint_lift'_closure_nhdsₓ'. -/
@@ -1594,7 +1594,7 @@ are useful without a `nonempty α` instance.
 
 /- warning: Lim_eq -> lim_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {f : Filter.{u1} α} {a : α} [_inst_3 : Filter.NeBot.{u1} α f], (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) f (nhds.{u1} α _inst_1 a)) -> (Eq.{succ u1} α (lim.{u1} α _inst_1 (Nonempty.intro.{succ u1} α a) f) a)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {f : Filter.{u1} α} {a : α} [_inst_3 : Filter.NeBot.{u1} α f], (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) f (nhds.{u1} α _inst_1 a)) -> (Eq.{succ u1} α (lim.{u1} α _inst_1 (Nonempty.intro.{succ u1} α a) f) a)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {f : Filter.{u1} α} {a : α} [_inst_3 : Filter.NeBot.{u1} α f], (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) f (nhds.{u1} α _inst_1 a)) -> (Eq.{succ u1} α (lim.{u1} α _inst_1 (Nonempty.intro.{succ u1} α a) f) a)
 Case conversion may be inaccurate. Consider using '#align Lim_eq lim_eqₓ'. -/
@@ -1604,7 +1604,7 @@ theorem lim_eq {a : α} [NeBot f] (h : f ≤ 𝓝 a) : @lim _ _ ⟨a⟩ f = a :=
 
 /- warning: Lim_eq_iff -> lim_eq_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {f : Filter.{u1} α} [_inst_3 : Filter.NeBot.{u1} α f], (Exists.{succ u1} α (fun (a : α) => LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) f (nhds.{u1} α _inst_1 a))) -> (forall {a : α}, Iff (Eq.{succ u1} α (lim.{u1} α _inst_1 (Nonempty.intro.{succ u1} α a) f) a) (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) f (nhds.{u1} α _inst_1 a)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {f : Filter.{u1} α} [_inst_3 : Filter.NeBot.{u1} α f], (Exists.{succ u1} α (fun (a : α) => LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) f (nhds.{u1} α _inst_1 a))) -> (forall {a : α}, Iff (Eq.{succ u1} α (lim.{u1} α _inst_1 (Nonempty.intro.{succ u1} α a) f) a) (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) f (nhds.{u1} α _inst_1 a)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {f : Filter.{u1} α} [_inst_3 : Filter.NeBot.{u1} α f], (Exists.{succ u1} α (fun (a : α) => LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) f (nhds.{u1} α _inst_1 a))) -> (forall {a : α}, Iff (Eq.{succ u1} α (lim.{u1} α _inst_1 (Nonempty.intro.{succ u1} α a) f) a) (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) f (nhds.{u1} α _inst_1 a)))
 Case conversion may be inaccurate. Consider using '#align Lim_eq_iff lim_eq_iffₓ'. -/
@@ -1614,7 +1614,7 @@ theorem lim_eq_iff [NeBot f] (h : ∃ a : α, f ≤ nhds a) {a} : @lim _ _ ⟨a
 
 /- warning: ultrafilter.Lim_eq_iff_le_nhds -> Ultrafilter.lim_eq_iff_le_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : CompactSpace.{u1} α _inst_1] {x : α} {F : Ultrafilter.{u1} α}, Iff (Eq.{succ u1} α (Ultrafilter.lim.{u1} α _inst_1 F) x) (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ultrafilter.{u1} α) (Filter.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (Ultrafilter.Filter.hasCoeT.{u1} α))) F) (nhds.{u1} α _inst_1 x))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : CompactSpace.{u1} α _inst_1] {x : α} {F : Ultrafilter.{u1} α}, Iff (Eq.{succ u1} α (Ultrafilter.lim.{u1} α _inst_1 F) x) (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ultrafilter.{u1} α) (Filter.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Ultrafilter.{u1} α) (Filter.{u1} α) (Ultrafilter.Filter.hasCoeT.{u1} α))) F) (nhds.{u1} α _inst_1 x))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] [_inst_3 : CompactSpace.{u1} α _inst_1] {x : α} {F : Ultrafilter.{u1} α}, Iff (Eq.{succ u1} α (Ultrafilter.lim.{u1} α _inst_1 F) x) (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Ultrafilter.toFilter.{u1} α F) (nhds.{u1} α _inst_1 x))
 Case conversion may be inaccurate. Consider using '#align ultrafilter.Lim_eq_iff_le_nhds Ultrafilter.lim_eq_iff_le_nhdsₓ'. -/
@@ -2177,7 +2177,7 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
 
 /- warning: regular_space_tfae -> regularSpace_TFAE is a dubious translation:
 lean 3 declaration is
-  forall (X : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} X], List.TFAE (List.cons.{0} Prop (RegularSpace.{u1} X _inst_2) (List.cons.{0} Prop (forall (s : Set.{u1} X) (a : X), (Not (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) a (closure.{u1} X _inst_2 s))) -> (Disjoint.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} X) (Filter.completeLattice.{u1} X))) (nhdsSet.{u1} X _inst_2 s) (nhds.{u1} X _inst_2 a))) (List.cons.{0} Prop (forall (a : X) (s : Set.{u1} X), Iff (Disjoint.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} X) (Filter.completeLattice.{u1} X))) (nhdsSet.{u1} X _inst_2 s) (nhds.{u1} X _inst_2 a)) (Not (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) a (closure.{u1} X _inst_2 s)))) (List.cons.{0} Prop (forall (a : X) (s : Set.{u1} X), (Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) s (nhds.{u1} X _inst_2 a)) -> (Exists.{succ u1} (Set.{u1} X) (fun (t : Set.{u1} X) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) t (nhds.{u1} X _inst_2 a)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) t (nhds.{u1} X _inst_2 a)) => And (IsClosed.{u1} X _inst_2 t) (HasSubset.Subset.{u1} (Set.{u1} X) (Set.hasSubset.{u1} X) t s))))) (List.cons.{0} Prop (forall (a : X), LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Filter.lift'.{u1, u1} X X (nhds.{u1} X _inst_2 a) (closure.{u1} X _inst_2)) (nhds.{u1} X _inst_2 a)) (List.cons.{0} Prop (forall (a : X), Eq.{succ u1} (Filter.{u1} X) (Filter.lift'.{u1, u1} X X (nhds.{u1} X _inst_2 a) (closure.{u1} X _inst_2)) (nhds.{u1} X _inst_2 a)) (List.nil.{0} Prop)))))))
+  forall (X : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} X], List.TFAE (List.cons.{0} Prop (RegularSpace.{u1} X _inst_2) (List.cons.{0} Prop (forall (s : Set.{u1} X) (a : X), (Not (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) a (closure.{u1} X _inst_2 s))) -> (Disjoint.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} X) (Filter.completeLattice.{u1} X))) (nhdsSet.{u1} X _inst_2 s) (nhds.{u1} X _inst_2 a))) (List.cons.{0} Prop (forall (a : X) (s : Set.{u1} X), Iff (Disjoint.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} X) (Filter.completeLattice.{u1} X))) (nhdsSet.{u1} X _inst_2 s) (nhds.{u1} X _inst_2 a)) (Not (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) a (closure.{u1} X _inst_2 s)))) (List.cons.{0} Prop (forall (a : X) (s : Set.{u1} X), (Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) s (nhds.{u1} X _inst_2 a)) -> (Exists.{succ u1} (Set.{u1} X) (fun (t : Set.{u1} X) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) t (nhds.{u1} X _inst_2 a)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (Filter.hasMem.{u1} X) t (nhds.{u1} X _inst_2 a)) => And (IsClosed.{u1} X _inst_2 t) (HasSubset.Subset.{u1} (Set.{u1} X) (Set.hasSubset.{u1} X) t s))))) (List.cons.{0} Prop (forall (a : X), LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Filter.lift'.{u1, u1} X X (nhds.{u1} X _inst_2 a) (closure.{u1} X _inst_2)) (nhds.{u1} X _inst_2 a)) (List.cons.{0} Prop (forall (a : X), Eq.{succ u1} (Filter.{u1} X) (Filter.lift'.{u1, u1} X X (nhds.{u1} X _inst_2 a) (closure.{u1} X _inst_2)) (nhds.{u1} X _inst_2 a)) (List.nil.{0} Prop)))))))
 but is expected to have type
   forall (X : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} X], List.TFAE (List.cons.{0} Prop (RegularSpace.{u1} X _inst_2) (List.cons.{0} Prop (forall (s : Set.{u1} X) (a : X), (Not (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) a (closure.{u1} X _inst_2 s))) -> (Disjoint.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} X) (Filter.instCompleteLatticeFilter.{u1} X))) (nhdsSet.{u1} X _inst_2 s) (nhds.{u1} X _inst_2 a))) (List.cons.{0} Prop (forall (a : X) (s : Set.{u1} X), Iff (Disjoint.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} X) (Filter.instCompleteLatticeFilter.{u1} X))) (nhdsSet.{u1} X _inst_2 s) (nhds.{u1} X _inst_2 a)) (Not (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) a (closure.{u1} X _inst_2 s)))) (List.cons.{0} Prop (forall (a : X) (s : Set.{u1} X), (Membership.mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (instMembershipSetFilter.{u1} X) s (nhds.{u1} X _inst_2 a)) -> (Exists.{succ u1} (Set.{u1} X) (fun (t : Set.{u1} X) => And (Membership.mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (instMembershipSetFilter.{u1} X) t (nhds.{u1} X _inst_2 a)) (And (IsClosed.{u1} X _inst_2 t) (HasSubset.Subset.{u1} (Set.{u1} X) (Set.instHasSubsetSet.{u1} X) t s))))) (List.cons.{0} Prop (forall (a : X), LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (Filter.lift'.{u1, u1} X X (nhds.{u1} X _inst_2 a) (closure.{u1} X _inst_2)) (nhds.{u1} X _inst_2 a)) (List.cons.{0} Prop (forall (a : X), Eq.{succ u1} (Filter.{u1} X) (Filter.lift'.{u1, u1} X X (nhds.{u1} X _inst_2 a) (closure.{u1} X _inst_2)) (nhds.{u1} X _inst_2 a)) (List.nil.{0} Prop)))))))
 Case conversion may be inaccurate. Consider using '#align regular_space_tfae regularSpace_TFAEₓ'. -/
@@ -2247,7 +2247,7 @@ variable [RegularSpace α] {a : α} {s : Set α}
 
 /- warning: disjoint_nhds_set_nhds -> disjoint_nhdsSet_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhds.{u1} α _inst_1 a)) (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α _inst_1 s)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhds.{u1} α _inst_1 a)) (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α _inst_1 s)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhdsSet.{u1} α _inst_1 s) (nhds.{u1} α _inst_1 a)) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α _inst_1 s)))
 Case conversion may be inaccurate. Consider using '#align disjoint_nhds_set_nhds disjoint_nhdsSet_nhdsₓ'. -/
@@ -2257,7 +2257,7 @@ theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 a) ↔ a ∉ closure s
 
 /- warning: disjoint_nhds_nhds_set -> disjoint_nhds_nhdsSet is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 a) (nhdsSet.{u1} α _inst_1 s)) (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α _inst_1 s)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 a) (nhdsSet.{u1} α _inst_1 s)) (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α _inst_1 s)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {s : Set.{u1} α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 a) (nhdsSet.{u1} α _inst_1 s)) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α _inst_1 s)))
 Case conversion may be inaccurate. Consider using '#align disjoint_nhds_nhds_set disjoint_nhds_nhdsSetₓ'. -/
@@ -2333,7 +2333,7 @@ theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set
 
 /- warning: disjoint_nhds_nhds_iff_not_specializes -> disjoint_nhds_nhds_iff_not_specializes is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {b : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 a) (nhds.{u1} α _inst_1 b)) (Not (Specializes.{u1} α _inst_1 a b))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {b : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_1 a) (nhds.{u1} α _inst_1 b)) (Not (Specializes.{u1} α _inst_1 a b))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : RegularSpace.{u1} α _inst_1] {a : α} {b : α}, Iff (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_1 a) (nhds.{u1} α _inst_1 b)) (Not (Specializes.{u1} α _inst_1 a b))
 Case conversion may be inaccurate. Consider using '#align disjoint_nhds_nhds_iff_not_specializes disjoint_nhds_nhds_iff_not_specializesₓ'. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -562,7 +562,7 @@ theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
 protected theorem Set.Finite.isClosed [T1Space α] {s : Set α} (hs : Set.Finite s) : IsClosed s :=
   by
   rw [← bUnion_of_singleton s]
-  exact isClosed_bunionᵢ hs fun i hi => isClosed_singleton
+  exact isClosed_biUnion hs fun i hi => isClosed_singleton
 #align set.finite.is_closed Set.Finite.isClosed
 -/
 
@@ -956,13 +956,13 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
 #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert
 -/
 
-/- warning: bInter_basis_nhds -> binterᵢ_basis_nhds is a dubious translation:
+/- warning: bInter_basis_nhds -> biInter_basis_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {ι : Sort.{u2}} {p : ι -> Prop} {s : ι -> (Set.{u1} α)} {x : α}, (Filter.HasBasis.{u1, u2} α ι (nhds.{u1} α _inst_1 x) p s) -> (Eq.{succ u1} (Set.{u1} α) (Set.interᵢ.{u1, u2} α ι (fun (i : ι) => Set.interᵢ.{u1, 0} α (p i) (fun (h : p i) => s i))) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {ι : Sort.{u2}} {p : ι -> Prop} {s : ι -> (Set.{u1} α)} {x : α}, (Filter.HasBasis.{u1, u2} α ι (nhds.{u1} α _inst_1 x) p s) -> (Eq.{succ u1} (Set.{u1} α) (Set.iInter.{u1, u2} α ι (fun (i : ι) => Set.iInter.{u1, 0} α (p i) (fun (h : p i) => s i))) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : T1Space.{u2} α _inst_1] {ι : Sort.{u1}} {p : ι -> Prop} {s : ι -> (Set.{u2} α)} {x : α}, (Filter.HasBasis.{u2, u1} α ι (nhds.{u2} α _inst_1 x) p s) -> (Eq.{succ u2} (Set.{u2} α) (Set.interᵢ.{u2, u1} α ι (fun (i : ι) => Set.interᵢ.{u2, 0} α (p i) (fun (h : p i) => s i))) (Singleton.singleton.{u2, u2} α (Set.{u2} α) (Set.instSingletonSet.{u2} α) x))
-Case conversion may be inaccurate. Consider using '#align bInter_basis_nhds binterᵢ_basis_nhdsₓ'. -/
-theorem binterᵢ_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι → Set α} {x : α}
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : T1Space.{u2} α _inst_1] {ι : Sort.{u1}} {p : ι -> Prop} {s : ι -> (Set.{u2} α)} {x : α}, (Filter.HasBasis.{u2, u1} α ι (nhds.{u2} α _inst_1 x) p s) -> (Eq.{succ u2} (Set.{u2} α) (Set.iInter.{u2, u1} α ι (fun (i : ι) => Set.iInter.{u2, 0} α (p i) (fun (h : p i) => s i))) (Singleton.singleton.{u2, u2} α (Set.{u2} α) (Set.instSingletonSet.{u2} α) x))
+Case conversion may be inaccurate. Consider using '#align bInter_basis_nhds biInter_basis_nhdsₓ'. -/
+theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι → Set α} {x : α}
     (h : (𝓝 x).HasBasis p s) : (⋂ (i) (h : p i), s i) = {x} :=
   by
   simp only [eq_singleton_iff_unique_mem, mem_Inter]
@@ -970,7 +970,7 @@ theorem binterᵢ_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : 
   contrapose! hy
   rcases h.mem_iff.1 (compl_singleton_mem_nhds hy.symm) with ⟨i, hi, hsub⟩
   exact ⟨i, hi, fun h => hsub h rfl⟩
-#align bInter_basis_nhds binterᵢ_basis_nhds
+#align bInter_basis_nhds biInter_basis_nhds
 
 /- warning: compl_singleton_mem_nhds_set_iff -> compl_singleton_mem_nhdsSet_iff is a dubious translation:
 lean 3 declaration is
@@ -1847,8 +1847,8 @@ theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ
     (hf₂ : ∀ y, Continuous fun x => f x y) (hg₁ : ∀ x, Continuous (g x))
     (hg₂ : ∀ y, Continuous fun x => g x y) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
   suffices closure s ⊆ ⋂ y ∈ closure t, { x | f x y = g x y } by simpa only [subset_def, mem_Inter]
-  (closure_minimal fun x hx => mem_interᵢ₂.2 <| Set.EqOn.closure (h x hx) (hf₁ _) (hg₁ _)) <|
-    isClosed_binterᵢ fun y hy => isClosed_eq (hf₂ _) (hg₂ _)
+  (closure_minimal fun x hx => mem_iInter₂.2 <| Set.EqOn.closure (h x hx) (hf₁ _) (hg₁ _)) <|
+    isClosed_biInter fun y hy => isClosed_eq (hf₂ _) (hg₂ _)
 #align eq_on_closure₂' eqOn_closure₂'
 
 /- warning: eq_on_closure₂ -> eqOn_closure₂ is a dubious translation:
@@ -1924,7 +1924,7 @@ theorem IsCompact.isClosed [T2Space α] {s : Set α} (hs : IsCompact s) : IsClos
 #print Filter.coclosedCompact_eq_cocompact /-
 @[simp]
 theorem Filter.coclosedCompact_eq_cocompact [T2Space α] : coclosedCompact α = cocompact α := by
-  simp [coclosed_compact, cocompact, infᵢ_and', and_iff_right_of_imp IsCompact.isClosed]
+  simp [coclosed_compact, cocompact, iInf_and', and_iff_right_of_imp IsCompact.isClosed]
 #align filter.coclosed_compact_eq_cocompact Filter.coclosedCompact_eq_cocompact
 -/
 
@@ -1938,9 +1938,9 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 
 /- warning: exists_subset_nhds_of_is_compact -> exists_subset_nhds_of_isCompact is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.iInter.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13050 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13052 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13050 x._@.Mathlib.Topology.Separation._hyg.13052) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13050 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13052 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13050 x._@.Mathlib.Topology.Separation._hyg.13052) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.iInter.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
 Case conversion may be inaccurate. Consider using '#align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompactₓ'. -/
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we
@@ -2031,9 +2031,9 @@ open Finset Function
 
 /- warning: is_compact.finite_compact_cover -> IsCompact.finite_compact_cover is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_1 s) -> (forall {ι : Type.{u2}} (t : Finset.{u2} ι) (U : ι -> (Set.{u1} α)), (forall (i : ι), (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) -> (IsOpen.{u1} α _inst_1 (U i))) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) => U i)))) -> (Exists.{max (succ u2) (succ u1)} (ι -> (Set.{u1} α)) (fun (K : ι -> (Set.{u1} α)) => And (forall (i : ι), IsCompact.{u1} α _inst_1 (K i)) (And (forall (i : ι), HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (K i) (U i)) (Eq.{succ u1} (Set.{u1} α) s (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) => K i))))))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {s : Set.{u1} α}, (IsCompact.{u1} α _inst_1 s) -> (forall {ι : Type.{u2}} (t : Finset.{u2} ι) (U : ι -> (Set.{u1} α)), (forall (i : ι), (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) -> (IsOpen.{u1} α _inst_1 (U i))) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) => U i)))) -> (Exists.{max (succ u2) (succ u1)} (ι -> (Set.{u1} α)) (fun (K : ι -> (Set.{u1} α)) => And (forall (i : ι), IsCompact.{u1} α _inst_1 (K i)) (And (forall (i : ι), HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (K i) (U i)) (Eq.{succ u1} (Set.{u1} α) s (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i t) => K i))))))))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {s : Set.{u2} α}, (IsCompact.{u2} α _inst_1 s) -> (forall {ι : Type.{u1}} (t : Finset.{u1} ι) (U : ι -> (Set.{u2} α)), (forall (i : ι), (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) -> (IsOpen.{u2} α _inst_1 (U i))) -> (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) s (Set.unionᵢ.{u2, succ u1} α ι (fun (i : ι) => Set.unionᵢ.{u2, 0} α (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) => U i)))) -> (Exists.{max (succ u2) (succ u1)} (ι -> (Set.{u2} α)) (fun (K : ι -> (Set.{u2} α)) => And (forall (i : ι), IsCompact.{u2} α _inst_1 (K i)) (And (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (K i) (U i)) (Eq.{succ u2} (Set.{u2} α) s (Set.unionᵢ.{u2, succ u1} α ι (fun (i : ι) => Set.unionᵢ.{u2, 0} α (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) => K i))))))))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {s : Set.{u2} α}, (IsCompact.{u2} α _inst_1 s) -> (forall {ι : Type.{u1}} (t : Finset.{u1} ι) (U : ι -> (Set.{u2} α)), (forall (i : ι), (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) -> (IsOpen.{u2} α _inst_1 (U i))) -> (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) s (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => Set.iUnion.{u2, 0} α (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) => U i)))) -> (Exists.{max (succ u2) (succ u1)} (ι -> (Set.{u2} α)) (fun (K : ι -> (Set.{u2} α)) => And (forall (i : ι), IsCompact.{u2} α _inst_1 (K i)) (And (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (K i) (U i)) (Eq.{succ u2} (Set.{u2} α) s (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => Set.iUnion.{u2, 0} α (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i t) => K i))))))))
 Case conversion may be inaccurate. Consider using '#align is_compact.finite_compact_cover IsCompact.finite_compact_coverₓ'. -/
 /-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
 theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact s) {ι} (t : Finset ι)
@@ -2043,10 +2043,10 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
     induction' t using Finset.induction with x t hx ih generalizing U hU s hs hsC
     · refine' ⟨fun _ => ∅, fun i => isCompact_empty, fun i => empty_subset _, _⟩
       simpa only [subset_empty_iff, Union_false, Union_empty] using hsC
-    simp only [Finset.set_bunionᵢ_insert] at hsC
+    simp only [Finset.set_biUnion_insert] at hsC
     simp only [Finset.mem_insert] at hU
     have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU i (Or.inr hi)
-    rcases hs.binary_compact_cover (hU x (Or.inl rfl)) (isOpen_bunionᵢ hU') hsC with
+    rcases hs.binary_compact_cover (hU x (Or.inl rfl)) (isOpen_biUnion hU') hsC with
       ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
     rcases ih U hU' h1K₂ h2K₂ with ⟨K, h1K, h2K, h3K⟩
     refine' ⟨update K x K₁, _, _, _⟩
@@ -2395,14 +2395,14 @@ theorem regularSpace_induced (f : β → α) : @RegularSpace β (induced f ‹_
 #align regular_space_induced regularSpace_induced
 -/
 
-/- warning: regular_space_Inf -> regularSpace_infₛ is a dubious translation:
+/- warning: regular_space_Inf -> regularSpace_sInf is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} {T : Set.{u1} (TopologicalSpace.{u1} X)}, (forall (t : TopologicalSpace.{u1} X), (Membership.Mem.{u1, u1} (TopologicalSpace.{u1} X) (Set.{u1} (TopologicalSpace.{u1} X)) (Set.hasMem.{u1} (TopologicalSpace.{u1} X)) t T) -> (RegularSpace.{u1} X t)) -> (RegularSpace.{u1} X (InfSet.infₛ.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toHasInf.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.completeLattice.{u1} X))) T))
+  forall {X : Type.{u1}} {T : Set.{u1} (TopologicalSpace.{u1} X)}, (forall (t : TopologicalSpace.{u1} X), (Membership.Mem.{u1, u1} (TopologicalSpace.{u1} X) (Set.{u1} (TopologicalSpace.{u1} X)) (Set.hasMem.{u1} (TopologicalSpace.{u1} X)) t T) -> (RegularSpace.{u1} X t)) -> (RegularSpace.{u1} X (InfSet.sInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toHasInf.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.completeLattice.{u1} X))) T))
 but is expected to have type
-  forall {X : Type.{u1}} {T : Set.{u1} (TopologicalSpace.{u1} X)}, (forall (t : TopologicalSpace.{u1} X), (Membership.mem.{u1, u1} (TopologicalSpace.{u1} X) (Set.{u1} (TopologicalSpace.{u1} X)) (Set.instMembershipSet.{u1} (TopologicalSpace.{u1} X)) t T) -> (RegularSpace.{u1} X t)) -> (RegularSpace.{u1} X (InfSet.infₛ.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toInfSet.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X))) T))
-Case conversion may be inaccurate. Consider using '#align regular_space_Inf regularSpace_infₛₓ'. -/
-theorem regularSpace_infₛ {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @RegularSpace X t) :
-    @RegularSpace X (infₛ T) := by
+  forall {X : Type.{u1}} {T : Set.{u1} (TopologicalSpace.{u1} X)}, (forall (t : TopologicalSpace.{u1} X), (Membership.mem.{u1, u1} (TopologicalSpace.{u1} X) (Set.{u1} (TopologicalSpace.{u1} X)) (Set.instMembershipSet.{u1} (TopologicalSpace.{u1} X)) t T) -> (RegularSpace.{u1} X t)) -> (RegularSpace.{u1} X (InfSet.sInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toInfSet.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X))) T))
+Case conversion may be inaccurate. Consider using '#align regular_space_Inf regularSpace_sInfₓ'. -/
+theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @RegularSpace X t) :
+    @RegularSpace X (sInf T) := by
   letI := Inf T
   have :
     ∀ a,
@@ -2412,22 +2412,22 @@ theorem regularSpace_infₛ {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T,
         fun If => ⋂ i : If.1, If.snd i :=
     by
     intro a
-    rw [nhds_infₛ, ← infᵢ_subtype'']
+    rw [nhds_sInf, ← iInf_subtype'']
     exact has_basis_infi fun t : T => @closed_nhds_basis X t (h t t.2) a
-  refine' RegularSpace.ofBasis this fun a If hIf => isClosed_interᵢ fun i => _
-  exact (hIf.2 i).2.mono (infₛ_le (i : T).2)
-#align regular_space_Inf regularSpace_infₛ
+  refine' RegularSpace.ofBasis this fun a If hIf => isClosed_iInter fun i => _
+  exact (hIf.2 i).2.mono (sInf_le (i : T).2)
+#align regular_space_Inf regularSpace_sInf
 
-/- warning: regular_space_infi -> regularSpace_infᵢ is a dubious translation:
+/- warning: regular_space_infi -> regularSpace_iInf is a dubious translation:
 lean 3 declaration is
-  forall {ι : Sort.{u1}} {X : Type.{u2}} {t : ι -> (TopologicalSpace.{u2} X)}, (forall (i : ι), RegularSpace.{u2} X (t i)) -> (RegularSpace.{u2} X (infᵢ.{u2, u1} (TopologicalSpace.{u2} X) (ConditionallyCompleteLattice.toHasInf.{u2} (TopologicalSpace.{u2} X) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} X) (TopologicalSpace.completeLattice.{u2} X))) ι t))
+  forall {ι : Sort.{u1}} {X : Type.{u2}} {t : ι -> (TopologicalSpace.{u2} X)}, (forall (i : ι), RegularSpace.{u2} X (t i)) -> (RegularSpace.{u2} X (iInf.{u2, u1} (TopologicalSpace.{u2} X) (ConditionallyCompleteLattice.toHasInf.{u2} (TopologicalSpace.{u2} X) (CompleteLattice.toConditionallyCompleteLattice.{u2} (TopologicalSpace.{u2} X) (TopologicalSpace.completeLattice.{u2} X))) ι t))
 but is expected to have type
-  forall {ι : Sort.{u2}} {X : Type.{u1}} {t : ι -> (TopologicalSpace.{u1} X)}, (forall (i : ι), RegularSpace.{u1} X (t i)) -> (RegularSpace.{u1} X (infᵢ.{u1, u2} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toInfSet.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X))) ι t))
-Case conversion may be inaccurate. Consider using '#align regular_space_infi regularSpace_infᵢₓ'. -/
-theorem regularSpace_infᵢ {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @RegularSpace X (t i)) :
-    @RegularSpace X (infᵢ t) :=
-  regularSpace_infₛ <| forall_range_iff.mpr h
-#align regular_space_infi regularSpace_infᵢ
+  forall {ι : Sort.{u2}} {X : Type.{u1}} {t : ι -> (TopologicalSpace.{u1} X)}, (forall (i : ι), RegularSpace.{u1} X (t i)) -> (RegularSpace.{u1} X (iInf.{u1, u2} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toInfSet.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X))) ι t))
+Case conversion may be inaccurate. Consider using '#align regular_space_infi regularSpace_iInfₓ'. -/
+theorem regularSpace_iInf {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @RegularSpace X (t i)) :
+    @RegularSpace X (iInf t) :=
+  regularSpace_sInf <| forall_range_iff.mpr h
+#align regular_space_infi regularSpace_iInf
 
 /- warning: regular_space.inf -> RegularSpace.inf is a dubious translation:
 lean 3 declaration is
@@ -2438,8 +2438,8 @@ Case conversion may be inaccurate. Consider using '#align regular_space.inf Regu
 theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSpace X t₁)
     (h₂ : @RegularSpace X t₂) : @RegularSpace X (t₁ ⊓ t₂) :=
   by
-  rw [inf_eq_infᵢ]
-  exact regularSpace_infᵢ (Bool.forall_bool.2 ⟨h₂, h₁⟩)
+  rw [inf_eq_iInf]
+  exact regularSpace_iInf (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align regular_space.inf RegularSpace.inf
 
 instance {p : α → Prop} : RegularSpace (Subtype p) :=
@@ -2450,7 +2450,7 @@ instance [TopologicalSpace β] [RegularSpace β] : RegularSpace (α × β) :=
 
 instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [∀ i, RegularSpace (π i)] :
     RegularSpace (∀ i, π i) :=
-  regularSpace_infᵢ fun i => regularSpace_induced _
+  regularSpace_iInf fun i => regularSpace_induced _
 
 end RegularSpace
 
@@ -2649,9 +2649,9 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
     refine' ⟨range V, _, forall_range_iff.2 <| Subtype.forall.2 hd, fun n => _⟩
     · rw [bUnion_range]
       exact fun x hx => mem_Union.2 ⟨⟨x, hx⟩, hxu x hx⟩
-    · simp only [← supr_eq_Union, supᵢ_and']
+    · simp only [← supr_eq_Union, iSup_and']
       exact
-        isClosed_bunionᵢ
+        isClosed_biUnion
           (((finite_le_nat n).preimage_embedding (Encodable.encode' _)).Subset <|
             inter_subset_right _ _)
           fun u hu => isClosed_closure
@@ -2661,8 +2661,8 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
   refine'
     ⟨⋃ u ∈ U, ↑u \ ⋃ (v ∈ V) (hv : Encodable.encode v ≤ Encodable.encode u), closure ↑v,
       ⋃ v ∈ V, ↑v \ ⋃ (u ∈ U) (hu : Encodable.encode u ≤ Encodable.encode v), closure ↑u,
-      isOpen_bunionᵢ fun u hu => (is_open_of_mem_countable_basis u.2).sdiff (hVc _),
-      isOpen_bunionᵢ fun v hv => (is_open_of_mem_countable_basis v.2).sdiff (hUc _), fun x hx => _,
+      isOpen_biUnion fun u hu => (is_open_of_mem_countable_basis u.2).sdiff (hVc _),
+      isOpen_biUnion fun v hv => (is_open_of_mem_countable_basis v.2).sdiff (hUc _), fun x hx => _,
       fun x hx => _, _⟩
   · rcases mem_Union₂.1 (hsU hx) with ⟨u, huU, hxu⟩
     refine' mem_bUnion huU ⟨hxu, _⟩
@@ -2746,20 +2746,20 @@ instance [T5Space α] : T5Space (SeparationQuotient α)
 
 end CompletelyNormal
 
-#print connectedComponent_eq_interᵢ_clopen /-
+#print connectedComponent_eq_iInter_clopen /-
 /-- In a compact t2 space, the connected component of a point equals the intersection of all
 its clopen neighbourhoods. -/
-theorem connectedComponent_eq_interᵢ_clopen [T2Space α] [CompactSpace α] (x : α) :
+theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x : α) :
     connectedComponent x = ⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, Z :=
   by
-  apply eq_of_subset_of_subset connectedComponent_subset_interᵢ_clopen
+  apply eq_of_subset_of_subset connectedComponent_subset_iInter_clopen
   -- Reduce to showing that the clopen intersection is connected.
   refine' IsPreconnected.subset_connectedComponent _ (mem_Inter.2 fun Z => Z.2.2)
   -- We do this by showing that any disjoint cover by two closed sets implies
   -- that one of these closed sets must contain our whole thing.
   -- To reduce to the case where the cover is disjoint on all of `α` we need that `s` is closed
   have hs : IsClosed (⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, Z : Set α) :=
-    isClosed_interᵢ fun Z => Z.2.1.2
+    isClosed_iInter fun Z => Z.2.1.2
   rw [isPreconnected_iff_subset_of_fully_disjoint_closed hs]
   intro a b ha hb hab ab_disj
   haveI := @normalOfCompactT2 α _ _ _
@@ -2810,15 +2810,15 @@ theorem connectedComponent_eq_interᵢ_clopen [T2Space α] [CompactSpace α] (x
   -- so there must be some finite intersection of clopen neighbourhoods of X disjoint to it,
   -- but a finite intersection of clopen sets is clopen so we let this be our Z.
   have H1 :=
-    (hu.union hv).isClosed_compl.IsCompact.inter_interᵢ_nonempty
+    (hu.union hv).isClosed_compl.IsCompact.inter_iInter_nonempty
       (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z) fun Z => Z.2.1.2
   rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, not_forall] at H1
   cases' H1 (disjoint_compl_left_iff_subset.2 <| hab.trans <| union_subset_union hau hbv) with Zi H2
   refine' ⟨⋂ U ∈ Zi, Subtype.val U, _, _, _⟩
-  · exact isClopen_binterᵢ_finset fun Z hZ => Z.2.1
+  · exact isClopen_biInter_finset fun Z hZ => Z.2.1
   · exact mem_Inter₂.2 fun Z hZ => Z.2.2
   · rwa [← disjoint_compl_left_iff_subset, disjoint_iff_inter_eq_empty, ← not_nonempty_iff_eq_empty]
-#align connected_component_eq_Inter_clopen connectedComponent_eq_interᵢ_clopen
+#align connected_component_eq_Inter_clopen connectedComponent_eq_iInter_clopen
 -/
 
 section Profinite
@@ -2852,7 +2852,7 @@ theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace α ↔ Totall
     intro hyp
     suffices x ∈ connectedComponent y by
       simpa [totallyDisconnectedSpace_iff_connectedComponent_singleton.1 h y, mem_singleton_iff]
-    rw [connectedComponent_eq_interᵢ_clopen, mem_Inter]
+    rw [connectedComponent_eq_iInter_clopen, mem_Inter]
     rintro ⟨w : Set α, hw : IsClopen w, hy : y ∈ w⟩
     by_contra hx
     exact
@@ -2870,7 +2870,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
     constructor
     · have : connectedComponent x = {x} :=
         totally_disconnected_space_iff_connected_component_singleton.mp ‹_› x
-      rw [connectedComponent_eq_interᵢ_clopen] at this
+      rw [connectedComponent_eq_iInter_clopen] at this
       intro hU
       let N := { Z // IsClopen Z ∧ x ∈ Z }
       rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ Z : N, Z.val ⊆ U
@@ -2991,7 +2991,7 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
   rw [ConnectedComponents.coe_ne_coe] at ne
   have h := connectedComponent_disjoint Ne
   -- write ↑b as the intersection of all clopen subsets containing it
-  rw [connectedComponent_eq_interᵢ_clopen b, disjoint_iff_inter_eq_empty] at h
+  rw [connectedComponent_eq_iInter_clopen b, disjoint_iff_inter_eq_empty] at h
   -- Now we show that this can be reduced to some clopen containing `↑b` being disjoint to `↑a`
   obtain ⟨U, V, hU, ha, hb, rfl⟩ :
     ∃ (U : Set α)(V : Set (ConnectedComponents α)),
@@ -3002,7 +3002,7 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
     · exact fun Z => Z.2.1.2
     -- This clopen and its complement will separate the connected components of `a` and `b`
     set U : Set α := ⋂ (i : { Z // IsClopen Z ∧ b ∈ Z }) (H : i ∈ fin_a), i
-    have hU : IsClopen U := isClopen_binterᵢ_finset fun i j => i.2.1
+    have hU : IsClopen U := isClopen_biInter_finset fun i j => i.2.1
     exact
       ⟨U, coe '' U, hU, ha, subset_Inter₂ fun Z _ => Z.2.1.connectedComponent_subset Z.2.2,
         (connectedComponents_preimage_image U).symm ▸ hU.bUnion_connected_component_eq⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/36b8aa61ea7c05727161f96a0532897bd72aedab

@@ -1940,7 +1940,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13060 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13062 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13060 x._@.Mathlib.Topology.Separation._hyg.13062) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13050 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13052 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13050 x._@.Mathlib.Topology.Separation._hyg.13052) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
 Case conversion may be inaccurate. Consider using '#align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompactₓ'. -/
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we

mathlib commit https://github.com/leanprover-community/mathlib/commit/17ad94b4953419f3e3ce3e77da3239c62d1d09f0

@@ -1940,7 +1940,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13084 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13086 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13084 x._@.Mathlib.Topology.Separation._hyg.13086) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13060 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13062 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13060 x._@.Mathlib.Topology.Separation._hyg.13062) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
 Case conversion may be inaccurate. Consider using '#align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompactₓ'. -/
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a4df69ca1a9a0e5e26bfe12e2b92814216016d0

@@ -1940,7 +1940,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13080 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13082 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13080 x._@.Mathlib.Topology.Separation._hyg.13082) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13084 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13086 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13084 x._@.Mathlib.Topology.Separation._hyg.13086) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
 Case conversion may be inaccurate. Consider using '#align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompactₓ'. -/
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -1940,7 +1940,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13052 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13054 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13052 x._@.Mathlib.Topology.Separation._hyg.13054) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13080 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13082 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13080 x._@.Mathlib.Topology.Separation._hyg.13082) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
 Case conversion may be inaccurate. Consider using '#align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompactₓ'. -/
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -630,7 +630,7 @@ protected theorem Finset.isClosed [T1Space α] (s : Finset α) : IsClosed (s : S
 lean 3 declaration is
   forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.topologicalSpace.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) => α -> (CofiniteTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_2 x)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α _inst_2 x)) => Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => Exists.{0} (IsOpen.{u1} α _inst_2 U) (fun (hU : IsOpen.{u1} α _inst_2 U) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x U) (Not (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
 but is expected to have type
-  forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_2 x)) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (IsOpen.{u1} α _inst_2 U) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x U) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
+  forall (α : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} α], List.TFAE (List.cons.{0} Prop (T1Space.{u1} α _inst_2) (List.cons.{0} Prop (forall (x : α), IsClosed.{u1} α _inst_2 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x)) (List.cons.{0} Prop (forall (x : α), IsOpen.{u1} α _inst_2 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))) (List.cons.{0} Prop (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) y)) (nhds.{u1} α _inst_2 x))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α _inst_2 x)) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y s))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U : Set.{u1} α) => And (IsOpen.{u1} α _inst_2 U) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x U) (Not (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) y U)))))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (nhds.{u1} α _inst_2 x) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Ne.{succ u1} α x y) -> (Disjoint.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} α x) (nhds.{u1} α _inst_2 y))) (List.cons.{0} Prop (forall {{x : α}} {{y : α}}, (Specializes.{u1} α _inst_2 x y) -> (Eq.{succ u1} α x y)) (List.nil.{0} Prop)))))))))))
 Case conversion may be inaccurate. Consider using '#align t1_space_tfae t1Space_TFAEₓ'. -/
 theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
     TFAE
@@ -672,7 +672,7 @@ theorem t1Space_TFAE (α : Type u) [TopologicalSpace α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_2) (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.topologicalSpace.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) => α -> (CofiniteTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_2) (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α)))
+  forall {α : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} α], Iff (T1Space.{u1} α _inst_2) (Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_2 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α)))
 Case conversion may be inaccurate. Consider using '#align t1_space_iff_continuous_cofinite_of t1Space_iff_continuous_cofinite_ofₓ'. -/
 theorem t1Space_iff_continuous_cofinite_of {α : Type _} [TopologicalSpace α] :
     T1Space α ↔ Continuous (@CofiniteTopology.of α) :=
@@ -683,7 +683,7 @@ theorem t1Space_iff_continuous_cofinite_of {α : Type _} [TopologicalSpace α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_1 (CofiniteTopology.topologicalSpace.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) => α -> (CofiniteTopology.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_1 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1], Continuous.{u1, u1} α (CofiniteTopology.{u1} α) _inst_1 (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => CofiniteTopology.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (CofiniteTopology.{u1} α)) (CofiniteTopology.of.{u1} α))
 Case conversion may be inaccurate. Consider using '#align cofinite_topology.continuous_of CofiniteTopology.continuous_ofₓ'. -/
 theorem CofiniteTopology.continuous_of [T1Space α] : Continuous (@CofiniteTopology.of α) :=
   t1Space_iff_continuous_cofinite_of.mp ‹_›
@@ -1940,7 +1940,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13047 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13049 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13047 x._@.Mathlib.Topology.Separation._hyg.13049) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13052 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13054 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13052 x._@.Mathlib.Topology.Separation._hyg.13054) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
 Case conversion may be inaccurate. Consider using '#align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompactₓ'. -/
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -252,32 +252,26 @@ theorem Inseparable.eq [T0Space α] {x y : α} (h : Inseparable x y) : x = y :=
 #align inseparable.eq Inseparable.eq
 -/
 
-/- warning: inducing.injective -> Inducing.injective is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : T0Space.{u1} α _inst_1] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_2 f) -> (Function.Injective.{succ u1, succ u2} α β f)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T0Space.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_3 f) -> (Function.Injective.{succ u1, succ u2} α β f)
-Case conversion may be inaccurate. Consider using '#align inducing.injective Inducing.injectiveₓ'. -/
+#print Inducing.injective /-
 protected theorem Inducing.injective [TopologicalSpace β] [T0Space α] {f : α → β}
     (hf : Inducing f) : Injective f := fun x y h =>
   Inseparable.eq <| hf.inseparable_iff.1 <| h ▸ Inseparable.refl _
 #align inducing.injective Inducing.injective
+-/
 
-/- warning: inducing.embedding -> Inducing.embedding is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : T0Space.{u1} α _inst_1] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_2 f) -> (Embedding.{u1, u2} α β _inst_1 _inst_2 f)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T0Space.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_3 f) -> (Embedding.{u1, u2} α β _inst_1 _inst_3 f)
-Case conversion may be inaccurate. Consider using '#align inducing.embedding Inducing.embeddingₓ'. -/
+#print Inducing.embedding /-
 protected theorem Inducing.embedding [TopologicalSpace β] [T0Space α] {f : α → β}
     (hf : Inducing f) : Embedding f :=
   ⟨hf, hf.Injective⟩
 #align inducing.embedding Inducing.embedding
+-/
 
+#print embedding_iff_inducing /-
 theorem embedding_iff_inducing [TopologicalSpace β] [T0Space α] {f : α → β} :
     Embedding f ↔ Inducing f :=
   ⟨Embedding.to_inducing, Inducing.embedding⟩
 #align embedding_iff_inducing embedding_iff_inducing
+-/
 
 #print t0Space_iff_nhds_injective /-
 theorem t0Space_iff_nhds_injective (α : Type u) [TopologicalSpace α] :
@@ -1946,7 +1940,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 lean 3 declaration is
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_4 : T2Space.{u1} α _inst_1] {ι : Type.{u2}} [_inst_5 : Nonempty.{succ u2} ι] {V : ι -> (Set.{u1} α)}, (Directed.{u1, succ u2} (Set.{u1} α) ι (Superset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α)) V) -> (forall (i : ι), IsCompact.{u1} α _inst_1 (V i)) -> (forall {U : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => V i))) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (nhds.{u1} α _inst_1 x))) -> (Exists.{succ u2} ι (fun (i : ι) => HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (V i) U)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13015 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13017 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13015 x._@.Mathlib.Topology.Separation._hyg.13017) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_4 : T2Space.{u2} α _inst_1] {ι : Type.{u1}} [_inst_5 : Nonempty.{succ u1} ι] {V : ι -> (Set.{u2} α)}, (Directed.{u2, succ u1} (Set.{u2} α) ι (fun (x._@.Mathlib.Topology.Separation._hyg.13047 : Set.{u2} α) (x._@.Mathlib.Topology.Separation._hyg.13049 : Set.{u2} α) => Superset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) x._@.Mathlib.Topology.Separation._hyg.13047 x._@.Mathlib.Topology.Separation._hyg.13049) V) -> (forall (i : ι), IsCompact.{u2} α _inst_1 (V i)) -> (forall {U : Set.{u2} α}, (forall (x : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => V i))) -> (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) U (nhds.{u2} α _inst_1 x))) -> (Exists.{succ u1} ι (fun (i : ι) => HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (V i) U)))
 Case conversion may be inaccurate. Consider using '#align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompactₓ'. -/
 /-- If `V : ι → set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 
 ! This file was ported from Lean 3 source module topology.separation
-! leanprover-community/mathlib commit 92ca63f0fb391a9ca5f22d2409a6080e786d99f7
+! leanprover-community/mathlib commit 195fcd60ff2bfe392543bceb0ec2adcdb472db4c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -252,6 +252,33 @@ theorem Inseparable.eq [T0Space α] {x y : α} (h : Inseparable x y) : x = y :=
 #align inseparable.eq Inseparable.eq
 -/
 
+/- warning: inducing.injective -> Inducing.injective is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : T0Space.{u1} α _inst_1] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_2 f) -> (Function.Injective.{succ u1, succ u2} α β f)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T0Space.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_3 f) -> (Function.Injective.{succ u1, succ u2} α β f)
+Case conversion may be inaccurate. Consider using '#align inducing.injective Inducing.injectiveₓ'. -/
+protected theorem Inducing.injective [TopologicalSpace β] [T0Space α] {f : α → β}
+    (hf : Inducing f) : Injective f := fun x y h =>
+  Inseparable.eq <| hf.inseparable_iff.1 <| h ▸ Inseparable.refl _
+#align inducing.injective Inducing.injective
+
+/- warning: inducing.embedding -> Inducing.embedding is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : T0Space.{u1} α _inst_1] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_2 f) -> (Embedding.{u1, u2} α β _inst_1 _inst_2 f)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T0Space.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_3 f) -> (Embedding.{u1, u2} α β _inst_1 _inst_3 f)
+Case conversion may be inaccurate. Consider using '#align inducing.embedding Inducing.embeddingₓ'. -/
+protected theorem Inducing.embedding [TopologicalSpace β] [T0Space α] {f : α → β}
+    (hf : Inducing f) : Embedding f :=
+  ⟨hf, hf.Injective⟩
+#align inducing.embedding Inducing.embedding
+
+theorem embedding_iff_inducing [TopologicalSpace β] [T0Space α] {f : α → β} :
+    Embedding f ↔ Inducing f :=
+  ⟨Embedding.to_inducing, Inducing.embedding⟩
+#align embedding_iff_inducing embedding_iff_inducing
+
 #print t0Space_iff_nhds_injective /-
 theorem t0Space_iff_nhds_injective (α : Type u) [TopologicalSpace α] :
     T0Space α ↔ Injective (𝓝 : α → Filter α) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -312,7 +312,7 @@ instance : T0Space (SeparationQuotient α) :=
     Quotient.inductionOn₂' x' y' fun x y h =>
       SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.inducing_mk.inseparable_iff.1 h⟩
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_subsingleton /-
 theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : s.Subsingleton :=
@@ -327,7 +327,7 @@ theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsC
 #align minimal_nonempty_closed_subsingleton minimal_nonempty_closed_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_closed_eq_singleton /-
 theorem minimal_nonempty_closed_eq_singleton [T0Space α] {s : Set α} (hs : IsClosed s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} :=
@@ -349,7 +349,7 @@ theorem IsClosed.exists_closed_singleton {α : Type _} [TopologicalSpace α] [T0
 #align is_closed.exists_closed_singleton IsClosed.exists_closed_singleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_subsingleton /-
 theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : s.Subsingleton :=
@@ -364,7 +364,7 @@ theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpe
 #align minimal_nonempty_open_subsingleton minimal_nonempty_open_subsingleton
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 #print minimal_nonempty_open_eq_singleton /-
 theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpen s)
     (hne : s.Nonempty) (hmin : ∀ (t) (_ : t ⊆ s), t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} :=
@@ -378,7 +378,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T0Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Finite.{u1} α s) -> (Set.Nonempty.{u1} α s) -> (IsOpen.{u1} α _inst_1 s) -> (Exists.{succ u1} α (fun (x : α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (IsOpen.{u1} α _inst_1 (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) x))))
 Case conversion may be inaccurate. Consider using '#align exists_open_singleton_of_open_finite exists_open_singleton_of_open_finiteₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (t «expr ⊂ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊂ » s) -/
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
 theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s.Finite)
     (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set α) :=
@@ -2160,7 +2160,7 @@ lean 3 declaration is
 but is expected to have type
   forall (X : Type.{u1}) [_inst_2 : TopologicalSpace.{u1} X], List.TFAE (List.cons.{0} Prop (RegularSpace.{u1} X _inst_2) (List.cons.{0} Prop (forall (s : Set.{u1} X) (a : X), (Not (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) a (closure.{u1} X _inst_2 s))) -> (Disjoint.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} X) (Filter.instCompleteLatticeFilter.{u1} X))) (nhdsSet.{u1} X _inst_2 s) (nhds.{u1} X _inst_2 a))) (List.cons.{0} Prop (forall (a : X) (s : Set.{u1} X), Iff (Disjoint.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X) (BoundedOrder.toOrderBot.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (CompleteLattice.toBoundedOrder.{u1} (Filter.{u1} X) (Filter.instCompleteLatticeFilter.{u1} X))) (nhdsSet.{u1} X _inst_2 s) (nhds.{u1} X _inst_2 a)) (Not (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) a (closure.{u1} X _inst_2 s)))) (List.cons.{0} Prop (forall (a : X) (s : Set.{u1} X), (Membership.mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (instMembershipSetFilter.{u1} X) s (nhds.{u1} X _inst_2 a)) -> (Exists.{succ u1} (Set.{u1} X) (fun (t : Set.{u1} X) => And (Membership.mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (instMembershipSetFilter.{u1} X) t (nhds.{u1} X _inst_2 a)) (And (IsClosed.{u1} X _inst_2 t) (HasSubset.Subset.{u1} (Set.{u1} X) (Set.instHasSubsetSet.{u1} X) t s))))) (List.cons.{0} Prop (forall (a : X), LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (Filter.lift'.{u1, u1} X X (nhds.{u1} X _inst_2 a) (closure.{u1} X _inst_2)) (nhds.{u1} X _inst_2 a)) (List.cons.{0} Prop (forall (a : X), Eq.{succ u1} (Filter.{u1} X) (Filter.lift'.{u1, u1} X X (nhds.{u1} X _inst_2 a) (closure.{u1} X _inst_2)) (nhds.{u1} X _inst_2 a)) (List.nil.{0} Prop)))))))
 Case conversion may be inaccurate. Consider using '#align regular_space_tfae regularSpace_TFAEₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a «expr ∉ » closure[closure] s) -/
 theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     TFAE
       [RegularSpace X, ∀ (s : Set X) (a) (_ : a ∉ closure s), Disjoint (𝓝ˢ s) (𝓝 a),
@@ -2483,8 +2483,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T3Space.{u1} α _inst_1] {x : α} {y : α}, (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} α) (fun (U₁ : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) U₁ (nhds.{u1} α _inst_1 x)) (Exists.{succ u1} (Set.{u1} α) (fun (V₁ : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) V₁ (nhds.{u1} α _inst_1 x)) (Exists.{succ u1} (Set.{u1} α) (fun (U₂ : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) U₂ (nhds.{u1} α _inst_1 y)) (Exists.{succ u1} (Set.{u1} α) (fun (V₂ : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) V₂ (nhds.{u1} α _inst_1 y)) (And (IsClosed.{u1} α _inst_1 V₁) (And (IsClosed.{u1} α _inst_1 V₂) (And (IsOpen.{u1} α _inst_1 U₁) (And (IsOpen.{u1} α _inst_1 U₂) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) V₁ U₁) (And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) V₂ U₂) (Disjoint.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} α) (Preorder.toLE.{u1} (Set.{u1} α) (PartialOrder.toPreorder.{u1} (Set.{u1} α) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} α) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} α) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} α) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} α) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} α) (Set.instCompleteBooleanAlgebraSet.{u1} α)))))) U₁ U₂)))))))))))))))
 Case conversion may be inaccurate. Consider using '#align disjoint_nested_nhds disjoint_nested_nhdsₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₁ V₁ «expr ∈ » nhds() x) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (U₂ V₂ «expr ∈ » nhds() y) -/
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
 with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. -/
 theorem disjoint_nested_nhds [T3Space α] {x y : α} (h : x ≠ y) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -1357,9 +1357,9 @@ instance (priority := 100) T2Space.t1Space [T2Space α] : T1Space α :=
 
 /- warning: t2_iff_nhds -> t2_iff_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall {x : α} {y : α}, (Filter.NeBot.{u1} α (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y))) -> (Eq.{succ u1} α x y))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall {x : α} {y : α}, (Filter.NeBot.{u1} α (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y))) -> (Eq.{succ u1} α x y))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall {x : α} {y : α}, (Filter.NeBot.{u1} α (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y))) -> (Eq.{succ u1} α x y))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], Iff (T2Space.{u1} α _inst_1) (forall {x : α} {y : α}, (Filter.NeBot.{u1} α (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y))) -> (Eq.{succ u1} α x y))
 Case conversion may be inaccurate. Consider using '#align t2_iff_nhds t2_iff_nhdsₓ'. -/
 /-- A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. -/
 theorem t2_iff_nhds : T2Space α ↔ ∀ {x y : α}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
@@ -1368,9 +1368,9 @@ theorem t2_iff_nhds : T2Space α ↔ ∀ {x y : α}, NeBot (𝓝 x ⊓ 𝓝 y) 
 
 /- warning: eq_of_nhds_ne_bot -> eq_of_nhds_neBot is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {x : α} {y : α}, (Filter.NeBot.{u1} α (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y))) -> (Eq.{succ u1} α x y)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {x : α} {y : α}, (Filter.NeBot.{u1} α (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y))) -> (Eq.{succ u1} α x y)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {x : α} {y : α}, (Filter.NeBot.{u1} α (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y))) -> (Eq.{succ u1} α x y)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T2Space.{u1} α _inst_1] {x : α} {y : α}, (Filter.NeBot.{u1} α (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhds.{u1} α _inst_1 x) (nhds.{u1} α _inst_1 y))) -> (Eq.{succ u1} α x y)
 Case conversion may be inaccurate. Consider using '#align eq_of_nhds_ne_bot eq_of_nhds_neBotₓ'. -/
 theorem eq_of_nhds_neBot [T2Space α] {x y : α} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y :=
   t2_iff_nhds.mp ‹_› h
@@ -2410,9 +2410,9 @@ theorem regularSpace_infᵢ {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @
 
 /- warning: regular_space.inf -> RegularSpace.inf is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} {t₁ : TopologicalSpace.{u1} X} {t₂ : TopologicalSpace.{u1} X}, (RegularSpace.{u1} X t₁) -> (RegularSpace.{u1} X t₂) -> (RegularSpace.{u1} X (HasInf.inf.{u1} (TopologicalSpace.{u1} X) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.{u1} X) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.completeLattice.{u1} X))))) t₁ t₂))
+  forall {X : Type.{u1}} {t₁ : TopologicalSpace.{u1} X} {t₂ : TopologicalSpace.{u1} X}, (RegularSpace.{u1} X t₁) -> (RegularSpace.{u1} X t₂) -> (RegularSpace.{u1} X (Inf.inf.{u1} (TopologicalSpace.{u1} X) (SemilatticeInf.toHasInf.{u1} (TopologicalSpace.{u1} X) (Lattice.toSemilatticeInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.completeLattice.{u1} X))))) t₁ t₂))
 but is expected to have type
-  forall {X : Type.{u1}} {t₁ : TopologicalSpace.{u1} X} {t₂ : TopologicalSpace.{u1} X}, (RegularSpace.{u1} X t₁) -> (RegularSpace.{u1} X t₂) -> (RegularSpace.{u1} X (HasInf.inf.{u1} (TopologicalSpace.{u1} X) (Lattice.toHasInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X)))) t₁ t₂))
+  forall {X : Type.{u1}} {t₁ : TopologicalSpace.{u1} X} {t₂ : TopologicalSpace.{u1} X}, (RegularSpace.{u1} X t₁) -> (RegularSpace.{u1} X t₂) -> (RegularSpace.{u1} X (Inf.inf.{u1} (TopologicalSpace.{u1} X) (Lattice.toInf.{u1} (TopologicalSpace.{u1} X) (ConditionallyCompleteLattice.toLattice.{u1} (TopologicalSpace.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} X) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u1} X)))) t₁ t₂))
 Case conversion may be inaccurate. Consider using '#align regular_space.inf RegularSpace.infₓ'. -/
 theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSpace X t₁)
     (h₂ : @RegularSpace X t₂) : @RegularSpace X (t₁ ⊓ t₂) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

This adds versions of the various Euler product statements in terms of the new topological products, namely HasProd (fun p : Primes ↦ ∑' e, f (p ^ e)) (∑' n, f n) and ∏' p : Primes, ∑' e, f (p ^ e) = ∑' n, f n (and similar for completely multiplicative f in terms of (1 - f p)⁻¹).

At the same time, I have reworked the proofs to some extent (in particular removing a few slow converts). I also added a bunch of local instances that speed up instance synthesis by a factor of 2 (from 1.4 to 0.7 seconds on my laptop).

Unfortunately, this means that the diff is fairly large.

See here and here on Zulip.

Co-authored-by: Michael Stoll <99838730+MichaelStollBayreuth@users.noreply.github.com>

@@ -2056,7 +2056,7 @@ a T₂.₅ space.  -/
 class T3Space (X : Type u) [TopologicalSpace X] extends T0Space X, RegularSpace X : Prop
 #align t3_space T3Space
 
-instance (priority := 90) [T0Space X] [RegularSpace X] : T3Space X := ⟨⟩
+instance (priority := 90) instT3Space [T0Space X] [RegularSpace X] : T3Space X := ⟨⟩
 
 theorem RegularSpace.t3Space_iff_t0Space [RegularSpace X] : T3Space X ↔ T0Space X := by
   constructor <;> intro <;> infer_instance

@@ -49,7 +49,7 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
 
 * `IsClosed.exists_closed_singleton`: Given a closed set `S` in a compact T₀ space,
   there is some `x ∈ S` such that `{x}` is closed.
-* `exists_open_singleton_of_open_finite`: Given an open finite set `S` in a T₀ space,
+* `exists_isOpen_singleton_of_isOpen_finite`: Given an open finite set `S` in a T₀ space,
   there is some `x ∈ S` such that `{x}` is open.
 
 ### T₁ spaces

@@ -1389,7 +1389,7 @@ theorem tendsto_nhds_unique_of_frequently_eq [T2Space X] {f g : Y → X} {l : Fi
 /-- If `s` and `t` are compact sets in a T₂ space, then the set neighborhoods filter of `s ∩ t`
 is the infimum of set neighborhoods filters for `s` and `t`.
 
-For general sets, only the `≤` inequality holds, see `nhdsSet_inf_le`. -/
+For general sets, only the `≤` inequality holds, see `nhdsSet_inter_le`. -/
 theorem IsCompact.nhdsSet_inter_eq [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
     𝓝ˢ (s ∩ t) = 𝓝ˢ s ⊓ 𝓝ˢ t := by
   refine le_antisymm (nhdsSet_inter_le _ _) ?_

@@ -1386,6 +1386,19 @@ theorem tendsto_nhds_unique_of_frequently_eq [T2Space X] {f g : Y → X} {l : Fi
   not_not.1 fun hne => this (isClosed_diagonal.isOpen_compl.mem_nhds hne)
 #align tendsto_nhds_unique_of_frequently_eq tendsto_nhds_unique_of_frequently_eq
 
+/-- If `s` and `t` are compact sets in a T₂ space, then the set neighborhoods filter of `s ∩ t`
+is the infimum of set neighborhoods filters for `s` and `t`.
+
+For general sets, only the `≤` inequality holds, see `nhdsSet_inf_le`. -/
+theorem IsCompact.nhdsSet_inter_eq [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
+    𝓝ˢ (s ∩ t) = 𝓝ˢ s ⊓ 𝓝ˢ t := by
+  refine le_antisymm (nhdsSet_inter_le _ _) ?_
+  simp_rw [hs.nhdsSet_inf_eq_biSup, ht.inf_nhdsSet_eq_biSup, nhdsSet, sSup_image]
+  refine iSup₂_le fun x hxs ↦ iSup₂_le fun y hyt ↦ ?_
+  rcases eq_or_ne x y with (rfl|hne)
+  · exact le_iSup₂_of_le x ⟨hxs, hyt⟩ (inf_idem _).le
+  · exact (disjoint_nhds_nhds.mpr hne).eq_bot ▸ bot_le
+
 /-- If a function `f` is
 
 - injective on a compact set `s`;

@@ -190,7 +190,7 @@ theorem t0Space_iff_inseparable (X : Type u) [TopologicalSpace X] :
 
 theorem t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] :
     T0Space X ↔ Pairwise fun x y : X => ¬Inseparable x y := by
-  simp only [t0Space_iff_inseparable, Ne.def, not_imp_not, Pairwise]
+  simp only [t0Space_iff_inseparable, Ne, not_imp_not, Pairwise]
 #align t0_space_iff_not_inseparable t0Space_iff_not_inseparable
 
 theorem Inseparable.eq [T0Space X] {x y : X} (h : Inseparable x y) : x = y :=
@@ -884,7 +884,7 @@ theorem PreconnectedSpace.trivial_of_discrete [PreconnectedSpace X] [DiscreteTop
     Subsingleton X := by
   rw [← not_nontrivial_iff_subsingleton]
   rintro ⟨x, y, hxy⟩
-  rw [Ne.def, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
+  rw [Ne, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
   exact hxy (mem_univ x)
 #align preconnected_space.trivial_of_discrete PreconnectedSpace.trivial_of_discrete
 
@@ -1325,7 +1325,7 @@ theorem R1Space.t2Space_iff_t0Space [R1Space X] : T2Space X ↔ T0Space X := by
 
 /-- A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. -/
 theorem t2_iff_nhds : T2Space X ↔ ∀ {x y : X}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
-  simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne.def, not_imp_comm, Pairwise]
+  simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne, not_imp_comm, Pairwise]
 #align t2_iff_nhds t2_iff_nhds
 
 theorem eq_of_nhds_neBot [T2Space X] {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y :=

I removed some redundant instance arguments throughout Mathlib. To do this, I used VS Code's regex search. See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/repeating.20instances.20from.20variable.20command I closed the previous PR for this and reopened it.

@@ -1992,7 +1992,7 @@ instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i,
 
 /-- In a regular space, if a compact set and a closed set are disjoint, then they have disjoint
 neighborhoods. -/
-lemma SeparatedNhds.of_isCompact_isClosed [RegularSpace X] {s t : Set X}
+lemma SeparatedNhds.of_isCompact_isClosed {s t : Set X}
     (hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t) : SeparatedNhds s t := by
   simpa only [separatedNhds_iff_disjoint, hs.disjoint_nhdsSet_left, disjoint_nhds_nhdsSet,
     ht.closure_eq, disjoint_left] using hst

Generalize coclosedCompact_eq_cocompact and relativelyCompact.

@@ -21,6 +21,8 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
   open sets.
 * `T0Space`: A T₀/Kolmogorov space is a space where, for every two points `x ≠ y`,
   there is an open set that contains one, but not the other.
+* `R0Space`: An R₀ space (sometimes called a *symmetric space*) is a topological space
+  such that the `Specializes` relation is symmetric.
 * `T1Space`: A T₁/Fréchet space is a space where every singleton set is closed.
   This is equivalent to, for every pair `x ≠ y`, there existing an open set containing `x`
   but not `y` (`t1Space_iff_exists_open` shows that these conditions are equivalent.)
@@ -381,6 +383,87 @@ theorem T0Space.of_open_cover (h : ∀ x, ∃ s : Set X, x ∈ s ∧ IsOpen s 
     ⟨s, hxs, (hxy.mem_open_iff hso).1 hxs, hs⟩
 #align t0_space.of_open_cover T0Space.of_open_cover
 
+/-- A topological space is called an R₀ space, if `Specializes` relation is symmetric.
+
+In other words, given two points `x y : X`,
+if every neighborhood of `y` contains `x`, then every neighborhood of `x` containx `y`. -/
+@[mk_iff]
+class R0Space (X : Type u) [TopologicalSpace X] : Prop where
+  /-- In an R₀ space, the `Specializes` relation is symmetric. -/
+  specializes_symmetric : Symmetric (Specializes : X → X → Prop)
+
+export R0Space (specializes_symmetric)
+
+section R0Space
+
+variable [R0Space X] {x y : X}
+
+/-- In an R₀ space, the `Specializes` relation is symmetric, dot notation version. -/
+theorem Specializes.symm (h : x ⤳ y) : y ⤳ x := specializes_symmetric h
+#align specializes.symm Specializes.symm
+
+/-- In an R₀ space, the `Specializes` relation is symmetric, `Iff` version. -/
+theorem specializes_comm : x ⤳ y ↔ y ⤳ x := ⟨Specializes.symm, Specializes.symm⟩
+#align specializes_comm specializes_comm
+
+/-- In an R₀ space, `Specializes` is equivalent to `Inseparable`. -/
+theorem specializes_iff_inseparable : x ⤳ y ↔ Inseparable x y :=
+  ⟨fun h ↦ h.antisymm h.symm, Inseparable.specializes⟩
+#align specializes_iff_inseparable specializes_iff_inseparable
+
+/-- In an R₀ space, `Specializes` implies `Inseparable`. -/
+alias ⟨Specializes.inseparable, _⟩ := specializes_iff_inseparable
+
+theorem Inducing.r0Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R0Space Y where
+  specializes_symmetric a b := by
+    simpa only [← hf.specializes_iff] using Specializes.symm
+
+instance {p : X → Prop} : R0Space {x // p x} := inducing_subtype_val.r0Space
+
+instance [TopologicalSpace Y] [R0Space Y] : R0Space (X × Y) where
+  specializes_symmetric _ _ h := h.fst.symm.prod h.snd.symm
+
+instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, R0Space (X i)] :
+    R0Space (∀ i, X i) where
+  specializes_symmetric _ _ h := specializes_pi.2 fun i ↦ (specializes_pi.1 h i).symm
+
+/-- In an R₀ space, the closure of a singleton is a compact set. -/
+theorem isCompact_closure_singleton : IsCompact (closure {x}) := by
+  refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_
+  obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <| hxU <| subset_closure rfl
+  refine ⟨{i}, fun y hy ↦ ?_⟩
+  rw [← specializes_iff_mem_closure, specializes_comm] at hy
+  simpa using hy.mem_open (hUo i) hi
+
+theorem Filter.coclosedCompact_le_cofinite : coclosedCompact X ≤ cofinite :=
+  le_cofinite_iff_compl_singleton_mem.2 fun _ ↦
+    compl_mem_coclosedCompact.2 isCompact_closure_singleton
+#align filter.coclosed_compact_le_cofinite Filter.coclosedCompact_le_cofinite
+
+variable (X)
+
+/-- In an R₀ space, relatively compact sets form a bornology.
+Its cobounded filter is `Filter.coclosedCompact`.
+See also `Bornology.inCompact` the bornology of sets contained in a compact set. -/
+def Bornology.relativelyCompact : Bornology X where
+  cobounded' := Filter.coclosedCompact X
+  le_cofinite' := Filter.coclosedCompact_le_cofinite
+#align bornology.relatively_compact Bornology.relativelyCompact
+
+variable {X}
+
+theorem Bornology.relativelyCompact.isBounded_iff {s : Set X} :
+    @Bornology.IsBounded _ (Bornology.relativelyCompact X) s ↔ IsCompact (closure s) :=
+  compl_mem_coclosedCompact
+#align bornology.relatively_compact.is_bounded_iff Bornology.relativelyCompact.isBounded_iff
+
+/-- In an R₀ space, the closure of a finite set is a compact set. -/
+theorem Set.Finite.isCompact_closure {s : Set X} (hs : s.Finite) : IsCompact (closure s) :=
+  let _ : Bornology X := .relativelyCompact X
+  Bornology.relativelyCompact.isBounded_iff.1 hs.isBounded
+
+end R0Space
+
 /-- A T₁ space, also known as a Fréchet space, is a topological space
   where every singleton set is closed. Equivalently, for every pair
   `x ≠ y`, there is an open set containing `x` and not `y`. -/
@@ -445,35 +528,6 @@ theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space X] {b : Se
   exact ⟨a, ab, xa, fun h => ha h rfl⟩
 #align topological_space.is_topological_basis.exists_mem_of_ne TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne
 
-theorem Filter.coclosedCompact_le_cofinite [T1Space X] :
-    Filter.coclosedCompact X ≤ Filter.cofinite := fun s hs =>
-  compl_compl s ▸ hs.isCompact.compl_mem_coclosedCompact_of_isClosed hs.isClosed
-#align filter.coclosed_compact_le_cofinite Filter.coclosedCompact_le_cofinite
-
-variable (X)
-
-/-- In a `T1Space`, relatively compact sets form a bornology. Its cobounded filter is
-`Filter.coclosedCompact`. See also `Bornology.inCompact` the bornology of sets contained
-in a compact set. -/
-def Bornology.relativelyCompact [T1Space X] : Bornology X where
-  cobounded' := Filter.coclosedCompact X
-  le_cofinite' := Filter.coclosedCompact_le_cofinite
-#align bornology.relatively_compact Bornology.relativelyCompact
-
-variable {X}
-
-theorem Bornology.relativelyCompact.isBounded_iff [T1Space X] {s : Set X} :
-    @Bornology.IsBounded _ (Bornology.relativelyCompact X) s ↔ IsCompact (closure s) := by
-  change sᶜ ∈ Filter.coclosedCompact X ↔ _
-  rw [Filter.mem_coclosedCompact]
-  constructor
-  · rintro ⟨t, ht₁, ht₂, hst⟩
-    rw [compl_subset_compl] at hst
-    exact ht₂.of_isClosed_subset isClosed_closure (closure_minimal hst ht₁)
-  · intro h
-    exact ⟨closure s, isClosed_closure, h, compl_subset_compl.mpr subset_closure⟩
-#align bornology.relatively_compact.is_bounded_iff Bornology.relativelyCompact.isBounded_iff
-
 protected theorem Finset.isClosed [T1Space X] (s : Finset X) : IsClosed (s : Set X) :=
   s.finite_toSet.isClosed
 #align finset.is_closed Finset.isClosed
@@ -572,6 +626,9 @@ theorem nhds_le_nhds_iff [T1Space X] {a b : X} : 𝓝 a ≤ 𝓝 b ↔ a = b :=
   specializes_iff_eq
 #align nhds_le_nhds_iff nhds_le_nhds_iff
 
+instance (priority := 100) [T1Space X] : R0Space X where
+  specializes_symmetric _ _ := by rw [specializes_iff_eq, specializes_iff_eq]; exact Eq.symm
+
 instance : T1Space (CofiniteTopology X) :=
   t1Space_iff_continuous_cofinite_of.mpr continuous_id
 
@@ -928,6 +985,10 @@ export R1Space (specializes_or_disjoint_nhds)
 
 variable [R1Space X] {x y : X}
 
+instance (priority := 100) : R0Space X where
+  specializes_symmetric _ _ h := (specializes_or_disjoint_nhds _ _).resolve_right <| fun hd ↦
+    h.not_disjoint hd.symm
+
 theorem disjoint_nhds_nhds_iff_not_specializes : Disjoint (𝓝 x) (𝓝 y) ↔ ¬x ⤳ y :=
   ⟨fun hd hspec ↦ hspec.not_disjoint hd, (specializes_or_disjoint_nhds _ _).resolve_left⟩
 #align disjoint_nhds_nhds_iff_not_specializes disjoint_nhds_nhds_iff_not_specializes
@@ -935,27 +996,6 @@ theorem disjoint_nhds_nhds_iff_not_specializes : Disjoint (𝓝 x) (𝓝 y) ↔
 theorem specializes_iff_not_disjoint : x ⤳ y ↔ ¬Disjoint (𝓝 x) (𝓝 y) :=
   disjoint_nhds_nhds_iff_not_specializes.not_left.symm
 
-/-- In an R₁ space, the `Specializes` relation is symmetric,
-i.e., an R₁ space is an R₀ space. -/
-theorem Specializes.symm (h : x ⤳ y) : y ⤳ x := by
-  simpa only [specializes_iff_not_disjoint, disjoint_comm] using h
-#align specializes.symm Specializes.symm
-
-/-- In an R₁ space, the `Specializes` relation is symmetric,
-i.e., an R₁ space is an R₀ space. -/
-theorem specializes_comm : x ⤳ y ↔ y ⤳ x := ⟨Specializes.symm, Specializes.symm⟩
-#align specializes_comm specializes_comm
-
-/-- In an R₁ space, `Specializes` is equivalent to `Inseparable`,
-i.e., an R₁ space is an R₀ space. -/
-theorem specializes_iff_inseparable : x ⤳ y ↔ Inseparable x y :=
-  ⟨fun h ↦ h.antisymm h.symm, Inseparable.specializes⟩
-#align specializes_iff_inseparable specializes_iff_inseparable
-
-/-- In an R₁ space, `Specializes` implies `Inseparable`,
-i.e., an R₁ space is an R₀ space. -/
-alias ⟨Specializes.inseparable, _⟩ := specializes_iff_inseparable
-
 theorem disjoint_nhds_nhds_iff_not_inseparable : Disjoint (𝓝 x) (𝓝 y) ↔ ¬Inseparable x y := by
   rw [disjoint_nhds_nhds_iff_not_specializes, specializes_iff_inseparable]
 
@@ -1152,6 +1192,21 @@ theorem IsCompact.isCompact_isClosed_basis_nhds {x : X} {L : Set X} (hLc : IsCom
     ⟨closure K, mem_of_superset hKx subset_closure, ⟨hKc.closure, isClosed_closure⟩,
       (hKc.closure_subset_of_isOpen isOpen_interior hKU).trans interior_subset⟩
 
+/-- In an R₁ space, the filters `coclosedCompact` and `cocompact` are equal. -/
+@[simp]
+theorem Filter.coclosedCompact_eq_cocompact : coclosedCompact X = cocompact X := by
+  refine le_antisymm ?_ cocompact_le_coclosedCompact
+  rw [hasBasis_coclosedCompact.le_basis_iff hasBasis_cocompact]
+  exact fun K hK ↦ ⟨closure K, ⟨isClosed_closure, hK.closure⟩, compl_subset_compl.2 subset_closure⟩
+#align filter.coclosed_compact_eq_cocompact Filter.coclosedCompact_eq_cocompact
+
+/-- In an R₁ space, the bornologies `relativelyCompact` and `inCompact` are equal. -/
+@[simp]
+theorem Bornology.relativelyCompact_eq_inCompact :
+    Bornology.relativelyCompact X = Bornology.inCompact X :=
+  Bornology.ext _ _ Filter.coclosedCompact_eq_cocompact
+#align bornology.relatively_compact_eq_in_compact Bornology.relativelyCompact_eq_inCompact
+
 /-!
 ### Lemmas about a weakly locally compact R₁ space
 
@@ -1675,18 +1730,6 @@ theorem IsCompact.isClosed [T2Space X] {s : Set X} (hs : IsCompact s) : IsClosed
     ⟨v, (uv.mono_left <| show s ≤ u from su).subset_compl_left, vo, by simpa using xv⟩
 #align is_compact.is_closed IsCompact.isClosed
 
-@[simp]
-theorem Filter.coclosedCompact_eq_cocompact [T2Space X] : coclosedCompact X = cocompact X := by
-  simp only [coclosedCompact, cocompact, iInf_and',
-    and_iff_right_of_imp (@IsCompact.isClosed X _ _ _)]
-#align filter.coclosed_compact_eq_cocompact Filter.coclosedCompact_eq_cocompact
-
-@[simp]
-theorem Bornology.relativelyCompact_eq_inCompact [T2Space X] :
-    Bornology.relativelyCompact X = Bornology.inCompact X :=
-  Bornology.ext _ _ Filter.coclosedCompact_eq_cocompact
-#align bornology.relatively_compact_eq_in_compact Bornology.relativelyCompact_eq_inCompact
-
 theorem IsCompact.preimage_continuous [CompactSpace X] [T2Space Y] {f : X → Y} {s : Set Y}
     (hs : IsCompact s) (hf : Continuous f) : IsCompact (f ⁻¹' s) :=
   (hs.isClosed.preimage hf).isCompact

All these lemmas refer to the range of some function being open/range (i.e. isOpen or isClosed).

@@ -1623,16 +1623,19 @@ theorem Set.EqOn.of_subset_closure [T2Space Y] {s t : Set X} {f g : X → Y} (h
       ((hg x hx).mono_left <| nhdsWithin_mono _ hst) (h.eventuallyEq_of_mem self_mem_nhdsWithin)
 #align set.eq_on.of_subset_closure Set.EqOn.of_subset_closure
 
-theorem Function.LeftInverse.closed_range [T2Space X] {f : X → Y} {g : Y → X}
+theorem Function.LeftInverse.isClosed_range [T2Space X] {f : X → Y} {g : Y → X}
     (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsClosed (range g) :=
   have : EqOn (g ∘ f) id (closure <| range g) :=
     h.rightInvOn_range.eqOn.closure (hg.comp hf) continuous_id
   isClosed_of_closure_subset fun x hx => ⟨f x, this hx⟩
-#align function.left_inverse.closed_range Function.LeftInverse.closed_range
+#align function.left_inverse.closed_range Function.LeftInverse.isClosed_range
+
+@[deprecated] alias Function.LeftInverse.closed_range :=
+  Function.LeftInverse.isClosed_range -- 2024-03-17
 
 theorem Function.LeftInverse.closedEmbedding [T2Space X] {f : X → Y} {g : Y → X}
     (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : ClosedEmbedding g :=
-  ⟨h.embedding hf hg, h.closed_range hf hg⟩
+  ⟨h.embedding hf hg, h.isClosed_range hf hg⟩
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 
 theorem SeparatedNhds.of_isCompact_isCompact [T2Space X] {s t : Set X} (hs : IsCompact s)

@@ -2343,8 +2343,8 @@ theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s
       rw [connectedComponent_eq_iInter_isClopen] at hx
       intro hU
       let N := { s // IsClopen s ∧ x ∈ s }
-      suffices ∃ s : N, s.val ⊆ U by
-        rcases this with ⟨⟨s, hs, hs'⟩, hs''⟩; exact ⟨s, ⟨hs', hs⟩, hs''⟩
+      rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ s : N, s.val ⊆ U
+      · exact ⟨s, ⟨hs', hs⟩, hs''⟩
       haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩
       have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.1
       have hdir : Directed Superset fun s : N => s.val := by

We had duplicated API between topological spaces and uniform spaces. In this PR I mostly deduplicate it with some exceptions:

• SeparationQuotient.lift' and SeparationQuotient.map are leftovers from the old version that are designed to work with uniform spaces;
• probably, some theorems/instances still assume UniformSpace when TopologicalSpace would work.

Outside of UniformSpace/Separation, I mostly changed SeparatedSpace to T0Space and separationRel to Inseparable. I also rewrote a few proofs that were broken by the API change.

Fixes #2031

@@ -262,7 +262,7 @@ def specializationOrder (X) [TopologicalSpace X] [T0Space X] : PartialOrder X :=
   { specializationPreorder X, PartialOrder.lift (OrderDual.toDual ∘ 𝓝) nhds_injective with }
 #align specialization_order specializationOrder
 
-instance : T0Space (SeparationQuotient X) :=
+instance SeparationQuotient.instT0Space : T0Space (SeparationQuotient X) :=
   ⟨fun x y => Quotient.inductionOn₂' x y fun _ _ h =>
     SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.inducing_mk.inseparable_iff.1 h⟩
 
@@ -1265,6 +1265,9 @@ instance (priority := 80) [R1Space X] [T0Space X] : T2Space X :=
   t2Space_iff_disjoint_nhds.2 fun _x _y hne ↦ disjoint_nhds_nhds_iff_not_inseparable.2 fun hxy ↦
     hne hxy.eq
 
+theorem R1Space.t2Space_iff_t0Space [R1Space X] : T2Space X ↔ T0Space X := by
+  constructor <;> intro <;> infer_instance
+
 /-- A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. -/
 theorem t2_iff_nhds : T2Space X ↔ ∀ {x y : X}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
   simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne.def, not_imp_comm, Pairwise]
@@ -1996,6 +1999,9 @@ class T3Space (X : Type u) [TopologicalSpace X] extends T0Space X, RegularSpace
 
 instance (priority := 90) [T0Space X] [RegularSpace X] : T3Space X := ⟨⟩
 
+theorem RegularSpace.t3Space_iff_t0Space [RegularSpace X] : T3Space X ↔ T0Space X := by
+  constructor <;> intro <;> infer_instance
+
 -- see Note [lower instance priority]
 instance (priority := 100) T3Space.t25Space [T3Space X] : T25Space X := by
   refine' ⟨fun x y hne => _⟩

1. ContinuousAt.comp₂, ContinuousAt.comp₂_continuousWithinAt, and _of_eq versions.
2. ContinuousAt.along_{fst,snd}: Continuous functions are continuous in their first and second arguments.

@@ -1605,8 +1605,7 @@ theorem eqOn_closure₂' [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y →
 theorem eqOn_closure₂ [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (uncurry f))
     (hg : Continuous (uncurry g)) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
-  eqOn_closure₂' h (fun x => continuous_uncurry_left x hf) (fun x => continuous_uncurry_right x hf)
-    (fun y => continuous_uncurry_left y hg) fun y => continuous_uncurry_right y hg
+  eqOn_closure₂' h hf.uncurry_left hf.uncurry_right hg.uncurry_left hg.uncurry_right
 #align eq_on_closure₂ eqOn_closure₂
 
 /-- If `f x = g x` for all `x ∈ s` and `f`, `g` are continuous on `t`, `s ⊆ t ⊆ closure s`, then

Classifies by adding issue number #11215 to porting notes claiming "TODO".

@@ -653,7 +653,7 @@ theorem closure_singleton [T1Space X] {x : X} : closure ({x} : Set X) = {x} :=
   isClosed_singleton.closure_eq
 #align closure_singleton closure_singleton
 
--- Porting note: todo: the proof was `hs.induction_on (by simp) fun x => by simp`
+-- Porting note (#11215): TODO: the proof was `hs.induction_on (by simp) fun x => by simp`
 theorem Set.Subsingleton.closure [T1Space X] {s : Set X} (hs : s.Subsingleton) :
     (closure s).Subsingleton := by
   rcases hs.eq_empty_or_singleton with (rfl | ⟨x, rfl⟩) <;> simp

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

@@ -1104,7 +1104,7 @@ protected theorem R1Space.sInf {X : Type*} {T : Set (TopologicalSpace X)}
 
 protected theorem R1Space.iInf {ι X : Type*} {t : ι → TopologicalSpace X}
     (ht : ∀ i, @R1Space X (t i)) : @R1Space X (iInf t) :=
-  .sInf <| forall_range_iff.2 ht
+  .sInf <| forall_mem_range.2 ht
 
 protected theorem R1Space.inf {X : Type*} {t₁ t₂ : TopologicalSpace X}
     (h₁ : @R1Space X t₁) (h₂ : @R1Space X t₂) : @R1Space X (t₁ ⊓ t₂) := by
@@ -1923,7 +1923,7 @@ theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @
 
 theorem regularSpace_iInf {ι X} {t : ι → TopologicalSpace X} (h : ∀ i, @RegularSpace X (t i)) :
     @RegularSpace X (iInf t) :=
-  regularSpace_sInf <| forall_range_iff.mpr h
+  regularSpace_sInf <| forall_mem_range.mpr h
 #align regular_space_infi regularSpace_iInf
 
 theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSpace X t₁)
@@ -2105,7 +2105,7 @@ instance (priority := 100) NormalSpace.of_regularSpace_secondCountableTopology
       exact ⟨u, hu, hxu, disjoint_left.2 hut⟩
     choose! U hu hxu hd using this
     set V : s → countableBasis X := MapsTo.restrict _ _ _ hu
-    refine' ⟨range V, _, forall_range_iff.2 <| Subtype.forall.2 hd, fun n => _⟩
+    refine' ⟨range V, _, forall_mem_range.2 <| Subtype.forall.2 hd, fun n => _⟩
     · rw [biUnion_range]
       exact fun x hx => mem_iUnion.2 ⟨⟨x, hx⟩, hxu x hx⟩
     · simp only [← iSup_eq_iUnion, iSup_and']

Add Filter.Tendsto.lift'_closure, R1Space.of_continuous_specializes_imp, and T25Space.of_injective_continuous.

@@ -1079,10 +1079,13 @@ theorem IsCompact.finite_compact_cover {s : Set X} (hs : IsCompact s) {ι : Type
   · simp only [Finset.set_biUnion_insert_update _ hx, hK, h3K]
 #align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 
-theorem Inducing.r1Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R1Space Y where
-  specializes_or_disjoint_nhds _ _ := by
-    simpa only [← hf.specializes_iff, hf.nhds_eq_comap, or_iff_not_imp_left,
-      ← disjoint_nhds_nhds_iff_not_specializes] using Filter.disjoint_comap
+theorem R1Space.of_continuous_specializes_imp [TopologicalSpace Y] {f : Y → X} (hc : Continuous f)
+    (hspec : ∀ x y, f x ⤳ f y → x ⤳ y) : R1Space Y where
+  specializes_or_disjoint_nhds x y := (specializes_or_disjoint_nhds (f x) (f y)).imp (hspec x y) <|
+    ((hc.tendsto _).disjoint · (hc.tendsto _))
+
+theorem Inducing.r1Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R1Space Y :=
+  .of_continuous_specializes_imp hf.continuous fun _ _ ↦ hf.specializes_iff.1
 
 protected theorem R1Space.induced (f : Y → X) : @R1Space Y (.induced f ‹_›) :=
   @Inducing.r1Space _ _ _ _ (.induced f _) f (inducing_induced f)
@@ -1396,6 +1399,14 @@ theorem exists_open_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
       (disjoint_lift'_closure_nhds.2 h)
 #align exists_open_nhds_disjoint_closure exists_open_nhds_disjoint_closure
 
+theorem T25Space.of_injective_continuous [TopologicalSpace Y] [T25Space Y] {f : X → Y}
+    (hinj : Injective f) (hcont : Continuous f) : T25Space X where
+  t2_5 x y hne := (tendsto_lift'_closure_nhds hcont x).disjoint (t2_5 <| hinj.ne hne)
+    (tendsto_lift'_closure_nhds hcont y)
+
+instance [T25Space X] {p : X → Prop} : T25Space {x // p x} :=
+  .of_injective_continuous Subtype.val_injective continuous_subtype_val
+
 section limUnder
 
 variable [T2Space X] {f : Filter X}

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -653,7 +653,7 @@ theorem closure_singleton [T1Space X] {x : X} : closure ({x} : Set X) = {x} :=
   isClosed_singleton.closure_eq
 #align closure_singleton closure_singleton
 
--- porting note: todo: the proof was `hs.induction_on (by simp) fun x => by simp`
+-- Porting note: todo: the proof was `hs.induction_on (by simp) fun x => by simp`
 theorem Set.Subsingleton.closure [T1Space X] {s : Set X} (hs : s.Subsingleton) :
     (closure s).Subsingleton := by
   rcases hs.eq_empty_or_singleton with (rfl | ⟨x, rfl⟩) <;> simp
@@ -1302,7 +1302,7 @@ theorem isClosed_diagonal [T2Space X] : IsClosed (diagonal X) :=
   t2_iff_isClosed_diagonal.mp ‹_›
 #align is_closed_diagonal isClosed_diagonal
 
--- porting note: 2 lemmas moved below
+-- Porting note: 2 lemmas moved below
 
 theorem tendsto_nhds_unique [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
     (ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b :=

Mathport chose wrong names for these constructors.

@@ -1802,24 +1802,33 @@ theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
   tfae_finish
 #align regular_space_tfae regularSpace_TFAE
 
-theorem RegularSpace.ofLift'_closure (h : ∀ x : X, (𝓝 x).lift' closure = 𝓝 x) : RegularSpace X :=
+theorem RegularSpace.of_lift'_closure (h : ∀ x : X, (𝓝 x).lift' closure = 𝓝 x) : RegularSpace X :=
   Iff.mpr ((regularSpace_TFAE X).out 0 5) h
-#align regular_space.of_lift'_closure RegularSpace.ofLift'_closure
+#align regular_space.of_lift'_closure RegularSpace.of_lift'_closure
 
-theorem RegularSpace.ofBasis {ι : X → Sort*} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set X}
+@[deprecated] -- 2024-02-28
+alias RegularSpace.ofLift'_closure := RegularSpace.of_lift'_closure
+
+theorem RegularSpace.of_hasBasis {ι : X → Sort*} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set X}
     (h₁ : ∀ a, (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ a i, p a i → IsClosed (s a i)) :
     RegularSpace X :=
-  RegularSpace.ofLift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a)
-#align regular_space.of_basis RegularSpace.ofBasis
+  .of_lift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a)
+#align regular_space.of_basis RegularSpace.of_hasBasis
+
+@[deprecated] -- 2024-02-28
+alias RegularSpace.ofBasis := RegularSpace.of_hasBasis
 
-theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
+theorem RegularSpace.of_exists_mem_nhds_isClosed_subset
     (h : ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) : RegularSpace X :=
   Iff.mpr ((regularSpace_TFAE X).out 0 3) h
-#align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.ofExistsMemNhdsIsClosedSubset
+#align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.of_exists_mem_nhds_isClosed_subset
+
+@[deprecated] -- 2024-02-28
+alias RegularSpace.ofExistsMemNhdsIsClosedSubset := RegularSpace.of_exists_mem_nhds_isClosed_subset
 
 /-- A weakly locally compact R₁ space is regular. -/
 instance (priority := 100) [WeaklyLocallyCompactSpace X] [R1Space X] : RegularSpace X :=
-  .ofBasis isCompact_isClosed_basis_nhds fun _ _ ⟨_, _, h⟩ ↦ h
+  .of_hasBasis isCompact_isClosed_basis_nhds fun _ _ ⟨_, _, h⟩ ↦ h
 
 variable [RegularSpace X] {x : X} {s : Set X}
 
@@ -1878,7 +1887,7 @@ theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set
 
 protected theorem Inducing.regularSpace [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) :
     RegularSpace Y :=
-  RegularSpace.ofBasis
+  .of_hasBasis
     (fun b => by rw [hf.nhds_eq_comap b]; exact (closed_nhds_basis _).comap _)
     fun b s hs => by exact hs.2.preimage hf.continuous
 #align inducing.regular_space Inducing.regularSpace
@@ -1892,13 +1901,12 @@ theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @
     @RegularSpace X (sInf T) := by
   let _ := sInf T
   have : ∀ a, (𝓝 a).HasBasis
-      (fun If : ΣI : Set T, I → Set X =>
+      (fun If : Σ I : Set T, I → Set X =>
         If.1.Finite ∧ ∀ i : If.1, If.2 i ∈ @nhds X i a ∧ @IsClosed X i (If.2 i))
-      fun If => ⋂ i : If.1, If.snd i := by
-    intro a
+      fun If => ⋂ i : If.1, If.snd i := fun a ↦ by
     rw [nhds_sInf, ← iInf_subtype'']
     exact hasBasis_iInf fun t : T => @closed_nhds_basis X t (h t t.2) a
-  refine' RegularSpace.ofBasis this fun a If hIf => isClosed_iInter fun i => _
+  refine .of_hasBasis this fun a If hIf => isClosed_iInter fun i => ?_
   exact (hIf.2 i).2.mono (sInf_le (i : T).2)
 #align regular_space_Inf regularSpace_sInf
 

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -2319,8 +2319,8 @@ theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s
       rw [connectedComponent_eq_iInter_isClopen] at hx
       intro hU
       let N := { s // IsClopen s ∧ x ∈ s }
-      suffices : ∃ s : N, s.val ⊆ U
-      · rcases this with ⟨⟨s, hs, hs'⟩, hs''⟩; exact ⟨s, ⟨hs', hs⟩, hs''⟩
+      suffices ∃ s : N, s.val ⊆ U by
+        rcases this with ⟨⟨s, hs, hs'⟩, hs''⟩; exact ⟨s, ⟨hs', hs⟩, hs''⟩
       haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩
       have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.1
       have hdir : Directed Superset fun s : N => s.val := by

This is partial work to make s ∩ . be consistently used for passing to a subset of s. This is sort of an adjoint to (Subtype.val : s -> _) '' ., except for the fact that it does not produce a Set s.

The main API changes are to Subtype.image_preimage_val and Subtype.preimage_val_eq_preimage_val_iff in Mathlib.Data.Set.Image. Changes in other modules are all proof fixups.

Zulip discussion

@@ -2376,12 +2376,12 @@ theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
       refine' ⟨f0, _⟩
       · have : Set.range ((↑) : u → H) = interior s := by
           rw [this, Set.range_comp, Subtype.range_coe, Subtype.image_preimage_coe]
-          apply Set.inter_eq_self_of_subset_left interior_subset
+          apply Set.inter_eq_self_of_subset_right interior_subset
         rw [this]
         apply isOpen_interior
     have f2 : IsOpen v := VisClopen.2.preimage continuous_subtype_val
     have f3 : ((↑) : s → H) '' V = ((↑) : u → H) '' v := by
-      rw [this, image_comp, Subtype.image_preimage_coe, inter_eq_self_of_subset_left V_sub]
+      rw [this, image_comp, Subtype.image_preimage_coe, inter_eq_self_of_subset_right V_sub]
     rw [f3]
     apply f1.isOpenMap v f2
   refine' ⟨(↑) '' V, VisClopen', by simp [Vx], Subset.trans _ sU⟩

Preliminaries for #6844 The way I prove Ascoli's theorem is by using equicontinuity to reduce it to the case of the product topology, where one can use Tychonoff, hence these variations.

@@ -1675,6 +1675,18 @@ theorem IsCompact.preimage_continuous [CompactSpace X] [T2Space Y] {f : X → Y}
     (hs : IsCompact s) (hf : Continuous f) : IsCompact (f ⁻¹' s) :=
   (hs.isClosed.preimage hf).isCompact
 
+lemma Pi.isCompact_iff {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
+    [∀ i, T2Space (π i)] {s : Set (Π i, π i)} :
+    IsCompact s ↔ IsClosed s ∧ ∀ i, IsCompact (eval i '' s):= by
+  constructor <;> intro H
+  · exact ⟨H.isClosed, fun i ↦ H.image <| continuous_apply i⟩
+  · exact IsCompact.of_isClosed_subset (isCompact_univ_pi H.2) H.1 (subset_pi_eval_image univ s)
+
+lemma Pi.isCompact_closure_iff {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
+    [∀ i, T2Space (π i)] {s : Set (Π i, π i)} :
+    IsCompact (closure s) ↔ ∀ i, IsCompact (closure <| eval i '' s) := by
+  simp_rw [← exists_isCompact_superset_iff, Pi.exists_compact_superset_iff, image_subset_iff]
+
 /-- If `V : ι → Set X` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhds_of_isCompact'` where we
 don't need to assume each `V i` closed because it follows from compactness since `X` is

Main API changes

• Define R1Space, a.k.a. preregular space.
• Drop T2OrLocallyCompactRegularSpace.
• Generalize all existing theorems about T2OrLocallyCompactRegularSpace to R1Space.
• Drop the [T2OrLocallyCompactRegularSpace _] assumption if the space is known to be regular for other reason (e.g., because it's a topological group).

New theorems

• Specializes.not_disjoint: if x ⤳ y, then 𝓝 x and 𝓝 y aren't disjoint;
• specializes_iff_not_disjoint, Specializes.inseparable, disjoint_nhds_nhds_iff_not_inseparable, r1Space_iff_inseparable_or_disjoint_nhds: basic API about R1Spaces;
• Inducing.r1Space, R1Space.induced, R1Space.sInf, R1Space.iInf, R1Space.inf, instances for Subtype _, X × Y, and ∀ i, X i: basic instances for R1Space;
• IsCompact.mem_closure_iff_exists_inseparable, IsCompact.closure_eq_biUnion_inseparable: characterizations of the closure of a compact set in a preregular space;
• Inseparable.mem_measurableSet_iff: topologically inseparable points can't be separated by a Borel measurable set;
• IsCompact.closure_subset_measurableSet, IsCompact.measure_closure: in a preregular space, a measurable superset of a compact set includes its closure as well; as a corollary, closure K has the same measure as K.
• exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds: an auxiliary lemma extracted from a LocallyCompactPair instance;
• IsCompact.isCompact_isClosed_basis_nhds: if x admits a compact neighborhood, then it admits a basis of compact closed neighborhoods; in particular, a weakly locally compact preregular space is a locally compact regular space;
• isCompact_isClosed_basis_nhds: a version of the previous theorem for weakly locally compact spaces;
• exists_mem_nhds_isCompact_isClosed: in a locally compact regular space, each point admits a compact closed neighborhood.

Deprecated theorems

Some theorems about topological groups are true for any (pre)regular space, so we deprecate the special cases.

• exists_isCompact_isClosed_subset_isCompact_nhds_one: use new IsCompact.isCompact_isClosed_basis_nhds instead;
• instLocallyCompactSpaceOfWeaklyOfGroup, instLocallyCompactSpaceOfWeaklyOfAddGroup: are now implied by WeaklyLocallyCompactSpace.locallyCompactSpace;
• local_isCompact_isClosed_nhds_of_group, local_isCompact_isClosed_nhds_of_addGroup: use isCompact_isClosed_basis_nhds instead;
• exists_isCompact_isClosed_nhds_one, exists_isCompact_isClosed_nhds_zero: use exists_mem_nhds_isCompact_isClosed instead.

Renamed/moved theorems

For each renamed theorem, the old theorem is redefined as a deprecated alias.

• isOpen_setOf_disjoint_nhds_nhds: moved to Constructions;
• isCompact_closure_of_subset_compact -> IsCompact.closure_of_subset;
• IsCompact.measure_eq_infi_isOpen -> IsCompact.measure_eq_iInf_isOpen;
• exists_compact_superset_iff -> exists_isCompact_superset_iff;
• separatedNhds_of_isCompact_isCompact_isClosed -> SeparatedNhds.of_isCompact_isCompact_isClosed;
• separatedNhds_of_isCompact_isCompact -> SeparatedNhds.of_isCompact_isCompact;
• separatedNhds_of_finset_finset -> SeparatedNhds.of_finset_finset;
• point_disjoint_finset_opens_of_t2 -> SeparatedNhds.of_singleton_finset;
• separatedNhds_of_isCompact_isClosed -> SeparatedNhds.of_isCompact_isClosed;
• exists_open_superset_and_isCompact_closure -> exists_isOpen_superset_and_isCompact_closure;
• exists_open_with_compact_closure -> exists_isOpen_mem_isCompact_closure;

@@ -24,6 +24,8 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
 * `T1Space`: A T₁/Fréchet space is a space where every singleton set is closed.
   This is equivalent to, for every pair `x ≠ y`, there existing an open set containing `x`
   but not `y` (`t1Space_iff_exists_open` shows that these conditions are equivalent.)
+* `R1Space`: An R₁/preregular space is a space where any two topologically distinguishable points
+  have disjoint neighbourhoods;
 * `T2Space`: A T₂/Hausdorff space is a space where, for every two points `x ≠ y`,
   there is two disjoint open sets, one containing `x`, and the other `y`.
 * `T25Space`: A T₂.₅/Urysohn space is a space where, for every two points `x ≠ y`,
@@ -912,6 +914,291 @@ theorem TopologicalSpace.subset_trans {s t : Set X} (ts : t ⊆ s) :
   (embedding_inclusion ts).induced
 #align topological_space.subset_trans TopologicalSpace.subset_trans
 
+/-! ### R₁ (preregular) spaces -/
+
+section R1Space
+
+/-- A topological space is called a *preregular* (a.k.a. R₁) space,
+if any two topologically distinguishable points have disjoint neighbourhoods. -/
+@[mk_iff r1Space_iff_specializes_or_disjoint_nhds]
+class R1Space (X : Type*) [TopologicalSpace X] : Prop where
+  specializes_or_disjoint_nhds (x y : X) : Specializes x y ∨ Disjoint (𝓝 x) (𝓝 y)
+
+export R1Space (specializes_or_disjoint_nhds)
+
+variable [R1Space X] {x y : X}
+
+theorem disjoint_nhds_nhds_iff_not_specializes : Disjoint (𝓝 x) (𝓝 y) ↔ ¬x ⤳ y :=
+  ⟨fun hd hspec ↦ hspec.not_disjoint hd, (specializes_or_disjoint_nhds _ _).resolve_left⟩
+#align disjoint_nhds_nhds_iff_not_specializes disjoint_nhds_nhds_iff_not_specializes
+
+theorem specializes_iff_not_disjoint : x ⤳ y ↔ ¬Disjoint (𝓝 x) (𝓝 y) :=
+  disjoint_nhds_nhds_iff_not_specializes.not_left.symm
+
+/-- In an R₁ space, the `Specializes` relation is symmetric,
+i.e., an R₁ space is an R₀ space. -/
+theorem Specializes.symm (h : x ⤳ y) : y ⤳ x := by
+  simpa only [specializes_iff_not_disjoint, disjoint_comm] using h
+#align specializes.symm Specializes.symm
+
+/-- In an R₁ space, the `Specializes` relation is symmetric,
+i.e., an R₁ space is an R₀ space. -/
+theorem specializes_comm : x ⤳ y ↔ y ⤳ x := ⟨Specializes.symm, Specializes.symm⟩
+#align specializes_comm specializes_comm
+
+/-- In an R₁ space, `Specializes` is equivalent to `Inseparable`,
+i.e., an R₁ space is an R₀ space. -/
+theorem specializes_iff_inseparable : x ⤳ y ↔ Inseparable x y :=
+  ⟨fun h ↦ h.antisymm h.symm, Inseparable.specializes⟩
+#align specializes_iff_inseparable specializes_iff_inseparable
+
+/-- In an R₁ space, `Specializes` implies `Inseparable`,
+i.e., an R₁ space is an R₀ space. -/
+alias ⟨Specializes.inseparable, _⟩ := specializes_iff_inseparable
+
+theorem disjoint_nhds_nhds_iff_not_inseparable : Disjoint (𝓝 x) (𝓝 y) ↔ ¬Inseparable x y := by
+  rw [disjoint_nhds_nhds_iff_not_specializes, specializes_iff_inseparable]
+
+theorem r1Space_iff_inseparable_or_disjoint_nhds {X : Type*} [TopologicalSpace X]:
+    R1Space X ↔ ∀ x y : X, Inseparable x y ∨ Disjoint (𝓝 x) (𝓝 y) :=
+  ⟨fun _h x y ↦ (specializes_or_disjoint_nhds x y).imp_left Specializes.inseparable, fun h ↦
+    ⟨fun x y ↦ (h x y).imp_left Inseparable.specializes⟩⟩
+
+theorem isClosed_setOf_specializes : IsClosed { p : X × X | p.1 ⤳ p.2 } := by
+  simp only [← isOpen_compl_iff, compl_setOf, ← disjoint_nhds_nhds_iff_not_specializes,
+    isOpen_setOf_disjoint_nhds_nhds]
+#align is_closed_set_of_specializes isClosed_setOf_specializes
+
+theorem isClosed_setOf_inseparable : IsClosed { p : X × X | Inseparable p.1 p.2 } := by
+  simp only [← specializes_iff_inseparable, isClosed_setOf_specializes]
+#align is_closed_set_of_inseparable isClosed_setOf_inseparable
+
+/-- In an R₁ space, a point belongs to the closure of a compact set `K`
+if and only if it is topologically inseparable from some point of `K`. -/
+theorem IsCompact.mem_closure_iff_exists_inseparable {K : Set X} (hK : IsCompact K) :
+    y ∈ closure K ↔ ∃ x ∈ K, Inseparable x y := by
+  refine ⟨fun hy ↦ ?_, fun ⟨x, hxK, hxy⟩ ↦
+    (hxy.mem_closed_iff isClosed_closure).1 <| subset_closure hxK⟩
+  contrapose! hy
+  have : Disjoint (𝓝 y) (𝓝ˢ K) := hK.disjoint_nhdsSet_right.2 fun x hx ↦
+    (disjoint_nhds_nhds_iff_not_inseparable.2 (hy x hx)).symm
+  simpa only [disjoint_iff, not_mem_closure_iff_nhdsWithin_eq_bot]
+    using this.mono_right principal_le_nhdsSet
+
+theorem IsCompact.closure_eq_biUnion_inseparable {K : Set X} (hK : IsCompact K) :
+    closure K = ⋃ x ∈ K, {y | Inseparable x y} := by
+  ext; simp [hK.mem_closure_iff_exists_inseparable]
+
+/-- In an R₁ space, the closure of a compact set is the union of the closures of its points. -/
+theorem IsCompact.closure_eq_biUnion_closure_singleton {K : Set X} (hK : IsCompact K) :
+    closure K = ⋃ x ∈ K, closure {x} := by
+  simp only [hK.closure_eq_biUnion_inseparable, ← specializes_iff_inseparable,
+    specializes_iff_mem_closure, setOf_mem_eq]
+
+/-- In an R₁ space, if a compact set `K` is contained in an open set `U`,
+then its closure is also contained in `U`. -/
+theorem IsCompact.closure_subset_of_isOpen {K : Set X} (hK : IsCompact K)
+    {U : Set X} (hU : IsOpen U) (hKU : K ⊆ U) : closure K ⊆ U := by
+  rw [hK.closure_eq_biUnion_inseparable, iUnion₂_subset_iff]
+  exact fun x hx y hxy ↦ (hxy.mem_open_iff hU).1 (hKU hx)
+
+/-- The closure of a compact set in an R₁ space is a compact set. -/
+protected theorem IsCompact.closure {K : Set X} (hK : IsCompact K) : IsCompact (closure K) := by
+  refine isCompact_of_finite_subcover fun U hUo hKU ↦ ?_
+  rcases hK.elim_finite_subcover U hUo (subset_closure.trans hKU) with ⟨t, ht⟩
+  exact ⟨t, hK.closure_subset_of_isOpen (isOpen_biUnion fun _ _ ↦ hUo _) ht⟩
+
+theorem IsCompact.closure_of_subset {s K : Set X} (hK : IsCompact K) (h : s ⊆ K) :
+    IsCompact (closure s) :=
+  hK.closure.of_isClosed_subset isClosed_closure (closure_mono h)
+#align is_compact_closure_of_subset_compact IsCompact.closure_of_subset
+
+@[deprecated] -- Since 28 Jan 2024
+alias isCompact_closure_of_subset_compact := IsCompact.closure_of_subset
+
+@[simp]
+theorem exists_isCompact_superset_iff {s : Set X} :
+    (∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) :=
+  ⟨fun ⟨_K, hK, hsK⟩ => hK.closure_of_subset hsK, fun h => ⟨closure s, h, subset_closure⟩⟩
+#align exists_compact_superset_iff exists_isCompact_superset_iff
+
+@[deprecated] -- Since 28 Jan 2024
+alias exists_compact_superset_iff := exists_isCompact_superset_iff
+
+/-- If `K` and `L` are disjoint compact sets in an R₁ topological space
+and `L` is also closed, then `K` and `L` have disjoint neighborhoods.  -/
+theorem SeparatedNhds.of_isCompact_isCompact_isClosed {K L : Set X} (hK : IsCompact K)
+    (hL : IsCompact L) (h'L : IsClosed L) (hd : Disjoint K L) : SeparatedNhds K L := by
+  simp_rw [separatedNhds_iff_disjoint, hK.disjoint_nhdsSet_left, hL.disjoint_nhdsSet_right,
+    disjoint_nhds_nhds_iff_not_inseparable]
+  intro x hx y hy h
+  exact absurd ((h.mem_closed_iff h'L).2 hy) <| disjoint_left.1 hd hx
+
+@[deprecated] -- Since 28 Jan 2024
+alias separatedNhds_of_isCompact_isCompact_isClosed := SeparatedNhds.of_isCompact_isCompact_isClosed
+
+/-- If a compact set is covered by two open sets, then we can cover it by two compact subsets. -/
+theorem IsCompact.binary_compact_cover {K U V : Set X}
+    (hK : IsCompact K) (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
+    ∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ := by
+  have hK' : IsCompact (closure K) := hK.closure
+  have : SeparatedNhds (closure K \ U) (closure K \ V) := by
+    apply SeparatedNhds.of_isCompact_isCompact_isClosed (hK'.diff hU) (hK'.diff hV)
+      (isClosed_closure.sdiff hV)
+    rw [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty]
+    exact hK.closure_subset_of_isOpen (hU.union hV) h2K
+  have : SeparatedNhds (K \ U) (K \ V) :=
+    this.mono (diff_subset_diff_left (subset_closure)) (diff_subset_diff_left (subset_closure))
+  rcases this with ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩
+  exact ⟨K \ O₁, K \ O₂, hK.diff h1O₁, hK.diff h1O₂, diff_subset_comm.mp h2O₁,
+    diff_subset_comm.mp h2O₂, by rw [← diff_inter, hO.inter_eq, diff_empty]⟩
+#align is_compact.binary_compact_cover IsCompact.binary_compact_cover
+
+/-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
+theorem IsCompact.finite_compact_cover {s : Set X} (hs : IsCompact s) {ι : Type*}
+    (t : Finset ι) (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
+    ∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
+  induction' t using Finset.induction with x t hx ih generalizing U s
+  · refine' ⟨fun _ => ∅, fun _ => isCompact_empty, fun i => empty_subset _, _⟩
+    simpa only [subset_empty_iff, Finset.not_mem_empty, iUnion_false, iUnion_empty] using hsC
+  simp only [Finset.set_biUnion_insert] at hsC
+  simp only [Finset.forall_mem_insert] at hU
+  have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU.2 i hi
+  rcases hs.binary_compact_cover hU.1 (isOpen_biUnion hU') hsC with
+    ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
+  rcases ih h1K₂ U hU' h2K₂ with ⟨K, h1K, h2K, h3K⟩
+  refine' ⟨update K x K₁, _, _, _⟩
+  · intro i
+    rcases eq_or_ne i x with rfl | hi
+    · simp only [update_same, h1K₁]
+    · simp only [update_noteq hi, h1K]
+  · intro i
+    rcases eq_or_ne i x with rfl | hi
+    · simp only [update_same, h2K₁]
+    · simp only [update_noteq hi, h2K]
+  · simp only [Finset.set_biUnion_insert_update _ hx, hK, h3K]
+#align is_compact.finite_compact_cover IsCompact.finite_compact_cover
+
+theorem Inducing.r1Space [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) : R1Space Y where
+  specializes_or_disjoint_nhds _ _ := by
+    simpa only [← hf.specializes_iff, hf.nhds_eq_comap, or_iff_not_imp_left,
+      ← disjoint_nhds_nhds_iff_not_specializes] using Filter.disjoint_comap
+
+protected theorem R1Space.induced (f : Y → X) : @R1Space Y (.induced f ‹_›) :=
+  @Inducing.r1Space _ _ _ _ (.induced f _) f (inducing_induced f)
+
+instance (p : X → Prop) : R1Space (Subtype p) := .induced _
+
+protected theorem R1Space.sInf {X : Type*} {T : Set (TopologicalSpace X)}
+    (hT : ∀ t ∈ T, @R1Space X t) : @R1Space X (sInf T) := by
+  let _ := sInf T
+  refine ⟨fun x y ↦ ?_⟩
+  simp only [Specializes, nhds_sInf]
+  rcases em (∃ t ∈ T, Disjoint (@nhds X t x) (@nhds X t y)) with ⟨t, htT, htd⟩ | hTd
+  · exact .inr <| htd.mono (iInf₂_le t htT) (iInf₂_le t htT)
+  · push_neg at hTd
+    exact .inl <| iInf₂_mono fun t ht ↦ ((hT t ht).1 x y).resolve_right (hTd t ht)
+
+protected theorem R1Space.iInf {ι X : Type*} {t : ι → TopologicalSpace X}
+    (ht : ∀ i, @R1Space X (t i)) : @R1Space X (iInf t) :=
+  .sInf <| forall_range_iff.2 ht
+
+protected theorem R1Space.inf {X : Type*} {t₁ t₂ : TopologicalSpace X}
+    (h₁ : @R1Space X t₁) (h₂ : @R1Space X t₂) : @R1Space X (t₁ ⊓ t₂) := by
+  rw [inf_eq_iInf]
+  apply R1Space.iInf
+  simp [*]
+
+instance [TopologicalSpace Y] [R1Space Y] : R1Space (X × Y) :=
+  .inf (.induced _) (.induced _)
+
+instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, R1Space (X i)] :
+    R1Space (∀ i, X i) :=
+  .iInf fun _ ↦ .induced _
+
+theorem exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds
+    {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [R1Space Y] {f : X → Y} {x : X}
+    {K : Set X} {s : Set Y} (hf : Continuous f) (hs : s ∈ 𝓝 (f x)) (hKc : IsCompact K)
+    (hKx : K ∈ 𝓝 x) : ∃ K ∈ 𝓝 x, IsCompact K ∧ MapsTo f K s := by
+  have hc : IsCompact (f '' K \ interior s) := (hKc.image hf).diff isOpen_interior
+  obtain ⟨U, V, Uo, Vo, hxU, hV, hd⟩ : SeparatedNhds {f x} (f '' K \ interior s) := by
+    simp_rw [separatedNhds_iff_disjoint, nhdsSet_singleton, hc.disjoint_nhdsSet_right,
+      disjoint_nhds_nhds_iff_not_inseparable]
+    rintro y ⟨-, hys⟩ hxy
+    refine hys <| (hxy.mem_open_iff isOpen_interior).1 ?_
+    rwa [mem_interior_iff_mem_nhds]
+  refine ⟨K \ f ⁻¹' V, diff_mem hKx ?_, hKc.diff <| Vo.preimage hf, fun y hy ↦ ?_⟩
+  · filter_upwards [hf.continuousAt <| Uo.mem_nhds (hxU rfl)] with x hx
+      using Set.disjoint_left.1 hd hx
+  · by_contra hys
+    exact hy.2 (hV ⟨mem_image_of_mem _ hy.1, not_mem_subset interior_subset hys⟩)
+
+instance (priority := 900) {X Y : Type*} [TopologicalSpace X] [WeaklyLocallyCompactSpace X]
+    [TopologicalSpace Y] [R1Space Y] : LocallyCompactPair X Y where
+  exists_mem_nhds_isCompact_mapsTo hf hs :=
+    let ⟨_K, hKc, hKx⟩ := exists_compact_mem_nhds _
+    exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds hf hs hKc hKx
+
+/-- If a point in an R₁ space has a compact neighborhood,
+then it has a basis of compact closed neighborhoods. -/
+theorem IsCompact.isCompact_isClosed_basis_nhds {x : X} {L : Set X} (hLc : IsCompact L)
+    (hxL : L ∈ 𝓝 x) : (𝓝 x).HasBasis (fun K ↦ K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) (·) :=
+  hasBasis_self.2 fun _U hU ↦
+    let ⟨K, hKx, hKc, hKU⟩ := exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds
+      continuous_id (interior_mem_nhds.2 hU) hLc hxL
+    ⟨closure K, mem_of_superset hKx subset_closure, ⟨hKc.closure, isClosed_closure⟩,
+      (hKc.closure_subset_of_isOpen isOpen_interior hKU).trans interior_subset⟩
+
+/-!
+### Lemmas about a weakly locally compact R₁ space
+
+In fact, a space with these properties is locally compact and regular.
+Some lemmas are formulated using the latter assumptions below.
+-/
+
+variable [WeaklyLocallyCompactSpace X]
+
+/-- In a (weakly) locally compact R₁ space, compact closed neighborhoods of a point `x`
+form a basis of neighborhoods of `x`. -/
+theorem isCompact_isClosed_basis_nhds (x : X) :
+    (𝓝 x).HasBasis (fun K => K ∈ 𝓝 x ∧ IsCompact K ∧ IsClosed K) (·) :=
+  let ⟨_L, hLc, hLx⟩ := exists_compact_mem_nhds x
+  hLc.isCompact_isClosed_basis_nhds hLx
+
+/-- In a (weakly) locally compact R₁ space, each point admits a compact closed neighborhood. -/
+theorem exists_mem_nhds_isCompact_isClosed (x : X) : ∃ K ∈ 𝓝 x, IsCompact K ∧ IsClosed K :=
+  (isCompact_isClosed_basis_nhds x).ex_mem
+
+-- see Note [lower instance priority]
+/-- A weakly locally compact R₁ space is locally compact. -/
+instance (priority := 80) WeaklyLocallyCompactSpace.locallyCompactSpace : LocallyCompactSpace X :=
+  .of_hasBasis isCompact_isClosed_basis_nhds fun _ _ ⟨_, h, _⟩ ↦ h
+#align locally_compact_of_compact_nhds WeaklyLocallyCompactSpace.locallyCompactSpace
+
+/-- In a weakly locally compact R₁ space,
+every compact set has an open neighborhood with compact closure. -/
+theorem exists_isOpen_superset_and_isCompact_closure {K : Set X} (hK : IsCompact K) :
+    ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
+  rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
+  exact ⟨interior K', isOpen_interior, hKK', hK'.closure_of_subset interior_subset⟩
+#align exists_open_superset_and_is_compact_closure exists_isOpen_superset_and_isCompact_closure
+
+@[deprecated] -- Since 28 Jan 2024
+alias exists_open_superset_and_isCompact_closure := exists_isOpen_superset_and_isCompact_closure
+
+/-- In a weakly locally compact R₁ space,
+every point has an open neighborhood with compact closure. -/
+theorem exists_isOpen_mem_isCompact_closure (x : X) :
+    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
+  simpa only [singleton_subset_iff]
+    using exists_isOpen_superset_and_isCompact_closure isCompact_singleton
+#align exists_open_with_compact_closure exists_isOpen_mem_isCompact_closure
+
+@[deprecated] -- Since 28 Jan 2024
+alias exists_open_with_compact_closure := exists_isOpen_mem_isCompact_closure
+
+end R1Space
+
 /-- A T₂ space, also known as a Hausdorff space, is one in which for every
   `x ≠ y` there exists disjoint open sets around `x` and `y`. This is
   the most widely used of the separation axioms. -/
@@ -953,22 +1240,28 @@ theorem Set.Finite.t2_separation [T2Space X] {s : Set X} (hs : s.Finite) :
   s.pairwiseDisjoint_nhds.exists_mem_filter_basis hs nhds_basis_opens
 #align set.finite.t2_separation Set.Finite.t2_separation
 
-theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : X × X | Disjoint (𝓝 p.1) (𝓝 p.2) } := by
-  simp only [isOpen_iff_mem_nhds, Prod.forall, mem_setOf_eq]
-  intro x y h
-  obtain ⟨U, hU, V, hV, hd⟩ := ((nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)).mp h
-  exact
-    mem_nhds_prod_iff.mpr
-      ⟨U, hU.2.mem_nhds hU.1, V, hV.2.mem_nhds hV.1, fun ⟨x', y'⟩ ⟨hx', hy'⟩ =>
-        disjoint_of_disjoint_of_mem hd (hU.2.mem_nhds hx') (hV.2.mem_nhds hy')⟩
-#align is_open_set_of_disjoint_nhds_nhds isOpen_setOf_disjoint_nhds_nhds
-
 -- see Note [lower instance priority]
 instance (priority := 100) T2Space.t1Space [T2Space X] : T1Space X :=
   t1Space_iff_disjoint_pure_nhds.mpr fun _ _ hne =>
     (disjoint_nhds_nhds.2 hne).mono_left <| pure_le_nhds _
 #align t2_space.t1_space T2Space.t1Space
 
+-- see Note [lower instance priority]
+instance (priority := 100) T2Space.r1Space [T2Space X] : R1Space X :=
+  ⟨fun x y ↦ (eq_or_ne x y).imp specializes_of_eq disjoint_nhds_nhds.2⟩
+
+theorem SeparationQuotient.t2Space_iff : T2Space (SeparationQuotient X) ↔ R1Space X := by
+  simp only [t2Space_iff_disjoint_nhds, Pairwise, surjective_mk.forall₂, ne_eq, mk_eq_mk,
+    r1Space_iff_inseparable_or_disjoint_nhds, ← disjoint_comap_iff surjective_mk, comap_mk_nhds_mk,
+    ← or_iff_not_imp_left]
+
+instance SeparationQuotient.t2Space [R1Space X] : T2Space (SeparationQuotient X) :=
+  t2Space_iff.2 ‹_›
+
+instance (priority := 80) [R1Space X] [T0Space X] : T2Space X :=
+  t2Space_iff_disjoint_nhds.2 fun _x _y hne ↦ disjoint_nhds_nhds_iff_not_inseparable.2 fun hxy ↦
+    hne hxy.eq
+
 /-- A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. -/
 theorem t2_iff_nhds : T2Space X ↔ ∀ {x y : X}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
   simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne.def, not_imp_comm, Pairwise]
@@ -1197,7 +1490,6 @@ Hausdorff spaces:
   We use this lemma to prove that topological spaces defined using `coinduced` are Hausdorff spaces.
 -/
 
-
 -- see Note [lower instance priority]
 instance (priority := 100) DiscreteTopology.toT2Space
     [DiscreteTopology X] : T2Space X :=
@@ -1219,13 +1511,10 @@ theorem separated_by_openEmbedding [TopologicalSpace Y] [T2Space X]
     mem_image_of_mem _ yv, disjoint_image_of_injective hf.inj uv⟩
 #align separated_by_open_embedding separated_by_openEmbedding
 
-instance {p : X → Prop} [T2Space X] : T2Space (Subtype p) :=
-  ⟨fun _ _ h => separated_by_continuous continuous_subtype_val (mt Subtype.eq h)⟩
+instance {p : X → Prop} [T2Space X] : T2Space (Subtype p) := inferInstance
 
 instance Prod.t2Space [T2Space X] [TopologicalSpace Y] [T2Space Y] : T2Space (X × Y) :=
-  ⟨fun _ _ h => Or.elim (not_and_or.mp (mt Prod.ext_iff.mpr h))
-    (fun h₁ => separated_by_continuous continuous_fst h₁) fun h₂ =>
-    separated_by_continuous continuous_snd h₂⟩
+  inferInstance
 
 /-- If the codomain of an injective continuous function is a Hausdorff space, then so is its
 domain. -/
@@ -1254,9 +1543,7 @@ instance [T2Space X] [TopologicalSpace Y] [T2Space Y] :
 
 instance Pi.t2Space {Y : X → Type v} [∀ a, TopologicalSpace (Y a)]
     [∀ a, T2Space (Y a)] : T2Space (∀ a, Y a) :=
-  ⟨fun _ _ h =>
-    let ⟨i, hi⟩ := not_forall.mp (mt funext h)
-    separated_by_continuous (continuous_apply i) hi⟩
+  inferInstance
 #align Pi.t2_space Pi.t2Space
 
 instance Sigma.t2Space {ι} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ a, T2Space (X a)] :
@@ -1335,24 +1622,32 @@ theorem Function.LeftInverse.closedEmbedding [T2Space X] {f : X → Y} {g : Y 
   ⟨h.embedding hf hg, h.closed_range hf hg⟩
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 
-theorem separatedNhds_of_isCompact_isCompact [T2Space X] {s t : Set X} (hs : IsCompact s)
+theorem SeparatedNhds.of_isCompact_isCompact [T2Space X] {s t : Set X} (hs : IsCompact s)
     (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t := by
   simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst ⊢
   exact generalized_tube_lemma hs ht isClosed_diagonal.isOpen_compl hst
-#align is_compact_is_compact_separated separatedNhds_of_isCompact_isCompact
+#align is_compact_is_compact_separated SeparatedNhds.of_isCompact_isCompact
+
+@[deprecated] -- Since 28 Jan 2024
+alias separatedNhds_of_isCompact_isCompact := SeparatedNhds.of_isCompact_isCompact
 
 section SeparatedFinset
 
-theorem separatedNhds_of_finset_finset [T2Space X] (s t : Finset X) (h : Disjoint s t) :
+theorem SeparatedNhds.of_finset_finset [T2Space X] (s t : Finset X) (h : Disjoint s t) :
     SeparatedNhds (s : Set X) t :=
-  separatedNhds_of_isCompact_isCompact s.finite_toSet.isCompact t.finite_toSet.isCompact
-    <| mod_cast h
-#align finset_disjoint_finset_opens_of_t2 separatedNhds_of_finset_finset
+  .of_isCompact_isCompact s.finite_toSet.isCompact t.finite_toSet.isCompact <| mod_cast h
+#align finset_disjoint_finset_opens_of_t2 SeparatedNhds.of_finset_finset
+
+@[deprecated] -- Since 28 Jan 2024
+alias separatedNhds_of_finset_finset := SeparatedNhds.of_finset_finset
 
-theorem point_disjoint_finset_opens_of_t2 [T2Space X] {x : X} {s : Finset X} (h : x ∉ s) :
+theorem SeparatedNhds.of_singleton_finset [T2Space X] {x : X} {s : Finset X} (h : x ∉ s) :
     SeparatedNhds ({x} : Set X) s :=
-  mod_cast separatedNhds_of_finset_finset {x} s (Finset.disjoint_singleton_left.mpr h)
-#align point_disjoint_finset_opens_of_t2 point_disjoint_finset_opens_of_t2
+  mod_cast .of_finset_finset {x} s (Finset.disjoint_singleton_left.mpr h)
+#align point_disjoint_finset_opens_of_t2 SeparatedNhds.of_singleton_finset
+
+@[deprecated] -- Since 28 Jan 2024
+alias point_disjoint_finset_opens_of_t2 := SeparatedNhds.of_singleton_finset
 
 end SeparatedFinset
 
@@ -1360,7 +1655,7 @@ end SeparatedFinset
 theorem IsCompact.isClosed [T2Space X] {s : Set X} (hs : IsCompact s) : IsClosed s :=
   isOpen_compl_iff.1 <| isOpen_iff_forall_mem_open.mpr fun x hx =>
     let ⟨u, v, _, vo, su, xv, uv⟩ :=
-      separatedNhds_of_isCompact_isCompact hs isCompact_singleton (disjoint_singleton_right.2 hx)
+      SeparatedNhds.of_isCompact_isCompact hs isCompact_singleton (disjoint_singleton_right.2 hx)
     ⟨v, (uv.mono_left <| show s ≤ u from su).subset_compl_left, vo, by simpa using xv⟩
 #align is_compact.is_closed IsCompact.isClosed
 
@@ -1422,21 +1717,6 @@ theorem QuotientMap.of_surjective_continuous [CompactSpace X] [T2Space Y] {f : X
   hcont.isClosedMap.to_quotientMap hcont hsurj
 #align quotient_map.of_surjective_continuous QuotientMap.of_surjective_continuous
 
-instance (priority := 900) [WeaklyLocallyCompactSpace X] [T2Space Y] : LocallyCompactPair X Y where
-  exists_mem_nhds_isCompact_mapsTo := by
-    intro f x s hf hs
-    rcases exists_compact_mem_nhds x with ⟨K, hKc, hKx⟩
-    have hc : IsCompact (f '' K \ interior s) := (hKc.image hf).diff isOpen_interior
-    have hd : Disjoint {f x} (f '' K \ interior s) := disjoint_singleton_left.2 fun h ↦
-      h.2 <| mem_interior_iff_mem_nhds.2 hs
-    rcases separatedNhds_of_isCompact_isCompact isCompact_singleton hc hd
-      with ⟨U, V, Uo, Vo, hxU, hV, hd⟩
-    refine ⟨K \ f ⁻¹' V, diff_mem hKx ?_, hKc.diff <| Vo.preimage hf, fun y hy ↦ ?_⟩
-    · filter_upwards [hf.continuousAt <| Uo.mem_nhds (hxU rfl)] with x hx
-        using Set.disjoint_left.1 hd hx
-    · by_contra hys
-      exact hy.2 (hV ⟨mem_image_of_mem _ hy.1, not_mem_subset interior_subset hys⟩)
-
 theorem isPreirreducible_iff_subsingleton [T2Space X] {S : Set X} :
     IsPreirreducible S ↔ S.Subsingleton := by
   refine' ⟨fun h x hx y hy => _, Set.Subsingleton.isPreirreducible⟩
@@ -1525,11 +1805,9 @@ theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
   Iff.mpr ((regularSpace_TFAE X).out 0 3) h
 #align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.ofExistsMemNhdsIsClosedSubset
 
-/-- A locally compact T2 space is regular. -/
-instance (priority := 100) [LocallyCompactSpace X] [T2Space X] : RegularSpace X := by
-  apply RegularSpace.ofExistsMemNhdsIsClosedSubset (fun x s hx ↦ ?_)
-  rcases local_compact_nhds hx with ⟨k, kx, ks, hk⟩
-  exact ⟨k, kx, hk.isClosed, ks⟩
+/-- A weakly locally compact R₁ space is regular. -/
+instance (priority := 100) [WeaklyLocallyCompactSpace X] [R1Space X] : RegularSpace X :=
+  .ofBasis isCompact_isClosed_basis_nhds fun _ _ ⟨_, _, h⟩ ↦ h
 
 variable [RegularSpace X] {x : X} {s : Set X}
 
@@ -1542,6 +1820,11 @@ theorem disjoint_nhds_nhdsSet : Disjoint (𝓝 x) (𝓝ˢ s) ↔ x ∉ closure s
   disjoint_comm.trans disjoint_nhdsSet_nhds
 #align disjoint_nhds_nhds_set disjoint_nhds_nhdsSet
 
+/-- A regular space is R₁. -/
+instance (priority := 100) : R1Space X where
+  specializes_or_disjoint_nhds _ _ := or_iff_not_imp_left.2 fun h ↦ by
+    rwa [← nhdsSet_singleton, disjoint_nhdsSet_nhds, ← specializes_iff_mem_closure]
+
 theorem exists_mem_nhds_isClosed_subset {x : X} {s : Set X} (h : s ∈ 𝓝 x) :
     ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s := by
   have h' := (regularSpace_TFAE X).out 0 3
@@ -1581,30 +1864,6 @@ theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set
   simpa only [exists_prop, and_assoc] using hB.nhds_hasBasis.nhds_closure.mem_iff.mp h
 #align topological_space.is_topological_basis.exists_closure_subset TopologicalSpace.IsTopologicalBasis.exists_closure_subset
 
-theorem disjoint_nhds_nhds_iff_not_specializes {a b : X} : Disjoint (𝓝 a) (𝓝 b) ↔ ¬a ⤳ b := by
-  rw [← nhdsSet_singleton, disjoint_nhdsSet_nhds, specializes_iff_mem_closure]
-#align disjoint_nhds_nhds_iff_not_specializes disjoint_nhds_nhds_iff_not_specializes
-
-theorem specializes_comm {a b : X} : a ⤳ b ↔ b ⤳ a := by
-  simp only [← (disjoint_nhds_nhds_iff_not_specializes (X := X)).not_left, disjoint_comm]
-#align specializes_comm specializes_comm
-
-alias ⟨Specializes.symm, _⟩ := specializes_comm
-#align specializes.symm Specializes.symm
-
-theorem specializes_iff_inseparable {a b : X} : a ⤳ b ↔ Inseparable a b :=
-  ⟨fun h => h.antisymm h.symm, le_of_eq⟩
-#align specializes_iff_inseparable specializes_iff_inseparable
-
-theorem isClosed_setOf_specializes : IsClosed { p : X × X | p.1 ⤳ p.2 } := by
-  simp only [← isOpen_compl_iff, compl_setOf, ← disjoint_nhds_nhds_iff_not_specializes,
-    isOpen_setOf_disjoint_nhds_nhds]
-#align is_closed_set_of_specializes isClosed_setOf_specializes
-
-theorem isClosed_setOf_inseparable : IsClosed { p : X × X | Inseparable p.1 p.2 } := by
-  simp only [← specializes_iff_inseparable, isClosed_setOf_specializes]
-#align is_closed_set_of_inseparable isClosed_setOf_inseparable
-
 protected theorem Inducing.regularSpace [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) :
     RegularSpace Y :=
   RegularSpace.ofBasis
@@ -1614,7 +1873,7 @@ protected theorem Inducing.regularSpace [TopologicalSpace Y] {f : Y → X} (hf :
 
 theorem regularSpace_induced (f : Y → X) : @RegularSpace Y (induced f ‹_›) :=
   letI := induced f ‹_›
-  Inducing.regularSpace ⟨rfl⟩
+  (inducing_induced f).regularSpace
 #align regular_space_induced regularSpace_induced
 
 theorem regularSpace_sInf {X} {T : Set (TopologicalSpace X)} (h : ∀ t ∈ T, @RegularSpace X t) :
@@ -1654,80 +1913,17 @@ instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i,
 
 /-- In a regular space, if a compact set and a closed set are disjoint, then they have disjoint
 neighborhoods. -/
-lemma separatedNhds_of_isCompact_isClosed [RegularSpace X] {s t : Set X}
+lemma SeparatedNhds.of_isCompact_isClosed [RegularSpace X] {s t : Set X}
     (hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t) : SeparatedNhds s t := by
-  apply hs.induction_on (p := fun u ↦ SeparatedNhds u t)
-  · simp
-  · intro s u su hu
-    exact hu.mono su Subset.rfl
-  · intro u v hu hv
-    exact hu.union_left hv
-  · intro x hx
-    have : tᶜ ∈ 𝓝 x := ht.isOpen_compl.mem_nhds (disjoint_left.1 hst hx)
-    rcases exists_mem_nhds_isClosed_subset this with ⟨u, u_mem, u_closed, hu⟩
-    refine ⟨interior u, mem_nhdsWithin_of_mem_nhds (interior_mem_nhds.2 u_mem), ?_⟩
-    exact ⟨interior u, uᶜ, isOpen_interior, u_closed.isOpen_compl, Subset.rfl,
-      subset_compl_comm.mp hu, disjoint_compl_right.mono_left (interior_subset)⟩
+  simpa only [separatedNhds_iff_disjoint, hs.disjoint_nhdsSet_left, disjoint_nhds_nhdsSet,
+    ht.closure_eq, disjoint_left] using hst
+
+@[deprecated] -- Since 28 Jan 2024
+alias separatedNhds_of_isCompact_isClosed := SeparatedNhds.of_isCompact_isClosed
 
 end RegularSpace
 
-section T2OrLocallyCompactRegularSpace
-
-/-- A topological space which is either T2 or locally compact regular. Equivalent to regularity
-among locally compact spaces. These two classes share the properties that are needed to develop a
-lot of measure theory, so it's worth defining a single class to avoid developing things twice. -/
-class T2OrLocallyCompactRegularSpace (X : Type*) [TopologicalSpace X] : Prop :=
-  out : T2Space X ∨ (LocallyCompactSpace X ∧ RegularSpace X)
-
-instance (priority := 100) [h : T2Space X] : T2OrLocallyCompactRegularSpace X := ⟨Or.inl h⟩
-instance (priority := 100) [h : LocallyCompactSpace X] [h' : RegularSpace X] :
-  T2OrLocallyCompactRegularSpace X := ⟨Or.inr ⟨h, h'⟩⟩
-
-/-- A locally compact space which is T2 or locally comapct regular is regular. Not an instance, as
-one should instead assume `LocallyCompactSpace X` and `RegularSpace X`. -/
-theorem RegularSpace.ofT2SpaceOrRegularSpace
-    [LocallyCompactSpace X] [h : T2OrLocallyCompactRegularSpace X] : RegularSpace X := by
-  rcases h.out with h'|⟨-, h'⟩ <;> infer_instance
-
-/-- In a space which is T2 or locally compact regular, if a compact set `s` is contained in an
-open set `u`, then its closure is also contained in `u`. -/
-theorem IsCompact.closure_subset_of_isOpen [hX : T2OrLocallyCompactRegularSpace X]
-    {s : Set X} (hs : IsCompact s) {u : Set X} (hu : IsOpen u) (h : s ⊆ u) :
-    closure s ⊆ u := by
-  rcases hX.out with h'|⟨-, h'⟩
-  · rwa [hs.isClosed.closure_eq]
-  obtain ⟨F, sF, F_closed, Fu⟩ : ∃ F, s ⊆ F ∧ IsClosed F ∧ F ⊆ u := by
-    apply hs.induction_on (p := fun t ↦ ∃ F, t ⊆ F ∧ IsClosed F ∧ F ⊆ u)
-    · exact ⟨∅, by simp⟩
-    · intro t' t ht't ⟨F, tF, F_closed, Fu⟩
-      exact ⟨F, ht't.trans tF, F_closed, Fu⟩
-    · intro t t' ⟨F, tF, F_closed, Fu⟩ ⟨F', t'F', F'_closed, F'u⟩
-      exact ⟨F ∪ F', union_subset_union tF t'F', F_closed.union F'_closed, union_subset Fu F'u⟩
-    · intro x hx
-      rcases exists_mem_nhds_isClosed_subset (hu.mem_nhds (h hx)) with ⟨F, F_mem, F_closed, Fu⟩
-      exact ⟨F, nhdsWithin_le_nhds F_mem, F, Subset.rfl, F_closed, Fu⟩
-  exact (closure_minimal sF F_closed).trans Fu
-
-theorem separatedNhds_of_isCompact_isCompact_isClosed [hX : T2OrLocallyCompactRegularSpace X]
-    {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) (h't : IsClosed t)
-    (hst : Disjoint s t) : SeparatedNhds s t := by
-  rcases hX.out with h'|⟨-, h'⟩
-  · exact separatedNhds_of_isCompact_isCompact hs ht hst
-  · exact separatedNhds_of_isCompact_isClosed hs h't hst
-
-protected theorem IsCompact.closure [hX : T2OrLocallyCompactRegularSpace X]
-    {K : Set X} (hK : IsCompact K) : IsCompact (closure K) := by
-  rcases hX.out with h'|⟨h', -⟩
-  · rwa [hK.isClosed.closure_eq]
-  rcases exists_compact_superset hK with ⟨L, L_comp, hL⟩
-  exact L_comp.of_isClosed_subset isClosed_closure
-    ((hK.closure_subset_of_isOpen isOpen_interior hL).trans interior_subset)
-
-theorem isCompact_closure_of_subset_compact [T2OrLocallyCompactRegularSpace X]
-    {s t : Set X} (ht : IsCompact t)
-    (h : s ⊆ t) : IsCompact (closure s) :=
-  ht.closure.of_isClosed_subset isClosed_closure (closure_mono h)
-#align is_compact_closure_of_subset_compact isCompact_closure_of_subset_compact
+section LocallyCompactRegularSpace
 
 /-- In a (possibly non-Hausdorff) locally compact regular space, for every containment `K ⊆ U` of
   a compact set `K` in an open set `U`, there is a compact closed neighborhood `L`
@@ -1735,18 +1931,10 @@ theorem isCompact_closure_of_subset_compact [T2OrLocallyCompactRegularSpace X]
   that `K ⊆ interior L` and `L ⊆ U`. -/
 theorem exists_compact_closed_between [LocallyCompactSpace X] [RegularSpace X]
     {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (h_KU : K ⊆ U) :
-    ∃ L, IsCompact L ∧ IsClosed L ∧ K ⊆ interior L ∧ L ⊆ U := by
-  rcases exists_compact_between hK hU h_KU with ⟨L, L_comp, KL, LU⟩
-  rcases exists_compact_between hK isOpen_interior KL with ⟨M, M_comp, KM, ML⟩
-  refine ⟨closure M, ?_, isClosed_closure, ?_, ?_⟩
-  · have : closure M ∩ L = closure M := by
-      apply inter_eq_self_of_subset_left
-      exact (M_comp.closure_subset_of_isOpen isOpen_interior ML).trans interior_subset
-    rw [← this]
-    apply L_comp.inter_left isClosed_closure
-  · exact KM.trans (interior_mono subset_closure)
-  · apply M_comp.closure_subset_of_isOpen hU
-    exact ML.trans (interior_subset.trans LU)
+    ∃ L, IsCompact L ∧ IsClosed L ∧ K ⊆ interior L ∧ L ⊆ U :=
+  let ⟨L, L_comp, KL, LU⟩ := exists_compact_between hK hU h_KU
+  ⟨closure L, L_comp.closure, isClosed_closure, KL.trans <| interior_mono subset_closure,
+    L_comp.closure_subset_of_isOpen hU LU⟩
 
 /-- In a locally compact regular space, given a compact set `K` inside an open set `U`, we can find
 an open set `V` between these sets with compact closure: `K ⊆ V` and the closure of `V` is
@@ -1758,99 +1946,16 @@ theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace X] [Regul
   have A : closure (interior L) ⊆ L := by
     apply (closure_mono interior_subset).trans (le_of_eq L_closed.closure_eq)
   refine ⟨interior L, isOpen_interior, KL, A.trans LU, ?_⟩
-  exact isCompact_closure_of_subset_compact L_compact interior_subset
+  exact L_compact.closure_of_subset interior_subset
 #align exists_open_between_and_is_compact_closure exists_open_between_and_isCompact_closure
 
-@[simp]
-theorem exists_compact_superset_iff [T2OrLocallyCompactRegularSpace X] {s : Set X} :
-    (∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) :=
-  ⟨fun ⟨_K, hK, hsK⟩ => isCompact_closure_of_subset_compact hK hsK, fun h =>
-    ⟨closure s, h, subset_closure⟩⟩
-#align exists_compact_superset_iff exists_compact_superset_iff
-
-/-- In a weakly locally compact space which is either T₂ or locally compact regular,
-every compact set has an open neighborhood with compact closure. -/
-theorem exists_open_superset_and_isCompact_closure
-    [WeaklyLocallyCompactSpace X] [T2OrLocallyCompactRegularSpace X]
-    {K : Set X} (hK : IsCompact K) : ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
-  rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
-  exact ⟨interior K', isOpen_interior, hKK',
-    isCompact_closure_of_subset_compact hK' interior_subset⟩
-#align exists_open_superset_and_is_compact_closure exists_open_superset_and_isCompact_closure
-
-/-- In a weakly locally compact which is either T₂ or locally compact regular,
-every point has an open neighborhood with compact closure. -/
-theorem exists_open_with_compact_closure
-    [WeaklyLocallyCompactSpace X] [T2OrLocallyCompactRegularSpace X] (x : X) :
-    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
-  simpa only [singleton_subset_iff]
-    using exists_open_superset_and_isCompact_closure isCompact_singleton
-#align exists_open_with_compact_closure exists_open_with_compact_closure
-
--- see Note [lower instance priority]
-/-- A weakly locally compact Hausdorff space is locally compact. -/
-instance (priority := 80) WeaklyLocallyCompactSpace.locallyCompactSpace
-    [WeaklyLocallyCompactSpace X] [hX : T2OrLocallyCompactRegularSpace X] :
-    LocallyCompactSpace X := by
-  rcases hX.out with h'X|⟨h'X, -⟩
-  · exact ⟨fun _ _ h =>
-      let ⟨K, hKx, hKc, hKs⟩ := exists_mem_nhds_isCompact_mapsTo continuous_id h
-      ⟨K, hKx, hKs, hKc⟩⟩
-  · exact h'X
-#align locally_compact_of_compact_nhds WeaklyLocallyCompactSpace.locallyCompactSpace
-
-@[deprecated WeaklyLocallyCompactSpace.locallyCompactSpace]
+@[deprecated WeaklyLocallyCompactSpace.locallyCompactSpace] -- 3 Sep 2023
 theorem locally_compact_of_compact [T2Space X] [CompactSpace X] :
     LocallyCompactSpace X :=
   inferInstance
 #align locally_compact_of_compact locally_compact_of_compact
 
-/-- If a compact set is covered by two open sets, then we can cover it by two compact subsets. -/
-theorem IsCompact.binary_compact_cover [T2OrLocallyCompactRegularSpace X] {K U V : Set X}
-    (hK : IsCompact K) (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
-    ∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ := by
-  have hK' : IsCompact (closure K) := hK.closure
-  have : SeparatedNhds (closure K \ U) (closure K \ V) := by
-    apply separatedNhds_of_isCompact_isCompact_isClosed (hK'.diff hU) (hK'.diff hV)
-      (isClosed_closure.sdiff hV)
-    rw [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty]
-    exact hK.closure_subset_of_isOpen (hU.union hV) h2K
-  have : SeparatedNhds (K \ U) (K \ V) :=
-    this.mono (diff_subset_diff_left (subset_closure)) (diff_subset_diff_left (subset_closure))
-  rcases this with ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩
-  exact ⟨K \ O₁, K \ O₂, hK.diff h1O₁, hK.diff h1O₂, diff_subset_comm.mp h2O₁,
-    diff_subset_comm.mp h2O₂, by rw [← diff_inter, hO.inter_eq, diff_empty]⟩
-#align is_compact.binary_compact_cover IsCompact.binary_compact_cover
-
-open Finset Function
-
-/-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
-theorem IsCompact.finite_compact_cover [T2OrLocallyCompactRegularSpace X]
-    {s : Set X} (hs : IsCompact s) {ι}
-    (t : Finset ι) (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
-    ∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
-  induction' t using Finset.induction with x t hx ih generalizing U s
-  · refine' ⟨fun _ => ∅, fun _ => isCompact_empty, fun i => empty_subset _, _⟩
-    simpa only [subset_empty_iff, Finset.not_mem_empty, iUnion_false, iUnion_empty] using hsC
-  simp only [Finset.set_biUnion_insert] at hsC
-  simp only [Finset.forall_mem_insert] at hU
-  have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU.2 i hi
-  rcases hs.binary_compact_cover hU.1 (isOpen_biUnion hU') hsC with
-    ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
-  rcases ih h1K₂ U hU' h2K₂ with ⟨K, h1K, h2K, h3K⟩
-  refine' ⟨update K x K₁, _, _, _⟩
-  · intro i
-    rcases eq_or_ne i x with rfl | hi
-    · simp only [update_same, h1K₁]
-    · simp only [update_noteq hi, h1K]
-  · intro i
-    rcases eq_or_ne i x with rfl | hi
-    · simp only [update_same, h2K₁]
-    · simp only [update_noteq hi, h2K]
-  · simp only [set_biUnion_insert_update _ hx, hK, h3K]
-#align is_compact.finite_compact_cover IsCompact.finite_compact_cover
-
-end T2OrLocallyCompactRegularSpace
+end LocallyCompactRegularSpace
 
 section T3
 
@@ -1948,7 +2053,13 @@ protected theorem ClosedEmbedding.normalSpace [TopologicalSpace Y] [NormalSpace
         (disjoint_image_of_injective hf.inj hst)
     exact (H.preimage hf.continuous).mono (subset_preimage_image _ _) (subset_preimage_image _ _)
 
-/-- A regular topological space with second countable topology is a normal space. -/
+instance (priority := 100) NormalSpace.of_compactSpace_r1Space [CompactSpace X] [R1Space X] :
+    NormalSpace X where
+  normal _s _t hs ht := .of_isCompact_isCompact_isClosed hs.isCompact ht.isCompact ht
+
+/-- A regular topological space with second countable topology is a normal space.
+
+TODO: The same is true for a regular Lindelöf space. -/
 instance (priority := 100) NormalSpace.of_regularSpace_secondCountableTopology
     [RegularSpace X] [SecondCountableTopology X] : NormalSpace X := by
   have key : ∀ {s t : Set X}, IsClosed t → Disjoint s t →
@@ -2009,9 +2120,9 @@ instance (priority := 100) T4Space.t3Space [T4Space X] : T3Space X where
     (disjoint_singleton_right.mpr hxs)).disjoint_nhdsSet
 #align normal_space.t3_space T4Space.t3Space
 
-instance (priority := 100) T4Space.of_compactSpace_t2Space [CompactSpace X] [T2Space X] :
-    T4Space X where
-  normal _s _t hs ht := separatedNhds_of_isCompact_isCompact hs.isCompact ht.isCompact
+@[deprecated inferInstance] -- Since 28 Jan 2024
+theorem T4Space.of_compactSpace_t2Space [CompactSpace X] [T2Space X] :
+    T4Space X := inferInstance
 #align normal_of_compact_t2 T4Space.of_compactSpace_t2Space
 
 /-- If the codomain of a closed embedding is a T₄ space, then so is the domain. -/

There is no NeBot in this lemma

@@ -1503,7 +1503,7 @@ theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     refine' disjoint_of_disjoint_of_mem disjoint_compl_left _ hU
     rwa [← subset_interior_iff_mem_nhdsSet, hUc.isOpen_compl.interior_eq, subset_compl_comm]
   tfae_have 2 → 3
-  · refine' fun H a s => ⟨fun hd has => mem_closure_iff_nhds_neBot.mp has _, H s a⟩
+  · refine' fun H a s => ⟨fun hd has => mem_closure_iff_nhds_ne_bot.mp has _, H s a⟩
     exact (hd.symm.mono_right <| @principal_le_nhdsSet _ _ s).eq_bot
   tfae_have 3 → 1
   · exact fun H => ⟨fun hs ha => (H _ _).mpr <| hs.closure_eq.symm ▸ ha⟩

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -2108,7 +2108,7 @@ theorem connectedComponent_eq_iInter_isClopen [T2Space X] [CompactSpace X] (x :
   -- that one of these closed sets must contain our whole thing.
   -- To reduce to the case where the cover is disjoint on all of `X` we need that `s` is closed
   have hs : @IsClosed X _ (⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s) :=
-    isClosed_iInter fun s => s.2.1.2
+    isClosed_iInter fun s => s.2.1.1
   rw [isPreconnected_iff_subset_of_fully_disjoint_closed hs]
   intro a b ha hb hab ab_disj
   -- Since our space is normal, we get two larger disjoint open sets containing the disjoint
@@ -2121,7 +2121,7 @@ theorem connectedComponent_eq_iInter_isClopen [T2Space X] [CompactSpace X] (x :
   `X` disjoint to it, but a finite intersection of clopen sets is clopen so we let this be our
   `s`. -/
   · have H1 := (hu.union hv).isClosed_compl.isCompact.inter_iInter_nonempty
-      (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s) fun s => s.2.1.2
+      (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s) fun s => s.2.1.1
     rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, not_forall] at H1
     cases' H1 (disjoint_compl_left_iff_subset.2 <| hab.trans <| union_subset_union hau hbv)
       with si H2
@@ -2178,7 +2178,7 @@ theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace X ↔ Totally
   rw [connectedComponent_eq_iInter_isClopen, mem_iInter]
   rintro ⟨w : Set X, hw : IsClopen w, hy : y ∈ w⟩
   by_contra hx
-  exact hyp wᶜ w hw.2.isOpen_compl hw.1 hx hy (@isCompl_compl _ w _).symm.codisjoint.top_le
+  exact hyp wᶜ w hw.1.isOpen_compl hw.2 hx hy (@isCompl_compl _ w _).symm.codisjoint.top_le
     disjoint_compl_left
 #align compact_t2_tot_disc_iff_tot_sep compact_t2_tot_disc_iff_tot_sep
 
@@ -2199,7 +2199,7 @@ theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s
       suffices : ∃ s : N, s.val ⊆ U
       · rcases this with ⟨⟨s, hs, hs'⟩, hs''⟩; exact ⟨s, ⟨hs', hs⟩, hs''⟩
       haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩
-      have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.2
+      have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.1
       have hdir : Directed Superset fun s : N => s.val := by
         rintro ⟨s, hs, hxs⟩ ⟨t, ht, hxt⟩
         exact ⟨⟨s ∩ t, hs.inter ht, ⟨hxs, hxt⟩⟩, inter_subset_left s t, inter_subset_right s t⟩
@@ -2207,13 +2207,13 @@ theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s
         erw [hx, mem_singleton_iff] at y_in
         rwa [y_in]
       exact exists_subset_nhds_of_compactSpace hdir hNcl h_nhd
-    · rintro ⟨V, ⟨hxV, V_op, -⟩, hUV : V ⊆ U⟩
+    · rintro ⟨V, ⟨hxV, -, V_op⟩, hUV : V ⊆ U⟩
       rw [mem_nhds_iff]
       exact ⟨V, hUV, V_op, hxV⟩⟩
 #align nhds_basis_clopen nhds_basis_clopen
 
 theorem isTopologicalBasis_isClopen : IsTopologicalBasis { s : Set X | IsClopen s } := by
-  apply isTopologicalBasis_of_isOpen_of_nhds fun U (hU : IsClopen U) => hU.1
+  apply isTopologicalBasis_of_isOpen_of_nhds fun U (hU : IsClopen U) => hU.2
   intro x U hxU U_op
   have : U ∈ 𝓝 x := IsOpen.mem_nhds U_op hxU
   rcases (nhds_basis_clopen x).mem_iff.mp this with ⟨V, ⟨hxV, hV⟩, hVU : V ⊆ U⟩
@@ -2237,7 +2237,7 @@ variable {H : Type*} [TopologicalSpace H] [LocallyCompactSpace H] [T2Space H]
 /-- A locally compact Hausdorff totally disconnected space has a basis with clopen elements. -/
 theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
     IsTopologicalBasis { s : Set H | IsClopen s } := by
-  refine isTopologicalBasis_of_isOpen_of_nhds (fun u hu => hu.1) fun x U memU hU => ?_
+  refine isTopologicalBasis_of_isOpen_of_nhds (fun u hu => hu.2) fun x U memU hU => ?_
   obtain ⟨s, comp, xs, sU⟩ := exists_compact_subset hU memU
   let u : Set s := ((↑) : s → H) ⁻¹' interior s
   have u_open_in_s : IsOpen u := isOpen_interior.preimage continuous_subtype_val
@@ -2245,7 +2245,7 @@ theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
   haveI : CompactSpace s := isCompact_iff_compactSpace.1 comp
   obtain ⟨V : Set s, VisClopen, Vx, V_sub⟩ := compact_exists_isClopen_in_isOpen u_open_in_s xs
   have VisClopen' : IsClopen (((↑) : s → H) '' V) := by
-    refine' ⟨_, comp.isClosed.closedEmbedding_subtype_val.closed_iff_image_closed.1 VisClopen.2⟩
+    refine' ⟨comp.isClosed.closedEmbedding_subtype_val.closed_iff_image_closed.1 VisClopen.1, _⟩
     let v : Set u := ((↑) : u → s) ⁻¹' V
     have : ((↑) : u → H) = ((↑) : s → H) ∘ ((↑) : u → s) := rfl
     have f0 : Embedding ((↑) : u → H) := embedding_subtype_val.comp embedding_subtype_val
@@ -2256,7 +2256,7 @@ theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
           apply Set.inter_eq_self_of_subset_left interior_subset
         rw [this]
         apply isOpen_interior
-    have f2 : IsOpen v := VisClopen.1.preimage continuous_subtype_val
+    have f2 : IsOpen v := VisClopen.2.preimage continuous_subtype_val
     have f3 : ((↑) : s → H) '' V = ((↑) : u → H) '' v := by
       rw [this, image_comp, Subtype.image_preimage_coe, inter_eq_self_of_subset_left V_sub]
     rw [f3]
@@ -2292,7 +2292,7 @@ instance ConnectedComponents.t2 [T2Space X] [CompactSpace X] : T2Space (Connecte
       IsClopen U ∧ connectedComponent a ∩ U = ∅ ∧ connectedComponent b ⊆ U ∧ (↑) ⁻¹' V = U := by
     have h :=
       (isClosed_connectedComponent (α := X)).isCompact.elim_finite_subfamily_closed
-        _ (fun s : { s : Set X // IsClopen s ∧ b ∈ s } => s.2.1.2) h
+        _ (fun s : { s : Set X // IsClopen s ∧ b ∈ s } => s.2.1.1) h
     cases' h with fin_a ha
     -- This clopen and its complement will separate the connected components of `a` and `b`
     set U : Set X := ⋂ (i : { s // IsClopen s ∧ b ∈ s }) (_ : i ∈ fin_a), i

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
 import Mathlib.Topology.Compactness.SigmaCompact
-import Mathlib.Topology.Connected.LocallyConnected
 import Mathlib.Topology.Connected.TotallyDisconnected
 import Mathlib.Topology.Inseparable
 

The construction we have is given in T2 spaces, but it works in non-Hausdorff spaces modulo a few modifications.

For this, we introduce an ad hoc class T2OrLocallyCompactRegularSpace, which is just enough to unify the arguments, as a replacement for the class ClosableCompactSubsetOpenSpace (which is not strong enough). In the file Separation.lean, we move some material that was only available on T2 spaces to this new class.

The construction is needed for a forthcoming improvement of uniqueness results for Haar measures, based on https://mathoverflow.net/questions/456670/uniqueness-of-left-invariant-borel-probability-measure-on-compact-groups.

@@ -59,7 +59,7 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
 * `t2_iff_nhds`: A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter.
 * `t2_iff_isClosed_diagonal`: A space is T₂ iff the `diagonal` of `X` (that is, the set of all
   points of the form `(a, a) : X × X`) is closed under the product topology.
-* `finset_disjoint_finset_opens_of_t2`: Any two disjoint finsets are `SeparatedNhds`.
+* `separatedNhds_of_finset_finset`: Any two disjoint finsets are `SeparatedNhds`.
 * Most topological constructions preserve Hausdorffness;
   these results are part of the typeclass inference system (e.g. `Embedding.t2Space`)
 * `Set.EqOn.closure`: If two functions are equal on some set `s`, they are equal on its closure.
@@ -149,11 +149,11 @@ theorem disjoint_closure_right (h : SeparatedNhds s t) : Disjoint s (closure t)
   h.symm.disjoint_closure_left.symm
 #align separated_nhds.disjoint_closure_right SeparatedNhds.disjoint_closure_right
 
-theorem empty_right (s : Set X) : SeparatedNhds s ∅ :=
+@[simp] theorem empty_right (s : Set X) : SeparatedNhds s ∅ :=
   ⟨_, _, isOpen_univ, isOpen_empty, fun a _ => mem_univ a, Subset.rfl, disjoint_empty _⟩
 #align separated_nhds.empty_right SeparatedNhds.empty_right
 
-theorem empty_left (s : Set X) : SeparatedNhds ∅ s :=
+@[simp] theorem empty_left (s : Set X) : SeparatedNhds ∅ s :=
   (empty_right _).symm
 #align separated_nhds.empty_left SeparatedNhds.empty_left
 
@@ -842,7 +842,7 @@ theorem ConnectedSpace.infinite [ConnectedSpace X] [Nontrivial X] [T1Space X] :
 #align connected_space.infinite ConnectedSpace.infinite
 
 /-- A non-trivial connected T1 space has no isolated points. -/
-instance ConnectedSpace.neBot_nhdsWithin_compl_of_nontrivial_of_t1space
+instance (priority := 100) ConnectedSpace.neBot_nhdsWithin_compl_of_nontrivial_of_t1space
     [ConnectedSpace X] [Nontrivial X] [T1Space X] (x : X) :
     NeBot (𝓝[≠] x) := by
   by_contra contra
@@ -1336,22 +1336,23 @@ theorem Function.LeftInverse.closedEmbedding [T2Space X] {f : X → Y} {g : Y 
   ⟨h.embedding hf hg, h.closed_range hf hg⟩
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 
-theorem isCompact_isCompact_separated [T2Space X] {s t : Set X} (hs : IsCompact s)
+theorem separatedNhds_of_isCompact_isCompact [T2Space X] {s t : Set X} (hs : IsCompact s)
     (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t := by
   simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst ⊢
   exact generalized_tube_lemma hs ht isClosed_diagonal.isOpen_compl hst
-#align is_compact_is_compact_separated isCompact_isCompact_separated
+#align is_compact_is_compact_separated separatedNhds_of_isCompact_isCompact
 
 section SeparatedFinset
 
-theorem finset_disjoint_finset_opens_of_t2 [T2Space X] (s t : Finset X) (h : Disjoint s t) :
+theorem separatedNhds_of_finset_finset [T2Space X] (s t : Finset X) (h : Disjoint s t) :
     SeparatedNhds (s : Set X) t :=
-  isCompact_isCompact_separated s.finite_toSet.isCompact t.finite_toSet.isCompact <| mod_cast h
-#align finset_disjoint_finset_opens_of_t2 finset_disjoint_finset_opens_of_t2
+  separatedNhds_of_isCompact_isCompact s.finite_toSet.isCompact t.finite_toSet.isCompact
+    <| mod_cast h
+#align finset_disjoint_finset_opens_of_t2 separatedNhds_of_finset_finset
 
 theorem point_disjoint_finset_opens_of_t2 [T2Space X] {x : X} {s : Finset X} (h : x ∉ s) :
     SeparatedNhds ({x} : Set X) s :=
-  mod_cast finset_disjoint_finset_opens_of_t2 {x} s (Finset.disjoint_singleton_left.mpr h)
+  mod_cast separatedNhds_of_finset_finset {x} s (Finset.disjoint_singleton_left.mpr h)
 #align point_disjoint_finset_opens_of_t2 point_disjoint_finset_opens_of_t2
 
 end SeparatedFinset
@@ -1360,7 +1361,7 @@ end SeparatedFinset
 theorem IsCompact.isClosed [T2Space X] {s : Set X} (hs : IsCompact s) : IsClosed s :=
   isOpen_compl_iff.1 <| isOpen_iff_forall_mem_open.mpr fun x hx =>
     let ⟨u, v, _, vo, su, xv, uv⟩ :=
-      isCompact_isCompact_separated hs isCompact_singleton (disjoint_singleton_right.2 hx)
+      separatedNhds_of_isCompact_isCompact hs isCompact_singleton (disjoint_singleton_right.2 hx)
     ⟨v, (uv.mono_left <| show s ≤ u from su).subset_compl_left, vo, by simpa using xv⟩
 #align is_compact.is_closed IsCompact.isClosed
 
@@ -1399,36 +1400,12 @@ theorem IsCompact.inter [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCom
   hs.inter_right <| ht.isClosed
 #align is_compact.inter IsCompact.inter
 
-theorem isCompact_closure_of_subset_compact [T2Space X] {s t : Set X} (ht : IsCompact t)
-    (h : s ⊆ t) : IsCompact (closure s) :=
-  ht.of_isClosed_subset isClosed_closure (closure_minimal h ht.isClosed)
-#align is_compact_closure_of_subset_compact isCompact_closure_of_subset_compact
-
-@[simp]
-theorem exists_compact_superset_iff [T2Space X] {s : Set X} :
-    (∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) :=
-  ⟨fun ⟨_K, hK, hsK⟩ => isCompact_closure_of_subset_compact hK hsK, fun h =>
-    ⟨closure s, h, subset_closure⟩⟩
-#align exists_compact_superset_iff exists_compact_superset_iff
-
 theorem image_closure_of_isCompact [T2Space Y] {s : Set X} (hs : IsCompact (closure s)) {f : X → Y}
     (hf : ContinuousOn f (closure s)) : f '' closure s = closure (f '' s) :=
   Subset.antisymm hf.image_closure <|
     closure_minimal (image_subset f subset_closure) (hs.image_of_continuousOn hf).isClosed
 #align image_closure_of_is_compact image_closure_of_isCompact
 
-/-- If a compact set is covered by two open sets, then we can cover it by two compact subsets. -/
-theorem IsCompact.binary_compact_cover [T2Space X] {K U V : Set X} (hK : IsCompact K)
-    (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
-    ∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ := by
-  obtain ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩ :=
-    isCompact_isCompact_separated (hK.diff hU) (hK.diff hV)
-      (by rwa [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty])
-  exact
-    ⟨_, _, hK.diff h1O₁, hK.diff h1O₂, by rwa [diff_subset_comm], by rwa [diff_subset_comm], by
-      rw [← diff_inter, hO.inter_eq, diff_empty]⟩
-#align is_compact.binary_compact_cover IsCompact.binary_compact_cover
-
 /-- A continuous map from a compact space to a Hausdorff space is a closed map. -/
 protected theorem Continuous.isClosedMap [CompactSpace X] [T2Space Y] {f : X → Y}
     (h : Continuous f) : IsClosedMap f := fun _s hs => (hs.isCompact.image h).isClosed
@@ -1446,37 +1423,6 @@ theorem QuotientMap.of_surjective_continuous [CompactSpace X] [T2Space Y] {f : X
   hcont.isClosedMap.to_quotientMap hcont hsurj
 #align quotient_map.of_surjective_continuous QuotientMap.of_surjective_continuous
 
-section
-
-open Finset Function
-
-/-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
-theorem IsCompact.finite_compact_cover [T2Space X] {s : Set X} (hs : IsCompact s) {ι} (t : Finset ι)
-    (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
-    ∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
-  induction' t using Finset.induction with x t hx ih generalizing U s
-  · refine' ⟨fun _ => ∅, fun _ => isCompact_empty, fun i => empty_subset _, _⟩
-    simpa only [subset_empty_iff, Finset.not_mem_empty, iUnion_false, iUnion_empty] using hsC
-  simp only [Finset.set_biUnion_insert] at hsC
-  simp only [Finset.forall_mem_insert] at hU
-  have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU.2 i hi
-  rcases hs.binary_compact_cover hU.1 (isOpen_biUnion hU') hsC with
-    ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
-  rcases ih h1K₂ U hU' h2K₂ with ⟨K, h1K, h2K, h3K⟩
-  refine' ⟨update K x K₁, _, _, _⟩
-  · intro i
-    rcases eq_or_ne i x with rfl | hi
-    · simp only [update_same, h1K₁]
-    · simp only [update_noteq hi, h1K]
-  · intro i
-    rcases eq_or_ne i x with rfl | hi
-    · simp only [update_same, h2K₁]
-    · simp only [update_noteq hi, h2K]
-  · simp only [set_biUnion_insert_update _ hx, hK, h3K]
-#align is_compact.finite_compact_cover IsCompact.finite_compact_cover
-
-end
-
 instance (priority := 900) [WeaklyLocallyCompactSpace X] [T2Space Y] : LocallyCompactPair X Y where
   exists_mem_nhds_isCompact_mapsTo := by
     intro f x s hf hs
@@ -1484,7 +1430,7 @@ instance (priority := 900) [WeaklyLocallyCompactSpace X] [T2Space Y] : LocallyCo
     have hc : IsCompact (f '' K \ interior s) := (hKc.image hf).diff isOpen_interior
     have hd : Disjoint {f x} (f '' K \ interior s) := disjoint_singleton_left.2 fun h ↦
       h.2 <| mem_interior_iff_mem_nhds.2 hs
-    rcases isCompact_isCompact_separated isCompact_singleton hc hd
+    rcases separatedNhds_of_isCompact_isCompact isCompact_singleton hc hd
       with ⟨U, V, Uo, Vo, hxU, hV, hd⟩
     refine ⟨K \ f ⁻¹' V, diff_mem hKx ?_, hKc.diff <| Vo.preimage hf, fun y hy ↦ ?_⟩
     · filter_upwards [hf.continuousAt <| Uo.mem_nhds (hxU rfl)] with x hx
@@ -1492,50 +1438,6 @@ instance (priority := 900) [WeaklyLocallyCompactSpace X] [T2Space Y] : LocallyCo
     · by_contra hys
       exact hy.2 (hV ⟨mem_image_of_mem _ hy.1, not_mem_subset interior_subset hys⟩)
 
--- see Note [lower instance priority]
-/-- A weakly locally compact Hausdorff space is locally compact. -/
-instance WeaklyLocallyCompactSpace.locallyCompactSpace [WeaklyLocallyCompactSpace X] [T2Space X] :
-    LocallyCompactSpace X :=
-  ⟨fun _ _ h =>
-    let ⟨K, hKx, hKc, hKs⟩ := exists_mem_nhds_isCompact_mapsTo continuous_id h
-    ⟨K, hKx, hKs, hKc⟩⟩
-#align locally_compact_of_compact_nhds WeaklyLocallyCompactSpace.locallyCompactSpace
-
-@[deprecated WeaklyLocallyCompactSpace.locallyCompactSpace]
-theorem locally_compact_of_compact [T2Space X] [CompactSpace X] :
-    LocallyCompactSpace X :=
-  inferInstance
-#align locally_compact_of_compact locally_compact_of_compact
-
-/-- In a weakly locally compact T₂ space,
-every compact set has an open neighborhood with compact closure. -/
-theorem exists_open_superset_and_isCompact_closure [WeaklyLocallyCompactSpace X] [T2Space X]
-    {K : Set X} (hK : IsCompact K) : ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
-  rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
-  exact ⟨interior K', isOpen_interior, hKK',
-    isCompact_closure_of_subset_compact hK' interior_subset⟩
-#align exists_open_superset_and_is_compact_closure exists_open_superset_and_isCompact_closure
-
-/-- In a weakly locally compact T₂ space,
-every point has an open neighborhood with compact closure. -/
-theorem exists_open_with_compact_closure [WeaklyLocallyCompactSpace X] [T2Space X] (x : X) :
-    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
-  simpa only [singleton_subset_iff]
-    using exists_open_superset_and_isCompact_closure isCompact_singleton
-#align exists_open_with_compact_closure exists_open_with_compact_closure
-
-/-- In a locally compact T₂ space, given a compact set `K` inside an open set `U`, we can find an
-open set `V` between these sets with compact closure: `K ⊆ V` and the closure of `V` is inside `U`.
--/
-theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace X] [T2Space X] {K U : Set X}
-    (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
-    ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (closure V) := by
-  rcases exists_compact_between hK hU hKU with ⟨V, hV, hKV, hVU⟩
-  exact
-    ⟨interior V, isOpen_interior, hKV, (closure_minimal interior_subset hV.isClosed).trans hVU,
-      isCompact_closure_of_subset_compact hV interior_subset⟩
-#align exists_open_between_and_is_compact_closure exists_open_between_and_isCompact_closure
-
 theorem isPreirreducible_iff_subsingleton [T2Space X] {S : Set X} :
     IsPreirreducible S ↔ S.Subsingleton := by
   refine' ⟨fun h x hx y hy => _, Set.Subsingleton.isPreirreducible⟩
@@ -1625,7 +1527,7 @@ theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
 #align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.ofExistsMemNhdsIsClosedSubset
 
 /-- A locally compact T2 space is regular. -/
-instance [LocallyCompactSpace X] [T2Space X] : RegularSpace X := by
+instance (priority := 100) [LocallyCompactSpace X] [T2Space X] : RegularSpace X := by
   apply RegularSpace.ofExistsMemNhdsIsClosedSubset (fun x s hx ↦ ?_)
   rcases local_compact_nhds hx with ⟨k, kx, ks, hk⟩
   exact ⟨k, kx, hk.isClosed, ks⟩
@@ -1751,29 +1653,50 @@ instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i,
     RegularSpace (∀ i, X i) :=
   regularSpace_iInf fun _ => regularSpace_induced _
 
+/-- In a regular space, if a compact set and a closed set are disjoint, then they have disjoint
+neighborhoods. -/
+lemma separatedNhds_of_isCompact_isClosed [RegularSpace X] {s t : Set X}
+    (hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t) : SeparatedNhds s t := by
+  apply hs.induction_on (p := fun u ↦ SeparatedNhds u t)
+  · simp
+  · intro s u su hu
+    exact hu.mono su Subset.rfl
+  · intro u v hu hv
+    exact hu.union_left hv
+  · intro x hx
+    have : tᶜ ∈ 𝓝 x := ht.isOpen_compl.mem_nhds (disjoint_left.1 hst hx)
+    rcases exists_mem_nhds_isClosed_subset this with ⟨u, u_mem, u_closed, hu⟩
+    refine ⟨interior u, mem_nhdsWithin_of_mem_nhds (interior_mem_nhds.2 u_mem), ?_⟩
+    exact ⟨interior u, uᶜ, isOpen_interior, u_closed.isOpen_compl, Subset.rfl,
+      subset_compl_comm.mp hu, disjoint_compl_right.mono_left (interior_subset)⟩
+
 end RegularSpace
 
-section ClosableCompactSubsetOpenSpace
+section T2OrLocallyCompactRegularSpace
 
-/-- A class of topological spaces in which, given a compact set included inside an open set, then
-the closure of the compact set is also included in the open set.
-Satisfied notably for T2 spaces and regular spaces, and useful when discussing classes of
-regular measures. Equivalent to regularity among locally compact spaces. -/
-class ClosableCompactSubsetOpenSpace (X : Type*) [TopologicalSpace X] : Prop :=
-  closure_subset_of_isOpen : ∀ (K U : Set X), IsCompact K → IsOpen U → K ⊆ U → closure K ⊆ U
+/-- A topological space which is either T2 or locally compact regular. Equivalent to regularity
+among locally compact spaces. These two classes share the properties that are needed to develop a
+lot of measure theory, so it's worth defining a single class to avoid developing things twice. -/
+class T2OrLocallyCompactRegularSpace (X : Type*) [TopologicalSpace X] : Prop :=
+  out : T2Space X ∨ (LocallyCompactSpace X ∧ RegularSpace X)
 
-theorem IsCompact.closure_subset_of_isOpen [ClosableCompactSubsetOpenSpace X]
-    {s : Set X} (hs : IsCompact s) {u : Set X} (hu : IsOpen u) (h : s ⊆ u) :
-    closure s ⊆ u :=
-  ClosableCompactSubsetOpenSpace.closure_subset_of_isOpen s u hs hu h
+instance (priority := 100) [h : T2Space X] : T2OrLocallyCompactRegularSpace X := ⟨Or.inl h⟩
+instance (priority := 100) [h : LocallyCompactSpace X] [h' : RegularSpace X] :
+  T2OrLocallyCompactRegularSpace X := ⟨Or.inr ⟨h, h'⟩⟩
 
-instance (priority := 150) [T2Space X] : ClosableCompactSubsetOpenSpace X :=
-  ⟨fun K _U K_comp _U_open KU ↦ by rwa [K_comp.isClosed.closure_eq]⟩
+/-- A locally compact space which is T2 or locally comapct regular is regular. Not an instance, as
+one should instead assume `LocallyCompactSpace X` and `RegularSpace X`. -/
+theorem RegularSpace.ofT2SpaceOrRegularSpace
+    [LocallyCompactSpace X] [h : T2OrLocallyCompactRegularSpace X] : RegularSpace X := by
+  rcases h.out with h'|⟨-, h'⟩ <;> infer_instance
 
-/-- In a (possibly non-Hausdorff) regular space, if a compact set `s` is contained in an
+/-- In a space which is T2 or locally compact regular, if a compact set `s` is contained in an
 open set `u`, then its closure is also contained in `u`. -/
-instance (priority := 150) [RegularSpace X] : ClosableCompactSubsetOpenSpace X := by
-  refine ⟨fun s u hs hu h ↦ ?_⟩
+theorem IsCompact.closure_subset_of_isOpen [hX : T2OrLocallyCompactRegularSpace X]
+    {s : Set X} (hs : IsCompact s) {u : Set X} (hu : IsOpen u) (h : s ⊆ u) :
+    closure s ⊆ u := by
+  rcases hX.out with h'|⟨-, h'⟩
+  · rwa [hs.isClosed.closure_eq]
   obtain ⟨F, sF, F_closed, Fu⟩ : ∃ F, s ⊆ F ∧ IsClosed F ∧ F ⊆ u := by
     apply hs.induction_on (p := fun t ↦ ∃ F, t ⊆ F ∧ IsClosed F ∧ F ⊆ u)
     · exact ⟨∅, by simp⟩
@@ -1786,11 +1709,32 @@ instance (priority := 150) [RegularSpace X] : ClosableCompactSubsetOpenSpace X :
       exact ⟨F, nhdsWithin_le_nhds F_mem, F, Subset.rfl, F_closed, Fu⟩
   exact (closure_minimal sF F_closed).trans Fu
 
-/-- In a (possibly non-Hausdorff) locally compact space with the `ClosableCompactSubsetOpenSpace`
-  property (for instance regular spaces), for every containment `K ⊆ U` of a compact set `K` in an
-  open set `U`, there is a compact closed neighborhood `L` such that `K ⊆ L ⊆ U`: equivalently,
-  there is a compact closed set `L` such that `K ⊆ interior L` and `L ⊆ U`. -/
-theorem exists_compact_closed_between [LocallyCompactSpace X] [ClosableCompactSubsetOpenSpace X]
+theorem separatedNhds_of_isCompact_isCompact_isClosed [hX : T2OrLocallyCompactRegularSpace X]
+    {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) (h't : IsClosed t)
+    (hst : Disjoint s t) : SeparatedNhds s t := by
+  rcases hX.out with h'|⟨-, h'⟩
+  · exact separatedNhds_of_isCompact_isCompact hs ht hst
+  · exact separatedNhds_of_isCompact_isClosed hs h't hst
+
+protected theorem IsCompact.closure [hX : T2OrLocallyCompactRegularSpace X]
+    {K : Set X} (hK : IsCompact K) : IsCompact (closure K) := by
+  rcases hX.out with h'|⟨h', -⟩
+  · rwa [hK.isClosed.closure_eq]
+  rcases exists_compact_superset hK with ⟨L, L_comp, hL⟩
+  exact L_comp.of_isClosed_subset isClosed_closure
+    ((hK.closure_subset_of_isOpen isOpen_interior hL).trans interior_subset)
+
+theorem isCompact_closure_of_subset_compact [T2OrLocallyCompactRegularSpace X]
+    {s t : Set X} (ht : IsCompact t)
+    (h : s ⊆ t) : IsCompact (closure s) :=
+  ht.closure.of_isClosed_subset isClosed_closure (closure_mono h)
+#align is_compact_closure_of_subset_compact isCompact_closure_of_subset_compact
+
+/-- In a (possibly non-Hausdorff) locally compact regular space, for every containment `K ⊆ U` of
+  a compact set `K` in an open set `U`, there is a compact closed neighborhood `L`
+  such that `K ⊆ L ⊆ U`: equivalently, there is a compact closed set `L` such
+  that `K ⊆ interior L` and `L ⊆ U`. -/
+theorem exists_compact_closed_between [LocallyCompactSpace X] [RegularSpace X]
     {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (h_KU : K ⊆ U) :
     ∃ L, IsCompact L ∧ IsClosed L ∧ K ⊆ interior L ∧ L ⊆ U := by
   rcases exists_compact_between hK hU h_KU with ⟨L, L_comp, KL, LU⟩
@@ -1805,23 +1749,109 @@ theorem exists_compact_closed_between [LocallyCompactSpace X] [ClosableCompactSu
   · apply M_comp.closure_subset_of_isOpen hU
     exact ML.trans (interior_subset.trans LU)
 
-/-- A locally compact space with the `ClosableCompactSubsetOpenSpace` is `Regular`. -/
-instance (priority := 80) [LocallyCompactSpace X] [ClosableCompactSubsetOpenSpace X] :
-    RegularSpace X := by
-  apply RegularSpace.ofExistsMemNhdsIsClosedSubset (fun x s hx ↦ ?_)
-  rcases _root_.mem_nhds_iff.1 hx with ⟨u, us, u_open, xu⟩
-  rcases exists_compact_closed_between (isCompact_singleton (x := x)) u_open (by simpa using xu)
-    with ⟨t, -, t_closed, xt, tu⟩
-  have : interior t ∈ 𝓝 x := isOpen_interior.mem_nhds (by simpa using xt)
-  exact ⟨t, interior_mem_nhds.mp this, t_closed, tu.trans us⟩
+/-- In a locally compact regular space, given a compact set `K` inside an open set `U`, we can find
+an open set `V` between these sets with compact closure: `K ⊆ V` and the closure of `V` is
+inside `U`. -/
+theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace X] [RegularSpace X]
+    {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
+    ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (closure V) := by
+  rcases exists_compact_closed_between hK hU hKU with ⟨L, L_compact, L_closed, KL, LU⟩
+  have A : closure (interior L) ⊆ L := by
+    apply (closure_mono interior_subset).trans (le_of_eq L_closed.closure_eq)
+  refine ⟨interior L, isOpen_interior, KL, A.trans LU, ?_⟩
+  exact isCompact_closure_of_subset_compact L_compact interior_subset
+#align exists_open_between_and_is_compact_closure exists_open_between_and_isCompact_closure
 
-protected theorem IsCompact.closure [WeaklyLocallyCompactSpace X] [ClosableCompactSubsetOpenSpace X]
-    {K : Set X} (hK : IsCompact K) : IsCompact (closure K) := by
-  rcases exists_compact_superset hK with ⟨L, L_comp, hL⟩
-  exact L_comp.of_isClosed_subset isClosed_closure
-    ((hK.closure_subset_of_isOpen isOpen_interior hL).trans interior_subset)
+@[simp]
+theorem exists_compact_superset_iff [T2OrLocallyCompactRegularSpace X] {s : Set X} :
+    (∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) :=
+  ⟨fun ⟨_K, hK, hsK⟩ => isCompact_closure_of_subset_compact hK hsK, fun h =>
+    ⟨closure s, h, subset_closure⟩⟩
+#align exists_compact_superset_iff exists_compact_superset_iff
+
+/-- In a weakly locally compact space which is either T₂ or locally compact regular,
+every compact set has an open neighborhood with compact closure. -/
+theorem exists_open_superset_and_isCompact_closure
+    [WeaklyLocallyCompactSpace X] [T2OrLocallyCompactRegularSpace X]
+    {K : Set X} (hK : IsCompact K) : ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
+  rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
+  exact ⟨interior K', isOpen_interior, hKK',
+    isCompact_closure_of_subset_compact hK' interior_subset⟩
+#align exists_open_superset_and_is_compact_closure exists_open_superset_and_isCompact_closure
+
+/-- In a weakly locally compact which is either T₂ or locally compact regular,
+every point has an open neighborhood with compact closure. -/
+theorem exists_open_with_compact_closure
+    [WeaklyLocallyCompactSpace X] [T2OrLocallyCompactRegularSpace X] (x : X) :
+    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
+  simpa only [singleton_subset_iff]
+    using exists_open_superset_and_isCompact_closure isCompact_singleton
+#align exists_open_with_compact_closure exists_open_with_compact_closure
+
+-- see Note [lower instance priority]
+/-- A weakly locally compact Hausdorff space is locally compact. -/
+instance (priority := 80) WeaklyLocallyCompactSpace.locallyCompactSpace
+    [WeaklyLocallyCompactSpace X] [hX : T2OrLocallyCompactRegularSpace X] :
+    LocallyCompactSpace X := by
+  rcases hX.out with h'X|⟨h'X, -⟩
+  · exact ⟨fun _ _ h =>
+      let ⟨K, hKx, hKc, hKs⟩ := exists_mem_nhds_isCompact_mapsTo continuous_id h
+      ⟨K, hKx, hKs, hKc⟩⟩
+  · exact h'X
+#align locally_compact_of_compact_nhds WeaklyLocallyCompactSpace.locallyCompactSpace
+
+@[deprecated WeaklyLocallyCompactSpace.locallyCompactSpace]
+theorem locally_compact_of_compact [T2Space X] [CompactSpace X] :
+    LocallyCompactSpace X :=
+  inferInstance
+#align locally_compact_of_compact locally_compact_of_compact
+
+/-- If a compact set is covered by two open sets, then we can cover it by two compact subsets. -/
+theorem IsCompact.binary_compact_cover [T2OrLocallyCompactRegularSpace X] {K U V : Set X}
+    (hK : IsCompact K) (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
+    ∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ := by
+  have hK' : IsCompact (closure K) := hK.closure
+  have : SeparatedNhds (closure K \ U) (closure K \ V) := by
+    apply separatedNhds_of_isCompact_isCompact_isClosed (hK'.diff hU) (hK'.diff hV)
+      (isClosed_closure.sdiff hV)
+    rw [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty]
+    exact hK.closure_subset_of_isOpen (hU.union hV) h2K
+  have : SeparatedNhds (K \ U) (K \ V) :=
+    this.mono (diff_subset_diff_left (subset_closure)) (diff_subset_diff_left (subset_closure))
+  rcases this with ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩
+  exact ⟨K \ O₁, K \ O₂, hK.diff h1O₁, hK.diff h1O₂, diff_subset_comm.mp h2O₁,
+    diff_subset_comm.mp h2O₂, by rw [← diff_inter, hO.inter_eq, diff_empty]⟩
+#align is_compact.binary_compact_cover IsCompact.binary_compact_cover
+
+open Finset Function
+
+/-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
+theorem IsCompact.finite_compact_cover [T2OrLocallyCompactRegularSpace X]
+    {s : Set X} (hs : IsCompact s) {ι}
+    (t : Finset ι) (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
+    ∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
+  induction' t using Finset.induction with x t hx ih generalizing U s
+  · refine' ⟨fun _ => ∅, fun _ => isCompact_empty, fun i => empty_subset _, _⟩
+    simpa only [subset_empty_iff, Finset.not_mem_empty, iUnion_false, iUnion_empty] using hsC
+  simp only [Finset.set_biUnion_insert] at hsC
+  simp only [Finset.forall_mem_insert] at hU
+  have hU' : ∀ i ∈ t, IsOpen (U i) := fun i hi => hU.2 i hi
+  rcases hs.binary_compact_cover hU.1 (isOpen_biUnion hU') hsC with
+    ⟨K₁, K₂, h1K₁, h1K₂, h2K₁, h2K₂, hK⟩
+  rcases ih h1K₂ U hU' h2K₂ with ⟨K, h1K, h2K, h3K⟩
+  refine' ⟨update K x K₁, _, _, _⟩
+  · intro i
+    rcases eq_or_ne i x with rfl | hi
+    · simp only [update_same, h1K₁]
+    · simp only [update_noteq hi, h1K]
+  · intro i
+    rcases eq_or_ne i x with rfl | hi
+    · simp only [update_same, h2K₁]
+    · simp only [update_noteq hi, h2K]
+  · simp only [set_biUnion_insert_update _ hx, hK, h3K]
+#align is_compact.finite_compact_cover IsCompact.finite_compact_cover
 
-end ClosableCompactSubsetOpenSpace
+end T2OrLocallyCompactRegularSpace
 
 section T3
 
@@ -1982,7 +2012,7 @@ instance (priority := 100) T4Space.t3Space [T4Space X] : T3Space X where
 
 instance (priority := 100) T4Space.of_compactSpace_t2Space [CompactSpace X] [T2Space X] :
     T4Space X where
-  normal _s _t hs ht := isCompact_isCompact_separated hs.isCompact ht.isCompact
+  normal _s _t hs ht := separatedNhds_of_isCompact_isCompact hs.isCompact ht.isCompact
 #align normal_of_compact_t2 T4Space.of_compactSpace_t2Space
 
 /-- If the codomain of a closed embedding is a T₄ space, then so is the domain. -/
@@ -2156,7 +2186,8 @@ theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace X ↔ Totally
 variable [TotallyDisconnectedSpace X]
 
 /-- A totally disconnected compact Hausdorff space is totally separated. -/
-instance : TotallySeparatedSpace X := compact_t2_tot_disc_iff_tot_sep.mp inferInstance
+instance (priority := 100) : TotallySeparatedSpace X :=
+  compact_t2_tot_disc_iff_tot_sep.mp inferInstance
 
 theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsClopen s) id :=
   ⟨fun U => by

See Zulip thread for the discussion.

@@ -237,7 +237,7 @@ theorem inseparable_eq_eq [T0Space X] : Inseparable = @Eq X :=
 theorem TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set X)}
     (hb : IsTopologicalBasis b) {x y : X} : Inseparable x y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
   ⟨fun h s hs ↦ inseparable_iff_forall_open.1 h _ (hb.isOpen hs),
-    fun h ↦ hb.nhds_hasBasis.eq_of_same_basis $ by
+    fun h ↦ hb.nhds_hasBasis.eq_of_same_basis <| by
       convert hb.nhds_hasBasis using 2
       exact and_congr_right (h _)⟩
 

• @[mk_iff] class MyPred now generates myPred_iff, not MyPred_iff
• add Lean.Name.decapitalize
• fix indentation and a few typos in the docs/comments.

Partially addresses issue #9129

@@ -916,7 +916,7 @@ theorem TopologicalSpace.subset_trans {s t : Set X} (ts : t ⊆ s) :
 /-- A T₂ space, also known as a Hausdorff space, is one in which for every
   `x ≠ y` there exists disjoint open sets around `x` and `y`. This is
   the most widely used of the separation axioms. -/
-@[mk_iff t2Space_iff]
+@[mk_iff]
 class T2Space (X : Type u) [TopologicalSpace X] : Prop where
   /-- Every two points in a Hausdorff space admit disjoint open neighbourhoods. -/
   t2 : Pairwise fun x y => ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
@@ -1568,7 +1568,7 @@ section RegularSpace
 /-- A topological space is called a *regular space* if for any closed set `s` and `a ∉ s`, there
 exist disjoint open sets `U ⊇ s` and `V ∋ a`. We formulate this condition in terms of `Disjoint`ness
 of filters `𝓝ˢ s` and `𝓝 a`. -/
-@[mk_iff regularSpace_iff]
+@[mk_iff]
 class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
   /-- If `a` is a point that does not belong to a closed set `s`, then `a` and `s` admit disjoint
   neighborhoods.  -/

Introduce a typeclass LocallyCompactPair that allows us to unify different versions of ContinuousMap.continuous_eval and similar lemmas.

@@ -1477,23 +1477,28 @@ theorem IsCompact.finite_compact_cover [T2Space X] {s : Set X} (hs : IsCompact s
 
 end
 
+instance (priority := 900) [WeaklyLocallyCompactSpace X] [T2Space Y] : LocallyCompactPair X Y where
+  exists_mem_nhds_isCompact_mapsTo := by
+    intro f x s hf hs
+    rcases exists_compact_mem_nhds x with ⟨K, hKc, hKx⟩
+    have hc : IsCompact (f '' K \ interior s) := (hKc.image hf).diff isOpen_interior
+    have hd : Disjoint {f x} (f '' K \ interior s) := disjoint_singleton_left.2 fun h ↦
+      h.2 <| mem_interior_iff_mem_nhds.2 hs
+    rcases isCompact_isCompact_separated isCompact_singleton hc hd
+      with ⟨U, V, Uo, Vo, hxU, hV, hd⟩
+    refine ⟨K \ f ⁻¹' V, diff_mem hKx ?_, hKc.diff <| Vo.preimage hf, fun y hy ↦ ?_⟩
+    · filter_upwards [hf.continuousAt <| Uo.mem_nhds (hxU rfl)] with x hx
+        using Set.disjoint_left.1 hd hx
+    · by_contra hys
+      exact hy.2 (hV ⟨mem_image_of_mem _ hy.1, not_mem_subset interior_subset hys⟩)
+
 -- see Note [lower instance priority]
 /-- A weakly locally compact Hausdorff space is locally compact. -/
 instance WeaklyLocallyCompactSpace.locallyCompactSpace [WeaklyLocallyCompactSpace X] [T2Space X] :
     LocallyCompactSpace X :=
-  ⟨fun x _n hn =>
-    let ⟨_u, un, uo, xu⟩ := mem_nhds_iff.mp hn
-    let ⟨k, kc, kx⟩ := exists_compact_mem_nhds x
-    -- K is compact but not necessarily contained in N.
-    -- K \ U is again compact and doesn't contain x, so
-    -- we may find open sets V, W separating x from K \ U.
-    -- Then K \ W is a compact neighborhood of x contained in U.
-    let ⟨v, w, vo, wo, xv, kuw, vw⟩ :=
-      isCompact_isCompact_separated isCompact_singleton (kc.diff uo)
-        (disjoint_singleton_left.2 fun h => h.2 xu)
-    have wn : wᶜ ∈ 𝓝 x :=
-      mem_nhds_iff.mpr ⟨v, vw.subset_compl_right, vo, singleton_subset_iff.mp xv⟩
-    ⟨k \ w, Filter.inter_mem kx wn, Subset.trans (diff_subset_comm.mp kuw) un, kc.diff wo⟩⟩
+  ⟨fun _ _ h =>
+    let ⟨K, hKx, hKc, hKs⟩ := exists_mem_nhds_isCompact_mapsTo continuous_id h
+    ⟨K, hKx, hKs, hKc⟩⟩
 #align locally_compact_of_compact_nhds WeaklyLocallyCompactSpace.locallyCompactSpace
 
 @[deprecated WeaklyLocallyCompactSpace.locallyCompactSpace]

Performed with a regex search for ∀ (.) (.), \1 ≠ \2 →, and a few variants to catch implicit binders and explicit types.

I have deliberately avoided trying to make the analogous Set.Pairwise transformation (or any Pairwise (foo on bar) transformations) in this PR, to keep the diff small.

Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>

@@ -186,8 +186,8 @@ theorem t0Space_iff_inseparable (X : Type u) [TopologicalSpace X] :
 #align t0_space_iff_inseparable t0Space_iff_inseparable
 
 theorem t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] :
-    T0Space X ↔ ∀ x y : X, x ≠ y → ¬Inseparable x y := by
-  simp only [t0Space_iff_inseparable, Ne.def, not_imp_not]
+    T0Space X ↔ Pairwise fun x y : X => ¬Inseparable x y := by
+  simp only [t0Space_iff_inseparable, Ne.def, not_imp_not, Pairwise]
 #align t0_space_iff_not_inseparable t0Space_iff_not_inseparable
 
 theorem Inseparable.eq [T0Space X] {x y : X} (h : Inseparable x y) : x = y :=
@@ -246,14 +246,14 @@ theorem TopologicalSpace.IsTopologicalBasis.eq_iff [T0Space X] {b : Set (Set X)}
   inseparable_iff_eq.symm.trans hb.inseparable_iff
 
 theorem t0Space_iff_exists_isOpen_xor'_mem (X : Type u) [TopologicalSpace X] :
-    T0Space X ↔ ∀ x y, x ≠ y → ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by
+    T0Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by
   simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,
-    inseparable_iff_forall_open]
+    inseparable_iff_forall_open, Pairwise]
 #align t0_space_iff_exists_is_open_xor_mem t0Space_iff_exists_isOpen_xor'_mem
 
 theorem exists_isOpen_xor'_mem [T0Space X] {x y : X} (h : x ≠ y) :
     ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) :=
-  (t0Space_iff_exists_isOpen_xor'_mem X).1 ‹_› x y h
+  (t0Space_iff_exists_isOpen_xor'_mem X).1 ‹_› h
 #align exists_is_open_xor_mem exists_isOpen_xor'_mem
 
 /-- Specialization forms a partial order on a t0 topological space. -/
@@ -349,7 +349,7 @@ instance Subtype.t0Space [T0Space X] {p : X → Prop} : T0Space (Subtype p) :=
 #align subtype.t0_space Subtype.t0Space
 
 theorem t0Space_iff_or_not_mem_closure (X : Type u) [TopologicalSpace X] :
-    T0Space X ↔ ∀ a b : X, a ≠ b → a ∉ closure ({b} : Set X) ∨ b ∉ closure ({a} : Set X) := by
+    T0Space X ↔ Pairwise fun a b : X => a ∉ closure ({b} : Set X) ∨ b ∉ closure ({a} : Set X) := by
   simp only [t0Space_iff_not_inseparable, inseparable_iff_mem_closure, not_and_or]
 #align t0_space_iff_or_not_mem_closure t0Space_iff_or_not_mem_closure
 
@@ -525,7 +525,7 @@ theorem CofiniteTopology.continuous_of [T1Space X] : Continuous (@CofiniteTopolo
 #align cofinite_topology.continuous_of CofiniteTopology.continuous_of
 
 theorem t1Space_iff_exists_open :
-    T1Space X ↔ ∀ x y, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U :=
+    T1Space X ↔ Pairwise fun x y => ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U :=
   (t1Space_TFAE X).out 0 6
 #align t1_space_iff_exists_open t1Space_iff_exists_open
 
@@ -919,17 +919,17 @@ theorem TopologicalSpace.subset_trans {s t : Set X} (ts : t ⊆ s) :
 @[mk_iff t2Space_iff]
 class T2Space (X : Type u) [TopologicalSpace X] : Prop where
   /-- Every two points in a Hausdorff space admit disjoint open neighbourhoods. -/
-  t2 : ∀ x y, x ≠ y → ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
+  t2 : Pairwise fun x y => ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
 #align t2_space T2Space
 
 /-- Two different points can be separated by open sets. -/
 theorem t2_separation [T2Space X] {x y : X} (h : x ≠ y) :
     ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
-  T2Space.t2 x y h
+  T2Space.t2 h
 #align t2_separation t2_separation
 
 -- todo: use this as a definition?
-theorem t2Space_iff_disjoint_nhds : T2Space X ↔ ∀ x y : X, x ≠ y → Disjoint (𝓝 x) (𝓝 y) := by
+theorem t2Space_iff_disjoint_nhds : T2Space X ↔ Pairwise fun x y : X => Disjoint (𝓝 x) (𝓝 y) := by
   refine (t2Space_iff X).trans (forall₃_congr fun x y _ => ?_)
   simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y), exists_prop, ← exists_and_left,
     and_assoc, and_comm, and_left_comm]
@@ -937,7 +937,7 @@ theorem t2Space_iff_disjoint_nhds : T2Space X ↔ ∀ x y : X, x ≠ y → Disjo
 
 @[simp]
 theorem disjoint_nhds_nhds [T2Space X] {x y : X} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y :=
-  ⟨fun hd he => by simp [he, nhds_neBot.ne] at hd, t2Space_iff_disjoint_nhds.mp ‹_› x y⟩
+  ⟨fun hd he => by simp [he, nhds_neBot.ne] at hd, (t2Space_iff_disjoint_nhds.mp ‹_› ·)⟩
 #align disjoint_nhds_nhds disjoint_nhds_nhds
 
 theorem pairwise_disjoint_nhds [T2Space X] : Pairwise (Disjoint on (𝓝 : X → Filter X)) := fun _ _ =>
@@ -972,15 +972,16 @@ instance (priority := 100) T2Space.t1Space [T2Space X] : T1Space X :=
 
 /-- A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. -/
 theorem t2_iff_nhds : T2Space X ↔ ∀ {x y : X}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
-  simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne.def, not_imp_comm]
+  simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne.def, not_imp_comm, Pairwise]
 #align t2_iff_nhds t2_iff_nhds
 
 theorem eq_of_nhds_neBot [T2Space X] {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y :=
   t2_iff_nhds.mp ‹_› h
 #align eq_of_nhds_ne_bot eq_of_nhds_neBot
 
-theorem t2Space_iff_nhds : T2Space X ↔ ∀ {x y : X}, x ≠ y → ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V := by
-  simp only [t2Space_iff_disjoint_nhds, Filter.disjoint_iff]
+theorem t2Space_iff_nhds :
+    T2Space X ↔ Pairwise fun x y : X => ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V := by
+  simp only [t2Space_iff_disjoint_nhds, Filter.disjoint_iff, Pairwise]
 #align t2_space_iff_nhds t2Space_iff_nhds
 
 theorem t2_separation_nhds [T2Space X] {x y : X} (h : x ≠ y) :
@@ -1002,7 +1003,7 @@ theorem t2_iff_ultrafilter :
 
 theorem t2_iff_isClosed_diagonal : T2Space X ↔ IsClosed (diagonal X) := by
   simp only [t2Space_iff_disjoint_nhds, ← isOpen_compl_iff, isOpen_iff_mem_nhds, Prod.forall,
-    nhds_prod_eq, compl_diagonal_mem_prod, mem_compl_iff, mem_diagonal_iff]
+    nhds_prod_eq, compl_diagonal_mem_prod, mem_compl_iff, mem_diagonal_iff, Pairwise]
 #align t2_iff_is_closed_diagonal t2_iff_isClosed_diagonal
 
 theorem isClosed_diagonal [T2Space X] : IsClosed (diagonal X) :=
@@ -1831,7 +1832,7 @@ instance (priority := 100) T3Space.t25Space [T3Space X] : T25Space X := by
   refine' ⟨fun x y hne => _⟩
   rw [lift'_nhds_closure, lift'_nhds_closure]
   have : x ∉ closure {y} ∨ y ∉ closure {x} :=
-    (t0Space_iff_or_not_mem_closure X).mp inferInstance x y hne
+    (t0Space_iff_or_not_mem_closure X).mp inferInstance hne
   simp only [← disjoint_nhds_nhdsSet, nhdsSet_singleton] at this
   exact this.elim id fun h => h.symm
 #align t3_space.t2_5_space T3Space.t25Space

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -1954,7 +1954,7 @@ instance (priority := 100) NormalSpace.of_regularSpace_secondCountableTopology
     exact (hUd u huU).le_bot ⟨hxu, hx⟩
   · simp only [disjoint_left, mem_iUnion, mem_diff, not_exists, not_and, not_forall, not_not]
     rintro a ⟨u, huU, hau, haV⟩ v hvV hav
-    cases' le_total (Encodable.encode u) (Encodable.encode v) with hle hle
+    rcases le_total (Encodable.encode u) (Encodable.encode v) with hle | hle
     exacts [⟨u, huU, hle, subset_closure hau⟩, (haV _ hvV hle <| subset_closure hav).elim]
 #align normal_space_of_t3_second_countable NormalSpace.of_regularSpace_secondCountableTopology
 

This also adds a handful of easy instances

Arguably the topological space instance should be defined via coinduced to make these true by Iff.rfl, but that creates a headache with constructors for normed spaces, so is not in this PR.

@@ -362,6 +362,9 @@ instance Pi.instT0Space {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace
   ⟨fun _ _ h => funext fun i => (h.map (continuous_apply i)).eq⟩
 #align pi.t0_space Pi.instT0Space
 
+instance ULift.instT0Space [T0Space X] : T0Space (ULift X) :=
+  embedding_uLift_down.t0Space
+
 theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set X, x ∈ s ∧ y ∈ s ∧ T0Space s) :
     T0Space X := by
   refine' ⟨fun x y hxy => _⟩
@@ -627,6 +630,9 @@ instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i,
     T1Space (∀ i, X i) :=
   ⟨fun f => univ_pi_singleton f ▸ isClosed_set_pi fun _ _ => isClosed_singleton⟩
 
+instance ULift.instT1Space [T1Space X] : T1Space (ULift X) :=
+  embedding_uLift_down.t1Space
+
 -- see Note [lower instance priority]
 instance (priority := 100) T1Space.t0Space [T1Space X] : T0Space X :=
   ⟨fun _ _ h => h.specializes.eq⟩
@@ -1234,6 +1240,9 @@ theorem Embedding.t2Space [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : E
   .of_injective_continuous hf.inj hf.continuous
 #align embedding.t2_space Embedding.t2Space
 
+instance ULift.instT2Space [T2Space X] : T2Space (ULift X) :=
+  embedding_uLift_down.t2Space
+
 instance [T2Space X] [TopologicalSpace Y] [T2Space Y] :
     T2Space (X ⊕ Y) := by
   constructor
@@ -1837,6 +1846,9 @@ instance Subtype.t3Space [T3Space X] {p : X → Prop} : T3Space (Subtype p) :=
   embedding_subtype_val.t3Space
 #align subtype.t3_space Subtype.t3Space
 
+instance ULift.instT3Space [T3Space X] : T3Space (ULift X) :=
+  embedding_uLift_down.t3Space
+
 instance [TopologicalSpace Y] [T3Space X] [T3Space Y] : T3Space (X × Y) := ⟨⟩
 
 instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, T3Space (X i)] :
@@ -1974,6 +1986,9 @@ protected theorem ClosedEmbedding.t4Space [TopologicalSpace Y] [T4Space Y] {f :
   toNormalSpace := hf.normalSpace
 #align closed_embedding.normal_space ClosedEmbedding.t4Space
 
+instance ULift.instT4Space [T4Space X] : T4Space (ULift X) :=
+  ULift.closedEmbedding_down.t4Space
+
 namespace SeparationQuotient
 
 /-- The `SeparationQuotient` of a normal space is a normal space. -/
@@ -2020,6 +2035,9 @@ theorem Embedding.t5Space [TopologicalSpace Y] [T5Space Y] {e : X → Y} (he : E
 instance [T5Space X] {p : X → Prop} : T5Space { x // p x } :=
   embedding_subtype_val.t5Space
 
+instance ULift.instT5Space [T5Space X] : T5Space (ULift X) :=
+  embedding_uLift_down.t5Space
+
 -- see Note [lower instance priority]
 /-- A `T₅` space is a `T₄` space. -/
 instance (priority := 100) T5Space.toT4Space [T5Space X] : T4Space X where

Mostly, this means replacing "of_open" by "of_isOpen". A few lemmas names were misleading and are corrected differently. Zulip discussion.

@@ -44,9 +44,9 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
 
 ### T₀ spaces
 
-* `IsClosed.exists_closed_singleton` Given a closed set `S` in a compact T₀ space,
+* `IsClosed.exists_closed_singleton`: Given a closed set `S` in a compact T₀ space,
   there is some `x ∈ S` such that `{x}` is closed.
-* `exists_open_singleton_of_open_finset` Given an open `Finset` `S` in a T₀ space,
+* `exists_open_singleton_of_open_finite`: Given an open finite set `S` in a T₀ space,
   there is some `x ∈ S` such that `{x}` is open.
 
 ### T₁ spaces
@@ -310,7 +310,7 @@ theorem minimal_nonempty_open_eq_singleton [T0Space X] {s : Set X} (hs : IsOpen
 #align minimal_nonempty_open_eq_singleton minimal_nonempty_open_eq_singleton
 
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
-theorem exists_open_singleton_of_open_finite [T0Space X] {s : Set X} (hfin : s.Finite)
+theorem exists_isOpen_singleton_of_isOpen_finite [T0Space X] {s : Set X} (hfin : s.Finite)
     (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set X) := by
   lift s to Finset X using hfin
   induction' s using Finset.strongInductionOn with s ihs
@@ -325,11 +325,12 @@ theorem exists_open_singleton_of_open_finite [T0Space X] {s : Set X} (hfin : s.F
     refine' fun t hts htne hto => of_not_not fun hts' => ht _
     lift t to Finset X using s.finite_toSet.subset hts
     exact ⟨t, ssubset_iff_subset_ne.2 ⟨hts, mt Finset.coe_inj.2 hts'⟩, htne, hto⟩
-#align exists_open_singleton_of_open_finite exists_open_singleton_of_open_finite
+#align exists_open_singleton_of_open_finite exists_isOpen_singleton_of_isOpen_finite
 
 theorem exists_open_singleton_of_finite [T0Space X] [Finite X] [Nonempty X] :
     ∃ x : X, IsOpen ({x} : Set X) :=
-  let ⟨x, _, h⟩ := exists_open_singleton_of_open_finite (Set.toFinite _) univ_nonempty isOpen_univ
+  let ⟨x, _, h⟩ := exists_isOpen_singleton_of_isOpen_finite (Set.toFinite _)
+    univ_nonempty isOpen_univ
   ⟨x, h⟩
 #align exists_open_singleton_of_fintype exists_open_singleton_of_finite
 
@@ -727,7 +728,7 @@ theorem nhds_le_nhdsSet_iff [T1Space X] {s : Set X} {x : X} : 𝓝 x ≤ 𝓝ˢ
 /-- Removing a non-isolated point from a dense set, one still obtains a dense set. -/
 theorem Dense.diff_singleton [T1Space X] {s : Set X} (hs : Dense s) (x : X) [NeBot (𝓝[≠] x)] :
     Dense (s \ {x}) :=
-  hs.inter_of_open_right (dense_compl_singleton x) isOpen_compl_singleton
+  hs.inter_of_isOpen_right (dense_compl_singleton x) isOpen_compl_singleton
 #align dense.diff_singleton Dense.diff_singleton
 
 /-- Removing a finset from a dense set in a space without isolated points, one still
@@ -2157,21 +2158,21 @@ theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s
       exact ⟨V, hUV, V_op, hxV⟩⟩
 #align nhds_basis_clopen nhds_basis_clopen
 
-theorem isTopologicalBasis_clopen : IsTopologicalBasis { s : Set X | IsClopen s } := by
-  apply isTopologicalBasis_of_open_of_nhds fun U (hU : IsClopen U) => hU.1
+theorem isTopologicalBasis_isClopen : IsTopologicalBasis { s : Set X | IsClopen s } := by
+  apply isTopologicalBasis_of_isOpen_of_nhds fun U (hU : IsClopen U) => hU.1
   intro x U hxU U_op
   have : U ∈ 𝓝 x := IsOpen.mem_nhds U_op hxU
   rcases (nhds_basis_clopen x).mem_iff.mp this with ⟨V, ⟨hxV, hV⟩, hVU : V ⊆ U⟩
   use V
   tauto
-#align is_topological_basis_clopen isTopologicalBasis_clopen
+#align is_topological_basis_clopen isTopologicalBasis_isClopen
 
 /-- Every member of an open set in a compact Hausdorff totally disconnected space
   is contained in a clopen set contained in the open set.  -/
-theorem compact_exists_clopen_in_open {x : X} {U : Set X} (is_open : IsOpen U) (memU : x ∈ U) :
+theorem compact_exists_isClopen_in_isOpen {x : X} {U : Set X} (is_open : IsOpen U) (memU : x ∈ U) :
     ∃ V : Set X, IsClopen V ∧ x ∈ V ∧ V ⊆ U :=
-  isTopologicalBasis_clopen.mem_nhds_iff.1 (is_open.mem_nhds memU)
-#align compact_exists_clopen_in_open compact_exists_clopen_in_open
+  isTopologicalBasis_isClopen.mem_nhds_iff.1 (is_open.mem_nhds memU)
+#align compact_exists_clopen_in_open compact_exists_isClopen_in_isOpen
 
 end Profinite
 
@@ -2182,15 +2183,15 @@ variable {H : Type*} [TopologicalSpace H] [LocallyCompactSpace H] [T2Space H]
 /-- A locally compact Hausdorff totally disconnected space has a basis with clopen elements. -/
 theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
     IsTopologicalBasis { s : Set H | IsClopen s } := by
-  refine isTopologicalBasis_of_open_of_nhds (fun u hu => hu.1) fun x U memU hU => ?_
+  refine isTopologicalBasis_of_isOpen_of_nhds (fun u hu => hu.1) fun x U memU hU => ?_
   obtain ⟨s, comp, xs, sU⟩ := exists_compact_subset hU memU
   let u : Set s := ((↑) : s → H) ⁻¹' interior s
   have u_open_in_s : IsOpen u := isOpen_interior.preimage continuous_subtype_val
   lift x to s using interior_subset xs
   haveI : CompactSpace s := isCompact_iff_compactSpace.1 comp
-  obtain ⟨V : Set s, clopen_in_s, Vx, V_sub⟩ := compact_exists_clopen_in_open u_open_in_s xs
-  have V_clopen : IsClopen (((↑) : s → H) '' V) := by
-    refine' ⟨_, comp.isClosed.closedEmbedding_subtype_val.closed_iff_image_closed.1 clopen_in_s.2⟩
+  obtain ⟨V : Set s, VisClopen, Vx, V_sub⟩ := compact_exists_isClopen_in_isOpen u_open_in_s xs
+  have VisClopen' : IsClopen (((↑) : s → H) '' V) := by
+    refine' ⟨_, comp.isClosed.closedEmbedding_subtype_val.closed_iff_image_closed.1 VisClopen.2⟩
     let v : Set u := ((↑) : u → s) ⁻¹' V
     have : ((↑) : u → H) = ((↑) : s → H) ∘ ((↑) : u → s) := rfl
     have f0 : Embedding ((↑) : u → H) := embedding_subtype_val.comp embedding_subtype_val
@@ -2201,12 +2202,12 @@ theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :
           apply Set.inter_eq_self_of_subset_left interior_subset
         rw [this]
         apply isOpen_interior
-    have f2 : IsOpen v := clopen_in_s.1.preimage continuous_subtype_val
+    have f2 : IsOpen v := VisClopen.1.preimage continuous_subtype_val
     have f3 : ((↑) : s → H) '' V = ((↑) : u → H) '' v := by
       rw [this, image_comp, Subtype.image_preimage_coe, inter_eq_self_of_subset_left V_sub]
     rw [f3]
     apply f1.isOpenMap v f2
-  refine' ⟨(↑) '' V, V_clopen, by simp [Vx], Subset.trans _ sU⟩
+  refine' ⟨(↑) '' V, VisClopen', by simp [Vx], Subset.trans _ sU⟩
   simp
 set_option linter.uppercaseLean3 false in
 #align loc_compact_Haus_tot_disc_of_zero_dim loc_compact_Haus_tot_disc_of_zero_dim

X, Y, Z are standard mathematical names for topological spaces. As discussed on zulip, let us rename them.

As a drive-by commit, re-use the declared variables $\iota$, $s$ and $t$ (more) when stating theorems.

Co-authored-by: grunweg <grunweg@posteo.de>

@@ -1542,7 +1542,7 @@ theorem isIrreducible_iff_singleton [T2Space X] {S : Set X} : IsIrreducible S 
 /-- There does not exist a nontrivial preirreducible T₂ space. -/
 theorem not_preirreducible_nontrivial_t2 (X) [TopologicalSpace X] [PreirreducibleSpace X]
     [Nontrivial X] [T2Space X] : False :=
-  (PreirreducibleSpace.isPreirreducible_univ (α := X)).subsingleton.not_nontrivial nontrivial_univ
+  (PreirreducibleSpace.isPreirreducible_univ (X := X)).subsingleton.not_nontrivial nontrivial_univ
 #align not_preirreducible_nontrivial_t2 not_preirreducible_nontrivial_t2
 
 end Separation
@@ -1794,7 +1794,7 @@ instance (priority := 80) [LocallyCompactSpace X] [ClosableCompactSubsetOpenSpac
     RegularSpace X := by
   apply RegularSpace.ofExistsMemNhdsIsClosedSubset (fun x s hx ↦ ?_)
   rcases _root_.mem_nhds_iff.1 hx with ⟨u, us, u_open, xu⟩
-  rcases exists_compact_closed_between (isCompact_singleton (a := x)) u_open (by simpa using xu)
+  rcases exists_compact_closed_between (isCompact_singleton (x := x)) u_open (by simpa using xu)
     with ⟨t, -, t_closed, xt, tu⟩
   have : interior t ∈ 𝓝 x := isOpen_interior.mem_nhds (by simpa using xt)
   exact ⟨t, interior_mem_nhds.mp this, t_closed, tu.trans us⟩

This PR renames the field Clopens.clopen' -> Clopens.isClopen', and the lemmas

• preimage_closed_of_closed -> ContinuousOn.preimage_isClosed_of_isClosed

as well as: ClopenUpperSet.clopen -> ClopenUpperSet.isClopen connectedComponent_eq_iInter_clopen -> connectedComponent_eq_iInter_isClopen connectedComponent_subset_iInter_clopen -> connectedComponent_subset_iInter_isClopen continuous_boolIndicator_iff_clopen -> continuous_boolIndicator_iff_isClopen continuousOn_boolIndicator_iff_clopen -> continuousOn_boolIndicator_iff_isClopen DiscreteQuotient.ofClopen -> DiscreteQuotient.ofIsClopen disjoint_or_subset_of_clopen -> disjoint_or_subset_of_isClopen exists_clopen_{lower,upper}of_not_le -> exists_isClopen{lower,upper}_of_not_le exists_clopen_of_cofiltered -> exists_isClopen_of_cofiltered exists_clopen_of_totally_separated -> exists_isClopen_of_totally_separated exists_clopen_upper_or_lower_of_ne -> exists_isClopen_upper_or_lower_of_ne IsPreconnected.subset_clopen -> IsPreconnected.subset_isClopen isTotallyDisconnected_of_clopen_set -> isTotallyDisconnected_of_isClopen_set LocallyConstant.ofClopen_fiber_one -> LocallyConstant.ofIsClopen_fiber_one LocallyConstant.ofClopen_fiber_zero -> LocallyConstant.ofIsClopen_fiber_zero LocallyConstant.ofClopen -> LocallyConstant.ofIsClopen preimage_clopen_of_clopen -> preimage_isClopen_of_isClopen TopologicalSpace.Clopens.clopen -> TopologicalSpace.Clopens.isClopen

@@ -75,7 +75,7 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
 If the space is also compact:
 
 * `normalOfCompactT2`: A compact T₂ space is a `NormalSpace`.
-* `connectedComponent_eq_iInter_clopen`: The connected component of a point
+* `connectedComponent_eq_iInter_isClopen`: The connected component of a point
   is the intersection of all its clopen neighbourhoods.
 * `compact_t2_tot_disc_iff_tot_sep`: Being a `TotallyDisconnectedSpace`
   is equivalent to being a `TotallySeparatedSpace`.
@@ -2045,9 +2045,9 @@ end CompletelyNormal
 
 /-- In a compact T₂ space, the connected component of a point equals the intersection of all
 its clopen neighbourhoods. -/
-theorem connectedComponent_eq_iInter_clopen [T2Space X] [CompactSpace X] (x : X) :
+theorem connectedComponent_eq_iInter_isClopen [T2Space X] [CompactSpace X] (x : X) :
     connectedComponent x = ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s := by
-  apply Subset.antisymm connectedComponent_subset_iInter_clopen
+  apply Subset.antisymm connectedComponent_subset_iInter_isClopen
   -- Reduce to showing that the clopen intersection is connected.
   refine' IsPreconnected.subset_connectedComponent _ (mem_iInter.2 fun s => s.2.2)
   -- We do this by showing that any disjoint cover by two closed sets implies
@@ -2097,7 +2097,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space X] [CompactSpace X] (x : X)
       · refine Subset.trans ?_ (inter_subset_right s v)
         exact iInter_subset (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s.1)
           ⟨s ∩ v, H2, mem_inter H.2.1 h1⟩
-#align connected_component_eq_Inter_clopen connectedComponent_eq_iInter_clopen
+#align connected_component_eq_Inter_clopen connectedComponent_eq_iInter_isClopen
 
 section Profinite
 
@@ -2121,7 +2121,7 @@ theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace X ↔ Totally
   intro hyp
   suffices x ∈ connectedComponent y by
     simpa [totallyDisconnectedSpace_iff_connectedComponent_singleton.1 h y, mem_singleton_iff]
-  rw [connectedComponent_eq_iInter_clopen, mem_iInter]
+  rw [connectedComponent_eq_iInter_isClopen, mem_iInter]
   rintro ⟨w : Set X, hw : IsClopen w, hy : y ∈ w⟩
   by_contra hx
   exact hyp wᶜ w hw.2.isOpen_compl hw.1 hx hy (@isCompl_compl _ w _).symm.codisjoint.top_le
@@ -2138,7 +2138,7 @@ theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s
     constructor
     · have hx : connectedComponent x = {x} :=
         totallyDisconnectedSpace_iff_connectedComponent_singleton.mp ‹_› x
-      rw [connectedComponent_eq_iInter_clopen] at hx
+      rw [connectedComponent_eq_iInter_isClopen] at hx
       intro hU
       let N := { s // IsClopen s ∧ x ∈ s }
       suffices : ∃ s : N, s.val ⊆ U
@@ -2231,7 +2231,7 @@ instance ConnectedComponents.t2 [T2Space X] [CompactSpace X] : T2Space (Connecte
   rw [ConnectedComponents.coe_ne_coe] at ne
   have h := connectedComponent_disjoint ne
   -- write ↑b as the intersection of all clopen subsets containing it
-  rw [connectedComponent_eq_iInter_clopen b, disjoint_iff_inter_eq_empty] at h
+  rw [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h
   -- Now we show that this can be reduced to some clopen containing `↑b` being disjoint to `↑a`
   obtain ⟨U, V, hU, ha, hb, rfl⟩ : ∃ (U : Set X) (V : Set (ConnectedComponents X)),
       IsClopen U ∧ connectedComponent a ∩ U = ∅ ∧ connectedComponent b ⊆ U ∧ (↑) ⁻¹' V = U := by

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -1335,13 +1335,12 @@ section SeparatedFinset
 
 theorem finset_disjoint_finset_opens_of_t2 [T2Space X] (s t : Finset X) (h : Disjoint s t) :
     SeparatedNhds (s : Set X) t :=
-  isCompact_isCompact_separated s.finite_toSet.isCompact t.finite_toSet.isCompact <| by
-    exact_mod_cast h
+  isCompact_isCompact_separated s.finite_toSet.isCompact t.finite_toSet.isCompact <| mod_cast h
 #align finset_disjoint_finset_opens_of_t2 finset_disjoint_finset_opens_of_t2
 
 theorem point_disjoint_finset_opens_of_t2 [T2Space X] {x : X} {s : Finset X} (h : x ∉ s) :
-    SeparatedNhds ({x} : Set X) s := by
-  exact_mod_cast finset_disjoint_finset_opens_of_t2 {x} s (Finset.disjoint_singleton_left.mpr h)
+    SeparatedNhds ({x} : Set X) s :=
+  mod_cast finset_disjoint_finset_opens_of_t2 {x} s (Finset.disjoint_singleton_left.mpr h)
 #align point_disjoint_finset_opens_of_t2 point_disjoint_finset_opens_of_t2
 
 end SeparatedFinset

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -585,8 +585,8 @@ theorem continuousWithinAt_update_of_ne [T1Space X] [DecidableEq X] [Topological
 
 theorem continuousAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y]
     {f : X → Y} {x x' : X} {y : Y} (hne : x' ≠ x) :
-    ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x' :=
-  by simp only [← continuousWithinAt_univ, continuousWithinAt_update_of_ne hne]
+    ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x' := by
+  simp only [← continuousWithinAt_univ, continuousWithinAt_update_of_ne hne]
 #align continuous_at_update_of_ne continuousAt_update_of_ne
 
 theorem continuousOn_update_iff [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}

@@ -873,6 +873,17 @@ theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s]
   simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx
 #align nhds_inter_eq_singleton_of_mem_discrete nhds_inter_eq_singleton_of_mem_discrete
 
+/-- Let `x` be a point in a discrete subset `s` of a topological space, then there exists an open
+set that only meets `s` at `x`.  -/
+theorem isOpen_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
+    (hx : x ∈ s) : ∃ U : Set X, IsOpen U ∧ U ∩ s = {x} := by
+  obtain ⟨U, hU_nhds, hU_inter⟩ := nhds_inter_eq_singleton_of_mem_discrete hx
+  obtain ⟨t, ht_sub, ht_open, ht_x⟩ := mem_nhds_iff.mp hU_nhds
+  refine ⟨t, ht_open, Set.Subset.antisymm ?_ ?_⟩
+  · exact hU_inter ▸ Set.inter_subset_inter_left s ht_sub
+  · rw [Set.subset_inter_iff, Set.singleton_subset_iff, Set.singleton_subset_iff]
+    exact ⟨ht_x, hx⟩
+
 /-- For point `x` in a discrete subset `s` of a topological space, there is a set `U`
 such that
 1. `U` is a punctured neighborhood of `x` (ie. `U ∪ {x}` is a neighbourhood of `x`),

Also change a few theorems to instances, as instance loops are not a problem any more.

@@ -1731,8 +1731,8 @@ section ClosableCompactSubsetOpenSpace
 
 /-- A class of topological spaces in which, given a compact set included inside an open set, then
 the closure of the compact set is also included in the open set.
-Satisfied notably for T2 spaces and regular spaces, and useful
-when discussing classes of regular measures. -/
+Satisfied notably for T2 spaces and regular spaces, and useful when discussing classes of
+regular measures. Equivalent to regularity among locally compact spaces. -/
 class ClosableCompactSubsetOpenSpace (X : Type*) [TopologicalSpace X] : Prop :=
   closure_subset_of_isOpen : ∀ (K U : Set X), IsCompact K → IsOpen U → K ⊆ U → closure K ⊆ U
 
@@ -1741,12 +1741,12 @@ theorem IsCompact.closure_subset_of_isOpen [ClosableCompactSubsetOpenSpace X]
     closure s ⊆ u :=
   ClosableCompactSubsetOpenSpace.closure_subset_of_isOpen s u hs hu h
 
-instance [T2Space X] : ClosableCompactSubsetOpenSpace X :=
+instance (priority := 150) [T2Space X] : ClosableCompactSubsetOpenSpace X :=
   ⟨fun K _U K_comp _U_open KU ↦ by rwa [K_comp.isClosed.closure_eq]⟩
 
 /-- In a (possibly non-Hausdorff) regular space, if a compact set `s` is contained in an
 open set `u`, then its closure is also contained in `u`. -/
-instance [RegularSpace X] : ClosableCompactSubsetOpenSpace X := by
+instance (priority := 150) [RegularSpace X] : ClosableCompactSubsetOpenSpace X := by
   refine ⟨fun s u hs hu h ↦ ?_⟩
   obtain ⟨F, sF, F_closed, Fu⟩ : ∃ F, s ⊆ F ∧ IsClosed F ∧ F ⊆ u := by
     apply hs.induction_on (p := fun t ↦ ∃ F, t ⊆ F ∧ IsClosed F ∧ F ⊆ u)
@@ -1779,6 +1779,16 @@ theorem exists_compact_closed_between [LocallyCompactSpace X] [ClosableCompactSu
   · apply M_comp.closure_subset_of_isOpen hU
     exact ML.trans (interior_subset.trans LU)
 
+/-- A locally compact space with the `ClosableCompactSubsetOpenSpace` is `Regular`. -/
+instance (priority := 80) [LocallyCompactSpace X] [ClosableCompactSubsetOpenSpace X] :
+    RegularSpace X := by
+  apply RegularSpace.ofExistsMemNhdsIsClosedSubset (fun x s hx ↦ ?_)
+  rcases _root_.mem_nhds_iff.1 hx with ⟨u, us, u_open, xu⟩
+  rcases exists_compact_closed_between (isCompact_singleton (a := x)) u_open (by simpa using xu)
+    with ⟨t, -, t_closed, xt, tu⟩
+  have : interior t ∈ 𝓝 x := isOpen_interior.mem_nhds (by simpa using xt)
+  exact ⟨t, interior_mem_nhds.mp this, t_closed, tu.trans us⟩
+
 protected theorem IsCompact.closure [WeaklyLocallyCompactSpace X] [ClosableCompactSubsetOpenSpace X]
     {K : Set X} (hK : IsCompact K) : IsCompact (closure K) := by
   rcases exists_compact_superset hK with ⟨L, L_comp, hL⟩

Useful to uniformize proofs for T2 spaces and regular spaces, notably to discuss regular measures in topological groups.

@@ -29,12 +29,16 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
   there is two disjoint open sets, one containing `x`, and the other `y`.
 * `T25Space`: A T₂.₅/Urysohn space is a space where, for every two points `x ≠ y`,
   there is two open sets, one containing `x`, and the other `y`, whose closures are disjoint.
-* `T3Space`: A T₃ space, is one where given any closed `C` and `x ∉ C`,
-  there is disjoint open sets containing `x` and `C` respectively. In `mathlib`, T₃ implies T₂.₅.
+* `RegularSpace`: A regular space is one where, given any closed `C` and `x ∉ C`,
+  there are disjoint open sets containing `x` and `C` respectively. Such a space is not necessarily
+  Hausdorff.
+* `T3Space`: A T₃ space is a T0 regular space. In `mathlib`, T₃ implies T₂.₅.
 * `NormalSpace`: A normal space, is one where given two disjoint closed sets,
   we can find two open sets that separate them.
 * `T4Space`: A T₄ space is a normal T₁ space. T₄ implies T₃.
-* `T5Space`: A T₅ space, also known as a *completely normal Hausdorff space*
+* `T5Space`: A T₅ space, also known as a *completely normal Hausdorff space*, is a space in which,
+  given two sets `s` and `t` such that the closure of `s` is disjoint from `t`, and conversely,
+  then `s` and `t` have disjoint neighborhoods.
 
 ## Main results
 
@@ -1594,6 +1598,12 @@ theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
   Iff.mpr ((regularSpace_TFAE X).out 0 3) h
 #align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.ofExistsMemNhdsIsClosedSubset
 
+/-- A locally compact T2 space is regular. -/
+instance [LocallyCompactSpace X] [T2Space X] : RegularSpace X := by
+  apply RegularSpace.ofExistsMemNhdsIsClosedSubset (fun x s hx ↦ ?_)
+  rcases local_compact_nhds hx with ⟨k, kx, ks, hk⟩
+  exact ⟨k, kx, hk.isClosed, ks⟩
+
 variable [RegularSpace X] {x : X} {s : Set X}
 
 theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s := by
@@ -1717,6 +1727,66 @@ instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i,
 
 end RegularSpace
 
+section ClosableCompactSubsetOpenSpace
+
+/-- A class of topological spaces in which, given a compact set included inside an open set, then
+the closure of the compact set is also included in the open set.
+Satisfied notably for T2 spaces and regular spaces, and useful
+when discussing classes of regular measures. -/
+class ClosableCompactSubsetOpenSpace (X : Type*) [TopologicalSpace X] : Prop :=
+  closure_subset_of_isOpen : ∀ (K U : Set X), IsCompact K → IsOpen U → K ⊆ U → closure K ⊆ U
+
+theorem IsCompact.closure_subset_of_isOpen [ClosableCompactSubsetOpenSpace X]
+    {s : Set X} (hs : IsCompact s) {u : Set X} (hu : IsOpen u) (h : s ⊆ u) :
+    closure s ⊆ u :=
+  ClosableCompactSubsetOpenSpace.closure_subset_of_isOpen s u hs hu h
+
+instance [T2Space X] : ClosableCompactSubsetOpenSpace X :=
+  ⟨fun K _U K_comp _U_open KU ↦ by rwa [K_comp.isClosed.closure_eq]⟩
+
+/-- In a (possibly non-Hausdorff) regular space, if a compact set `s` is contained in an
+open set `u`, then its closure is also contained in `u`. -/
+instance [RegularSpace X] : ClosableCompactSubsetOpenSpace X := by
+  refine ⟨fun s u hs hu h ↦ ?_⟩
+  obtain ⟨F, sF, F_closed, Fu⟩ : ∃ F, s ⊆ F ∧ IsClosed F ∧ F ⊆ u := by
+    apply hs.induction_on (p := fun t ↦ ∃ F, t ⊆ F ∧ IsClosed F ∧ F ⊆ u)
+    · exact ⟨∅, by simp⟩
+    · intro t' t ht't ⟨F, tF, F_closed, Fu⟩
+      exact ⟨F, ht't.trans tF, F_closed, Fu⟩
+    · intro t t' ⟨F, tF, F_closed, Fu⟩ ⟨F', t'F', F'_closed, F'u⟩
+      exact ⟨F ∪ F', union_subset_union tF t'F', F_closed.union F'_closed, union_subset Fu F'u⟩
+    · intro x hx
+      rcases exists_mem_nhds_isClosed_subset (hu.mem_nhds (h hx)) with ⟨F, F_mem, F_closed, Fu⟩
+      exact ⟨F, nhdsWithin_le_nhds F_mem, F, Subset.rfl, F_closed, Fu⟩
+  exact (closure_minimal sF F_closed).trans Fu
+
+/-- In a (possibly non-Hausdorff) locally compact space with the `ClosableCompactSubsetOpenSpace`
+  property (for instance regular spaces), for every containment `K ⊆ U` of a compact set `K` in an
+  open set `U`, there is a compact closed neighborhood `L` such that `K ⊆ L ⊆ U`: equivalently,
+  there is a compact closed set `L` such that `K ⊆ interior L` and `L ⊆ U`. -/
+theorem exists_compact_closed_between [LocallyCompactSpace X] [ClosableCompactSubsetOpenSpace X]
+    {K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (h_KU : K ⊆ U) :
+    ∃ L, IsCompact L ∧ IsClosed L ∧ K ⊆ interior L ∧ L ⊆ U := by
+  rcases exists_compact_between hK hU h_KU with ⟨L, L_comp, KL, LU⟩
+  rcases exists_compact_between hK isOpen_interior KL with ⟨M, M_comp, KM, ML⟩
+  refine ⟨closure M, ?_, isClosed_closure, ?_, ?_⟩
+  · have : closure M ∩ L = closure M := by
+      apply inter_eq_self_of_subset_left
+      exact (M_comp.closure_subset_of_isOpen isOpen_interior ML).trans interior_subset
+    rw [← this]
+    apply L_comp.inter_left isClosed_closure
+  · exact KM.trans (interior_mono subset_closure)
+  · apply M_comp.closure_subset_of_isOpen hU
+    exact ML.trans (interior_subset.trans LU)
+
+protected theorem IsCompact.closure [WeaklyLocallyCompactSpace X] [ClosableCompactSubsetOpenSpace X]
+    {K : Set X} (hK : IsCompact K) : IsCompact (closure K) := by
+  rcases exists_compact_superset hK with ⟨L, L_comp, hL⟩
+  exact L_comp.of_isClosed_subset isClosed_closure
+    ((hK.closure_subset_of_isOpen isOpen_interior hL).trans interior_subset)
+
+end ClosableCompactSubsetOpenSpace
+
 section T3
 
 /-- A T₃ space is a T₀ space which is a regular space. Any T₃ space is a T₁ space, a T₂ space, and

This file was using a mix of Greek letters and standard math convention (X, Y, Z). connectedComponent_eq_iInter_clopen was even using X in comments and $\alpha$ in the code.

As discussed on zulip, standardize on the latter.

@@ -53,8 +53,8 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
 ### T₂ spaces
 
 * `t2_iff_nhds`: A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter.
-* `t2_iff_isClosed_diagonal`: A space is T₂ iff the `diagonal` of `α` (that is, the set of all
-  points of the form `(a, a) : α × α`) is closed under the product topology.
+* `t2_iff_isClosed_diagonal`: A space is T₂ iff the `diagonal` of `X` (that is, the set of all
+  points of the form `(a, a) : X × X`) is closed under the product topology.
 * `finset_disjoint_finset_opens_of_t2`: Any two disjoint finsets are `SeparatedNhds`.
 * Most topological constructions preserve Hausdorffness;
   these results are part of the typeclass inference system (e.g. `Embedding.t2Space`)
@@ -63,7 +63,7 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
 * `WeaklyLocallyCompactSpace.locallyCompactSpace`: If a topological space is both
   weakly locally compact (i.e., each point has a compact neighbourhood)
   and is T₂, then it is locally compact.
-* `totallySeparatedSpace_of_t1_of_basis_clopen`: If `α` has a clopen basis, then
+* `totallySeparatedSpace_of_t1_of_basis_clopen`: If `X` has a clopen basis, then
   it is a `TotallySeparatedSpace`.
 * `loc_compact_t2_tot_disc_iff_tot_sep`: A locally compact T₂ space is totally disconnected iff
   it is totally separated.
@@ -75,7 +75,7 @@ If the space is also compact:
   is the intersection of all its clopen neighbourhoods.
 * `compact_t2_tot_disc_iff_tot_sep`: Being a `TotallyDisconnectedSpace`
   is equivalent to being a `TotallySeparatedSpace`.
-* `ConnectedComponents.t2`: `ConnectedComponents α` is T₂ for `α` T₂ and compact.
+* `ConnectedComponents.t2`: `ConnectedComponents X` is T₂ for `X` T₂ and compact.
 
 ### T₃ spaces
 
@@ -93,7 +93,7 @@ open scoped Classical
 
 universe u v
 
-variable {α : Type u} {β : Type v} [TopologicalSpace α]
+variable {X : Type*} {Y : Type*} [TopologicalSpace X]
 
 section Separation
 
@@ -101,11 +101,11 @@ section Separation
 `SeparatedNhds` is a predicate on pairs of sub`Set`s of a topological space.  It holds if the two
 sub`Set`s are contained in disjoint open sets.
 -/
-def SeparatedNhds : Set α → Set α → Prop := fun s t : Set α =>
-  ∃ U V : Set α, IsOpen U ∧ IsOpen V ∧ s ⊆ U ∧ t ⊆ V ∧ Disjoint U V
+def SeparatedNhds : Set X → Set X → Prop := fun s t : Set X =>
+  ∃ U V : Set X, IsOpen U ∧ IsOpen V ∧ s ⊆ U ∧ t ⊆ V ∧ Disjoint U V
 #align separated_nhds SeparatedNhds
 
-theorem separatedNhds_iff_disjoint {s t : Set α} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by
+theorem separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by
   simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, ←
     exists_and_left, and_assoc, and_comm, and_left_comm]
 #align separated_nhds_iff_disjoint separatedNhds_iff_disjoint
@@ -114,18 +114,18 @@ alias ⟨SeparatedNhds.disjoint_nhdsSet, _⟩ := separatedNhds_iff_disjoint
 
 namespace SeparatedNhds
 
-variable {s s₁ s₂ t t₁ t₂ u : Set α}
+variable {s s₁ s₂ t t₁ t₂ u : Set X}
 
 @[symm]
 theorem symm : SeparatedNhds s t → SeparatedNhds t s := fun ⟨U, V, oU, oV, aU, bV, UV⟩ =>
   ⟨V, U, oV, oU, bV, aU, Disjoint.symm UV⟩
 #align separated_nhds.symm SeparatedNhds.symm
 
-theorem comm (s t : Set α) : SeparatedNhds s t ↔ SeparatedNhds t s :=
+theorem comm (s t : Set X) : SeparatedNhds s t ↔ SeparatedNhds t s :=
   ⟨symm, symm⟩
 #align separated_nhds.comm SeparatedNhds.comm
 
-theorem preimage [TopologicalSpace β] {f : α → β} {s t : Set β} (h : SeparatedNhds s t)
+theorem preimage [TopologicalSpace Y] {f : X → Y} {s t : Set Y} (h : SeparatedNhds s t)
     (hf : Continuous f) : SeparatedNhds (f ⁻¹' s) (f ⁻¹' t) :=
   let ⟨U, V, oU, oV, sU, tV, UV⟩ := h
   ⟨f ⁻¹' U, f ⁻¹' V, oU.preimage hf, oV.preimage hf, preimage_mono sU, preimage_mono tV,
@@ -145,11 +145,11 @@ theorem disjoint_closure_right (h : SeparatedNhds s t) : Disjoint s (closure t)
   h.symm.disjoint_closure_left.symm
 #align separated_nhds.disjoint_closure_right SeparatedNhds.disjoint_closure_right
 
-theorem empty_right (s : Set α) : SeparatedNhds s ∅ :=
+theorem empty_right (s : Set X) : SeparatedNhds s ∅ :=
   ⟨_, _, isOpen_univ, isOpen_empty, fun a _ => mem_univ a, Subset.rfl, disjoint_empty _⟩
 #align separated_nhds.empty_right SeparatedNhds.empty_right
 
-theorem empty_left (s : Set α) : SeparatedNhds ∅ s :=
+theorem empty_left (s : Set X) : SeparatedNhds ∅ s :=
   (empty_right _).symm
 #align separated_nhds.empty_left SeparatedNhds.empty_left
 
@@ -171,99 +171,99 @@ end SeparatedNhds
 /-- A T₀ space, also known as a Kolmogorov space, is a topological space such that for every pair
 `x ≠ y`, there is an open set containing one but not the other. We formulate the definition in terms
 of the `Inseparable` relation.  -/
-class T0Space (α : Type u) [TopologicalSpace α] : Prop where
+class T0Space (X : Type u) [TopologicalSpace X] : Prop where
   /-- Two inseparable points in a T₀ space are equal. -/
-  t0 : ∀ ⦃x y : α⦄, Inseparable x y → x = y
+  t0 : ∀ ⦃x y : X⦄, Inseparable x y → x = y
 #align t0_space T0Space
 
-theorem t0Space_iff_inseparable (α : Type u) [TopologicalSpace α] :
-    T0Space α ↔ ∀ x y : α, Inseparable x y → x = y :=
+theorem t0Space_iff_inseparable (X : Type u) [TopologicalSpace X] :
+    T0Space X ↔ ∀ x y : X, Inseparable x y → x = y :=
   ⟨fun ⟨h⟩ => h, fun h => ⟨h⟩⟩
 #align t0_space_iff_inseparable t0Space_iff_inseparable
 
-theorem t0Space_iff_not_inseparable (α : Type u) [TopologicalSpace α] :
-    T0Space α ↔ ∀ x y : α, x ≠ y → ¬Inseparable x y := by
+theorem t0Space_iff_not_inseparable (X : Type u) [TopologicalSpace X] :
+    T0Space X ↔ ∀ x y : X, x ≠ y → ¬Inseparable x y := by
   simp only [t0Space_iff_inseparable, Ne.def, not_imp_not]
 #align t0_space_iff_not_inseparable t0Space_iff_not_inseparable
 
-theorem Inseparable.eq [T0Space α] {x y : α} (h : Inseparable x y) : x = y :=
+theorem Inseparable.eq [T0Space X] {x y : X} (h : Inseparable x y) : x = y :=
   T0Space.t0 h
 #align inseparable.eq Inseparable.eq
 
 /-- A topology `Inducing` map from a T₀ space is injective. -/
-protected theorem Inducing.injective [TopologicalSpace β] [T0Space α] {f : α → β}
+protected theorem Inducing.injective [TopologicalSpace Y] [T0Space X] {f : X → Y}
     (hf : Inducing f) : Injective f := fun _ _ h =>
   (hf.inseparable_iff.1 <| .of_eq h).eq
 #align inducing.injective Inducing.injective
 
 /-- A topology `Inducing` map from a T₀ space is a topological embedding. -/
-protected theorem Inducing.embedding [TopologicalSpace β] [T0Space α] {f : α → β}
+protected theorem Inducing.embedding [TopologicalSpace Y] [T0Space X] {f : X → Y}
     (hf : Inducing f) : Embedding f :=
   ⟨hf, hf.injective⟩
 #align inducing.embedding Inducing.embedding
 
-lemma embedding_iff_inducing [TopologicalSpace β] [T0Space α] {f : α → β} :
+lemma embedding_iff_inducing [TopologicalSpace Y] [T0Space X] {f : X → Y} :
     Embedding f ↔ Inducing f :=
   ⟨Embedding.toInducing, Inducing.embedding⟩
 #align embedding_iff_inducing embedding_iff_inducing
 
-theorem t0Space_iff_nhds_injective (α : Type u) [TopologicalSpace α] :
-    T0Space α ↔ Injective (𝓝 : α → Filter α) :=
-  t0Space_iff_inseparable α
+theorem t0Space_iff_nhds_injective (X : Type u) [TopologicalSpace X] :
+    T0Space X ↔ Injective (𝓝 : X → Filter X) :=
+  t0Space_iff_inseparable X
 #align t0_space_iff_nhds_injective t0Space_iff_nhds_injective
 
-theorem nhds_injective [T0Space α] : Injective (𝓝 : α → Filter α) :=
-  (t0Space_iff_nhds_injective α).1 ‹_›
+theorem nhds_injective [T0Space X] : Injective (𝓝 : X → Filter X) :=
+  (t0Space_iff_nhds_injective X).1 ‹_›
 #align nhds_injective nhds_injective
 
-theorem inseparable_iff_eq [T0Space α] {x y : α} : Inseparable x y ↔ x = y :=
+theorem inseparable_iff_eq [T0Space X] {x y : X} : Inseparable x y ↔ x = y :=
   nhds_injective.eq_iff
 #align inseparable_iff_eq inseparable_iff_eq
 
 @[simp]
-theorem nhds_eq_nhds_iff [T0Space α] {a b : α} : 𝓝 a = 𝓝 b ↔ a = b :=
+theorem nhds_eq_nhds_iff [T0Space X] {a b : X} : 𝓝 a = 𝓝 b ↔ a = b :=
   nhds_injective.eq_iff
 #align nhds_eq_nhds_iff nhds_eq_nhds_iff
 
 @[simp]
-theorem inseparable_eq_eq [T0Space α] : Inseparable = @Eq α :=
+theorem inseparable_eq_eq [T0Space X] : Inseparable = @Eq X :=
   funext₂ fun _ _ => propext inseparable_iff_eq
 #align inseparable_eq_eq inseparable_eq_eq
 
-theorem TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set α)}
-    (hb : IsTopologicalBasis b) {x y : α} : Inseparable x y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
+theorem TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set X)}
+    (hb : IsTopologicalBasis b) {x y : X} : Inseparable x y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
   ⟨fun h s hs ↦ inseparable_iff_forall_open.1 h _ (hb.isOpen hs),
     fun h ↦ hb.nhds_hasBasis.eq_of_same_basis $ by
       convert hb.nhds_hasBasis using 2
       exact and_congr_right (h _)⟩
 
-theorem TopologicalSpace.IsTopologicalBasis.eq_iff [T0Space α] {b : Set (Set α)}
-    (hb : IsTopologicalBasis b) {x y : α} : x = y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
+theorem TopologicalSpace.IsTopologicalBasis.eq_iff [T0Space X] {b : Set (Set X)}
+    (hb : IsTopologicalBasis b) {x y : X} : x = y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
   inseparable_iff_eq.symm.trans hb.inseparable_iff
 
-theorem t0Space_iff_exists_isOpen_xor'_mem (α : Type u) [TopologicalSpace α] :
-    T0Space α ↔ ∀ x y, x ≠ y → ∃ U : Set α, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by
+theorem t0Space_iff_exists_isOpen_xor'_mem (X : Type u) [TopologicalSpace X] :
+    T0Space X ↔ ∀ x y, x ≠ y → ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by
   simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,
     inseparable_iff_forall_open]
 #align t0_space_iff_exists_is_open_xor_mem t0Space_iff_exists_isOpen_xor'_mem
 
-theorem exists_isOpen_xor'_mem [T0Space α] {x y : α} (h : x ≠ y) :
-    ∃ U : Set α, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) :=
-  (t0Space_iff_exists_isOpen_xor'_mem α).1 ‹_› x y h
+theorem exists_isOpen_xor'_mem [T0Space X] {x y : X} (h : x ≠ y) :
+    ∃ U : Set X, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) :=
+  (t0Space_iff_exists_isOpen_xor'_mem X).1 ‹_› x y h
 #align exists_is_open_xor_mem exists_isOpen_xor'_mem
 
 /-- Specialization forms a partial order on a t0 topological space. -/
-def specializationOrder (α : Type*) [TopologicalSpace α] [T0Space α] : PartialOrder α :=
-  { specializationPreorder α, PartialOrder.lift (OrderDual.toDual ∘ 𝓝) nhds_injective with }
+def specializationOrder (X) [TopologicalSpace X] [T0Space X] : PartialOrder X :=
+  { specializationPreorder X, PartialOrder.lift (OrderDual.toDual ∘ 𝓝) nhds_injective with }
 #align specialization_order specializationOrder
 
-instance : T0Space (SeparationQuotient α) :=
+instance : T0Space (SeparationQuotient X) :=
   ⟨fun x y => Quotient.inductionOn₂' x y fun _ _ h =>
     SeparationQuotient.mk_eq_mk.2 <| SeparationQuotient.inducing_mk.inseparable_iff.1 h⟩
 
-theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsClosed s)
+theorem minimal_nonempty_closed_subsingleton [T0Space X] {s : Set X} (hs : IsClosed s)
     (hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : s.Subsingleton := by
-  clear β -- Porting note: added
+  clear Y -- Porting note: added
   refine' fun x hx y hy => of_not_not fun hxy => _
   rcases exists_isOpen_xor'_mem hxy with ⟨U, hUo, hU⟩
   wlog h : x ∈ U ∧ y ∉ U
@@ -273,7 +273,7 @@ theorem minimal_nonempty_closed_subsingleton [T0Space α] {s : Set α} (hs : IsC
   exact (this.symm.subset hx).2 hxU
 #align minimal_nonempty_closed_subsingleton minimal_nonempty_closed_subsingleton
 
-theorem minimal_nonempty_closed_eq_singleton [T0Space α] {s : Set α} (hs : IsClosed s)
+theorem minimal_nonempty_closed_eq_singleton [T0Space X] {s : Set X} (hs : IsClosed s)
     (hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsClosed t → t = s) : ∃ x, s = {x} :=
   exists_eq_singleton_iff_nonempty_subsingleton.2
     ⟨hne, minimal_nonempty_closed_subsingleton hs hmin⟩
@@ -281,17 +281,16 @@ theorem minimal_nonempty_closed_eq_singleton [T0Space α] {s : Set α} (hs : IsC
 
 /-- Given a closed set `S` in a compact T₀ space, there is some `x ∈ S` such that `{x}` is
 closed. -/
-theorem IsClosed.exists_closed_singleton {α : Type*} [TopologicalSpace α] [T0Space α]
-    [CompactSpace α] {S : Set α} (hS : IsClosed S) (hne : S.Nonempty) :
-    ∃ x : α, x ∈ S ∧ IsClosed ({x} : Set α) := by
+theorem IsClosed.exists_closed_singleton [T0Space X] [CompactSpace X] {S : Set X}
+    (hS : IsClosed S) (hne : S.Nonempty) : ∃ x : X, x ∈ S ∧ IsClosed ({x} : Set X) := by
   obtain ⟨V, Vsub, Vne, Vcls, hV⟩ := hS.exists_minimal_nonempty_closed_subset hne
   rcases minimal_nonempty_closed_eq_singleton Vcls Vne hV with ⟨x, rfl⟩
   exact ⟨x, Vsub (mem_singleton x), Vcls⟩
 #align is_closed.exists_closed_singleton IsClosed.exists_closed_singleton
 
-theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpen s)
+theorem minimal_nonempty_open_subsingleton [T0Space X] {s : Set X} (hs : IsOpen s)
     (hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : s.Subsingleton := by
-  clear β -- Porting note: added
+  clear Y -- Porting note: added
   refine' fun x hx y hy => of_not_not fun hxy => _
   rcases exists_isOpen_xor'_mem hxy with ⟨U, hUo, hU⟩
   wlog h : x ∈ U ∧ y ∉ U
@@ -301,17 +300,17 @@ theorem minimal_nonempty_open_subsingleton [T0Space α] {s : Set α} (hs : IsOpe
   exact hyU (this.symm.subset hy).2
 #align minimal_nonempty_open_subsingleton minimal_nonempty_open_subsingleton
 
-theorem minimal_nonempty_open_eq_singleton [T0Space α] {s : Set α} (hs : IsOpen s)
+theorem minimal_nonempty_open_eq_singleton [T0Space X] {s : Set X} (hs : IsOpen s)
     (hne : s.Nonempty) (hmin : ∀ t, t ⊆ s → t.Nonempty → IsOpen t → t = s) : ∃ x, s = {x} :=
   exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨hne, minimal_nonempty_open_subsingleton hs hmin⟩
 #align minimal_nonempty_open_eq_singleton minimal_nonempty_open_eq_singleton
 
 /-- Given an open finite set `S` in a T₀ space, there is some `x ∈ S` such that `{x}` is open. -/
-theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s.Finite)
-    (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set α) := by
-  lift s to Finset α using hfin
+theorem exists_open_singleton_of_open_finite [T0Space X] {s : Set X} (hfin : s.Finite)
+    (hne : s.Nonempty) (ho : IsOpen s) : ∃ x ∈ s, IsOpen ({x} : Set X) := by
+  lift s to Finset X using hfin
   induction' s using Finset.strongInductionOn with s ihs
-  rcases em (∃ t, t ⊂ s ∧ t.Nonempty ∧ IsOpen (t : Set α)) with (⟨t, hts, htne, hto⟩ | ht)
+  rcases em (∃ t, t ⊂ s ∧ t.Nonempty ∧ IsOpen (t : Set X)) with (⟨t, hts, htne, hto⟩ | ht)
   · rcases ihs t hts htne hto with ⟨x, hxt, hxo⟩
     exact ⟨x, hts.1 hxt, hxo⟩
   · -- Porting note: was `rcases minimal_nonempty_open_eq_singleton ho hne _ with ⟨x, hx⟩`
@@ -320,46 +319,46 @@ theorem exists_open_singleton_of_open_finite [T0Space α] {s : Set α} (hfin : s
     · exact ⟨x, hx.symm ▸ rfl, hx ▸ ho⟩
     refine minimal_nonempty_open_eq_singleton ho hne ?_
     refine' fun t hts htne hto => of_not_not fun hts' => ht _
-    lift t to Finset α using s.finite_toSet.subset hts
+    lift t to Finset X using s.finite_toSet.subset hts
     exact ⟨t, ssubset_iff_subset_ne.2 ⟨hts, mt Finset.coe_inj.2 hts'⟩, htne, hto⟩
 #align exists_open_singleton_of_open_finite exists_open_singleton_of_open_finite
 
-theorem exists_open_singleton_of_finite [T0Space α] [Finite α] [Nonempty α] :
-    ∃ x : α, IsOpen ({x} : Set α) :=
+theorem exists_open_singleton_of_finite [T0Space X] [Finite X] [Nonempty X] :
+    ∃ x : X, IsOpen ({x} : Set X) :=
   let ⟨x, _, h⟩ := exists_open_singleton_of_open_finite (Set.toFinite _) univ_nonempty isOpen_univ
   ⟨x, h⟩
 #align exists_open_singleton_of_fintype exists_open_singleton_of_finite
 
-theorem t0Space_of_injective_of_continuous [TopologicalSpace β] {f : α → β}
-    (hf : Function.Injective f) (hf' : Continuous f) [T0Space β] : T0Space α :=
+theorem t0Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y}
+    (hf : Function.Injective f) (hf' : Continuous f) [T0Space Y] : T0Space X :=
   ⟨fun _ _ h => hf <| (h.map hf').eq⟩
 #align t0_space_of_injective_of_continuous t0Space_of_injective_of_continuous
 
-protected theorem Embedding.t0Space [TopologicalSpace β] [T0Space β] {f : α → β}
-    (hf : Embedding f) : T0Space α :=
+protected theorem Embedding.t0Space [TopologicalSpace Y] [T0Space Y] {f : X → Y}
+    (hf : Embedding f) : T0Space X :=
   t0Space_of_injective_of_continuous hf.inj hf.continuous
 #align embedding.t0_space Embedding.t0Space
 
-instance Subtype.t0Space [T0Space α] {p : α → Prop} : T0Space (Subtype p) :=
+instance Subtype.t0Space [T0Space X] {p : X → Prop} : T0Space (Subtype p) :=
   embedding_subtype_val.t0Space
 #align subtype.t0_space Subtype.t0Space
 
-theorem t0Space_iff_or_not_mem_closure (α : Type u) [TopologicalSpace α] :
-    T0Space α ↔ ∀ a b : α, a ≠ b → a ∉ closure ({b} : Set α) ∨ b ∉ closure ({a} : Set α) := by
+theorem t0Space_iff_or_not_mem_closure (X : Type u) [TopologicalSpace X] :
+    T0Space X ↔ ∀ a b : X, a ≠ b → a ∉ closure ({b} : Set X) ∨ b ∉ closure ({a} : Set X) := by
   simp only [t0Space_iff_not_inseparable, inseparable_iff_mem_closure, not_and_or]
 #align t0_space_iff_or_not_mem_closure t0Space_iff_or_not_mem_closure
 
-instance Prod.instT0Space [TopologicalSpace β] [T0Space α] [T0Space β] : T0Space (α × β) :=
+instance Prod.instT0Space [TopologicalSpace Y] [T0Space X] [T0Space Y] : T0Space (X × Y) :=
   ⟨fun _ _ h => Prod.ext (h.map continuous_fst).eq (h.map continuous_snd).eq⟩
 
-instance Pi.instT0Space {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
-    [∀ i, T0Space (π i)] :
-    T0Space (∀ i, π i) :=
+instance Pi.instT0Space {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
+    [∀ i, T0Space (X i)] :
+    T0Space (∀ i, X i) :=
   ⟨fun _ _ h => funext fun i => (h.map (continuous_apply i)).eq⟩
 #align pi.t0_space Pi.instT0Space
 
-theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set α, x ∈ s ∧ y ∈ s ∧ T0Space s) :
-    T0Space α := by
+theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set X, x ∈ s ∧ y ∈ s ∧ T0Space s) :
+    T0Space X := by
   refine' ⟨fun x y hxy => _⟩
   rcases h x y hxy with ⟨s, hxs, hys, hs⟩
   lift x to s using hxs; lift y to s using hys
@@ -367,7 +366,7 @@ theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set α, x ∈
   exact congr_arg Subtype.val hxy.eq
 #align t0_space.of_cover T0Space.of_cover
 
-theorem T0Space.of_open_cover (h : ∀ x, ∃ s : Set α, x ∈ s ∧ IsOpen s ∧ T0Space s) : T0Space α :=
+theorem T0Space.of_open_cover (h : ∀ x, ∃ s : Set X, x ∈ s ∧ IsOpen s ∧ T0Space s) : T0Space X :=
   T0Space.of_cover fun x _ hxy =>
     let ⟨s, hxs, hso, hs⟩ := h x
     ⟨s, hxs, (hxy.mem_open_iff hso).1 hxs, hs⟩
@@ -376,47 +375,47 @@ theorem T0Space.of_open_cover (h : ∀ x, ∃ s : Set α, x ∈ s ∧ IsOpen s 
 /-- A T₁ space, also known as a Fréchet space, is a topological space
   where every singleton set is closed. Equivalently, for every pair
   `x ≠ y`, there is an open set containing `x` and not `y`. -/
-class T1Space (α : Type u) [TopologicalSpace α] : Prop where
+class T1Space (X : Type u) [TopologicalSpace X] : Prop where
   /-- A singleton in a T₁ space is a closed set. -/
-  t1 : ∀ x, IsClosed ({x} : Set α)
+  t1 : ∀ x, IsClosed ({x} : Set X)
 #align t1_space T1Space
 
-theorem isClosed_singleton [T1Space α] {x : α} : IsClosed ({x} : Set α) :=
+theorem isClosed_singleton [T1Space X] {x : X} : IsClosed ({x} : Set X) :=
   T1Space.t1 x
 #align is_closed_singleton isClosed_singleton
 
-theorem isOpen_compl_singleton [T1Space α] {x : α} : IsOpen ({x}ᶜ : Set α) :=
+theorem isOpen_compl_singleton [T1Space X] {x : X} : IsOpen ({x}ᶜ : Set X) :=
   isClosed_singleton.isOpen_compl
 #align is_open_compl_singleton isOpen_compl_singleton
 
-theorem isOpen_ne [T1Space α] {x : α} : IsOpen { y | y ≠ x } :=
+theorem isOpen_ne [T1Space X] {x : X} : IsOpen { y | y ≠ x } :=
   isOpen_compl_singleton
 #align is_open_ne isOpen_ne
 
 @[to_additive]
-theorem Continuous.isOpen_mulSupport [T1Space α] [One α] [TopologicalSpace β] {f : β → α}
+theorem Continuous.isOpen_mulSupport [T1Space X] [One X] [TopologicalSpace Y] {f : Y → X}
     (hf : Continuous f) : IsOpen (mulSupport f) :=
   isOpen_ne.preimage hf
 #align continuous.is_open_mul_support Continuous.isOpen_mulSupport
 #align continuous.is_open_support Continuous.isOpen_support
 
-theorem Ne.nhdsWithin_compl_singleton [T1Space α] {x y : α} (h : x ≠ y) : 𝓝[{y}ᶜ] x = 𝓝 x :=
+theorem Ne.nhdsWithin_compl_singleton [T1Space X] {x y : X} (h : x ≠ y) : 𝓝[{y}ᶜ] x = 𝓝 x :=
   isOpen_ne.nhdsWithin_eq h
 #align ne.nhds_within_compl_singleton Ne.nhdsWithin_compl_singleton
 
-theorem Ne.nhdsWithin_diff_singleton [T1Space α] {x y : α} (h : x ≠ y) (s : Set α) :
+theorem Ne.nhdsWithin_diff_singleton [T1Space X] {x y : X} (h : x ≠ y) (s : Set X) :
     𝓝[s \ {y}] x = 𝓝[s] x := by
   rw [diff_eq, inter_comm, nhdsWithin_inter_of_mem]
   exact mem_nhdsWithin_of_mem_nhds (isOpen_ne.mem_nhds h)
 #align ne.nhds_within_diff_singleton Ne.nhdsWithin_diff_singleton
 
-lemma nhdsWithin_compl_singleton_le [T1Space α] (x y : α) : 𝓝[{x}ᶜ] x ≤ 𝓝[{y}ᶜ] x := by
+lemma nhdsWithin_compl_singleton_le [T1Space X] (x y : X) : 𝓝[{x}ᶜ] x ≤ 𝓝[{y}ᶜ] x := by
   rcases eq_or_ne x y with rfl|hy
   · exact Eq.le rfl
   · rw [Ne.nhdsWithin_compl_singleton hy]
     exact nhdsWithin_le_nhds
 
-theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
+theorem isOpen_setOf_eventually_nhdsWithin [T1Space X] {p : X → Prop} :
     IsOpen { x | ∀ᶠ y in 𝓝[≠] x, p y } := by
   refine' isOpen_iff_mem_nhds.mpr fun a ha => _
   filter_upwards [eventually_nhds_nhdsWithin.mpr ha] with b hb
@@ -426,37 +425,37 @@ theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
     exact hb.filter_mono nhdsWithin_le_nhds
 #align is_open_set_of_eventually_nhds_within isOpen_setOf_eventually_nhdsWithin
 
-protected theorem Set.Finite.isClosed [T1Space α] {s : Set α} (hs : Set.Finite s) : IsClosed s := by
+protected theorem Set.Finite.isClosed [T1Space X] {s : Set X} (hs : Set.Finite s) : IsClosed s := by
   rw [← biUnion_of_singleton s]
   exact hs.isClosed_biUnion fun i _ => isClosed_singleton
 #align set.finite.is_closed Set.Finite.isClosed
 
-theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space α] {b : Set (Set α)}
-    (hb : IsTopologicalBasis b) {x y : α} (h : x ≠ y) : ∃ a ∈ b, x ∈ a ∧ y ∉ a := by
+theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space X] {b : Set (Set X)}
+    (hb : IsTopologicalBasis b) {x y : X} (h : x ≠ y) : ∃ a ∈ b, x ∈ a ∧ y ∉ a := by
   rcases hb.isOpen_iff.1 isOpen_ne x h with ⟨a, ab, xa, ha⟩
   exact ⟨a, ab, xa, fun h => ha h rfl⟩
 #align topological_space.is_topological_basis.exists_mem_of_ne TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne
 
-theorem Filter.coclosedCompact_le_cofinite [T1Space α] :
-    Filter.coclosedCompact α ≤ Filter.cofinite := fun s hs =>
+theorem Filter.coclosedCompact_le_cofinite [T1Space X] :
+    Filter.coclosedCompact X ≤ Filter.cofinite := fun s hs =>
   compl_compl s ▸ hs.isCompact.compl_mem_coclosedCompact_of_isClosed hs.isClosed
 #align filter.coclosed_compact_le_cofinite Filter.coclosedCompact_le_cofinite
 
-variable (α)
+variable (X)
 
 /-- In a `T1Space`, relatively compact sets form a bornology. Its cobounded filter is
 `Filter.coclosedCompact`. See also `Bornology.inCompact` the bornology of sets contained
 in a compact set. -/
-def Bornology.relativelyCompact [T1Space α] : Bornology α where
-  cobounded' := Filter.coclosedCompact α
+def Bornology.relativelyCompact [T1Space X] : Bornology X where
+  cobounded' := Filter.coclosedCompact X
   le_cofinite' := Filter.coclosedCompact_le_cofinite
 #align bornology.relatively_compact Bornology.relativelyCompact
 
-variable {α}
+variable {X}
 
-theorem Bornology.relativelyCompact.isBounded_iff [T1Space α] {s : Set α} :
-    @Bornology.IsBounded _ (Bornology.relativelyCompact α) s ↔ IsCompact (closure s) := by
-  change sᶜ ∈ Filter.coclosedCompact α ↔ _
+theorem Bornology.relativelyCompact.isBounded_iff [T1Space X] {s : Set X} :
+    @Bornology.IsBounded _ (Bornology.relativelyCompact X) s ↔ IsCompact (closure s) := by
+  change sᶜ ∈ Filter.coclosedCompact X ↔ _
   rw [Filter.mem_coclosedCompact]
   constructor
   · rintro ⟨t, ht₁, ht₂, hst⟩
@@ -466,21 +465,21 @@ theorem Bornology.relativelyCompact.isBounded_iff [T1Space α] {s : Set α} :
     exact ⟨closure s, isClosed_closure, h, compl_subset_compl.mpr subset_closure⟩
 #align bornology.relatively_compact.is_bounded_iff Bornology.relativelyCompact.isBounded_iff
 
-protected theorem Finset.isClosed [T1Space α] (s : Finset α) : IsClosed (s : Set α) :=
+protected theorem Finset.isClosed [T1Space X] (s : Finset X) : IsClosed (s : Set X) :=
   s.finite_toSet.isClosed
 #align finset.is_closed Finset.isClosed
 
-theorem t1Space_TFAE (α : Type u) [ TopologicalSpace α ] :
-    List.TFAE [T1Space α,
-      ∀ x, IsClosed ({ x } : Set α),
-      ∀ x, IsOpen ({ x }ᶜ : Set α),
-      Continuous (@CofiniteTopology.of α),
-      ∀ ⦃x y : α⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
-      ∀ ⦃x y : α⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
-      ∀ ⦃x y : α⦄, x ≠ y → ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ y ∉ U,
-      ∀ ⦃x y : α⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
-      ∀ ⦃x y : α⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
-      ∀ ⦃x y : α⦄, x ⤳ y → x = y] := by
+theorem t1Space_TFAE (X : Type u) [TopologicalSpace X] :
+    List.TFAE [T1Space X,
+      ∀ x, IsClosed ({ x } : Set X),
+      ∀ x, IsOpen ({ x }ᶜ : Set X),
+      Continuous (@CofiniteTopology.of X),
+      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
+      ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
+      ∀ ⦃x y : X⦄, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U,
+      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
+      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
+      ∀ ⦃x y : X⦄, x ⤳ y → x = y] := by
   tfae_have 1 ↔ 2
   · exact ⟨fun h => h.1, fun h => ⟨h⟩⟩
   tfae_have 2 ↔ 3
@@ -509,85 +508,85 @@ theorem t1Space_TFAE (α : Type u) [ TopologicalSpace α ] :
   tfae_finish
 #align t1_space_tfae t1Space_TFAE
 
-theorem t1Space_iff_continuous_cofinite_of {α : Type*} [TopologicalSpace α] :
-    T1Space α ↔ Continuous (@CofiniteTopology.of α) :=
-  (t1Space_TFAE α).out 0 3
+theorem t1Space_iff_continuous_cofinite_of : T1Space X ↔ Continuous (@CofiniteTopology.of X) :=
+  (t1Space_TFAE X).out 0 3
 #align t1_space_iff_continuous_cofinite_of t1Space_iff_continuous_cofinite_of
 
-theorem CofiniteTopology.continuous_of [T1Space α] : Continuous (@CofiniteTopology.of α) :=
+theorem CofiniteTopology.continuous_of [T1Space X] : Continuous (@CofiniteTopology.of X) :=
   t1Space_iff_continuous_cofinite_of.mp ‹_›
 #align cofinite_topology.continuous_of CofiniteTopology.continuous_of
 
 theorem t1Space_iff_exists_open :
-    T1Space α ↔ ∀ x y, x ≠ y → ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ y ∉ U :=
-  (t1Space_TFAE α).out 0 6
+    T1Space X ↔ ∀ x y, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U :=
+  (t1Space_TFAE X).out 0 6
 #align t1_space_iff_exists_open t1Space_iff_exists_open
 
-theorem t1Space_iff_disjoint_pure_nhds : T1Space α ↔ ∀ ⦃x y : α⦄, x ≠ y → Disjoint (pure x) (𝓝 y) :=
-  (t1Space_TFAE α).out 0 8
+theorem t1Space_iff_disjoint_pure_nhds : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y) :=
+  (t1Space_TFAE X).out 0 8
 #align t1_space_iff_disjoint_pure_nhds t1Space_iff_disjoint_pure_nhds
 
-theorem t1Space_iff_disjoint_nhds_pure : T1Space α ↔ ∀ ⦃x y : α⦄, x ≠ y → Disjoint (𝓝 x) (pure y) :=
-  (t1Space_TFAE α).out 0 7
+theorem t1Space_iff_disjoint_nhds_pure : T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y) :=
+  (t1Space_TFAE X).out 0 7
 #align t1_space_iff_disjoint_nhds_pure t1Space_iff_disjoint_nhds_pure
 
-theorem t1Space_iff_specializes_imp_eq : T1Space α ↔ ∀ ⦃x y : α⦄, x ⤳ y → x = y :=
-  (t1Space_TFAE α).out 0 9
+theorem t1Space_iff_specializes_imp_eq : T1Space X ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y :=
+  (t1Space_TFAE X).out 0 9
 #align t1_space_iff_specializes_imp_eq t1Space_iff_specializes_imp_eq
 
-theorem disjoint_pure_nhds [T1Space α] {x y : α} (h : x ≠ y) : Disjoint (pure x) (𝓝 y) :=
+theorem disjoint_pure_nhds [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (pure x) (𝓝 y) :=
   t1Space_iff_disjoint_pure_nhds.mp ‹_› h
 #align disjoint_pure_nhds disjoint_pure_nhds
 
-theorem disjoint_nhds_pure [T1Space α] {x y : α} (h : x ≠ y) : Disjoint (𝓝 x) (pure y) :=
+theorem disjoint_nhds_pure [T1Space X] {x y : X} (h : x ≠ y) : Disjoint (𝓝 x) (pure y) :=
   t1Space_iff_disjoint_nhds_pure.mp ‹_› h
 #align disjoint_nhds_pure disjoint_nhds_pure
 
-theorem Specializes.eq [T1Space α] {x y : α} (h : x ⤳ y) : x = y :=
+theorem Specializes.eq [T1Space X] {x y : X} (h : x ⤳ y) : x = y :=
   t1Space_iff_specializes_imp_eq.1 ‹_› h
 #align specializes.eq Specializes.eq
 
-theorem specializes_iff_eq [T1Space α] {x y : α} : x ⤳ y ↔ x = y :=
+theorem specializes_iff_eq [T1Space X] {x y : X} : x ⤳ y ↔ x = y :=
   ⟨Specializes.eq, fun h => h ▸ specializes_rfl⟩
 #align specializes_iff_eq specializes_iff_eq
 
-@[simp] theorem specializes_eq_eq [T1Space α] : (· ⤳ ·) = @Eq α :=
+@[simp] theorem specializes_eq_eq [T1Space X] : (· ⤳ ·) = @Eq X :=
   funext₂ fun _ _ => propext specializes_iff_eq
 #align specializes_eq_eq specializes_eq_eq
 
 @[simp]
-theorem pure_le_nhds_iff [T1Space α] {a b : α} : pure a ≤ 𝓝 b ↔ a = b :=
+theorem pure_le_nhds_iff [T1Space X] {a b : X} : pure a ≤ 𝓝 b ↔ a = b :=
   specializes_iff_pure.symm.trans specializes_iff_eq
 #align pure_le_nhds_iff pure_le_nhds_iff
 
 @[simp]
-theorem nhds_le_nhds_iff [T1Space α] {a b : α} : 𝓝 a ≤ 𝓝 b ↔ a = b :=
+theorem nhds_le_nhds_iff [T1Space X] {a b : X} : 𝓝 a ≤ 𝓝 b ↔ a = b :=
   specializes_iff_eq
 #align nhds_le_nhds_iff nhds_le_nhds_iff
 
-instance {α : Type*} : T1Space (CofiniteTopology α) :=
+instance : T1Space (CofiniteTopology X) :=
   t1Space_iff_continuous_cofinite_of.mpr continuous_id
 
-theorem t1Space_antitone {α : Type*} : Antitone (@T1Space α) := fun a _ h _ =>
+theorem t1Space_antitone : Antitone (@T1Space X) := fun a _ h _ =>
   @T1Space.mk _ a fun x => (T1Space.t1 x).mono h
 #align t1_space_antitone t1Space_antitone
 
-theorem continuousWithinAt_update_of_ne [T1Space α] [DecidableEq α] [TopologicalSpace β] {f : α → β}
-    {s : Set α} {x y : α} {z : β} (hne : y ≠ x) :
-    ContinuousWithinAt (Function.update f x z) s y ↔ ContinuousWithinAt f s y :=
+theorem continuousWithinAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
+    {s : Set X} {x x' : X} {y : Y} (hne : x' ≠ x) :
+    ContinuousWithinAt (Function.update f x y) s x' ↔ ContinuousWithinAt f s x' :=
   EventuallyEq.congr_continuousWithinAt
     (mem_nhdsWithin_of_mem_nhds <| mem_of_superset (isOpen_ne.mem_nhds hne) fun _y' hy' =>
       Function.update_noteq hy' _ _)
     (Function.update_noteq hne _ _)
 #align continuous_within_at_update_of_ne continuousWithinAt_update_of_ne
 
-theorem continuousAt_update_of_ne [T1Space α] [DecidableEq α] [TopologicalSpace β] {f : α → β}
-    {x y : α} {z : β} (hne : y ≠ x) : ContinuousAt (Function.update f x z) y ↔ ContinuousAt f y :=
+theorem continuousAt_update_of_ne [T1Space X] [DecidableEq X] [TopologicalSpace Y]
+    {f : X → Y} {x x' : X} {y : Y} (hne : x' ≠ x) :
+    ContinuousAt (Function.update f x y) x' ↔ ContinuousAt f x' :=
   by simp only [← continuousWithinAt_univ, continuousWithinAt_update_of_ne hne]
 #align continuous_at_update_of_ne continuousAt_update_of_ne
 
-theorem continuousOn_update_iff [T1Space α] [DecidableEq α] [TopologicalSpace β] {f : α → β}
-    {s : Set α} {x : α} {y : β} :
+theorem continuousOn_update_iff [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
+    {s : Set X} {x : X} {y : Y} :
     ContinuousOn (Function.update f x y) s ↔
       ContinuousOn f (s \ {x}) ∧ (x ∈ s → Tendsto f (𝓝[s \ {x}] x) (𝓝 y)) := by
   rw [ContinuousOn, ← and_forall_ne x, and_comm]
@@ -601,64 +600,64 @@ theorem continuousOn_update_iff [T1Space α] [DecidableEq α] [TopologicalSpace
   · exact continuousWithinAt_update_same
 #align continuous_on_update_iff continuousOn_update_iff
 
-theorem t1Space_of_injective_of_continuous [TopologicalSpace β] {f : α → β}
-    (hf : Function.Injective f) (hf' : Continuous f) [T1Space β] : T1Space α :=
+theorem t1Space_of_injective_of_continuous [TopologicalSpace Y] {f : X → Y}
+    (hf : Function.Injective f) (hf' : Continuous f) [T1Space Y] : T1Space X :=
   t1Space_iff_specializes_imp_eq.2 fun _ _ h => hf (h.map hf').eq
 #align t1_space_of_injective_of_continuous t1Space_of_injective_of_continuous
 
-protected theorem Embedding.t1Space [TopologicalSpace β] [T1Space β] {f : α → β}
-    (hf : Embedding f) : T1Space α :=
+protected theorem Embedding.t1Space [TopologicalSpace Y] [T1Space Y] {f : X → Y}
+    (hf : Embedding f) : T1Space X :=
   t1Space_of_injective_of_continuous hf.inj hf.continuous
 #align embedding.t1_space Embedding.t1Space
 
-instance Subtype.t1Space {α : Type u} [TopologicalSpace α] [T1Space α] {p : α → Prop} :
+instance Subtype.t1Space {X : Type u} [TopologicalSpace X] [T1Space X] {p : X → Prop} :
     T1Space (Subtype p) :=
   embedding_subtype_val.t1Space
 #align subtype.t1_space Subtype.t1Space
 
-instance [TopologicalSpace β] [T1Space α] [T1Space β] : T1Space (α × β) :=
+instance [TopologicalSpace Y] [T1Space X] [T1Space Y] : T1Space (X × Y) :=
   ⟨fun ⟨a, b⟩ => @singleton_prod_singleton _ _ a b ▸ isClosed_singleton.prod isClosed_singleton⟩
 
-instance {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)] [∀ i, T1Space (π i)] :
-    T1Space (∀ i, π i) :=
+instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, T1Space (X i)] :
+    T1Space (∀ i, X i) :=
   ⟨fun f => univ_pi_singleton f ▸ isClosed_set_pi fun _ _ => isClosed_singleton⟩
 
 -- see Note [lower instance priority]
-instance (priority := 100) T1Space.t0Space [T1Space α] : T0Space α :=
+instance (priority := 100) T1Space.t0Space [T1Space X] : T0Space X :=
   ⟨fun _ _ h => h.specializes.eq⟩
 #align t1_space.t0_space T1Space.t0Space
 
 @[simp]
-theorem compl_singleton_mem_nhds_iff [T1Space α] {x y : α} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x :=
+theorem compl_singleton_mem_nhds_iff [T1Space X] {x y : X} : {x}ᶜ ∈ 𝓝 y ↔ y ≠ x :=
   isOpen_compl_singleton.mem_nhds_iff
 #align compl_singleton_mem_nhds_iff compl_singleton_mem_nhds_iff
 
-theorem compl_singleton_mem_nhds [T1Space α] {x y : α} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y :=
+theorem compl_singleton_mem_nhds [T1Space X] {x y : X} (h : y ≠ x) : {x}ᶜ ∈ 𝓝 y :=
   compl_singleton_mem_nhds_iff.mpr h
 #align compl_singleton_mem_nhds compl_singleton_mem_nhds
 
 @[simp]
-theorem closure_singleton [T1Space α] {a : α} : closure ({a} : Set α) = {a} :=
+theorem closure_singleton [T1Space X] {x : X} : closure ({x} : Set X) = {x} :=
   isClosed_singleton.closure_eq
 #align closure_singleton closure_singleton
 
 -- porting note: todo: the proof was `hs.induction_on (by simp) fun x => by simp`
-theorem Set.Subsingleton.closure [T1Space α] {s : Set α} (hs : s.Subsingleton) :
+theorem Set.Subsingleton.closure [T1Space X] {s : Set X} (hs : s.Subsingleton) :
     (closure s).Subsingleton := by
   rcases hs.eq_empty_or_singleton with (rfl | ⟨x, rfl⟩) <;> simp
 #align set.subsingleton.closure Set.Subsingleton.closure
 
 @[simp]
-theorem subsingleton_closure [T1Space α] {s : Set α} : (closure s).Subsingleton ↔ s.Subsingleton :=
+theorem subsingleton_closure [T1Space X] {s : Set X} : (closure s).Subsingleton ↔ s.Subsingleton :=
   ⟨fun h => h.anti subset_closure, fun h => h.closure⟩
 #align subsingleton_closure subsingleton_closure
 
-theorem isClosedMap_const {α β} [TopologicalSpace α] [TopologicalSpace β] [T1Space β] {y : β} :
-    IsClosedMap (Function.const α y) :=
+theorem isClosedMap_const {X Y} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {y : Y} :
+    IsClosedMap (Function.const X y) :=
   IsClosedMap.of_nonempty fun s _ h2s => by simp_rw [const, h2s.image_const, isClosed_singleton]
 #align is_closed_map_const isClosedMap_const
 
-theorem nhdsWithin_insert_of_ne [T1Space α] {x y : α} {s : Set α} (hxy : x ≠ y) :
+theorem nhdsWithin_insert_of_ne [T1Space X] {x y : X} {s : Set X} (hxy : x ≠ y) :
     𝓝[insert y s] x = 𝓝[s] x := by
   refine' le_antisymm (Filter.le_def.2 fun t ht => _) (nhdsWithin_mono x <| subset_insert y s)
   obtain ⟨o, ho, hxo, host⟩ := mem_nhdsWithin.mp ht
@@ -669,7 +668,7 @@ theorem nhdsWithin_insert_of_ne [T1Space α] {x y : α} {s : Set α} (hxy : x 
 
 /-- If `t` is a subset of `s`, except for one point,
 then `insert x s` is a neighborhood of `x` within `t`. -/
-theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Set α}
+theorem insert_mem_nhdsWithin_of_subset_insert [T1Space X] {x y : X} {s t : Set X}
     (hu : t ⊆ insert y s) : insert x s ∈ 𝓝[t] x := by
   rcases eq_or_ne x y with (rfl | h)
   · exact mem_of_superset self_mem_nhdsWithin hu
@@ -679,21 +678,21 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
 #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert
 
 @[simp]
-theorem ker_nhds [T1Space α] (x : α) : (𝓝 x).ker = {x} := by
+theorem ker_nhds [T1Space X] (x : X) : (𝓝 x).ker = {x} := by
   simp [ker_nhds_eq_specializes]
 
-theorem biInter_basis_nhds [T1Space α] {ι : Sort*} {p : ι → Prop} {s : ι → Set α} {x : α}
+theorem biInter_basis_nhds [T1Space X] {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {x : X}
     (h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} := by
   rw [← h.ker, ker_nhds]
 #align bInter_basis_nhds biInter_basis_nhds
 
 @[simp]
-theorem compl_singleton_mem_nhdsSet_iff [T1Space α] {x : α} {s : Set α} : {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s := by
+theorem compl_singleton_mem_nhdsSet_iff [T1Space X] {x : X} {s : Set X} : {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s := by
   rw [isOpen_compl_singleton.mem_nhdsSet, subset_compl_singleton_iff]
 #align compl_singleton_mem_nhds_set_iff compl_singleton_mem_nhdsSet_iff
 
 @[simp]
-theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t := by
+theorem nhdsSet_le_iff [T1Space X] {s t : Set X} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t := by
   refine' ⟨_, fun h => monotone_nhdsSet h⟩
   simp_rw [Filter.le_def]; intro h x hx
   specialize h {x}ᶜ
@@ -703,34 +702,34 @@ theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s
 #align nhds_set_le_iff nhdsSet_le_iff
 
 @[simp]
-theorem nhdsSet_inj_iff [T1Space α] {s t : Set α} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t := by
+theorem nhdsSet_inj_iff [T1Space X] {s t : Set X} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t := by
   simp_rw [le_antisymm_iff]
   exact and_congr nhdsSet_le_iff nhdsSet_le_iff
 #align nhds_set_inj_iff nhdsSet_inj_iff
 
-theorem injective_nhdsSet [T1Space α] : Function.Injective (𝓝ˢ : Set α → Filter α) := fun _ _ hst =>
+theorem injective_nhdsSet [T1Space X] : Function.Injective (𝓝ˢ : Set X → Filter X) := fun _ _ hst =>
   nhdsSet_inj_iff.mp hst
 #align injective_nhds_set injective_nhdsSet
 
-theorem strictMono_nhdsSet [T1Space α] : StrictMono (𝓝ˢ : Set α → Filter α) :=
+theorem strictMono_nhdsSet [T1Space X] : StrictMono (𝓝ˢ : Set X → Filter X) :=
   monotone_nhdsSet.strictMono_of_injective injective_nhdsSet
 #align strict_mono_nhds_set strictMono_nhdsSet
 
 @[simp]
-theorem nhds_le_nhdsSet_iff [T1Space α] {s : Set α} {x : α} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s := by
+theorem nhds_le_nhdsSet_iff [T1Space X] {s : Set X} {x : X} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s := by
   rw [← nhdsSet_singleton, nhdsSet_le_iff, singleton_subset_iff]
 #align nhds_le_nhds_set_iff nhds_le_nhdsSet_iff
 
 /-- Removing a non-isolated point from a dense set, one still obtains a dense set. -/
-theorem Dense.diff_singleton [T1Space α] {s : Set α} (hs : Dense s) (x : α) [NeBot (𝓝[≠] x)] :
+theorem Dense.diff_singleton [T1Space X] {s : Set X} (hs : Dense s) (x : X) [NeBot (𝓝[≠] x)] :
     Dense (s \ {x}) :=
   hs.inter_of_open_right (dense_compl_singleton x) isOpen_compl_singleton
 #align dense.diff_singleton Dense.diff_singleton
 
 /-- Removing a finset from a dense set in a space without isolated points, one still
 obtains a dense set. -/
-theorem Dense.diff_finset [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Set α} (hs : Dense s)
-    (t : Finset α) : Dense (s \ t) := by
+theorem Dense.diff_finset [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
+    (t : Finset X) : Dense (s \ t) := by
   induction t using Finset.induction_on with
   | empty => simpa using hs
   | insert _ ih =>
@@ -740,54 +739,54 @@ theorem Dense.diff_finset [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Se
 
 /-- Removing a finite set from a dense set in a space without isolated points, one still
 obtains a dense set. -/
-theorem Dense.diff_finite [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Set α} (hs : Dense s)
-    {t : Set α} (ht : t.Finite) : Dense (s \ t) := by
+theorem Dense.diff_finite [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
+    {t : Set X} (ht : t.Finite) : Dense (s \ t) := by
   convert hs.diff_finset ht.toFinset
   exact (Finite.coe_toFinset _).symm
 #align dense.diff_finite Dense.diff_finite
 
-/-- If a function to a `T1Space` tends to some limit `b` at some point `a`, then necessarily
-`b = f a`. -/
-theorem eq_of_tendsto_nhds [TopologicalSpace β] [T1Space β] {f : α → β} {a : α} {b : β}
-    (h : Tendsto f (𝓝 a) (𝓝 b)) : f a = b :=
-  by_contra fun hfa : f a ≠ b =>
-    have fact₁ : {f a}ᶜ ∈ 𝓝 b := compl_singleton_mem_nhds hfa.symm
-    have fact₂ : Tendsto f (pure a) (𝓝 b) := h.comp (tendsto_id'.2 <| pure_le_nhds a)
-    fact₂ fact₁ (Eq.refl <| f a)
+/-- If a function to a `T1Space` tends to some limit `y` at some point `x`, then necessarily
+`y = f x`. -/
+theorem eq_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
+    (h : Tendsto f (𝓝 x) (𝓝 y)) : f x = y :=
+  by_contra fun hfa : f x ≠ y =>
+    have fact₁ : {f x}ᶜ ∈ 𝓝 y := compl_singleton_mem_nhds hfa.symm
+    have fact₂ : Tendsto f (pure x) (𝓝 y) := h.comp (tendsto_id'.2 <| pure_le_nhds x)
+    fact₂ fact₁ (Eq.refl <| f x)
 #align eq_of_tendsto_nhds eq_of_tendsto_nhds
 
-theorem Filter.Tendsto.eventually_ne [TopologicalSpace β] [T1Space β] {α : Type*} {g : α → β}
-    {l : Filter α} {b₁ b₂ : β} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ :=
+theorem Filter.Tendsto.eventually_ne [TopologicalSpace Y] [T1Space Y] {g : X → Y}
+    {l : Filter X} {b₁ b₂ : Y} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ :=
   hg.eventually (isOpen_compl_singleton.eventually_mem hb)
 #align filter.tendsto.eventually_ne Filter.Tendsto.eventually_ne
 
-theorem ContinuousAt.eventually_ne [TopologicalSpace β] [T1Space β] {g : α → β} {a : α} {b : β}
-    (hg1 : ContinuousAt g a) (hg2 : g a ≠ b) : ∀ᶠ z in 𝓝 a, g z ≠ b :=
+theorem ContinuousAt.eventually_ne [TopologicalSpace Y] [T1Space Y] {g : X → Y} {x : X} {y : Y}
+    (hg1 : ContinuousAt g x) (hg2 : g x ≠ y) : ∀ᶠ z in 𝓝 x, g z ≠ y :=
   hg1.tendsto.eventually_ne hg2
 #align continuous_at.eventually_ne ContinuousAt.eventually_ne
 
-theorem eventually_ne_nhds [T1Space α] {a b : α} (h : a ≠ b) : ∀ᶠ x in 𝓝 a, x ≠ b :=
+theorem eventually_ne_nhds [T1Space X] {a b : X} (h : a ≠ b) : ∀ᶠ x in 𝓝 a, x ≠ b :=
   IsOpen.eventually_mem isOpen_ne h
 
-theorem eventually_ne_nhdsWithin [T1Space α] {a b : α} {s : Set α} (h : a ≠ b) :
+theorem eventually_ne_nhdsWithin [T1Space X] {a b : X} {s : Set X} (h : a ≠ b) :
     ∀ᶠ x in 𝓝[s] a, x ≠ b :=
   Filter.Eventually.filter_mono nhdsWithin_le_nhds <| eventually_ne_nhds h
 
-/-- To prove a function to a `T1Space` is continuous at some point `a`, it suffices to prove that
-`f` admits *some* limit at `a`. -/
-theorem continuousAt_of_tendsto_nhds [TopologicalSpace β] [T1Space β] {f : α → β} {a : α} {b : β}
-    (h : Tendsto f (𝓝 a) (𝓝 b)) : ContinuousAt f a := by
+/-- To prove a function to a `T1Space` is continuous at some point `x`, it suffices to prove that
+`f` admits *some* limit at `x`. -/
+theorem continuousAt_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
+    (h : Tendsto f (𝓝 x) (𝓝 y)) : ContinuousAt f x := by
   rwa [ContinuousAt, eq_of_tendsto_nhds h]
 #align continuous_at_of_tendsto_nhds continuousAt_of_tendsto_nhds
 
 @[simp]
-theorem tendsto_const_nhds_iff [T1Space α] {l : Filter β} [NeBot l] {c d : α} :
+theorem tendsto_const_nhds_iff [T1Space X] {l : Filter Y} [NeBot l] {c d : X} :
     Tendsto (fun _ => c) l (𝓝 d) ↔ c = d := by simp_rw [Tendsto, Filter.map_const, pure_le_nhds_iff]
 #align tendsto_const_nhds_iff tendsto_const_nhds_iff
 
 /-- A point with a finite neighborhood has to be isolated. -/
-theorem isOpen_singleton_of_finite_mem_nhds {α : Type*} [TopologicalSpace α] [T1Space α] (x : α)
-    {s : Set α} (hs : s ∈ 𝓝 x) (hsf : s.Finite) : IsOpen ({x} : Set α) := by
+theorem isOpen_singleton_of_finite_mem_nhds [T1Space X] (x : X)
+    {s : Set X} (hs : s ∈ 𝓝 x) (hsf : s.Finite) : IsOpen ({x} : Set X) := by
   have A : {x} ⊆ s := by simp only [singleton_subset_iff, mem_of_mem_nhds hs]
   have B : IsClosed (s \ {x}) := (hsf.subset (diff_subset _ _)).isClosed
   have C : (s \ {x})ᶜ ∈ 𝓝 x := B.isOpen_compl.mem_nhds fun h => h.2 rfl
@@ -797,14 +796,14 @@ theorem isOpen_singleton_of_finite_mem_nhds {α : Type*} [TopologicalSpace α] [
 
 /-- If the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is
 infinite. -/
-theorem infinite_of_mem_nhds {α} [TopologicalSpace α] [T1Space α] (x : α) [hx : NeBot (𝓝[≠] x)]
-    {s : Set α} (hs : s ∈ 𝓝 x) : Set.Infinite s := by
+theorem infinite_of_mem_nhds {X} [TopologicalSpace X] [T1Space X] (x : X) [hx : NeBot (𝓝[≠] x)]
+    {s : Set X} (hs : s ∈ 𝓝 x) : Set.Infinite s := by
   refine' fun hsf => hx.1 _
   rw [← isOpen_singleton_iff_punctured_nhds]
   exact isOpen_singleton_of_finite_mem_nhds x hs hsf
 #align infinite_of_mem_nhds infinite_of_mem_nhds
 
-theorem discrete_of_t1_of_finite {X : Type*} [TopologicalSpace X] [T1Space X] [Finite X] :
+theorem discrete_of_t1_of_finite [T1Space X] [Finite X] :
     DiscreteTopology X := by
   apply singletons_open_iff_discrete.mp
   intro x
@@ -812,28 +811,28 @@ theorem discrete_of_t1_of_finite {X : Type*} [TopologicalSpace X] [T1Space X] [F
   exact (Set.toFinite _).isClosed
 #align discrete_of_t1_of_finite discrete_of_t1_of_finite
 
-theorem PreconnectedSpace.trivial_of_discrete [PreconnectedSpace α] [DiscreteTopology α] :
-    Subsingleton α := by
+theorem PreconnectedSpace.trivial_of_discrete [PreconnectedSpace X] [DiscreteTopology X] :
+    Subsingleton X := by
   rw [← not_nontrivial_iff_subsingleton]
   rintro ⟨x, y, hxy⟩
   rw [Ne.def, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
   exact hxy (mem_univ x)
 #align preconnected_space.trivial_of_discrete PreconnectedSpace.trivial_of_discrete
 
-theorem IsPreconnected.infinite_of_nontrivial [T1Space α] {s : Set α} (h : IsPreconnected s)
+theorem IsPreconnected.infinite_of_nontrivial [T1Space X] {s : Set X} (h : IsPreconnected s)
     (hs : s.Nontrivial) : s.Infinite := by
   refine' mt (fun hf => (subsingleton_coe s).mp _) (not_subsingleton_iff.mpr hs)
   haveI := @discrete_of_t1_of_finite s _ _ hf.to_subtype
   exact @PreconnectedSpace.trivial_of_discrete _ _ (Subtype.preconnectedSpace h) _
 #align is_preconnected.infinite_of_nontrivial IsPreconnected.infinite_of_nontrivial
 
-theorem ConnectedSpace.infinite [ConnectedSpace α] [Nontrivial α] [T1Space α] : Infinite α :=
+theorem ConnectedSpace.infinite [ConnectedSpace X] [Nontrivial X] [T1Space X] : Infinite X :=
   infinite_univ_iff.mp <| isPreconnected_univ.infinite_of_nontrivial nontrivial_univ
 #align connected_space.infinite ConnectedSpace.infinite
 
 /-- A non-trivial connected T1 space has no isolated points. -/
 instance ConnectedSpace.neBot_nhdsWithin_compl_of_nontrivial_of_t1space
-    [ConnectedSpace α] [Nontrivial α] [T1Space α] (x : α) :
+    [ConnectedSpace X] [Nontrivial X] [T1Space X] (x : X) :
     NeBot (𝓝[≠] x) := by
   by_contra contra
   rw [not_neBot, ← isOpen_singleton_iff_punctured_nhds] at contra
@@ -841,22 +840,22 @@ instance ConnectedSpace.neBot_nhdsWithin_compl_of_nontrivial_of_t1space
     contra (compl_union_self _) (Set.nonempty_compl_of_nontrivial _) (singleton_nonempty _)
   simp [compl_inter_self {x}] at contra
 
-theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α}
+theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
     (hx : x ∈ s) : {x} ∈ 𝓝[s] x := by
   have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete]
   simpa only [nhdsWithin_eq_map_subtype_coe hx, image_singleton] using
-    @image_mem_map _ _ _ ((↑) : s → α) _ this
+    @image_mem_map _ _ _ ((↑) : s → X) _ this
 #align singleton_mem_nhds_within_of_mem_discrete singleton_mem_nhdsWithin_of_mem_discrete
 
 /-- The neighbourhoods filter of `x` within `s`, under the discrete topology, is equal to
 the pure `x` filter (which is the principal filter at the singleton `{x}`.) -/
-theorem nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α} (hx : x ∈ s) :
+theorem nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) :
     𝓝[s] x = pure x :=
   le_antisymm (le_pure_iff.2 <| singleton_mem_nhdsWithin_of_mem_discrete hx) (pure_le_nhdsWithin hx)
 #align nhds_within_of_mem_discrete nhdsWithin_of_mem_discrete
 
 theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type*} {p : ι → Prop}
-    {t : ι → Set α} {s : Set α} [DiscreteTopology s] {x : α} (hb : (𝓝 x).HasBasis p t)
+    {t : ι → Set X} {s : Set X} [DiscreteTopology s] {x : X} (hb : (𝓝 x).HasBasis p t)
     (hx : x ∈ s) : ∃ i, p i ∧ t i ∩ s = {x} := by
   rcases (nhdsWithin_hasBasis hb s).mem_iff.1 (singleton_mem_nhdsWithin_of_mem_discrete hx) with
     ⟨i, hi, hix⟩
@@ -865,7 +864,7 @@ theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type*} {
 
 /-- A point `x` in a discrete subset `s` of a topological space admits a neighbourhood
 that only meets `s` at `x`.  -/
-theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α}
+theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
     (hx : x ∈ s) : ∃ U ∈ 𝓝 x, U ∩ s = {x} := by
   simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx
 #align nhds_inter_eq_singleton_of_mem_discrete nhds_inter_eq_singleton_of_mem_discrete
@@ -875,7 +874,7 @@ such that
 1. `U` is a punctured neighborhood of `x` (ie. `U ∪ {x}` is a neighbourhood of `x`),
 2. `U` is disjoint from `s`.
 -/
-theorem disjoint_nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α} (hx : x ∈ s) :
+theorem disjoint_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X} (hx : x ∈ s) :
     ∃ U ∈ 𝓝[≠] x, Disjoint U s :=
   let ⟨V, h, h'⟩ := nhds_inter_eq_singleton_of_mem_discrete hx
   ⟨{x}ᶜ ∩ V, inter_mem_nhdsWithin _ h,
@@ -886,8 +885,7 @@ theorem disjoint_nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x
 `t ⊆ s`, then the topological space structure on `t` induced by `X` is the same as the one
 obtained by the induced topological space structure on `s`. Use `embedding_inclusion` instead. -/
 @[deprecated embedding_inclusion]
-theorem TopologicalSpace.subset_trans {X : Type*} [TopologicalSpace X] {s t : Set X}
-    (ts : t ⊆ s) :
+theorem TopologicalSpace.subset_trans {s t : Set X} (ts : t ⊆ s) :
     (instTopologicalSpaceSubtype : TopologicalSpace t) =
       (instTopologicalSpaceSubtype : TopologicalSpace s).induced (Set.inclusion ts) :=
   (embedding_inclusion ts).induced
@@ -897,44 +895,44 @@ theorem TopologicalSpace.subset_trans {X : Type*} [TopologicalSpace X] {s t : Se
   `x ≠ y` there exists disjoint open sets around `x` and `y`. This is
   the most widely used of the separation axioms. -/
 @[mk_iff t2Space_iff]
-class T2Space (α : Type u) [TopologicalSpace α] : Prop where
+class T2Space (X : Type u) [TopologicalSpace X] : Prop where
   /-- Every two points in a Hausdorff space admit disjoint open neighbourhoods. -/
-  t2 : ∀ x y, x ≠ y → ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
+  t2 : ∀ x y, x ≠ y → ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
 #align t2_space T2Space
 
 /-- Two different points can be separated by open sets. -/
-theorem t2_separation [T2Space α] {x y : α} (h : x ≠ y) :
-    ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
+theorem t2_separation [T2Space X] {x y : X} (h : x ≠ y) :
+    ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
   T2Space.t2 x y h
 #align t2_separation t2_separation
 
 -- todo: use this as a definition?
-theorem t2Space_iff_disjoint_nhds : T2Space α ↔ ∀ x y : α, x ≠ y → Disjoint (𝓝 x) (𝓝 y) := by
-  refine (t2Space_iff α).trans (forall₃_congr fun x y _ => ?_)
+theorem t2Space_iff_disjoint_nhds : T2Space X ↔ ∀ x y : X, x ≠ y → Disjoint (𝓝 x) (𝓝 y) := by
+  refine (t2Space_iff X).trans (forall₃_congr fun x y _ => ?_)
   simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y), exists_prop, ← exists_and_left,
     and_assoc, and_comm, and_left_comm]
 #align t2_space_iff_disjoint_nhds t2Space_iff_disjoint_nhds
 
 @[simp]
-theorem disjoint_nhds_nhds [T2Space α] {x y : α} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y :=
+theorem disjoint_nhds_nhds [T2Space X] {x y : X} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y :=
   ⟨fun hd he => by simp [he, nhds_neBot.ne] at hd, t2Space_iff_disjoint_nhds.mp ‹_› x y⟩
 #align disjoint_nhds_nhds disjoint_nhds_nhds
 
-theorem pairwise_disjoint_nhds [T2Space α] : Pairwise (Disjoint on (𝓝 : α → Filter α)) := fun _ _ =>
+theorem pairwise_disjoint_nhds [T2Space X] : Pairwise (Disjoint on (𝓝 : X → Filter X)) := fun _ _ =>
   disjoint_nhds_nhds.2
 #align pairwise_disjoint_nhds pairwise_disjoint_nhds
 
-protected theorem Set.pairwiseDisjoint_nhds [T2Space α] (s : Set α) : s.PairwiseDisjoint 𝓝 :=
+protected theorem Set.pairwiseDisjoint_nhds [T2Space X] (s : Set X) : s.PairwiseDisjoint 𝓝 :=
   pairwise_disjoint_nhds.set_pairwise s
 #align set.pairwise_disjoint_nhds Set.pairwiseDisjoint_nhds
 
 /-- Points of a finite set can be separated by open sets from each other. -/
-theorem Set.Finite.t2_separation [T2Space α] {s : Set α} (hs : s.Finite) :
-    ∃ U : α → Set α, (∀ x, x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U :=
+theorem Set.Finite.t2_separation [T2Space X] {s : Set X} (hs : s.Finite) :
+    ∃ U : X → Set X, (∀ x, x ∈ U x ∧ IsOpen (U x)) ∧ s.PairwiseDisjoint U :=
   s.pairwiseDisjoint_nhds.exists_mem_filter_basis hs nhds_basis_opens
 #align set.finite.t2_separation Set.Finite.t2_separation
 
-theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : α × α | Disjoint (𝓝 p.1) (𝓝 p.2) } := by
+theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : X × X | Disjoint (𝓝 p.1) (𝓝 p.2) } := by
   simp only [isOpen_iff_mem_nhds, Prod.forall, mem_setOf_eq]
   intro x y h
   obtain ⟨U, hU, V, hV, hd⟩ := ((nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)).mp h
@@ -945,70 +943,70 @@ theorem isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : α × α | Disjoint (𝓝
 #align is_open_set_of_disjoint_nhds_nhds isOpen_setOf_disjoint_nhds_nhds
 
 -- see Note [lower instance priority]
-instance (priority := 100) T2Space.t1Space [T2Space α] : T1Space α :=
+instance (priority := 100) T2Space.t1Space [T2Space X] : T1Space X :=
   t1Space_iff_disjoint_pure_nhds.mpr fun _ _ hne =>
     (disjoint_nhds_nhds.2 hne).mono_left <| pure_le_nhds _
 #align t2_space.t1_space T2Space.t1Space
 
 /-- A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. -/
-theorem t2_iff_nhds : T2Space α ↔ ∀ {x y : α}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
+theorem t2_iff_nhds : T2Space X ↔ ∀ {x y : X}, NeBot (𝓝 x ⊓ 𝓝 y) → x = y := by
   simp only [t2Space_iff_disjoint_nhds, disjoint_iff, neBot_iff, Ne.def, not_imp_comm]
 #align t2_iff_nhds t2_iff_nhds
 
-theorem eq_of_nhds_neBot [T2Space α] {x y : α} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y :=
+theorem eq_of_nhds_neBot [T2Space X] {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) : x = y :=
   t2_iff_nhds.mp ‹_› h
 #align eq_of_nhds_ne_bot eq_of_nhds_neBot
 
-theorem t2Space_iff_nhds : T2Space α ↔ ∀ {x y : α}, x ≠ y → ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V := by
+theorem t2Space_iff_nhds : T2Space X ↔ ∀ {x y : X}, x ≠ y → ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V := by
   simp only [t2Space_iff_disjoint_nhds, Filter.disjoint_iff]
 #align t2_space_iff_nhds t2Space_iff_nhds
 
-theorem t2_separation_nhds [T2Space α] {x y : α} (h : x ≠ y) :
+theorem t2_separation_nhds [T2Space X] {x y : X} (h : x ≠ y) :
     ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ Disjoint u v :=
   let ⟨u, v, open_u, open_v, x_in, y_in, huv⟩ := t2_separation h
   ⟨u, v, open_u.mem_nhds x_in, open_v.mem_nhds y_in, huv⟩
 #align t2_separation_nhds t2_separation_nhds
 
-theorem t2_separation_compact_nhds [LocallyCompactSpace α] [T2Space α] {x y : α} (h : x ≠ y) :
+theorem t2_separation_compact_nhds [LocallyCompactSpace X] [T2Space X] {x y : X} (h : x ≠ y) :
     ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v := by
   simpa only [exists_prop, ← exists_and_left, and_comm, and_assoc, and_left_comm] using
     ((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h)
 #align t2_separation_compact_nhds t2_separation_compact_nhds
 
 theorem t2_iff_ultrafilter :
-    T2Space α ↔ ∀ {x y : α} (f : Ultrafilter α), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y :=
+    T2Space X ↔ ∀ {x y : X} (f : Ultrafilter X), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y :=
   t2_iff_nhds.trans <| by simp only [← exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp]
 #align t2_iff_ultrafilter t2_iff_ultrafilter
 
-theorem t2_iff_isClosed_diagonal : T2Space α ↔ IsClosed (diagonal α) := by
+theorem t2_iff_isClosed_diagonal : T2Space X ↔ IsClosed (diagonal X) := by
   simp only [t2Space_iff_disjoint_nhds, ← isOpen_compl_iff, isOpen_iff_mem_nhds, Prod.forall,
     nhds_prod_eq, compl_diagonal_mem_prod, mem_compl_iff, mem_diagonal_iff]
 #align t2_iff_is_closed_diagonal t2_iff_isClosed_diagonal
 
-theorem isClosed_diagonal [T2Space α] : IsClosed (diagonal α) :=
+theorem isClosed_diagonal [T2Space X] : IsClosed (diagonal X) :=
   t2_iff_isClosed_diagonal.mp ‹_›
 #align is_closed_diagonal isClosed_diagonal
 
 -- porting note: 2 lemmas moved below
 
-theorem tendsto_nhds_unique [T2Space α] {f : β → α} {l : Filter β} {a b : α} [NeBot l]
+theorem tendsto_nhds_unique [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
     (ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b :=
   eq_of_nhds_neBot <| neBot_of_le <| le_inf ha hb
 #align tendsto_nhds_unique tendsto_nhds_unique
 
-theorem tendsto_nhds_unique' [T2Space α] {f : β → α} {l : Filter β} {a b : α} (_ : NeBot l)
+theorem tendsto_nhds_unique' [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} (_ : NeBot l)
     (ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b :=
   eq_of_nhds_neBot <| neBot_of_le <| le_inf ha hb
 #align tendsto_nhds_unique' tendsto_nhds_unique'
 
-theorem tendsto_nhds_unique_of_eventuallyEq [T2Space α] {f g : β → α} {l : Filter β} {a b : α}
+theorem tendsto_nhds_unique_of_eventuallyEq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
     [NeBot l] (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : f =ᶠ[l] g) : a = b :=
   tendsto_nhds_unique (ha.congr' hfg) hb
 #align tendsto_nhds_unique_of_eventually_eq tendsto_nhds_unique_of_eventuallyEq
 
-theorem tendsto_nhds_unique_of_frequently_eq [T2Space α] {f g : β → α} {l : Filter β} {a b : α}
+theorem tendsto_nhds_unique_of_frequently_eq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
     (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : ∃ᶠ x in l, f x = g x) : a = b :=
-  have : ∃ᶠ z : α × α in 𝓝 (a, b), z.1 = z.2 := (ha.prod_mk_nhds hb).frequently hfg
+  have : ∃ᶠ z : X × X in 𝓝 (a, b), z.1 = z.2 := (ha.prod_mk_nhds hb).frequently hfg
   not_not.1 fun hne => this (isClosed_diagonal.isOpen_compl.mem_nhds hne)
 #align tendsto_nhds_unique_of_frequently_eq tendsto_nhds_unique_of_frequently_eq
 
@@ -1051,33 +1049,33 @@ theorem Set.InjOn.exists_isOpen_superset {X Y : Type*} [TopologicalSpace X] [Top
 /-- A T₂.₅ space, also known as a Urysohn space, is a topological space
   where for every pair `x ≠ y`, there are two open sets, with the intersection of closures
   empty, one containing `x` and the other `y` . -/
-class T25Space (α : Type u) [TopologicalSpace α] : Prop where
+class T25Space (X : Type u) [TopologicalSpace X] : Prop where
   /-- Given two distinct points in a T₂.₅ space, their filters of closed neighborhoods are
   disjoint. -/
-  t2_5 : ∀ ⦃x y : α⦄, x ≠ y → Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure)
+  t2_5 : ∀ ⦃x y : X⦄, x ≠ y → Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure)
 #align t2_5_space T25Space
 
 @[simp]
-theorem disjoint_lift'_closure_nhds [T25Space α] {x y : α} :
+theorem disjoint_lift'_closure_nhds [T25Space X] {x y : X} :
     Disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure) ↔ x ≠ y :=
   ⟨fun h hxy => by simp [hxy, nhds_neBot.ne] at h, fun h => T25Space.t2_5 h⟩
 #align disjoint_lift'_closure_nhds disjoint_lift'_closure_nhds
 
 -- see Note [lower instance priority]
-instance (priority := 100) T25Space.t2Space [T25Space α] : T2Space α :=
+instance (priority := 100) T25Space.t2Space [T25Space X] : T2Space X :=
   t2Space_iff_disjoint_nhds.2 fun _ _ hne =>
     (disjoint_lift'_closure_nhds.2 hne).mono (le_lift'_closure _) (le_lift'_closure _)
 #align t2_5_space.t2_space T25Space.t2Space
 
-theorem exists_nhds_disjoint_closure [T25Space α] {x y : α} (h : x ≠ y) :
+theorem exists_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
     ∃ s ∈ 𝓝 x, ∃ t ∈ 𝓝 y, Disjoint (closure s) (closure t) :=
   ((𝓝 x).basis_sets.lift'_closure.disjoint_iff (𝓝 y).basis_sets.lift'_closure).1 <|
     disjoint_lift'_closure_nhds.2 h
 #align exists_nhds_disjoint_closure exists_nhds_disjoint_closure
 
-theorem exists_open_nhds_disjoint_closure [T25Space α] {x y : α} (h : x ≠ y) :
-    ∃ u : Set α,
-      x ∈ u ∧ IsOpen u ∧ ∃ v : Set α, y ∈ v ∧ IsOpen v ∧ Disjoint (closure u) (closure v) := by
+theorem exists_open_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
+    ∃ u : Set X,
+      x ∈ u ∧ IsOpen u ∧ ∃ v : Set X, y ∈ v ∧ IsOpen v ∧ Disjoint (closure u) (closure v) := by
   simpa only [exists_prop, and_assoc] using
     ((nhds_basis_opens x).lift'_closure.disjoint_iff (nhds_basis_opens y).lift'_closure).1
       (disjoint_lift'_closure_nhds.2 h)
@@ -1085,34 +1083,34 @@ theorem exists_open_nhds_disjoint_closure [T25Space α] {x y : α} (h : x ≠ y)
 
 section limUnder
 
-variable [T2Space α] {f : Filter α}
+variable [T2Space X] {f : Filter X}
 
 /-!
 ### Properties of `lim` and `limUnder`
 
-In this section we use explicit `Nonempty α` instances for `lim` and `limUnder`. This way the lemmas
-are useful without a `Nonempty α` instance.
+In this section we use explicit `Nonempty X` instances for `lim` and `limUnder`. This way the lemmas
+are useful without a `Nonempty X` instance.
 -/
 
 
-theorem lim_eq {a : α} [NeBot f] (h : f ≤ 𝓝 a) : @lim _ _ ⟨a⟩ f = a :=
-  tendsto_nhds_unique (le_nhds_lim ⟨a, h⟩) h
+theorem lim_eq {x : X} [NeBot f] (h : f ≤ 𝓝 x) : @lim _ _ ⟨x⟩ f = x :=
+  tendsto_nhds_unique (le_nhds_lim ⟨x, h⟩) h
 set_option linter.uppercaseLean3 false in
 #align Lim_eq lim_eq
 
-theorem lim_eq_iff [NeBot f] (h : ∃ a : α, f ≤ nhds a) {a} : @lim _ _ ⟨a⟩ f = a ↔ f ≤ 𝓝 a :=
+theorem lim_eq_iff [NeBot f] (h : ∃ x : X, f ≤ 𝓝 x) {x} : @lim _ _ ⟨x⟩ f = x ↔ f ≤ 𝓝 x :=
   ⟨fun c => c ▸ le_nhds_lim h, lim_eq⟩
 set_option linter.uppercaseLean3 false in
 #align Lim_eq_iff lim_eq_iff
 
-theorem Ultrafilter.lim_eq_iff_le_nhds [CompactSpace α] {x : α} {F : Ultrafilter α} :
+theorem Ultrafilter.lim_eq_iff_le_nhds [CompactSpace X] {x : X} {F : Ultrafilter X} :
     F.lim = x ↔ ↑F ≤ 𝓝 x :=
   ⟨fun h => h ▸ F.le_nhds_lim, lim_eq⟩
 set_option linter.uppercaseLean3 false in
 #align ultrafilter.Lim_eq_iff_le_nhds Ultrafilter.lim_eq_iff_le_nhds
 
-theorem isOpen_iff_ultrafilter' [CompactSpace α] (U : Set α) :
-    IsOpen U ↔ ∀ F : Ultrafilter α, F.lim ∈ U → U ∈ F.1 := by
+theorem isOpen_iff_ultrafilter' [CompactSpace X] (U : Set X) :
+    IsOpen U ↔ ∀ F : Ultrafilter X, F.lim ∈ U → U ∈ F.1 := by
   rw [isOpen_iff_ultrafilter]
   refine' ⟨fun h F hF => h F.lim hF F F.le_nhds_lim, _⟩
   intro cond x hx f h
@@ -1120,42 +1118,42 @@ theorem isOpen_iff_ultrafilter' [CompactSpace α] (U : Set α) :
   exact cond _ hx
 #align is_open_iff_ultrafilter' isOpen_iff_ultrafilter'
 
-theorem Filter.Tendsto.limUnder_eq {a : α} {f : Filter β} [NeBot f] {g : β → α}
-    (h : Tendsto g f (𝓝 a)) : @limUnder _ _ _ ⟨a⟩ f g = a :=
+theorem Filter.Tendsto.limUnder_eq {x : X} {f : Filter Y} [NeBot f] {g : Y → X}
+    (h : Tendsto g f (𝓝 x)) : @limUnder _ _ _ ⟨x⟩ f g = x :=
   lim_eq h
 #align filter.tendsto.lim_eq Filter.Tendsto.limUnder_eq
 
-theorem Filter.limUnder_eq_iff {f : Filter β} [NeBot f] {g : β → α} (h : ∃ a, Tendsto g f (𝓝 a))
-    {a} : @limUnder _ _ _ ⟨a⟩ f g = a ↔ Tendsto g f (𝓝 a) :=
+theorem Filter.limUnder_eq_iff {f : Filter Y} [NeBot f] {g : Y → X} (h : ∃ x, Tendsto g f (𝓝 x))
+    {x} : @limUnder _ _ _ ⟨x⟩ f g = x ↔ Tendsto g f (𝓝 x) :=
   ⟨fun c => c ▸ tendsto_nhds_limUnder h, Filter.Tendsto.limUnder_eq⟩
 #align filter.lim_eq_iff Filter.limUnder_eq_iff
 
-theorem Continuous.limUnder_eq [TopologicalSpace β] {f : β → α} (h : Continuous f) (a : β) :
-    @limUnder _ _ _ ⟨f a⟩ (𝓝 a) f = f a :=
-  (h.tendsto a).limUnder_eq
+theorem Continuous.limUnder_eq [TopologicalSpace Y] {f : Y → X} (h : Continuous f) (y : Y) :
+    @limUnder _ _ _ ⟨f y⟩ (𝓝 y) f = f y :=
+  (h.tendsto y).limUnder_eq
 #align continuous.lim_eq Continuous.limUnder_eq
 
 @[simp]
-theorem lim_nhds (a : α) : @lim _ _ ⟨a⟩ (𝓝 a) = a :=
+theorem lim_nhds (x : X) : @lim _ _ ⟨x⟩ (𝓝 x) = x :=
   lim_eq le_rfl
 set_option linter.uppercaseLean3 false in
 #align Lim_nhds lim_nhds
 
 @[simp]
-theorem limUnder_nhds_id (a : α) : @limUnder _ _ _ ⟨a⟩ (𝓝 a) id = a :=
-  lim_nhds a
+theorem limUnder_nhds_id (x : X) : @limUnder _ _ _ ⟨x⟩ (𝓝 x) id = x :=
+  lim_nhds x
 #align lim_nhds_id limUnder_nhds_id
 
 @[simp]
-theorem lim_nhdsWithin {a : α} {s : Set α} (h : a ∈ closure s) : @lim _ _ ⟨a⟩ (𝓝[s] a) = a :=
-  haveI : NeBot (𝓝[s] a) := mem_closure_iff_clusterPt.1 h
+theorem lim_nhdsWithin {x : X} {s : Set X} (h : x ∈ closure s) : @lim _ _ ⟨x⟩ (𝓝[s] x) = x :=
+  haveI : NeBot (𝓝[s] x) := mem_closure_iff_clusterPt.1 h
   lim_eq inf_le_left
 set_option linter.uppercaseLean3 false in
 #align Lim_nhds_within lim_nhdsWithin
 
 @[simp]
-theorem limUnder_nhdsWithin_id {a : α} {s : Set α} (h : a ∈ closure s) :
-    @limUnder _ _ _ ⟨a⟩ (𝓝[s] a) id = a :=
+theorem limUnder_nhdsWithin_id {x : X} {s : Set X} (h : x ∈ closure s) :
+    @limUnder _ _ _ ⟨x⟩ (𝓝[s] x) id = x :=
   lim_nhdsWithin h
 #align lim_nhds_within_id limUnder_nhdsWithin_id
 
@@ -1167,62 +1165,61 @@ end limUnder
 We use two lemmas to prove that various standard constructions generate Hausdorff spaces from
 Hausdorff spaces:
 
-* `separated_by_continuous` says that two points `x y : α` can be separated by open neighborhoods
-  provided that there exists a continuous map `f : α → β` with a Hausdorff codomain such that
+* `separated_by_continuous` says that two points `x y : X` can be separated by open neighborhoods
+  provided that there exists a continuous map `f : X → Y` with a Hausdorff codomain such that
   `f x ≠ f y`. We use this lemma to prove that topological spaces defined using `induced` are
   Hausdorff spaces.
 
-* `separated_by_openEmbedding` says that for an open embedding `f : α → β` of a Hausdorff space
-  `α`, the images of two distinct points `x y : α`, `x ≠ y` can be separated by open neighborhoods.
+* `separated_by_openEmbedding` says that for an open embedding `f : X → Y` of a Hausdorff space
+  `X`, the images of two distinct points `x y : X`, `x ≠ y` can be separated by open neighborhoods.
   We use this lemma to prove that topological spaces defined using `coinduced` are Hausdorff spaces.
 -/
 
 
 -- see Note [lower instance priority]
-instance (priority := 100) DiscreteTopology.toT2Space {α : Type*} [TopologicalSpace α]
-    [DiscreteTopology α] : T2Space α :=
+instance (priority := 100) DiscreteTopology.toT2Space
+    [DiscreteTopology X] : T2Space X :=
   ⟨fun x y h => ⟨{x}, {y}, isOpen_discrete _, isOpen_discrete _, rfl, rfl, disjoint_singleton.2 h⟩⟩
 #align discrete_topology.to_t2_space DiscreteTopology.toT2Space
 
-theorem separated_by_continuous {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β]
-    [T2Space β] {f : α → β} (hf : Continuous f) {x y : α} (h : f x ≠ f y) :
-    ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
+theorem separated_by_continuous [TopologicalSpace Y] [T2Space Y]
+    {f : X → Y} (hf : Continuous f) {x y : X} (h : f x ≠ f y) :
+    ∃ u v : Set X, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
   let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
   ⟨f ⁻¹' u, f ⁻¹' v, uo.preimage hf, vo.preimage hf, xu, yv, uv.preimage _⟩
 #align separated_by_continuous separated_by_continuous
 
-theorem separated_by_openEmbedding {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
-    [T2Space α] {f : α → β} (hf : OpenEmbedding f) {x y : α} (h : x ≠ y) :
-    ∃ u v : Set β, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v :=
+theorem separated_by_openEmbedding [TopologicalSpace Y] [T2Space X]
+    {f : X → Y} (hf : OpenEmbedding f) {x y : X} (h : x ≠ y) :
+    ∃ u v : Set Y, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v :=
   let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
   ⟨f '' u, f '' v, hf.isOpenMap _ uo, hf.isOpenMap _ vo, mem_image_of_mem _ xu,
     mem_image_of_mem _ yv, disjoint_image_of_injective hf.inj uv⟩
 #align separated_by_open_embedding separated_by_openEmbedding
 
-instance {α : Type*} {p : α → Prop} [TopologicalSpace α] [T2Space α] : T2Space (Subtype p) :=
+instance {p : X → Prop} [T2Space X] : T2Space (Subtype p) :=
   ⟨fun _ _ h => separated_by_continuous continuous_subtype_val (mt Subtype.eq h)⟩
 
-instance Prod.t2Space {α : Type*} {β : Type*} [TopologicalSpace α] [T2Space α]
-    [TopologicalSpace β] [T2Space β] : T2Space (α × β) :=
+instance Prod.t2Space [T2Space X] [TopologicalSpace Y] [T2Space Y] : T2Space (X × Y) :=
   ⟨fun _ _ h => Or.elim (not_and_or.mp (mt Prod.ext_iff.mpr h))
     (fun h₁ => separated_by_continuous continuous_fst h₁) fun h₂ =>
     separated_by_continuous continuous_snd h₂⟩
 
 /-- If the codomain of an injective continuous function is a Hausdorff space, then so is its
 domain. -/
-theorem T2Space.of_injective_continuous [TopologicalSpace β] [T2Space β] {f : α → β}
-    (hinj : Injective f) (hc : Continuous f) : T2Space α :=
+theorem T2Space.of_injective_continuous [TopologicalSpace Y] [T2Space Y] {f : X → Y}
+    (hinj : Injective f) (hc : Continuous f) : T2Space X :=
   ⟨fun _ _ h => separated_by_continuous hc (hinj.ne h)⟩
 
 /-- If the codomain of a topological embedding is a Hausdorff space, then so is its domain.
 See also `T2Space.of_continuous_injective`. -/
-theorem Embedding.t2Space [TopologicalSpace β] [T2Space β] {f : α → β} (hf : Embedding f) :
-    T2Space α :=
+theorem Embedding.t2Space [TopologicalSpace Y] [T2Space Y] {f : X → Y} (hf : Embedding f) :
+    T2Space X :=
   .of_injective_continuous hf.inj hf.continuous
 #align embedding.t2_space Embedding.t2Space
 
-instance {α β : Type*} [TopologicalSpace α] [T2Space α] [TopologicalSpace β] [T2Space β] :
-    T2Space (α ⊕ β) := by
+instance [T2Space X] [TopologicalSpace Y] [T2Space Y] :
+    T2Space (X ⊕ Y) := by
   constructor
   rintro (x | x) (y | y) h
   · exact separated_by_openEmbedding openEmbedding_inl <| ne_of_apply_ne _ h
@@ -1230,15 +1227,15 @@ instance {α β : Type*} [TopologicalSpace α] [T2Space α] [TopologicalSpace β
   · exact separated_by_continuous continuous_isLeft <| by simp
   · exact separated_by_openEmbedding openEmbedding_inr <| ne_of_apply_ne _ h
 
-instance Pi.t2Space {α : Type*} {β : α → Type v} [∀ a, TopologicalSpace (β a)]
-    [∀ a, T2Space (β a)] : T2Space (∀ a, β a) :=
+instance Pi.t2Space {Y : X → Type v} [∀ a, TopologicalSpace (Y a)]
+    [∀ a, T2Space (Y a)] : T2Space (∀ a, Y a) :=
   ⟨fun _ _ h =>
     let ⟨i, hi⟩ := not_forall.mp (mt funext h)
     separated_by_continuous (continuous_apply i) hi⟩
 #align Pi.t2_space Pi.t2Space
 
-instance Sigma.t2Space {ι} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] [∀ a, T2Space (α a)] :
-    T2Space (Σi, α i) := by
+instance Sigma.t2Space {ι} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ a, T2Space (X a)] :
+    T2Space (Σi, X i) := by
   constructor
   rintro ⟨i, x⟩ ⟨j, y⟩ neq
   rcases eq_or_ne i j with (rfl | h)
@@ -1248,32 +1245,32 @@ instance Sigma.t2Space {ι} {α : ι → Type*} [∀ i, TopologicalSpace (α i)]
     exact separated_by_continuous (continuous_def.2 fun u _ => isOpen_sigma_fst_preimage u) h
 #align sigma.t2_space Sigma.t2Space
 
-variable {γ : Type*} [TopologicalSpace β] [TopologicalSpace γ]
+variable {Z : Type*} [TopologicalSpace Y] [TopologicalSpace Z]
 
-theorem isClosed_eq [T2Space α] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
-    IsClosed { x : β | f x = g x } :=
+theorem isClosed_eq [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g) :
+    IsClosed { y : Y | f y = g y } :=
   continuous_iff_isClosed.mp (hf.prod_mk hg) _ isClosed_diagonal
 #align is_closed_eq isClosed_eq
 
-theorem isOpen_ne_fun [T2Space α] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
-    IsOpen { x : β | f x ≠ g x } :=
+theorem isOpen_ne_fun [T2Space X] {f g : Y → X} (hf : Continuous f) (hg : Continuous g) :
+    IsOpen { y : Y | f y ≠ g y } :=
   isOpen_compl_iff.mpr <| isClosed_eq hf hg
 #align is_open_ne_fun isOpen_ne_fun
 
 /-- If two continuous maps are equal on `s`, then they are equal on the closure of `s`. See also
 `Set.EqOn.of_subset_closure` for a more general version. -/
-protected theorem Set.EqOn.closure [T2Space α] {s : Set β} {f g : β → α} (h : EqOn f g s)
+protected theorem Set.EqOn.closure [T2Space X] {s : Set Y} {f g : Y → X} (h : EqOn f g s)
     (hf : Continuous f) (hg : Continuous g) : EqOn f g (closure s) :=
   closure_minimal h (isClosed_eq hf hg)
 #align set.eq_on.closure Set.EqOn.closure
 
 /-- If two continuous functions are equal on a dense set, then they are equal. -/
-theorem Continuous.ext_on [T2Space α] {s : Set β} (hs : Dense s) {f g : β → α} (hf : Continuous f)
+theorem Continuous.ext_on [T2Space X] {s : Set Y} (hs : Dense s) {f g : Y → X} (hf : Continuous f)
     (hg : Continuous g) (h : EqOn f g s) : f = g :=
   funext fun x => h.closure hf hg (hs x)
 #align continuous.ext_on Continuous.ext_on
 
-theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ → α}
+theorem eqOn_closure₂' [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf₁ : ∀ x, Continuous (f x))
     (hf₂ : ∀ y, Continuous fun x => f x y) (hg₁ : ∀ x, Continuous (g x))
     (hg₂ : ∀ y, Continuous fun x => g x y) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
@@ -1282,7 +1279,7 @@ theorem eqOn_closure₂' [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ
     isClosed_biInter fun y _ => isClosed_eq (hf₂ _) (hg₂ _)
 #align eq_on_closure₂' eqOn_closure₂'
 
-theorem eqOn_closure₂ [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ → α}
+theorem eqOn_closure₂ [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (uncurry f))
     (hg : Continuous (uncurry g)) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
   eqOn_closure₂' h (fun x => continuous_uncurry_left x hf) (fun x => continuous_uncurry_right x hf)
@@ -1291,7 +1288,7 @@ theorem eqOn_closure₂ [T2Space α] {s : Set β} {t : Set γ} {f g : β → γ
 
 /-- If `f x = g x` for all `x ∈ s` and `f`, `g` are continuous on `t`, `s ⊆ t ⊆ closure s`, then
 `f x = g x` for all `x ∈ t`. See also `Set.EqOn.closure`. -/
-theorem Set.EqOn.of_subset_closure [T2Space α] {s t : Set β} {f g : β → α} (h : EqOn f g s)
+theorem Set.EqOn.of_subset_closure [T2Space Y] {s t : Set X} {f g : X → Y} (h : EqOn f g s)
     (hf : ContinuousOn f t) (hg : ContinuousOn g t) (hst : s ⊆ t) (hts : t ⊆ closure s) :
     EqOn f g t := by
   intro x hx
@@ -1301,19 +1298,19 @@ theorem Set.EqOn.of_subset_closure [T2Space α] {s t : Set β} {f g : β → α}
       ((hg x hx).mono_left <| nhdsWithin_mono _ hst) (h.eventuallyEq_of_mem self_mem_nhdsWithin)
 #align set.eq_on.of_subset_closure Set.EqOn.of_subset_closure
 
-theorem Function.LeftInverse.closed_range [T2Space α] {f : α → β} {g : β → α}
+theorem Function.LeftInverse.closed_range [T2Space X] {f : X → Y} {g : Y → X}
     (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsClosed (range g) :=
   have : EqOn (g ∘ f) id (closure <| range g) :=
     h.rightInvOn_range.eqOn.closure (hg.comp hf) continuous_id
   isClosed_of_closure_subset fun x hx => ⟨f x, this hx⟩
 #align function.left_inverse.closed_range Function.LeftInverse.closed_range
 
-theorem Function.LeftInverse.closedEmbedding [T2Space α] {f : α → β} {g : β → α}
+theorem Function.LeftInverse.closedEmbedding [T2Space X] {f : X → Y} {g : Y → X}
     (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : ClosedEmbedding g :=
   ⟨h.embedding hf hg, h.closed_range hf hg⟩
 #align function.left_inverse.closed_embedding Function.LeftInverse.closedEmbedding
 
-theorem isCompact_isCompact_separated [T2Space α] {s t : Set α} (hs : IsCompact s)
+theorem isCompact_isCompact_separated [T2Space X] {s t : Set X} (hs : IsCompact s)
     (ht : IsCompact t) (hst : Disjoint s t) : SeparatedNhds s t := by
   simp only [SeparatedNhds, prod_subset_compl_diagonal_iff_disjoint.symm] at hst ⊢
   exact generalized_tube_lemma hs ht isClosed_diagonal.isOpen_compl hst
@@ -1321,21 +1318,21 @@ theorem isCompact_isCompact_separated [T2Space α] {s t : Set α} (hs : IsCompac
 
 section SeparatedFinset
 
-theorem finset_disjoint_finset_opens_of_t2 [T2Space α] (s t : Finset α) (h : Disjoint s t) :
-    SeparatedNhds (s : Set α) t :=
+theorem finset_disjoint_finset_opens_of_t2 [T2Space X] (s t : Finset X) (h : Disjoint s t) :
+    SeparatedNhds (s : Set X) t :=
   isCompact_isCompact_separated s.finite_toSet.isCompact t.finite_toSet.isCompact <| by
     exact_mod_cast h
 #align finset_disjoint_finset_opens_of_t2 finset_disjoint_finset_opens_of_t2
 
-theorem point_disjoint_finset_opens_of_t2 [T2Space α] {x : α} {s : Finset α} (h : x ∉ s) :
-    SeparatedNhds ({x} : Set α) s := by
+theorem point_disjoint_finset_opens_of_t2 [T2Space X] {x : X} {s : Finset X} (h : x ∉ s) :
+    SeparatedNhds ({x} : Set X) s := by
   exact_mod_cast finset_disjoint_finset_opens_of_t2 {x} s (Finset.disjoint_singleton_left.mpr h)
 #align point_disjoint_finset_opens_of_t2 point_disjoint_finset_opens_of_t2
 
 end SeparatedFinset
 
 /-- In a `T2Space`, every compact set is closed. -/
-theorem IsCompact.isClosed [T2Space α] {s : Set α} (hs : IsCompact s) : IsClosed s :=
+theorem IsCompact.isClosed [T2Space X] {s : Set X} (hs : IsCompact s) : IsClosed s :=
   isOpen_compl_iff.1 <| isOpen_iff_forall_mem_open.mpr fun x hx =>
     let ⟨u, v, _, vo, su, xv, uv⟩ :=
       isCompact_isCompact_separated hs isCompact_singleton (disjoint_singleton_right.2 hx)
@@ -1343,62 +1340,62 @@ theorem IsCompact.isClosed [T2Space α] {s : Set α} (hs : IsCompact s) : IsClos
 #align is_compact.is_closed IsCompact.isClosed
 
 @[simp]
-theorem Filter.coclosedCompact_eq_cocompact [T2Space α] : coclosedCompact α = cocompact α := by
+theorem Filter.coclosedCompact_eq_cocompact [T2Space X] : coclosedCompact X = cocompact X := by
   simp only [coclosedCompact, cocompact, iInf_and',
-    and_iff_right_of_imp (@IsCompact.isClosed α _ _ _)]
+    and_iff_right_of_imp (@IsCompact.isClosed X _ _ _)]
 #align filter.coclosed_compact_eq_cocompact Filter.coclosedCompact_eq_cocompact
 
 @[simp]
-theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
-    Bornology.relativelyCompact α = Bornology.inCompact α :=
+theorem Bornology.relativelyCompact_eq_inCompact [T2Space X] :
+    Bornology.relativelyCompact X = Bornology.inCompact X :=
   Bornology.ext _ _ Filter.coclosedCompact_eq_cocompact
 #align bornology.relatively_compact_eq_in_compact Bornology.relativelyCompact_eq_inCompact
 
-theorem IsCompact.preimage_continuous [CompactSpace α] [T2Space β] {f : α → β} {s : Set β}
+theorem IsCompact.preimage_continuous [CompactSpace X] [T2Space Y] {f : X → Y} {s : Set Y}
     (hs : IsCompact s) (hf : Continuous f) : IsCompact (f ⁻¹' s) :=
   (hs.isClosed.preimage hf).isCompact
 
-/-- If `V : ι → Set α` is a decreasing family of compact sets then any neighborhood of
+/-- If `V : ι → Set X` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhds_of_isCompact'` where we
-don't need to assume each `V i` closed because it follows from compactness since `α` is
+don't need to assume each `V i` closed because it follows from compactness since `X` is
 assumed to be Hausdorff. -/
-theorem exists_subset_nhds_of_isCompact [T2Space α] {ι : Type*} [Nonempty ι] {V : ι → Set α}
-    (hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) {U : Set α}
+theorem exists_subset_nhds_of_isCompact [T2Space X] {ι : Type*} [Nonempty ι] {V : ι → Set X}
+    (hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) {U : Set X}
     (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U :=
   exists_subset_nhds_of_isCompact' hV hV_cpct (fun i => (hV_cpct i).isClosed) hU
 #align exists_subset_nhds_of_is_compact exists_subset_nhds_of_isCompact
 
-theorem CompactExhaustion.isClosed [T2Space α] (K : CompactExhaustion α) (n : ℕ) : IsClosed (K n) :=
+theorem CompactExhaustion.isClosed [T2Space X] (K : CompactExhaustion X) (n : ℕ) : IsClosed (K n) :=
   (K.isCompact n).isClosed
 #align compact_exhaustion.is_closed CompactExhaustion.isClosed
 
-theorem IsCompact.inter [T2Space α] {s t : Set α} (hs : IsCompact s) (ht : IsCompact t) :
+theorem IsCompact.inter [T2Space X] {s t : Set X} (hs : IsCompact s) (ht : IsCompact t) :
     IsCompact (s ∩ t) :=
   hs.inter_right <| ht.isClosed
 #align is_compact.inter IsCompact.inter
 
-theorem isCompact_closure_of_subset_compact [T2Space α] {s t : Set α} (ht : IsCompact t)
+theorem isCompact_closure_of_subset_compact [T2Space X] {s t : Set X} (ht : IsCompact t)
     (h : s ⊆ t) : IsCompact (closure s) :=
   ht.of_isClosed_subset isClosed_closure (closure_minimal h ht.isClosed)
 #align is_compact_closure_of_subset_compact isCompact_closure_of_subset_compact
 
 @[simp]
-theorem exists_compact_superset_iff [T2Space α] {s : Set α} :
+theorem exists_compact_superset_iff [T2Space X] {s : Set X} :
     (∃ K, IsCompact K ∧ s ⊆ K) ↔ IsCompact (closure s) :=
   ⟨fun ⟨_K, hK, hsK⟩ => isCompact_closure_of_subset_compact hK hsK, fun h =>
     ⟨closure s, h, subset_closure⟩⟩
 #align exists_compact_superset_iff exists_compact_superset_iff
 
-theorem image_closure_of_isCompact [T2Space β] {s : Set α} (hs : IsCompact (closure s)) {f : α → β}
+theorem image_closure_of_isCompact [T2Space Y] {s : Set X} (hs : IsCompact (closure s)) {f : X → Y}
     (hf : ContinuousOn f (closure s)) : f '' closure s = closure (f '' s) :=
   Subset.antisymm hf.image_closure <|
     closure_minimal (image_subset f subset_closure) (hs.image_of_continuousOn hf).isClosed
 #align image_closure_of_is_compact image_closure_of_isCompact
 
 /-- If a compact set is covered by two open sets, then we can cover it by two compact subsets. -/
-theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCompact K)
+theorem IsCompact.binary_compact_cover [T2Space X] {K U V : Set X} (hK : IsCompact K)
     (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
-    ∃ K₁ K₂ : Set α, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ := by
+    ∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ := by
   obtain ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩ :=
     isCompact_isCompact_separated (hK.diff hU) (hK.diff hV)
       (by rwa [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty])
@@ -1408,18 +1405,18 @@ theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCom
 #align is_compact.binary_compact_cover IsCompact.binary_compact_cover
 
 /-- A continuous map from a compact space to a Hausdorff space is a closed map. -/
-protected theorem Continuous.isClosedMap [CompactSpace α] [T2Space β] {f : α → β}
+protected theorem Continuous.isClosedMap [CompactSpace X] [T2Space Y] {f : X → Y}
     (h : Continuous f) : IsClosedMap f := fun _s hs => (hs.isCompact.image h).isClosed
 #align continuous.is_closed_map Continuous.isClosedMap
 
 /-- A continuous injective map from a compact space to a Hausdorff space is a closed embedding. -/
-theorem Continuous.closedEmbedding [CompactSpace α] [T2Space β] {f : α → β} (h : Continuous f)
+theorem Continuous.closedEmbedding [CompactSpace X] [T2Space Y] {f : X → Y} (h : Continuous f)
     (hf : Function.Injective f) : ClosedEmbedding f :=
   closedEmbedding_of_continuous_injective_closed h hf h.isClosedMap
 #align continuous.closed_embedding Continuous.closedEmbedding
 
 /-- A continuous surjective map from a compact space to a Hausdorff space is a quotient map. -/
-theorem QuotientMap.of_surjective_continuous [CompactSpace α] [T2Space β] {f : α → β}
+theorem QuotientMap.of_surjective_continuous [CompactSpace X] [T2Space Y] {f : X → Y}
     (hsurj : Surjective f) (hcont : Continuous f) : QuotientMap f :=
   hcont.isClosedMap.to_quotientMap hcont hsurj
 #align quotient_map.of_surjective_continuous QuotientMap.of_surjective_continuous
@@ -1429,9 +1426,9 @@ section
 open Finset Function
 
 /-- For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. -/
-theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact s) {ι} (t : Finset ι)
-    (U : ι → Set α) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
-    ∃ K : ι → Set α, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
+theorem IsCompact.finite_compact_cover [T2Space X] {s : Set X} (hs : IsCompact s) {ι} (t : Finset ι)
+    (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
+    ∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i := by
   induction' t using Finset.induction with x t hx ih generalizing U s
   · refine' ⟨fun _ => ∅, fun _ => isCompact_empty, fun i => empty_subset _, _⟩
     simpa only [subset_empty_iff, Finset.not_mem_empty, iUnion_false, iUnion_empty] using hsC
@@ -1457,8 +1454,8 @@ end
 
 -- see Note [lower instance priority]
 /-- A weakly locally compact Hausdorff space is locally compact. -/
-instance WeaklyLocallyCompactSpace.locallyCompactSpace [WeaklyLocallyCompactSpace α] [T2Space α] :
-    LocallyCompactSpace α :=
+instance WeaklyLocallyCompactSpace.locallyCompactSpace [WeaklyLocallyCompactSpace X] [T2Space X] :
+    LocallyCompactSpace X :=
   ⟨fun x _n hn =>
     let ⟨_u, un, uo, xu⟩ := mem_nhds_iff.mp hn
     let ⟨k, kc, kx⟩ := exists_compact_mem_nhds x
@@ -1475,15 +1472,15 @@ instance WeaklyLocallyCompactSpace.locallyCompactSpace [WeaklyLocallyCompactSpac
 #align locally_compact_of_compact_nhds WeaklyLocallyCompactSpace.locallyCompactSpace
 
 @[deprecated WeaklyLocallyCompactSpace.locallyCompactSpace]
-theorem locally_compact_of_compact [T2Space α] [CompactSpace α] :
-    LocallyCompactSpace α :=
+theorem locally_compact_of_compact [T2Space X] [CompactSpace X] :
+    LocallyCompactSpace X :=
   inferInstance
 #align locally_compact_of_compact locally_compact_of_compact
 
 /-- In a weakly locally compact T₂ space,
 every compact set has an open neighborhood with compact closure. -/
-theorem exists_open_superset_and_isCompact_closure [WeaklyLocallyCompactSpace α] [T2Space α]
-    {K : Set α} (hK : IsCompact K) : ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
+theorem exists_open_superset_and_isCompact_closure [WeaklyLocallyCompactSpace X] [T2Space X]
+    {K : Set X} (hK : IsCompact K) : ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
   rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
   exact ⟨interior K', isOpen_interior, hKK',
     isCompact_closure_of_subset_compact hK' interior_subset⟩
@@ -1491,8 +1488,8 @@ theorem exists_open_superset_and_isCompact_closure [WeaklyLocallyCompactSpace α
 
 /-- In a weakly locally compact T₂ space,
 every point has an open neighborhood with compact closure. -/
-theorem exists_open_with_compact_closure [WeaklyLocallyCompactSpace α] [T2Space α] (x : α) :
-    ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
+theorem exists_open_with_compact_closure [WeaklyLocallyCompactSpace X] [T2Space X] (x : X) :
+    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
   simpa only [singleton_subset_iff]
     using exists_open_superset_and_isCompact_closure isCompact_singleton
 #align exists_open_with_compact_closure exists_open_with_compact_closure
@@ -1500,7 +1497,7 @@ theorem exists_open_with_compact_closure [WeaklyLocallyCompactSpace α] [T2Space
 /-- In a locally compact T₂ space, given a compact set `K` inside an open set `U`, we can find an
 open set `V` between these sets with compact closure: `K ⊆ V` and the closure of `V` is inside `U`.
 -/
-theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace α] [T2Space α] {K U : Set α}
+theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace X] [T2Space X] {K U : Set X}
     (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
     ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (closure V) := by
   rcases exists_compact_between hK hU hKU with ⟨V, hV, hKV, hVU⟩
@@ -1509,7 +1506,7 @@ theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace α] [T2Sp
       isCompact_closure_of_subset_compact hV interior_subset⟩
 #align exists_open_between_and_is_compact_closure exists_open_between_and_isCompact_closure
 
-theorem isPreirreducible_iff_subsingleton [T2Space α] {S : Set α} :
+theorem isPreirreducible_iff_subsingleton [T2Space X] {S : Set X} :
     IsPreirreducible S ↔ S.Subsingleton := by
   refine' ⟨fun h x hx y hy => _, Set.Subsingleton.isPreirreducible⟩
   by_contra e
@@ -1518,20 +1515,20 @@ theorem isPreirreducible_iff_subsingleton [T2Space α] {S : Set α} :
 #align is_preirreducible_iff_subsingleton isPreirreducible_iff_subsingleton
 
 -- todo: use `alias` + `attribute [protected]` once we get `attribute [protected]`
-protected lemma IsPreirreducible.subsingleton [T2Space α] {S : Set α} (h : IsPreirreducible S) :
+protected lemma IsPreirreducible.subsingleton [T2Space X] {S : Set X} (h : IsPreirreducible S) :
     S.Subsingleton :=
   isPreirreducible_iff_subsingleton.1 h
 #align is_preirreducible.subsingleton IsPreirreducible.subsingleton
 
-theorem isIrreducible_iff_singleton [T2Space α] {S : Set α} : IsIrreducible S ↔ ∃ x, S = {x} := by
+theorem isIrreducible_iff_singleton [T2Space X] {S : Set X} : IsIrreducible S ↔ ∃ x, S = {x} := by
   rw [IsIrreducible, isPreirreducible_iff_subsingleton,
     exists_eq_singleton_iff_nonempty_subsingleton]
 #align is_irreducible_iff_singleton isIrreducible_iff_singleton
 
 /-- There does not exist a nontrivial preirreducible T₂ space. -/
-theorem not_preirreducible_nontrivial_t2 (α) [TopologicalSpace α] [PreirreducibleSpace α]
-    [Nontrivial α] [T2Space α] : False :=
-  (PreirreducibleSpace.isPreirreducible_univ (α := α)).subsingleton.not_nontrivial nontrivial_univ
+theorem not_preirreducible_nontrivial_t2 (X) [TopologicalSpace X] [PreirreducibleSpace X]
+    [Nontrivial X] [T2Space X] : False :=
+  (PreirreducibleSpace.isPreirreducible_univ (α := X)).subsingleton.not_nontrivial nontrivial_univ
 #align not_preirreducible_nontrivial_t2 not_preirreducible_nontrivial_t2
 
 end Separation
@@ -1548,13 +1545,13 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
   regular : ∀ {s : Set X} {a}, IsClosed s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)
 #align regular_space RegularSpace
 
-theorem regularSpace_TFAE (X : Type u) [ TopologicalSpace X ] :
+theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
     List.TFAE [RegularSpace X,
-      ∀ (s : Set X) a, a ∉ closure s → Disjoint (𝓝ˢ s) (𝓝 a),
-      ∀ (a : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 a) ↔ a ∉ closure s,
-      ∀ (a : X) (s : Set X), s ∈ 𝓝 a → ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s,
-      ∀ a : X, (𝓝 a).lift' closure ≤ 𝓝 a,
-      ∀ a : X , (𝓝 a).lift' closure = 𝓝 a] := by
+      ∀ (s : Set X) x, x ∉ closure s → Disjoint (𝓝ˢ s) (𝓝 x),
+      ∀ (x : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s,
+      ∀ (x : X) (s : Set X), s ∈ 𝓝 x → ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
+      ∀ x : X, (𝓝 x).lift' closure ≤ 𝓝 x,
+      ∀ x : X , (𝓝 x).lift' closure = 𝓝 x] := by
   tfae_have 1 ↔ 5
   · rw [regularSpace_iff, (@compl_surjective (Set X) _).forall, forall_swap]
     simp only [isClosed_compl_iff, mem_compl_iff, Classical.not_not, @and_comm (_ ∈ _),
@@ -1582,103 +1579,103 @@ theorem regularSpace_TFAE (X : Type u) [ TopologicalSpace X ] :
   tfae_finish
 #align regular_space_tfae regularSpace_TFAE
 
-theorem RegularSpace.ofLift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 𝓝 a) : RegularSpace α :=
-  Iff.mpr ((regularSpace_TFAE α).out 0 5) h
+theorem RegularSpace.ofLift'_closure (h : ∀ x : X, (𝓝 x).lift' closure = 𝓝 x) : RegularSpace X :=
+  Iff.mpr ((regularSpace_TFAE X).out 0 5) h
 #align regular_space.of_lift'_closure RegularSpace.ofLift'_closure
 
-theorem RegularSpace.ofBasis {ι : α → Sort*} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set α}
+theorem RegularSpace.ofBasis {ι : X → Sort*} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set X}
     (h₁ : ∀ a, (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ a i, p a i → IsClosed (s a i)) :
-    RegularSpace α :=
+    RegularSpace X :=
   RegularSpace.ofLift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a)
 #align regular_space.of_basis RegularSpace.ofBasis
 
 theorem RegularSpace.ofExistsMemNhdsIsClosedSubset
-    (h : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s) : RegularSpace α :=
-  Iff.mpr ((regularSpace_TFAE α).out 0 3) h
+    (h : ∀ (x : X), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s) : RegularSpace X :=
+  Iff.mpr ((regularSpace_TFAE X).out 0 3) h
 #align regular_space.of_exists_mem_nhds_is_closed_subset RegularSpace.ofExistsMemNhdsIsClosedSubset
 
-variable [RegularSpace α] {a : α} {s : Set α}
+variable [RegularSpace X] {x : X} {s : Set X}
 
-theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 a) ↔ a ∉ closure s := by
-  have h := (regularSpace_TFAE α).out 0 2
+theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s := by
+  have h := (regularSpace_TFAE X).out 0 2
   exact h.mp ‹_› _ _
 #align disjoint_nhds_set_nhds disjoint_nhdsSet_nhds
 
-theorem disjoint_nhds_nhdsSet : Disjoint (𝓝 a) (𝓝ˢ s) ↔ a ∉ closure s :=
+theorem disjoint_nhds_nhdsSet : Disjoint (𝓝 x) (𝓝ˢ s) ↔ x ∉ closure s :=
   disjoint_comm.trans disjoint_nhdsSet_nhds
 #align disjoint_nhds_nhds_set disjoint_nhds_nhdsSet
 
-theorem exists_mem_nhds_isClosed_subset {a : α} {s : Set α} (h : s ∈ 𝓝 a) :
-    ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s := by
-  have h' := (regularSpace_TFAE α).out 0 3
+theorem exists_mem_nhds_isClosed_subset {x : X} {s : Set X} (h : s ∈ 𝓝 x) :
+    ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s := by
+  have h' := (regularSpace_TFAE X).out 0 3
   exact h'.mp ‹_› _ _ h
 #align exists_mem_nhds_is_closed_subset exists_mem_nhds_isClosed_subset
 
-theorem closed_nhds_basis (a : α) : (𝓝 a).HasBasis (fun s : Set α => s ∈ 𝓝 a ∧ IsClosed s) id :=
+theorem closed_nhds_basis (x : X) : (𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsClosed s) id :=
   hasBasis_self.2 fun _ => exists_mem_nhds_isClosed_subset
 #align closed_nhds_basis closed_nhds_basis
 
-theorem lift'_nhds_closure (a : α) : (𝓝 a).lift' closure = 𝓝 a :=
-  (closed_nhds_basis a).lift'_closure_eq_self fun _ => And.right
+theorem lift'_nhds_closure (x : X) : (𝓝 x).lift' closure = 𝓝 x :=
+  (closed_nhds_basis x).lift'_closure_eq_self fun _ => And.right
 #align lift'_nhds_closure lift'_nhds_closure
 
-theorem Filter.HasBasis.nhds_closure {ι : Sort*} {a : α} {p : ι → Prop} {s : ι → Set α}
-    (h : (𝓝 a).HasBasis p s) : (𝓝 a).HasBasis p fun i => closure (s i) :=
-  lift'_nhds_closure a ▸ h.lift'_closure
+theorem Filter.HasBasis.nhds_closure {ι : Sort*} {x : X} {p : ι → Prop} {s : ι → Set X}
+    (h : (𝓝 x).HasBasis p s) : (𝓝 x).HasBasis p fun i => closure (s i) :=
+  lift'_nhds_closure x ▸ h.lift'_closure
 #align filter.has_basis.nhds_closure Filter.HasBasis.nhds_closure
 
-theorem hasBasis_nhds_closure (a : α) : (𝓝 a).HasBasis (fun s => s ∈ 𝓝 a) closure :=
-  (𝓝 a).basis_sets.nhds_closure
+theorem hasBasis_nhds_closure (x : X) : (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x) closure :=
+  (𝓝 x).basis_sets.nhds_closure
 #align has_basis_nhds_closure hasBasis_nhds_closure
 
-theorem hasBasis_opens_closure (a : α) : (𝓝 a).HasBasis (fun s => a ∈ s ∧ IsOpen s) closure :=
-  (nhds_basis_opens a).nhds_closure
+theorem hasBasis_opens_closure (x : X) : (𝓝 x).HasBasis (fun s => x ∈ s ∧ IsOpen s) closure :=
+  (nhds_basis_opens x).nhds_closure
 #align has_basis_opens_closure hasBasis_opens_closure
 
-theorem TopologicalSpace.IsTopologicalBasis.nhds_basis_closure {B : Set (Set α)}
-    (hB : IsTopologicalBasis B) (a : α) :
-    (𝓝 a).HasBasis (fun s : Set α => a ∈ s ∧ s ∈ B) closure := by
+theorem TopologicalSpace.IsTopologicalBasis.nhds_basis_closure {B : Set (Set X)}
+    (hB : IsTopologicalBasis B) (x : X) :
+    (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ s ∈ B) closure := by
   simpa only [and_comm] using hB.nhds_hasBasis.nhds_closure
 #align topological_space.is_topological_basis.nhds_basis_closure TopologicalSpace.IsTopologicalBasis.nhds_basis_closure
 
-theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set α)}
-    (hB : IsTopologicalBasis B) {a : α} {s : Set α} (h : s ∈ 𝓝 a) :
-    ∃ t ∈ B, a ∈ t ∧ closure t ⊆ s := by
+theorem TopologicalSpace.IsTopologicalBasis.exists_closure_subset {B : Set (Set X)}
+    (hB : IsTopologicalBasis B) {x : X} {s : Set X} (h : s ∈ 𝓝 x) :
+    ∃ t ∈ B, x ∈ t ∧ closure t ⊆ s := by
   simpa only [exists_prop, and_assoc] using hB.nhds_hasBasis.nhds_closure.mem_iff.mp h
 #align topological_space.is_topological_basis.exists_closure_subset TopologicalSpace.IsTopologicalBasis.exists_closure_subset
 
-theorem disjoint_nhds_nhds_iff_not_specializes {a b : α} : Disjoint (𝓝 a) (𝓝 b) ↔ ¬a ⤳ b := by
+theorem disjoint_nhds_nhds_iff_not_specializes {a b : X} : Disjoint (𝓝 a) (𝓝 b) ↔ ¬a ⤳ b := by
   rw [← nhdsSet_singleton, disjoint_nhdsSet_nhds, specializes_iff_mem_closure]
 #align disjoint_nhds_nhds_iff_not_specializes disjoint_nhds_nhds_iff_not_specializes
 
-theorem specializes_comm {a b : α} : a ⤳ b ↔ b ⤳ a := by
-  simp only [← (disjoint_nhds_nhds_iff_not_specializes (α := α)).not_left, disjoint_comm]
+theorem specializes_comm {a b : X} : a ⤳ b ↔ b ⤳ a := by
+  simp only [← (disjoint_nhds_nhds_iff_not_specializes (X := X)).not_left, disjoint_comm]
 #align specializes_comm specializes_comm
 
 alias ⟨Specializes.symm, _⟩ := specializes_comm
 #align specializes.symm Specializes.symm
 
-theorem specializes_iff_inseparable {a b : α} : a ⤳ b ↔ Inseparable a b :=
+theorem specializes_iff_inseparable {a b : X} : a ⤳ b ↔ Inseparable a b :=
   ⟨fun h => h.antisymm h.symm, le_of_eq⟩
 #align specializes_iff_inseparable specializes_iff_inseparable
 
-theorem isClosed_setOf_specializes : IsClosed { p : α × α | p.1 ⤳ p.2 } := by
+theorem isClosed_setOf_specializes : IsClosed { p : X × X | p.1 ⤳ p.2 } := by
   simp only [← isOpen_compl_iff, compl_setOf, ← disjoint_nhds_nhds_iff_not_specializes,
     isOpen_setOf_disjoint_nhds_nhds]
 #align is_closed_set_of_specializes isClosed_setOf_specializes
 
-theorem isClosed_setOf_inseparable : IsClosed { p : α × α | Inseparable p.1 p.2 } := by
+theorem isClosed_setOf_inseparable : IsClosed { p : X × X | Inseparable p.1 p.2 } := by
   simp only [← specializes_iff_inseparable, isClosed_setOf_specializes]
 #align is_closed_set_of_inseparable isClosed_setOf_inseparable
 
-protected theorem Inducing.regularSpace [TopologicalSpace β] {f : β → α} (hf : Inducing f) :
-    RegularSpace β :=
+protected theorem Inducing.regularSpace [TopologicalSpace Y] {f : Y → X} (hf : Inducing f) :
+    RegularSpace Y :=
   RegularSpace.ofBasis
     (fun b => by rw [hf.nhds_eq_comap b]; exact (closed_nhds_basis _).comap _)
     fun b s hs => by exact hs.2.preimage hf.continuous
 #align inducing.regular_space Inducing.regularSpace
 
-theorem regularSpace_induced (f : β → α) : @RegularSpace β (induced f ‹_›) :=
+theorem regularSpace_induced (f : Y → X) : @RegularSpace Y (induced f ‹_›) :=
   letI := induced f ‹_›
   Inducing.regularSpace ⟨rfl⟩
 #align regular_space_induced regularSpace_induced
@@ -1708,14 +1705,14 @@ theorem RegularSpace.inf {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @RegularSp
   exact regularSpace_iInf (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align regular_space.inf RegularSpace.inf
 
-instance {p : α → Prop} : RegularSpace (Subtype p) :=
+instance {p : X → Prop} : RegularSpace (Subtype p) :=
   embedding_subtype_val.toInducing.regularSpace
 
-instance [TopologicalSpace β] [RegularSpace β] : RegularSpace (α × β) :=
-  (regularSpace_induced (@Prod.fst α β)).inf (regularSpace_induced (@Prod.snd α β))
+instance [TopologicalSpace Y] [RegularSpace Y] : RegularSpace (X × Y) :=
+  (regularSpace_induced (@Prod.fst X Y)).inf (regularSpace_induced (@Prod.snd X Y))
 
-instance {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)] [∀ i, RegularSpace (π i)] :
-    RegularSpace (∀ i, π i) :=
+instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, RegularSpace (X i)] :
+    RegularSpace (∀ i, X i) :=
   regularSpace_iInf fun _ => regularSpace_induced _
 
 end RegularSpace
@@ -1724,39 +1721,39 @@ section T3
 
 /-- A T₃ space is a T₀ space which is a regular space. Any T₃ space is a T₁ space, a T₂ space, and
 a T₂.₅ space.  -/
-class T3Space (α : Type u) [TopologicalSpace α] extends T0Space α, RegularSpace α : Prop
+class T3Space (X : Type u) [TopologicalSpace X] extends T0Space X, RegularSpace X : Prop
 #align t3_space T3Space
 
-instance (priority := 90) [T0Space α] [RegularSpace α] : T3Space α := ⟨⟩
+instance (priority := 90) [T0Space X] [RegularSpace X] : T3Space X := ⟨⟩
 
 -- see Note [lower instance priority]
-instance (priority := 100) T3Space.t25Space [T3Space α] : T25Space α := by
+instance (priority := 100) T3Space.t25Space [T3Space X] : T25Space X := by
   refine' ⟨fun x y hne => _⟩
   rw [lift'_nhds_closure, lift'_nhds_closure]
   have : x ∉ closure {y} ∨ y ∉ closure {x} :=
-    (t0Space_iff_or_not_mem_closure α).mp inferInstance x y hne
+    (t0Space_iff_or_not_mem_closure X).mp inferInstance x y hne
   simp only [← disjoint_nhds_nhdsSet, nhdsSet_singleton] at this
   exact this.elim id fun h => h.symm
 #align t3_space.t2_5_space T3Space.t25Space
 
-protected theorem Embedding.t3Space [TopologicalSpace β] [T3Space β] {f : α → β}
-    (hf : Embedding f) : T3Space α :=
+protected theorem Embedding.t3Space [TopologicalSpace Y] [T3Space Y] {f : X → Y}
+    (hf : Embedding f) : T3Space X :=
   { toT0Space := hf.t0Space
     toRegularSpace := hf.toInducing.regularSpace }
 #align embedding.t3_space Embedding.t3Space
 
-instance Subtype.t3Space [T3Space α] {p : α → Prop} : T3Space (Subtype p) :=
+instance Subtype.t3Space [T3Space X] {p : X → Prop} : T3Space (Subtype p) :=
   embedding_subtype_val.t3Space
 #align subtype.t3_space Subtype.t3Space
 
-instance [TopologicalSpace β] [T3Space α] [T3Space β] : T3Space (α × β) := ⟨⟩
+instance [TopologicalSpace Y] [T3Space X] [T3Space Y] : T3Space (X × Y) := ⟨⟩
 
-instance {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)] [∀ i, T3Space (π i)] :
-    T3Space (∀ i, π i) := ⟨⟩
+instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, T3Space (X i)] :
+    T3Space (∀ i, X i) := ⟨⟩
 
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
 with the `Vₖ` closed and the `Uₖ` open, such that the `Uₖ` are disjoint. -/
-theorem disjoint_nested_nhds [T3Space α] {x y : α} (h : x ≠ y) :
+theorem disjoint_nested_nhds [T3Space X] {x y : X} (h : x ≠ y) :
     ∃ U₁ ∈ 𝓝 x, ∃ V₁ ∈ 𝓝 x, ∃ U₂ ∈ 𝓝 y, ∃ V₂ ∈ 𝓝 y,
       IsClosed V₁ ∧ IsClosed V₂ ∧ IsOpen U₁ ∧ IsOpen U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Disjoint U₁ U₂ := by
   rcases t2_separation h with ⟨U₁, U₂, U₁_op, U₂_op, x_in, y_in, H⟩
@@ -1769,7 +1766,7 @@ theorem disjoint_nested_nhds [T3Space α] {x y : α} (h : x ≠ y) :
 open SeparationQuotient
 
 /-- The `SeparationQuotient` of a regular space is a T₃ space. -/
-instance [RegularSpace α] : T3Space (SeparationQuotient α) where
+instance [RegularSpace X] : T3Space (SeparationQuotient X) where
   regular {s a} hs ha := by
     rcases surjective_mk a with ⟨a, rfl⟩
     rw [← disjoint_comap_iff surjective_mk, comap_mk_nhds_mk, comap_mk_nhdsSet]
@@ -1785,16 +1782,16 @@ class NormalSpace (X : Type u) [TopologicalSpace X] : Prop where
   /-- Two disjoint sets in a normal space admit disjoint neighbourhoods. -/
   normal : ∀ s t : Set X, IsClosed s → IsClosed t → Disjoint s t → SeparatedNhds s t
 
-theorem normal_separation [NormalSpace α] {s t : Set α} (H1 : IsClosed s) (H2 : IsClosed t)
+theorem normal_separation [NormalSpace X] {s t : Set X} (H1 : IsClosed s) (H2 : IsClosed t)
     (H3 : Disjoint s t) : SeparatedNhds s t :=
   NormalSpace.normal s t H1 H2 H3
 #align normal_separation normal_separation
 
-theorem disjoint_nhdsSet_nhdsSet [NormalSpace α] {s t : Set α} (hs : IsClosed s) (ht : IsClosed t)
+theorem disjoint_nhdsSet_nhdsSet [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsClosed t)
     (hd : Disjoint s t) : Disjoint (𝓝ˢ s) (𝓝ˢ t) :=
   (normal_separation hs ht hd).disjoint_nhdsSet
 
-theorem normal_exists_closure_subset [NormalSpace α] {s t : Set α} (hs : IsClosed s) (ht : IsOpen t)
+theorem normal_exists_closure_subset [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsOpen t)
     (hst : s ⊆ t) : ∃ u, IsOpen u ∧ s ⊆ u ∧ closure u ⊆ t := by
   have : Disjoint s tᶜ := Set.disjoint_left.mpr fun x hxs hxt => hxt (hst hxs)
   rcases normal_separation hs (isClosed_compl_iff.2 ht) this with
@@ -1805,8 +1802,8 @@ theorem normal_exists_closure_subset [NormalSpace α] {s t : Set α} (hs : IsClo
 #align normal_exists_closure_subset normal_exists_closure_subset
 
 /-- If the codomain of a closed embedding is a normal space, then so is the domain. -/
-protected theorem ClosedEmbedding.normalSpace [TopologicalSpace β] [NormalSpace β] {f : α → β}
-    (hf : ClosedEmbedding f) : NormalSpace α where
+protected theorem ClosedEmbedding.normalSpace [TopologicalSpace Y] [NormalSpace Y] {f : X → Y}
+    (hf : ClosedEmbedding f) : NormalSpace X where
   normal s t hs ht hst := by
     have H : SeparatedNhds (f '' s) (f '' t) :=
       NormalSpace.normal (f '' s) (f '' t) (hf.isClosedMap s hs) (hf.isClosedMap t ht)
@@ -1815,19 +1812,19 @@ protected theorem ClosedEmbedding.normalSpace [TopologicalSpace β] [NormalSpace
 
 /-- A regular topological space with second countable topology is a normal space. -/
 instance (priority := 100) NormalSpace.of_regularSpace_secondCountableTopology
-    [RegularSpace α] [SecondCountableTopology α] : NormalSpace α := by
-  have key : ∀ {s t : Set α}, IsClosed t → Disjoint s t →
-    ∃ U : Set (countableBasis α), (s ⊆ ⋃ u ∈ U, ↑u) ∧ (∀ u ∈ U, Disjoint (closure ↑u) t) ∧
-      ∀ n : ℕ, IsClosed (⋃ (u ∈ U) (_ : Encodable.encode u ≤ n), closure (u : Set α)) := by
+    [RegularSpace X] [SecondCountableTopology X] : NormalSpace X := by
+  have key : ∀ {s t : Set X}, IsClosed t → Disjoint s t →
+    ∃ U : Set (countableBasis X), (s ⊆ ⋃ u ∈ U, ↑u) ∧ (∀ u ∈ U, Disjoint (closure ↑u) t) ∧
+      ∀ n : ℕ, IsClosed (⋃ (u ∈ U) (_ : Encodable.encode u ≤ n), closure (u : Set X)) := by
     intro s t hc hd
     rw [disjoint_left] at hd
-    have : ∀ x ∈ s, ∃ U ∈ countableBasis α, x ∈ U ∧ Disjoint (closure U) t := by
+    have : ∀ x ∈ s, ∃ U ∈ countableBasis X, x ∈ U ∧ Disjoint (closure U) t := by
       intro x hx
-      rcases (isBasis_countableBasis α).exists_closure_subset (hc.compl_mem_nhds (hd hx))
+      rcases (isBasis_countableBasis X).exists_closure_subset (hc.compl_mem_nhds (hd hx))
         with ⟨u, hu, hxu, hut⟩
       exact ⟨u, hu, hxu, disjoint_left.2 hut⟩
     choose! U hu hxu hd using this
-    set V : s → countableBasis α := MapsTo.restrict _ _ _ hu
+    set V : s → countableBasis X := MapsTo.restrict _ _ _ hu
     refine' ⟨range V, _, forall_range_iff.2 <| Subtype.forall.2 hd, fun n => _⟩
     · rw [biUnion_range]
       exact fun x hx => mem_iUnion.2 ⟨⟨x, hx⟩, hxu x hx⟩
@@ -1863,25 +1860,25 @@ end NormalSpace
 section Normality
 
 /-- A T₄ space is a normal T₁ space. -/
-class T4Space (α : Type u) [TopologicalSpace α] extends T1Space α, NormalSpace α : Prop
+class T4Space (X : Type u) [TopologicalSpace X] extends T1Space X, NormalSpace X : Prop
 #align normal_space NormalSpace
 
-instance (priority := 100) [T1Space α] [NormalSpace α] : T4Space α := ⟨⟩
+instance (priority := 100) [T1Space X] [NormalSpace X] : T4Space X := ⟨⟩
 
 -- see Note [lower instance priority]
-instance (priority := 100) T4Space.t3Space [T4Space α] : T3Space α where
+instance (priority := 100) T4Space.t3Space [T4Space X] : T3Space X where
   regular hs hxs := by simpa only [nhdsSet_singleton] using (normal_separation hs isClosed_singleton
     (disjoint_singleton_right.mpr hxs)).disjoint_nhdsSet
 #align normal_space.t3_space T4Space.t3Space
 
-instance (priority := 100) T4Space.of_compactSpace_t2Space [CompactSpace α] [T2Space α] :
-    T4Space α where
+instance (priority := 100) T4Space.of_compactSpace_t2Space [CompactSpace X] [T2Space X] :
+    T4Space X where
   normal _s _t hs ht := isCompact_isCompact_separated hs.isCompact ht.isCompact
 #align normal_of_compact_t2 T4Space.of_compactSpace_t2Space
 
 /-- If the codomain of a closed embedding is a T₄ space, then so is the domain. -/
-protected theorem ClosedEmbedding.t4Space [TopologicalSpace β] [T4Space β] {f : α → β}
-    (hf : ClosedEmbedding f) : T4Space α where
+protected theorem ClosedEmbedding.t4Space [TopologicalSpace Y] [T4Space Y] {f : X → Y}
+    (hf : ClosedEmbedding f) : T4Space X where
   toT1Space := hf.toEmbedding.t1Space
   toNormalSpace := hf.normalSpace
 #align closed_embedding.normal_space ClosedEmbedding.t4Space
@@ -1889,7 +1886,7 @@ protected theorem ClosedEmbedding.t4Space [TopologicalSpace β] [T4Space β] {f
 namespace SeparationQuotient
 
 /-- The `SeparationQuotient` of a normal space is a normal space. -/
-instance [NormalSpace α] : NormalSpace (SeparationQuotient α) where
+instance [NormalSpace X] : NormalSpace (SeparationQuotient X) where
   normal s t hs ht hd := separatedNhds_iff_disjoint.2 <| by
     rw [← disjoint_comap_iff surjective_mk, comap_mk_nhdsSet, comap_mk_nhdsSet]
     exact disjoint_nhdsSet_nhdsSet (hs.preimage continuous_mk) (ht.preimage continuous_mk)
@@ -1897,27 +1894,27 @@ instance [NormalSpace α] : NormalSpace (SeparationQuotient α) where
 
 end SeparationQuotient
 
-variable (α)
+variable (X)
 
 end Normality
 
 section CompletelyNormal
 
-/-- A topological space `α` is a *completely normal Hausdorff space* if each subspace `s : Set α` is
-a normal Hausdorff space. Equivalently, `α` is a `T₁` space and for any two sets `s`, `t` such that
+/-- A topological space `X` is a *completely normal Hausdorff space* if each subspace `s : Set X` is
+a normal Hausdorff space. Equivalently, `X` is a `T₁` space and for any two sets `s`, `t` such that
 `closure s` is disjoint with `t` and `s` is disjoint with `closure t`, there exist disjoint
 neighbourhoods of `s` and `t`. -/
-class T5Space (α : Type u) [TopologicalSpace α] extends T1Space α : Prop where
+class T5Space (X : Type u) [TopologicalSpace X] extends T1Space X : Prop where
   /-- If `closure s` is disjoint with `t` and `s` is disjoint with `closure t`, then `s` and `t`
   admit disjoint neighbourhoods. -/
   completely_normal :
-    ∀ ⦃s t : Set α⦄, Disjoint (closure s) t → Disjoint s (closure t) → Disjoint (𝓝ˢ s) (𝓝ˢ t)
+    ∀ ⦃s t : Set X⦄, Disjoint (closure s) t → Disjoint s (closure t) → Disjoint (𝓝ˢ s) (𝓝ˢ t)
 #align t5_space T5Space
 
 export T5Space (completely_normal)
 
-theorem Embedding.t5Space [TopologicalSpace β] [T5Space β] {e : α → β} (he : Embedding e) :
-    T5Space α := by
+theorem Embedding.t5Space [TopologicalSpace Y] [T5Space Y] {e : X → Y} (he : Embedding e) :
+    T5Space X := by
   haveI := he.t1Space
   refine' ⟨fun s t hd₁ hd₂ => _⟩
   simp only [he.toInducing.nhdsSet_eq_comap]
@@ -1929,12 +1926,12 @@ theorem Embedding.t5Space [TopologicalSpace β] [T5Space β] {e : α → β} (he
 #align embedding.t5_space Embedding.t5Space
 
 /-- A subspace of a `T₅` space is a `T₅` space. -/
-instance [T5Space α] {p : α → Prop} : T5Space { x // p x } :=
+instance [T5Space X] {p : X → Prop} : T5Space { x // p x } :=
   embedding_subtype_val.t5Space
 
 -- see Note [lower instance priority]
 /-- A `T₅` space is a `T₄` space. -/
-instance (priority := 100) T5Space.toT4Space [T5Space α] : T4Space α where
+instance (priority := 100) T5Space.toT4Space [T5Space X] : T4Space X where
   normal s t hs ht hd := separatedNhds_iff_disjoint.2 <|
     completely_normal (by rwa [hs.closure_eq]) (by rwa [ht.closure_eq])
 #align t5_space.to_normal_space T5Space.toT4Space
@@ -1945,10 +1942,10 @@ open SeparationQuotient
 We don't have typeclasses for completely normal spaces (without T₁ assumption) and R₀ spaces,
 so we use `T5Space` for assumption and for conclusion.
 
-One can prove this using a homeomorphism between `α` and `SeparationQuotient α`.
+One can prove this using a homeomorphism between `X` and `SeparationQuotient X`.
 We give an alternative proof of the `completely_normal` axiom
-that works without assuming that `α` is a T₁ space. -/
-instance [T5Space α] : T5Space (SeparationQuotient α) where
+that works without assuming that `X` is a T₁ space. -/
+instance [T5Space X] : T5Space (SeparationQuotient X) where
   completely_normal s t hd₁ hd₂ := by
     rw [← disjoint_comap_iff surjective_mk, comap_mk_nhdsSet, comap_mk_nhdsSet]
     apply T5Space.completely_normal <;> rw [← preimage_mk_closure]
@@ -1956,40 +1953,40 @@ instance [T5Space α] : T5Space (SeparationQuotient α) where
 
 end CompletelyNormal
 
-/-- In a compact t2 space, the connected component of a point equals the intersection of all
+/-- In a compact T₂ space, the connected component of a point equals the intersection of all
 its clopen neighbourhoods. -/
-theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x : α) :
-    connectedComponent x = ⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, Z := by
+theorem connectedComponent_eq_iInter_clopen [T2Space X] [CompactSpace X] (x : X) :
+    connectedComponent x = ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s := by
   apply Subset.antisymm connectedComponent_subset_iInter_clopen
   -- Reduce to showing that the clopen intersection is connected.
-  refine' IsPreconnected.subset_connectedComponent _ (mem_iInter.2 fun Z => Z.2.2)
+  refine' IsPreconnected.subset_connectedComponent _ (mem_iInter.2 fun s => s.2.2)
   -- We do this by showing that any disjoint cover by two closed sets implies
   -- that one of these closed sets must contain our whole thing.
-  -- To reduce to the case where the cover is disjoint on all of `α` we need that `s` is closed
-  have hs : @IsClosed α _ (⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, Z) :=
-    isClosed_iInter fun Z => Z.2.1.2
+  -- To reduce to the case where the cover is disjoint on all of `X` we need that `s` is closed
+  have hs : @IsClosed X _ (⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s) :=
+    isClosed_iInter fun s => s.2.1.2
   rw [isPreconnected_iff_subset_of_fully_disjoint_closed hs]
   intro a b ha hb hab ab_disj
   -- Since our space is normal, we get two larger disjoint open sets containing the disjoint
   -- closed sets. If we can show that our intersection is a subset of any of these we can then
   -- "descend" this to show that it is a subset of either a or b.
   rcases normal_separation ha hb ab_disj with ⟨u, v, hu, hv, hau, hbv, huv⟩
-  obtain ⟨Z, H⟩ : ∃ Z : Set α, IsClopen Z ∧ x ∈ Z ∧ Z ⊆ u ∪ v
-  /- Now we find a clopen set `Z` around `x`, contained in `u ∪ v`. We utilize the fact that
+  obtain ⟨s, H⟩ : ∃ s : Set X, IsClopen s ∧ x ∈ s ∧ s ⊆ u ∪ v
+  /- Now we find a clopen set `s` around `x`, contained in `u ∪ v`. We utilize the fact that
   `X \ u ∪ v` will be compact, so there must be some finite intersection of clopen neighbourhoods of
   `X` disjoint to it, but a finite intersection of clopen sets is clopen so we let this be our
-  `Z`. -/
+  `s`. -/
   · have H1 := (hu.union hv).isClosed_compl.isCompact.inter_iInter_nonempty
-      (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z) fun Z => Z.2.1.2
+      (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s) fun s => s.2.1.2
     rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, not_forall] at H1
     cases' H1 (disjoint_compl_left_iff_subset.2 <| hab.trans <| union_subset_union hau hbv)
-      with Zi H2
-    refine' ⟨⋂ U ∈ Zi, Subtype.val U, _, _, _⟩
-    · exact isClopen_biInter_finset fun Z _ => Z.2.1
-    · exact mem_iInter₂.2 fun Z _ => Z.2.2
+      with si H2
+    refine' ⟨⋂ U ∈ si, Subtype.val U, _, _, _⟩
+    · exact isClopen_biInter_finset fun s _ => s.2.1
+    · exact mem_iInter₂.2 fun s _ => s.2.2
     · rwa [← disjoint_compl_left_iff_subset, disjoint_iff_inter_eq_empty,
         ← not_nonempty_iff_eq_empty]
-  -- So, we get a disjoint decomposition `Z = Z ∩ u ∪ Z ∩ v` of clopen sets. The intersection of all
+  -- So, we get a disjoint decomposition `s = s ∩ u ∪ s ∩ v` of clopen sets. The intersection of all
   -- clopen neighbourhoods will then lie in whichever of u or v x lies in and hence will be a subset
   -- of either a or b.
   · have H1 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hu hv huv
@@ -1997,26 +1994,26 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
     have H2 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hv hu huv.symm
     by_cases hxu : x ∈ u <;> [left; right]
     -- The x ∈ u case.
-    · suffices ⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, ↑Z ⊆ u
+    · suffices ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, ↑s ⊆ u
         from Disjoint.left_le_of_le_sup_right hab (huv.mono this hbv)
-      · apply Subset.trans _ (inter_subset_right Z u)
-        exact iInter_subset (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z.1)
-          ⟨Z ∩ u, H1, mem_inter H.2.1 hxu⟩
+      · apply Subset.trans _ (inter_subset_right s u)
+        exact iInter_subset (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s.1)
+          ⟨s ∩ u, H1, mem_inter H.2.1 hxu⟩
     -- If x ∉ u, we get x ∈ v since x ∈ u ∪ v. The rest is then like the x ∈ u case.
     · have h1 : x ∈ v :=
         (hab.trans (union_subset_union hau hbv) (mem_iInter.2 fun i => i.2.2)).resolve_left hxu
-      suffices ⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, ↑Z ⊆ v
+      suffices ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, ↑s ⊆ v
         from (huv.symm.mono this hau).left_le_of_le_sup_left hab
-      · refine Subset.trans ?_ (inter_subset_right Z v)
-        exact iInter_subset (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z.1)
-          ⟨Z ∩ v, H2, mem_inter H.2.1 h1⟩
+      · refine Subset.trans ?_ (inter_subset_right s v)
+        exact iInter_subset (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s.1)
+          ⟨s ∩ v, H2, mem_inter H.2.1 h1⟩
 #align connected_component_eq_Inter_clopen connectedComponent_eq_iInter_clopen
 
 section Profinite
 
 /-- A T1 space with a clopen basis is totally separated. -/
-theorem totallySeparatedSpace_of_t1_of_basis_clopen [T1Space α]
-    (h : IsTopologicalBasis { s : Set α | IsClopen s }) : TotallySeparatedSpace α := by
+theorem totallySeparatedSpace_of_t1_of_basis_clopen [T1Space X]
+    (h : IsTopologicalBasis { s : Set X | IsClopen s }) : TotallySeparatedSpace X := by
   constructor
   rintro x - y - hxy
   rcases h.mem_nhds_iff.mp (isOpen_ne.mem_nhds hxy) with ⟨U, hU, hxU, hyU⟩
@@ -2024,44 +2021,44 @@ theorem totallySeparatedSpace_of_t1_of_basis_clopen [T1Space α]
     disjoint_compl_right⟩
 #align totally_separated_space_of_t1_of_basis_clopen totallySeparatedSpace_of_t1_of_basis_clopen
 
-variable [T2Space α] [CompactSpace α]
+variable [T2Space X] [CompactSpace X]
 
 /-- A compact Hausdorff space is totally disconnected if and only if it is totally separated, this
   is also true for locally compact spaces. -/
-theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace α ↔ TotallySeparatedSpace α := by
+theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace X ↔ TotallySeparatedSpace X := by
   refine ⟨fun h => ⟨fun x _ y _ => ?_⟩, @TotallySeparatedSpace.totallyDisconnectedSpace _ _⟩
   contrapose!
   intro hyp
   suffices x ∈ connectedComponent y by
     simpa [totallyDisconnectedSpace_iff_connectedComponent_singleton.1 h y, mem_singleton_iff]
   rw [connectedComponent_eq_iInter_clopen, mem_iInter]
-  rintro ⟨w : Set α, hw : IsClopen w, hy : y ∈ w⟩
+  rintro ⟨w : Set X, hw : IsClopen w, hy : y ∈ w⟩
   by_contra hx
   exact hyp wᶜ w hw.2.isOpen_compl hw.1 hx hy (@isCompl_compl _ w _).symm.codisjoint.top_le
     disjoint_compl_left
 #align compact_t2_tot_disc_iff_tot_sep compact_t2_tot_disc_iff_tot_sep
 
-variable [TotallyDisconnectedSpace α]
+variable [TotallyDisconnectedSpace X]
 
 /-- A totally disconnected compact Hausdorff space is totally separated. -/
-instance : TotallySeparatedSpace α := compact_t2_tot_disc_iff_tot_sep.mp inferInstance
+instance : TotallySeparatedSpace X := compact_t2_tot_disc_iff_tot_sep.mp inferInstance
 
-theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈ s ∧ IsClopen s) id :=
+theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsClopen s) id :=
   ⟨fun U => by
     constructor
     · have hx : connectedComponent x = {x} :=
         totallyDisconnectedSpace_iff_connectedComponent_singleton.mp ‹_› x
       rw [connectedComponent_eq_iInter_clopen] at hx
       intro hU
-      let N := { Z // IsClopen Z ∧ x ∈ Z }
-      suffices : ∃ Z : N, Z.val ⊆ U
+      let N := { s // IsClopen s ∧ x ∈ s }
+      suffices : ∃ s : N, s.val ⊆ U
       · rcases this with ⟨⟨s, hs, hs'⟩, hs''⟩; exact ⟨s, ⟨hs', hs⟩, hs''⟩
       haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩
-      have hNcl : ∀ Z : N, IsClosed Z.val := fun Z => Z.property.1.2
-      have hdir : Directed Superset fun Z : N => Z.val := by
+      have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.2
+      have hdir : Directed Superset fun s : N => s.val := by
         rintro ⟨s, hs, hxs⟩ ⟨t, ht, hxt⟩
         exact ⟨⟨s ∩ t, hs.inter ht, ⟨hxs, hxt⟩⟩, inter_subset_left s t, inter_subset_right s t⟩
-      have h_nhd : ∀ y ∈ ⋂ Z : N, Z.val, U ∈ 𝓝 y := fun y y_in => by
+      have h_nhd : ∀ y ∈ ⋂ s : N, s.val, U ∈ 𝓝 y := fun y y_in => by
         erw [hx, mem_singleton_iff] at y_in
         rwa [y_in]
       exact exists_subset_nhds_of_compactSpace hdir hNcl h_nhd
@@ -2070,7 +2067,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
       exact ⟨V, hUV, V_op, hxV⟩⟩
 #align nhds_basis_clopen nhds_basis_clopen
 
-theorem isTopologicalBasis_clopen : IsTopologicalBasis { s : Set α | IsClopen s } := by
+theorem isTopologicalBasis_clopen : IsTopologicalBasis { s : Set X | IsClopen s } := by
   apply isTopologicalBasis_of_open_of_nhds fun U (hU : IsClopen U) => hU.1
   intro x U hxU U_op
   have : U ∈ 𝓝 x := IsOpen.mem_nhds U_op hxU
@@ -2081,8 +2078,8 @@ theorem isTopologicalBasis_clopen : IsTopologicalBasis { s : Set α | IsClopen s
 
 /-- Every member of an open set in a compact Hausdorff totally disconnected space
   is contained in a clopen set contained in the open set.  -/
-theorem compact_exists_clopen_in_open {x : α} {U : Set α} (is_open : IsOpen U) (memU : x ∈ U) :
-    ∃ V : Set α, IsClopen V ∧ x ∈ V ∧ V ⊆ U :=
+theorem compact_exists_clopen_in_open {x : X} {U : Set X} (is_open : IsOpen U) (memU : x ∈ U) :
+    ∃ V : Set X, IsClopen V ∧ x ∈ V ∧ V ⊆ U :=
   isTopologicalBasis_clopen.mem_nhds_iff.1 (is_open.mem_nhds memU)
 #align compact_exists_clopen_in_open compact_exists_clopen_in_open
 
@@ -2136,8 +2133,8 @@ theorem loc_compact_t2_tot_disc_iff_tot_sep :
 
 end LocallyCompact
 
-/-- `ConnectedComponents α` is Hausdorff when `α` is Hausdorff and compact -/
-instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (ConnectedComponents α) := by
+/-- `ConnectedComponents X` is Hausdorff when `X` is Hausdorff and compact -/
+instance ConnectedComponents.t2 [T2Space X] [CompactSpace X] : T2Space (ConnectedComponents X) := by
   -- Proof follows that of: https://stacks.math.columbia.edu/tag/0900
   -- Fix 2 distinct connected components, with points a and b
   refine ⟨ConnectedComponents.surjective_coe.forall₂.2 fun a b ne => ?_⟩
@@ -2146,16 +2143,16 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
   -- write ↑b as the intersection of all clopen subsets containing it
   rw [connectedComponent_eq_iInter_clopen b, disjoint_iff_inter_eq_empty] at h
   -- Now we show that this can be reduced to some clopen containing `↑b` being disjoint to `↑a`
-  obtain ⟨U, V, hU, ha, hb, rfl⟩ : ∃ (U : Set α) (V : Set (ConnectedComponents α)),
+  obtain ⟨U, V, hU, ha, hb, rfl⟩ : ∃ (U : Set X) (V : Set (ConnectedComponents X)),
       IsClopen U ∧ connectedComponent a ∩ U = ∅ ∧ connectedComponent b ⊆ U ∧ (↑) ⁻¹' V = U := by
     have h :=
-      (isClosed_connectedComponent (α := α)).isCompact.elim_finite_subfamily_closed
-        _ (fun Z : { Z : Set α // IsClopen Z ∧ b ∈ Z } => Z.2.1.2) h
+      (isClosed_connectedComponent (α := X)).isCompact.elim_finite_subfamily_closed
+        _ (fun s : { s : Set X // IsClopen s ∧ b ∈ s } => s.2.1.2) h
     cases' h with fin_a ha
     -- This clopen and its complement will separate the connected components of `a` and `b`
-    set U : Set α := ⋂ (i : { Z // IsClopen Z ∧ b ∈ Z }) (_ : i ∈ fin_a), i
+    set U : Set X := ⋂ (i : { s // IsClopen s ∧ b ∈ s }) (_ : i ∈ fin_a), i
     have hU : IsClopen U := isClopen_biInter_finset fun i _ => i.2.1
-    exact ⟨U, (↑) '' U, hU, ha, subset_iInter₂ fun Z _ => Z.2.1.connectedComponent_subset Z.2.2,
+    exact ⟨U, (↑) '' U, hU, ha, subset_iInter₂ fun s _ => s.2.1.connectedComponent_subset s.2.2,
       (connectedComponents_preimage_image U).symm ▸ hU.biUnion_connectedComponent_eq⟩
   rw [ConnectedComponents.quotientMap_coe.isClopen_preimage] at hU
   refine' ⟨Vᶜ, V, hU.compl.isOpen, hU.isOpen, _, hb mem_connectedComponent, disjoint_compl_left⟩

A continuous function injective on a compact set and injective on a neighborhood of each point of this set is injective on a neighborhood of this set.

From the Mandelbrot set connectedness project.

Co-authored-by: @girving

@@ -1012,6 +1012,42 @@ theorem tendsto_nhds_unique_of_frequently_eq [T2Space α] {f g : β → α} {l :
   not_not.1 fun hne => this (isClosed_diagonal.isOpen_compl.mem_nhds hne)
 #align tendsto_nhds_unique_of_frequently_eq tendsto_nhds_unique_of_frequently_eq
 
+/-- If a function `f` is
+
+- injective on a compact set `s`;
+- continuous at every point of this set;
+- injective on a neighborhood of each point of this set,
+
+then it is injective on a neighborhood of this set. -/
+theorem Set.InjOn.exists_mem_nhdsSet {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    [T2Space Y] {f : X → Y} {s : Set X} (inj : InjOn f s) (sc : IsCompact s)
+    (fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u) :
+    ∃ t ∈ 𝓝ˢ s, InjOn f t := by
+  have : ∀ x ∈ s ×ˢ s, ∀ᶠ y in 𝓝 x, f y.1 = f y.2 → y.1 = y.2 := fun (x, y) ⟨hx, hy⟩ ↦ by
+    rcases eq_or_ne x y with rfl | hne
+    · rcases loc x hx with ⟨u, hu, hf⟩
+      exact Filter.mem_of_superset (prod_mem_nhds hu hu) <| forall_prod_set.2 hf
+    · suffices ∀ᶠ z in 𝓝 (x, y), f z.1 ≠ f z.2 from this.mono fun _ hne h ↦ absurd h hne
+      refine (fc x hx).prod_map' (fc y hy) <| isClosed_diagonal.isOpen_compl.mem_nhds ?_
+      exact inj.ne hx hy hne
+  rw [← eventually_nhdsSet_iff_forall, sc.nhdsSet_prod_eq sc] at this
+  exact eventually_prod_self_iff.1 this
+
+/-- If a function `f` is
+
+- injective on a compact set `s`;
+- continuous at every point of this set;
+- injective on a neighborhood of each point of this set,
+
+then it is injective on an open neighborhood of this set. -/
+theorem Set.InjOn.exists_isOpen_superset {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    [T2Space Y] {f : X → Y} {s : Set X} (inj : InjOn f s) (sc : IsCompact s)
+    (fc : ∀ x ∈ s, ContinuousAt f x) (loc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u) :
+    ∃ t, IsOpen t ∧ s ⊆ t ∧ InjOn f t :=
+  let ⟨_t, hst, ht⟩ := inj.exists_mem_nhdsSet sc fc loc
+  let ⟨u, huo, hsu, hut⟩ := mem_nhdsSet_iff_exists.1 hst
+  ⟨u, huo, hsu, ht.mono hut⟩
+
 /-- A T₂.₅ space, also known as a Urysohn space, is a topological space
   where for every pair `x ≠ y`, there are two open sets, with the intersection of closures
   empty, one containing `x` and the other `y` . -/

In the last step, I have removed redundant imports: those which are implied by the other imports. I can revert those changes if desired/if this seems too brittle.

@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
 import Mathlib.Topology.Compactness.SigmaCompact
-import Mathlib.Topology.Connected
-import Mathlib.Topology.NhdsSet
+import Mathlib.Topology.Connected.LocallyConnected
+import Mathlib.Topology.Connected.TotallyDisconnected
 import Mathlib.Topology.Inseparable
 
 #align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -1530,7 +1530,7 @@ theorem regularSpace_TFAE (X : Type u) [ TopologicalSpace X ] :
   tfae_have 6 → 4
   · intro H a s hs
     rw [← H] at hs
-    rcases(𝓝 a).basis_sets.lift'_closure.mem_iff.mp hs with ⟨U, hU, hUs⟩
+    rcases (𝓝 a).basis_sets.lift'_closure.mem_iff.mp hs with ⟨U, hU, hUs⟩
     exact ⟨closure U, mem_of_superset hU subset_closure, isClosed_closure, hUs⟩
   tfae_have 4 → 2
   · intro H s a ha

Split up the 2000-line Topology/SubsetProperties.lean into several smaller files. Not only is it too huge, but the name is very unhelpful, since actually about 90% of the file is about compactness; I've moved this material into various files inside a new subdirectory Topology/Compactness/.

@@ -3,7 +3,7 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
-import Mathlib.Topology.SubsetProperties
+import Mathlib.Topology.Compactness.SigmaCompact
 import Mathlib.Topology.Connected
 import Mathlib.Topology.NhdsSet
 import Mathlib.Topology.Inseparable

@@ -71,7 +71,7 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
 If the space is also compact:
 
 * `normalOfCompactT2`: A compact T₂ space is a `NormalSpace`.
-* `connected_components_eq_Inter_clopen`: The connected component of a point
+* `connectedComponent_eq_iInter_clopen`: The connected component of a point
   is the intersection of all its clopen neighbourhoods.
 * `compact_t2_tot_disc_iff_tot_sep`: Being a `TotallyDisconnectedSpace`
   is equivalent to being a `TotallySeparatedSpace`.

As discussed on Zulip.

Co-authored-by: grunweg <grunweg@posteo.de>

@@ -461,7 +461,7 @@ theorem Bornology.relativelyCompact.isBounded_iff [T1Space α] {s : Set α} :
   constructor
   · rintro ⟨t, ht₁, ht₂, hst⟩
     rw [compl_subset_compl] at hst
-    exact isCompact_of_isClosed_subset ht₂ isClosed_closure (closure_minimal hst ht₁)
+    exact ht₂.of_isClosed_subset isClosed_closure (closure_minimal hst ht₁)
   · intro h
     exact ⟨closure s, isClosed_closure, h, compl_subset_compl.mpr subset_closure⟩
 #align bornology.relatively_compact.is_bounded_iff Bornology.relativelyCompact.isBounded_iff
@@ -1343,7 +1343,7 @@ theorem IsCompact.inter [T2Space α] {s t : Set α} (hs : IsCompact s) (ht : IsC
 
 theorem isCompact_closure_of_subset_compact [T2Space α] {s t : Set α} (ht : IsCompact t)
     (h : s ⊆ t) : IsCompact (closure s) :=
-  isCompact_of_isClosed_subset ht isClosed_closure (closure_minimal h ht.isClosed)
+  ht.of_isClosed_subset isClosed_closure (closure_minimal h ht.isClosed)
 #align is_compact_closure_of_subset_compact isCompact_closure_of_subset_compact
 
 @[simp]

In an Alexandrov-discrete space, every set has a smallest neighborhood. We call this neighborhood the exterior of the set. It is completely analogous to the interior, except that all inclusions are reversed.

@@ -679,12 +679,12 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
 #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert
 
 @[simp]
-theorem sInter_sets_nhds [T1Space α] (x : α) : ⋂₀ (𝓝 x).sets = {x} := by
-  simp [sInter_nhds_sets_eq_specializes]
+theorem ker_nhds [T1Space α] (x : α) : (𝓝 x).ker = {x} := by
+  simp [ker_nhds_eq_specializes]
 
 theorem biInter_basis_nhds [T1Space α] {ι : Sort*} {p : ι → Prop} {s : ι → Set α} {x : α}
     (h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} := by
-  rw [← h.sInter_sets, sInter_sets_nhds]
+  rw [← h.ker, ker_nhds]
 #align bInter_basis_nhds biInter_basis_nhds
 
 @[simp]

@@ -410,6 +410,12 @@ theorem Ne.nhdsWithin_diff_singleton [T1Space α] {x y : α} (h : x ≠ y) (s :
   exact mem_nhdsWithin_of_mem_nhds (isOpen_ne.mem_nhds h)
 #align ne.nhds_within_diff_singleton Ne.nhdsWithin_diff_singleton
 
+lemma nhdsWithin_compl_singleton_le [T1Space α] (x y : α) : 𝓝[{x}ᶜ] x ≤ 𝓝[{y}ᶜ] x := by
+  rcases eq_or_ne x y with rfl|hy
+  · exact Eq.le rfl
+  · rw [Ne.nhdsWithin_compl_singleton hy]
+    exact nhdsWithin_le_nhds
+
 theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
     IsOpen { x | ∀ᶠ y in 𝓝[≠] x, p y } := by
   refine' isOpen_iff_mem_nhds.mpr fun a ha => _

• Rename NormalSpace to T4Space.
• Add NormalSpace, a version without the T1Space assumption.
• Adjust some theorems.
• Supersedes thus closes #6892.
• Add some instance cycles, see #2030

@@ -31,9 +31,9 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
   there is two open sets, one containing `x`, and the other `y`, whose closures are disjoint.
 * `T3Space`: A T₃ space, is one where given any closed `C` and `x ∉ C`,
   there is disjoint open sets containing `x` and `C` respectively. In `mathlib`, T₃ implies T₂.₅.
-* `NormalSpace`: A T₄ space (sometimes referred to as normal, but authors vary on
-  whether this includes T₂; `mathlib` does), is one where given two disjoint closed sets,
-  we can find two open sets that separate them. In `mathlib`, T₄ implies T₃.
+* `NormalSpace`: A normal space, is one where given two disjoint closed sets,
+  we can find two open sets that separate them.
+* `T4Space`: A T₄ space is a normal T₁ space. T₄ implies T₃.
 * `T5Space`: A T₅ space, also known as a *completely normal Hausdorff space*
 
 ## Main results
@@ -1735,22 +1735,23 @@ instance [RegularSpace α] : T3Space (SeparationQuotient α) where
 
 end T3
 
-section Normality
+section NormalSpace
 
--- todo: rename this to `T4Space`, introduce `NormalSpace` without `T1Space` assumption
-/-- A T₄ space, also known as a normal space (although this condition sometimes
-  omits T₂), is one in which for every pair of disjoint closed sets `C` and `D`,
-  there exist disjoint open sets containing `C` and `D` respectively. -/
-class NormalSpace (α : Type u) [TopologicalSpace α] extends T1Space α : Prop where
+/-- A topological space is said to be a *normal space* if any two disjoint closed sets
+have disjoint open neighborhoods. -/
+class NormalSpace (X : Type u) [TopologicalSpace X] : Prop where
   /-- Two disjoint sets in a normal space admit disjoint neighbourhoods. -/
-  normal : ∀ s t : Set α, IsClosed s → IsClosed t → Disjoint s t → SeparatedNhds s t
-#align normal_space NormalSpace
+  normal : ∀ s t : Set X, IsClosed s → IsClosed t → Disjoint s t → SeparatedNhds s t
 
 theorem normal_separation [NormalSpace α] {s t : Set α} (H1 : IsClosed s) (H2 : IsClosed t)
     (H3 : Disjoint s t) : SeparatedNhds s t :=
   NormalSpace.normal s t H1 H2 H3
 #align normal_separation normal_separation
 
+theorem disjoint_nhdsSet_nhdsSet [NormalSpace α] {s t : Set α} (hs : IsClosed s) (ht : IsClosed t)
+    (hd : Disjoint s t) : Disjoint (𝓝ˢ s) (𝓝ˢ t) :=
+  (normal_separation hs ht hd).disjoint_nhdsSet
+
 theorem normal_exists_closure_subset [NormalSpace α] {s t : Set α} (hs : IsClosed s) (ht : IsOpen t)
     (hst : s ⊆ t) : ∃ u, IsOpen u ∧ s ⊆ u ∧ closure u ⊆ t := by
   have : Disjoint s tᶜ := Set.disjoint_left.mpr fun x hxs hxt => hxt (hst hxs)
@@ -1761,49 +1762,18 @@ theorem normal_exists_closure_subset [NormalSpace α] {s t : Set α} (hs : IsClo
   exact fun x hxs hxt => hs't'.le_bot ⟨hxs, hxt⟩
 #align normal_exists_closure_subset normal_exists_closure_subset
 
--- see Note [lower instance priority]
-instance (priority := 100) NormalSpace.t3Space [NormalSpace α] : T3Space α where
-  regular hs hxs := by simpa only [nhdsSet_singleton] using (normal_separation hs isClosed_singleton
-    (disjoint_singleton_right.mpr hxs)).disjoint_nhdsSet
-#align normal_space.t3_space NormalSpace.t3Space
-
--- We can't make this an instance because it could cause an instance loop.
--- porting note: todo: now we can
-theorem normalOfCompactT2 [CompactSpace α] [T2Space α] : NormalSpace α :=
-  ⟨fun _s _t hs ht => isCompact_isCompact_separated hs.isCompact ht.isCompact⟩
-#align normal_of_compact_t2 normalOfCompactT2
-
+/-- If the codomain of a closed embedding is a normal space, then so is the domain. -/
 protected theorem ClosedEmbedding.normalSpace [TopologicalSpace β] [NormalSpace β] {f : α → β}
     (hf : ClosedEmbedding f) : NormalSpace α where
-  toT1Space := hf.toEmbedding.t1Space
   normal s t hs ht hst := by
     have H : SeparatedNhds (f '' s) (f '' t) :=
       NormalSpace.normal (f '' s) (f '' t) (hf.isClosedMap s hs) (hf.isClosedMap t ht)
         (disjoint_image_of_injective hf.inj hst)
     exact (H.preimage hf.continuous).mono (subset_preimage_image _ _) (subset_preimage_image _ _)
-#align closed_embedding.normal_space ClosedEmbedding.normalSpace
-
-namespace SeparationQuotient
 
-/-- The `SeparationQuotient` of a normal space is a T₄ space. We don't have separate typeclasses
-for normal spaces (without T₁ assumption) and T₄ spaces, so we use the same class for assumption
-and for conclusion.
-
-One can prove this using a homeomorphism between `α` and `SeparationQuotient α`. We give an
-alternative proof that works without assuming that `α` is a T₁ space. -/
-instance [NormalSpace α] : NormalSpace (SeparationQuotient α) where
-  normal s t hs ht hd := separatedNhds_iff_disjoint.2 <| by
-    rw [← disjoint_comap_iff surjective_mk, comap_mk_nhdsSet, comap_mk_nhdsSet]
-    exact (normal_separation (hs.preimage continuous_mk) (ht.preimage continuous_mk)
-      (hd.preimage mk)).disjoint_nhdsSet
-
-end SeparationQuotient
-
-variable (α)
-
-/-- A T₃ topological space with second countable topology is a normal space.
-This lemma is not an instance to avoid a loop. -/
-theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α] : NormalSpace α := by
+/-- A regular topological space with second countable topology is a normal space. -/
+instance (priority := 100) NormalSpace.of_regularSpace_secondCountableTopology
+    [RegularSpace α] [SecondCountableTopology α] : NormalSpace α := by
   have key : ∀ {s t : Set α}, IsClosed t → Disjoint s t →
     ∃ U : Set (countableBasis α), (s ⊆ ⋃ u ∈ U, ↑u) ∧ (∀ u ∈ U, Disjoint (closure ↑u) t) ∧
       ∀ n : ℕ, IsClosed (⋃ (u ∈ U) (_ : Encodable.encode u ≤ n), closure (u : Set α)) := by
@@ -1822,7 +1792,7 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
     · simp only [← iSup_eq_iUnion, iSup_and']
       exact (((finite_le_nat n).preimage_embedding (Encodable.encode' _)).subset <|
         inter_subset_right _ _).isClosed_biUnion fun u _ => isClosed_closure
-  refine' ⟨fun s t hs ht hd => _⟩
+  refine' { normal := fun s t hs ht hd => _ }
   rcases key ht hd with ⟨U, hsU, hUd, hUc⟩
   rcases key hs hd.symm with ⟨V, htV, hVd, hVc⟩
   refine ⟨⋃ u ∈ U, ↑u \ ⋃ (v ∈ V) (_ : Encodable.encode v ≤ Encodable.encode u), closure ↑v,
@@ -1844,7 +1814,48 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
     rintro a ⟨u, huU, hau, haV⟩ v hvV hav
     cases' le_total (Encodable.encode u) (Encodable.encode v) with hle hle
     exacts [⟨u, huU, hle, subset_closure hau⟩, (haV _ hvV hle <| subset_closure hav).elim]
-#align normal_space_of_t3_second_countable normalSpaceOfT3SecondCountable
+#align normal_space_of_t3_second_countable NormalSpace.of_regularSpace_secondCountableTopology
+
+end NormalSpace
+
+section Normality
+
+/-- A T₄ space is a normal T₁ space. -/
+class T4Space (α : Type u) [TopologicalSpace α] extends T1Space α, NormalSpace α : Prop
+#align normal_space NormalSpace
+
+instance (priority := 100) [T1Space α] [NormalSpace α] : T4Space α := ⟨⟩
+
+-- see Note [lower instance priority]
+instance (priority := 100) T4Space.t3Space [T4Space α] : T3Space α where
+  regular hs hxs := by simpa only [nhdsSet_singleton] using (normal_separation hs isClosed_singleton
+    (disjoint_singleton_right.mpr hxs)).disjoint_nhdsSet
+#align normal_space.t3_space T4Space.t3Space
+
+instance (priority := 100) T4Space.of_compactSpace_t2Space [CompactSpace α] [T2Space α] :
+    T4Space α where
+  normal _s _t hs ht := isCompact_isCompact_separated hs.isCompact ht.isCompact
+#align normal_of_compact_t2 T4Space.of_compactSpace_t2Space
+
+/-- If the codomain of a closed embedding is a T₄ space, then so is the domain. -/
+protected theorem ClosedEmbedding.t4Space [TopologicalSpace β] [T4Space β] {f : α → β}
+    (hf : ClosedEmbedding f) : T4Space α where
+  toT1Space := hf.toEmbedding.t1Space
+  toNormalSpace := hf.normalSpace
+#align closed_embedding.normal_space ClosedEmbedding.t4Space
+
+namespace SeparationQuotient
+
+/-- The `SeparationQuotient` of a normal space is a normal space. -/
+instance [NormalSpace α] : NormalSpace (SeparationQuotient α) where
+  normal s t hs ht hd := separatedNhds_iff_disjoint.2 <| by
+    rw [← disjoint_comap_iff surjective_mk, comap_mk_nhdsSet, comap_mk_nhdsSet]
+    exact disjoint_nhdsSet_nhdsSet (hs.preimage continuous_mk) (ht.preimage continuous_mk)
+      (hd.preimage mk)
+
+end SeparationQuotient
+
+variable (α)
 
 end Normality
 
@@ -1881,19 +1892,20 @@ instance [T5Space α] {p : α → Prop} : T5Space { x // p x } :=
 
 -- see Note [lower instance priority]
 /-- A `T₅` space is a `T₄` space. -/
-instance (priority := 100) T5Space.toNormalSpace [T5Space α] : NormalSpace α :=
-  ⟨fun s t hs ht hd => separatedNhds_iff_disjoint.2 <|
-    completely_normal (by rwa [hs.closure_eq]) (by rwa [ht.closure_eq])⟩
-#align t5_space.to_normal_space T5Space.toNormalSpace
+instance (priority := 100) T5Space.toT4Space [T5Space α] : T4Space α where
+  normal s t hs ht hd := separatedNhds_iff_disjoint.2 <|
+    completely_normal (by rwa [hs.closure_eq]) (by rwa [ht.closure_eq])
+#align t5_space.to_normal_space T5Space.toT4Space
 
 open SeparationQuotient
 
-/-- The `SeparationQuotient` of a completely normal space is a T₅ space. We don't have separate
-typeclasses for completely normal spaces (without T₁ assumption) and T₅ spaces, so we use the same
-class for assumption and for conclusion.
+/-- The `SeparationQuotient` of a completely normal R₀ space is a T₅ space.
+We don't have typeclasses for completely normal spaces (without T₁ assumption) and R₀ spaces,
+so we use `T5Space` for assumption and for conclusion.
 
-One can prove this using a homeomorphism between `α` and `SeparationQuotient α`. We give an
-alternative proof that works without assuming that `α` is a T₁ space. -/
+One can prove this using a homeomorphism between `α` and `SeparationQuotient α`.
+We give an alternative proof of the `completely_normal` axiom
+that works without assuming that `α` is a T₁ space. -/
 instance [T5Space α] : T5Space (SeparationQuotient α) where
   completely_normal s t hd₁ hd₂ := by
     rw [← disjoint_comap_iff surjective_mk, comap_mk_nhdsSet, comap_mk_nhdsSet]
@@ -1916,13 +1928,11 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
     isClosed_iInter fun Z => Z.2.1.2
   rw [isPreconnected_iff_subset_of_fully_disjoint_closed hs]
   intro a b ha hb hab ab_disj
-  haveI := @normalOfCompactT2 α _ _ _
   -- Since our space is normal, we get two larger disjoint open sets containing the disjoint
   -- closed sets. If we can show that our intersection is a subset of any of these we can then
   -- "descend" this to show that it is a subset of either a or b.
   rcases normal_separation ha hb ab_disj with ⟨u, v, hu, hv, hau, hbv, huv⟩
   obtain ⟨Z, H⟩ : ∃ Z : Set α, IsClopen Z ∧ x ∈ Z ∧ Z ⊆ u ∪ v
-
   /- Now we find a clopen set `Z` around `x`, contained in `u ∪ v`. We utilize the fact that
   `X \ u ∪ v` will be compact, so there must be some finite intersection of clopen neighbourhoods of
   `X` disjoint to it, but a finite intersection of clopen sets is clopen so we let this be our

Add an instance [T0Space α] [RegularSpace α] : T3Space α, drop now unneeded lemmas like TopologicalGroup.t3Space.

Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -349,7 +349,7 @@ theorem t0Space_iff_or_not_mem_closure (α : Type u) [TopologicalSpace α] :
   simp only [t0Space_iff_not_inseparable, inseparable_iff_mem_closure, not_and_or]
 #align t0_space_iff_or_not_mem_closure t0Space_iff_or_not_mem_closure
 
-instance [TopologicalSpace β] [T0Space α] [T0Space β] : T0Space (α × β) :=
+instance Prod.instT0Space [TopologicalSpace β] [T0Space α] [T0Space β] : T0Space (α × β) :=
   ⟨fun _ _ h => Prod.ext (h.map continuous_fst).eq (h.map continuous_snd).eq⟩
 
 instance Pi.instT0Space {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
@@ -1685,6 +1685,8 @@ a T₂.₅ space.  -/
 class T3Space (α : Type u) [TopologicalSpace α] extends T0Space α, RegularSpace α : Prop
 #align t3_space T3Space
 
+instance (priority := 90) [T0Space α] [RegularSpace α] : T3Space α := ⟨⟩
+
 -- see Note [lower instance priority]
 instance (priority := 100) T3Space.t25Space [T3Space α] : T25Space α := by
   refine' ⟨fun x y hne => _⟩

We define Alexandrov-discrete spaces as topological spaces where the intersection of a family of open sets is open.

This PR only gives a minimal API because the goal is to ensure that lemma names like isOpen_sInter are free to use for AlexandrovDiscrete. The existing lemmas are getting prefixed by Set.Finite or suffixed by _of_finite.

@@ -422,7 +422,7 @@ theorem isOpen_setOf_eventually_nhdsWithin [T1Space α] {p : α → Prop} :
 
 protected theorem Set.Finite.isClosed [T1Space α] {s : Set α} (hs : Set.Finite s) : IsClosed s := by
   rw [← biUnion_of_singleton s]
-  exact isClosed_biUnion hs fun i _ => isClosed_singleton
+  exact hs.isClosed_biUnion fun i _ => isClosed_singleton
 #align set.finite.is_closed Set.Finite.isClosed
 
 theorem TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne [T1Space α] {b : Set (Set α)}
@@ -1818,8 +1818,8 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
     · rw [biUnion_range]
       exact fun x hx => mem_iUnion.2 ⟨⟨x, hx⟩, hxu x hx⟩
     · simp only [← iSup_eq_iUnion, iSup_and']
-      exact isClosed_biUnion (((finite_le_nat n).preimage_embedding (Encodable.encode' _)).subset <|
-        inter_subset_right _ _) fun u _ => isClosed_closure
+      exact (((finite_le_nat n).preimage_embedding (Encodable.encode' _)).subset <|
+        inter_subset_right _ _).isClosed_biUnion fun u _ => isClosed_closure
   refine' ⟨fun s t hs ht hd => _⟩
   rcases key ht hd with ⟨U, hsU, hUd, hUc⟩
   rcases key hs hd.symm with ⟨V, htV, hVd, hVc⟩

@@ -60,8 +60,9 @@ This file defines the predicate `SeparatedNhds`, and common separation axioms
   these results are part of the typeclass inference system (e.g. `Embedding.t2Space`)
 * `Set.EqOn.closure`: If two functions are equal on some set `s`, they are equal on its closure.
 * `IsCompact.isClosed`: All compact sets are closed.
-* `locally_compact_of_compact_nhds`: If every point has a compact neighbourhood,
-  then the space is locally compact.
+* `WeaklyLocallyCompactSpace.locallyCompactSpace`: If a topological space is both
+  weakly locally compact (i.e., each point has a compact neighbourhood)
+  and is T₂, then it is locally compact.
 * `totallySeparatedSpace_of_t1_of_basis_clopen`: If `α` has a clopen basis, then
   it is a `TotallySeparatedSpace`.
 * `loc_compact_t2_tot_disc_iff_tot_sep`: A locally compact T₂ space is totally disconnected iff
@@ -1412,13 +1413,13 @@ theorem IsCompact.finite_compact_cover [T2Space α] {s : Set α} (hs : IsCompact
 
 end
 
-/-- If every points of a Hausdorff space admits a compact neighborhood, then this space is locally
-compact. -/
-theorem locally_compact_of_compact_nhds [T2Space α] (h : ∀ x : α, ∃ s, s ∈ 𝓝 x ∧ IsCompact s) :
+-- see Note [lower instance priority]
+/-- A weakly locally compact Hausdorff space is locally compact. -/
+instance WeaklyLocallyCompactSpace.locallyCompactSpace [WeaklyLocallyCompactSpace α] [T2Space α] :
     LocallyCompactSpace α :=
   ⟨fun x _n hn =>
     let ⟨_u, un, uo, xu⟩ := mem_nhds_iff.mp hn
-    let ⟨k, kx, kc⟩ := h x
+    let ⟨k, kc, kx⟩ := exists_compact_mem_nhds x
     -- K is compact but not necessarily contained in N.
     -- K \ U is again compact and doesn't contain x, so
     -- we may find open sets V, W separating x from K \ U.
@@ -1429,31 +1430,31 @@ theorem locally_compact_of_compact_nhds [T2Space α] (h : ∀ x : α, ∃ s, s 
     have wn : wᶜ ∈ 𝓝 x :=
       mem_nhds_iff.mpr ⟨v, vw.subset_compl_right, vo, singleton_subset_iff.mp xv⟩
     ⟨k \ w, Filter.inter_mem kx wn, Subset.trans (diff_subset_comm.mp kuw) un, kc.diff wo⟩⟩
-#align locally_compact_of_compact_nhds locally_compact_of_compact_nhds
+#align locally_compact_of_compact_nhds WeaklyLocallyCompactSpace.locallyCompactSpace
 
--- see Note [lower instance priority]
-instance (priority := 100) locally_compact_of_compact [T2Space α] [CompactSpace α] :
+@[deprecated WeaklyLocallyCompactSpace.locallyCompactSpace]
+theorem locally_compact_of_compact [T2Space α] [CompactSpace α] :
     LocallyCompactSpace α :=
-  locally_compact_of_compact_nhds fun _ => ⟨univ, isOpen_univ.mem_nhds trivial, isCompact_univ⟩
+  inferInstance
 #align locally_compact_of_compact locally_compact_of_compact
 
-/-- In a locally compact T₂ space, every point has an open neighborhood with compact closure -/
-theorem exists_open_with_compact_closure [LocallyCompactSpace α] [T2Space α] (x : α) :
-    ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
-  rcases exists_compact_mem_nhds x with ⟨K, hKc, hxK⟩
-  rcases mem_nhds_iff.1 hxK with ⟨t, h1t, h2t, h3t⟩
-  exact ⟨t, h2t, h3t, isCompact_closure_of_subset_compact hKc h1t⟩
-#align exists_open_with_compact_closure exists_open_with_compact_closure
-
-/-- In a locally compact T₂ space, every compact set has an open neighborhood with compact closure.
--/
-theorem exists_open_superset_and_isCompact_closure [LocallyCompactSpace α] [T2Space α] {K : Set α}
-    (hK : IsCompact K) : ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
+/-- In a weakly locally compact T₂ space,
+every compact set has an open neighborhood with compact closure. -/
+theorem exists_open_superset_and_isCompact_closure [WeaklyLocallyCompactSpace α] [T2Space α]
+    {K : Set α} (hK : IsCompact K) : ∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) := by
   rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
-  refine'
-    ⟨interior K', isOpen_interior, hKK', isCompact_closure_of_subset_compact hK' interior_subset⟩
+  exact ⟨interior K', isOpen_interior, hKK',
+    isCompact_closure_of_subset_compact hK' interior_subset⟩
 #align exists_open_superset_and_is_compact_closure exists_open_superset_and_isCompact_closure
 
+/-- In a weakly locally compact T₂ space,
+every point has an open neighborhood with compact closure. -/
+theorem exists_open_with_compact_closure [WeaklyLocallyCompactSpace α] [T2Space α] (x : α) :
+    ∃ U : Set α, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
+  simpa only [singleton_subset_iff]
+    using exists_open_superset_and_isCompact_closure isCompact_singleton
+#align exists_open_with_compact_closure exists_open_with_compact_closure
+
 /-- In a locally compact T₂ space, given a compact set `K` inside an open set `U`, we can find an
 open set `V` between these sets with compact closure: `K ⊆ V` and the closure of `V` is inside `U`.
 -/

@@ -1311,6 +1311,10 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
   Bornology.ext _ _ Filter.coclosedCompact_eq_cocompact
 #align bornology.relatively_compact_eq_in_compact Bornology.relativelyCompact_eq_inCompact
 
+theorem IsCompact.preimage_continuous [CompactSpace α] [T2Space β] {f : α → β} {s : Set β}
+    (hs : IsCompact s) (hf : Continuous f) : IsCompact (f ⁻¹' s) :=
+  (hs.isClosed.preimage hf).isCompact
+
 /-- If `V : ι → Set α` is a decreasing family of compact sets then any neighborhood of
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhds_of_isCompact'` where we
 don't need to assume each `V i` closed because it follows from compactness since `α` is

Adds an instance TotallySeparatedSpace for totally disconnected compact Hausdorff spaces. This direction of compact_t2_tot_disc_iff_tot_sep was missing an instance.

@@ -1984,6 +1984,9 @@ theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace α ↔ Totall
 
 variable [TotallyDisconnectedSpace α]
 
+/-- A totally disconnected compact Hausdorff space is totally separated. -/
+instance : TotallySeparatedSpace α := compact_t2_tot_disc_iff_tot_sep.mp inferInstance
+
 theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈ s ∧ IsClopen s) id :=
   ⟨fun U => by
     constructor

@@ -109,7 +109,7 @@ theorem separatedNhds_iff_disjoint {s t : Set α} : SeparatedNhds s t ↔ Disjoi
     exists_and_left, and_assoc, and_comm, and_left_comm]
 #align separated_nhds_iff_disjoint separatedNhds_iff_disjoint
 
-alias separatedNhds_iff_disjoint ↔ SeparatedNhds.disjoint_nhdsSet _
+alias ⟨SeparatedNhds.disjoint_nhdsSet, _⟩ := separatedNhds_iff_disjoint
 
 namespace SeparatedNhds
 
@@ -1608,7 +1608,7 @@ theorem specializes_comm {a b : α} : a ⤳ b ↔ b ⤳ a := by
   simp only [← (disjoint_nhds_nhds_iff_not_specializes (α := α)).not_left, disjoint_comm]
 #align specializes_comm specializes_comm
 
-alias specializes_comm ↔ Specializes.symm _
+alias ⟨Specializes.symm, _⟩ := specializes_comm
 #align specializes.symm Specializes.symm
 
 theorem specializes_iff_inseparable {a b : α} : a ⤳ b ↔ Inseparable a b :=

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -252,7 +252,7 @@ theorem exists_isOpen_xor'_mem [T0Space α] {x y : α} (h : x ≠ y) :
 #align exists_is_open_xor_mem exists_isOpen_xor'_mem
 
 /-- Specialization forms a partial order on a t0 topological space. -/
-def specializationOrder (α : Type _) [TopologicalSpace α] [T0Space α] : PartialOrder α :=
+def specializationOrder (α : Type*) [TopologicalSpace α] [T0Space α] : PartialOrder α :=
   { specializationPreorder α, PartialOrder.lift (OrderDual.toDual ∘ 𝓝) nhds_injective with }
 #align specialization_order specializationOrder
 
@@ -280,7 +280,7 @@ theorem minimal_nonempty_closed_eq_singleton [T0Space α] {s : Set α} (hs : IsC
 
 /-- Given a closed set `S` in a compact T₀ space, there is some `x ∈ S` such that `{x}` is
 closed. -/
-theorem IsClosed.exists_closed_singleton {α : Type _} [TopologicalSpace α] [T0Space α]
+theorem IsClosed.exists_closed_singleton {α : Type*} [TopologicalSpace α] [T0Space α]
     [CompactSpace α] {S : Set α} (hS : IsClosed S) (hne : S.Nonempty) :
     ∃ x : α, x ∈ S ∧ IsClosed ({x} : Set α) := by
   obtain ⟨V, Vsub, Vne, Vcls, hV⟩ := hS.exists_minimal_nonempty_closed_subset hne
@@ -351,7 +351,7 @@ theorem t0Space_iff_or_not_mem_closure (α : Type u) [TopologicalSpace α] :
 instance [TopologicalSpace β] [T0Space α] [T0Space β] : T0Space (α × β) :=
   ⟨fun _ _ h => Prod.ext (h.map continuous_fst).eq (h.map continuous_snd).eq⟩
 
-instance Pi.instT0Space {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)]
+instance Pi.instT0Space {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
     [∀ i, T0Space (π i)] :
     T0Space (∀ i, π i) :=
   ⟨fun _ _ h => funext fun i => (h.map (continuous_apply i)).eq⟩
@@ -502,7 +502,7 @@ theorem t1Space_TFAE (α : Type u) [ TopologicalSpace α ] :
   tfae_finish
 #align t1_space_tfae t1Space_TFAE
 
-theorem t1Space_iff_continuous_cofinite_of {α : Type _} [TopologicalSpace α] :
+theorem t1Space_iff_continuous_cofinite_of {α : Type*} [TopologicalSpace α] :
     T1Space α ↔ Continuous (@CofiniteTopology.of α) :=
   (t1Space_TFAE α).out 0 3
 #align t1_space_iff_continuous_cofinite_of t1Space_iff_continuous_cofinite_of
@@ -558,10 +558,10 @@ theorem nhds_le_nhds_iff [T1Space α] {a b : α} : 𝓝 a ≤ 𝓝 b ↔ a = b :
   specializes_iff_eq
 #align nhds_le_nhds_iff nhds_le_nhds_iff
 
-instance {α : Type _} : T1Space (CofiniteTopology α) :=
+instance {α : Type*} : T1Space (CofiniteTopology α) :=
   t1Space_iff_continuous_cofinite_of.mpr continuous_id
 
-theorem t1Space_antitone {α : Type _} : Antitone (@T1Space α) := fun a _ h _ =>
+theorem t1Space_antitone {α : Type*} : Antitone (@T1Space α) := fun a _ h _ =>
   @T1Space.mk _ a fun x => (T1Space.t1 x).mono h
 #align t1_space_antitone t1Space_antitone
 
@@ -612,7 +612,7 @@ instance Subtype.t1Space {α : Type u} [TopologicalSpace α] [T1Space α] {p : 
 instance [TopologicalSpace β] [T1Space α] [T1Space β] : T1Space (α × β) :=
   ⟨fun ⟨a, b⟩ => @singleton_prod_singleton _ _ a b ▸ isClosed_singleton.prod isClosed_singleton⟩
 
-instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [∀ i, T1Space (π i)] :
+instance {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)] [∀ i, T1Space (π i)] :
     T1Space (∀ i, π i) :=
   ⟨fun f => univ_pi_singleton f ▸ isClosed_set_pi fun _ _ => isClosed_singleton⟩
 
@@ -675,7 +675,7 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
 theorem sInter_sets_nhds [T1Space α] (x : α) : ⋂₀ (𝓝 x).sets = {x} := by
   simp [sInter_nhds_sets_eq_specializes]
 
-theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι → Set α} {x : α}
+theorem biInter_basis_nhds [T1Space α] {ι : Sort*} {p : ι → Prop} {s : ι → Set α} {x : α}
     (h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} := by
   rw [← h.sInter_sets, sInter_sets_nhds]
 #align bInter_basis_nhds biInter_basis_nhds
@@ -749,7 +749,7 @@ theorem eq_of_tendsto_nhds [TopologicalSpace β] [T1Space β] {f : α → β} {a
     fact₂ fact₁ (Eq.refl <| f a)
 #align eq_of_tendsto_nhds eq_of_tendsto_nhds
 
-theorem Filter.Tendsto.eventually_ne [TopologicalSpace β] [T1Space β] {α : Type _} {g : α → β}
+theorem Filter.Tendsto.eventually_ne [TopologicalSpace β] [T1Space β] {α : Type*} {g : α → β}
     {l : Filter α} {b₁ b₂ : β} (hg : Tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) : ∀ᶠ z in l, g z ≠ b₂ :=
   hg.eventually (isOpen_compl_singleton.eventually_mem hb)
 #align filter.tendsto.eventually_ne Filter.Tendsto.eventually_ne
@@ -779,7 +779,7 @@ theorem tendsto_const_nhds_iff [T1Space α] {l : Filter β} [NeBot l] {c d : α}
 #align tendsto_const_nhds_iff tendsto_const_nhds_iff
 
 /-- A point with a finite neighborhood has to be isolated. -/
-theorem isOpen_singleton_of_finite_mem_nhds {α : Type _} [TopologicalSpace α] [T1Space α] (x : α)
+theorem isOpen_singleton_of_finite_mem_nhds {α : Type*} [TopologicalSpace α] [T1Space α] (x : α)
     {s : Set α} (hs : s ∈ 𝓝 x) (hsf : s.Finite) : IsOpen ({x} : Set α) := by
   have A : {x} ⊆ s := by simp only [singleton_subset_iff, mem_of_mem_nhds hs]
   have B : IsClosed (s \ {x}) := (hsf.subset (diff_subset _ _)).isClosed
@@ -797,7 +797,7 @@ theorem infinite_of_mem_nhds {α} [TopologicalSpace α] [T1Space α] (x : α) [h
   exact isOpen_singleton_of_finite_mem_nhds x hs hsf
 #align infinite_of_mem_nhds infinite_of_mem_nhds
 
-theorem discrete_of_t1_of_finite {X : Type _} [TopologicalSpace X] [T1Space X] [Finite X] :
+theorem discrete_of_t1_of_finite {X : Type*} [TopologicalSpace X] [T1Space X] [Finite X] :
     DiscreteTopology X := by
   apply singletons_open_iff_discrete.mp
   intro x
@@ -848,7 +848,7 @@ theorem nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α} (h
   le_antisymm (le_pure_iff.2 <| singleton_mem_nhdsWithin_of_mem_discrete hx) (pure_le_nhdsWithin hx)
 #align nhds_within_of_mem_discrete nhdsWithin_of_mem_discrete
 
-theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type _} {p : ι → Prop}
+theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type*} {p : ι → Prop}
     {t : ι → Set α} {s : Set α} [DiscreteTopology s] {x : α} (hb : (𝓝 x).HasBasis p t)
     (hx : x ∈ s) : ∃ i, p i ∧ t i ∩ s = {x} := by
   rcases (nhdsWithin_hasBasis hb s).mem_iff.1 (singleton_mem_nhdsWithin_of_mem_discrete hx) with
@@ -879,7 +879,7 @@ theorem disjoint_nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x
 `t ⊆ s`, then the topological space structure on `t` induced by `X` is the same as the one
 obtained by the induced topological space structure on `s`. Use `embedding_inclusion` instead. -/
 @[deprecated embedding_inclusion]
-theorem TopologicalSpace.subset_trans {X : Type _} [TopologicalSpace X] {s t : Set X}
+theorem TopologicalSpace.subset_trans {X : Type*} [TopologicalSpace X] {s t : Set X}
     (ts : t ⊆ s) :
     (instTopologicalSpaceSubtype : TopologicalSpace t) =
       (instTopologicalSpaceSubtype : TopologicalSpace s).induced (Set.inclusion ts) :=
@@ -1136,19 +1136,19 @@ Hausdorff spaces:
 
 
 -- see Note [lower instance priority]
-instance (priority := 100) DiscreteTopology.toT2Space {α : Type _} [TopologicalSpace α]
+instance (priority := 100) DiscreteTopology.toT2Space {α : Type*} [TopologicalSpace α]
     [DiscreteTopology α] : T2Space α :=
   ⟨fun x y h => ⟨{x}, {y}, isOpen_discrete _, isOpen_discrete _, rfl, rfl, disjoint_singleton.2 h⟩⟩
 #align discrete_topology.to_t2_space DiscreteTopology.toT2Space
 
-theorem separated_by_continuous {α : Type _} {β : Type _} [TopologicalSpace α] [TopologicalSpace β]
+theorem separated_by_continuous {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β]
     [T2Space β] {f : α → β} (hf : Continuous f) {x y : α} (h : f x ≠ f y) :
     ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v :=
   let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
   ⟨f ⁻¹' u, f ⁻¹' v, uo.preimage hf, vo.preimage hf, xu, yv, uv.preimage _⟩
 #align separated_by_continuous separated_by_continuous
 
-theorem separated_by_openEmbedding {α β : Type _} [TopologicalSpace α] [TopologicalSpace β]
+theorem separated_by_openEmbedding {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
     [T2Space α] {f : α → β} (hf : OpenEmbedding f) {x y : α} (h : x ≠ y) :
     ∃ u v : Set β, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ f y ∈ v ∧ Disjoint u v :=
   let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h
@@ -1156,10 +1156,10 @@ theorem separated_by_openEmbedding {α β : Type _} [TopologicalSpace α] [Topol
     mem_image_of_mem _ yv, disjoint_image_of_injective hf.inj uv⟩
 #align separated_by_open_embedding separated_by_openEmbedding
 
-instance {α : Type _} {p : α → Prop} [TopologicalSpace α] [T2Space α] : T2Space (Subtype p) :=
+instance {α : Type*} {p : α → Prop} [TopologicalSpace α] [T2Space α] : T2Space (Subtype p) :=
   ⟨fun _ _ h => separated_by_continuous continuous_subtype_val (mt Subtype.eq h)⟩
 
-instance Prod.t2Space {α : Type _} {β : Type _} [TopologicalSpace α] [T2Space α]
+instance Prod.t2Space {α : Type*} {β : Type*} [TopologicalSpace α] [T2Space α]
     [TopologicalSpace β] [T2Space β] : T2Space (α × β) :=
   ⟨fun _ _ h => Or.elim (not_and_or.mp (mt Prod.ext_iff.mpr h))
     (fun h₁ => separated_by_continuous continuous_fst h₁) fun h₂ =>
@@ -1178,7 +1178,7 @@ theorem Embedding.t2Space [TopologicalSpace β] [T2Space β] {f : α → β} (hf
   .of_injective_continuous hf.inj hf.continuous
 #align embedding.t2_space Embedding.t2Space
 
-instance {α β : Type _} [TopologicalSpace α] [T2Space α] [TopologicalSpace β] [T2Space β] :
+instance {α β : Type*} [TopologicalSpace α] [T2Space α] [TopologicalSpace β] [T2Space β] :
     T2Space (α ⊕ β) := by
   constructor
   rintro (x | x) (y | y) h
@@ -1187,14 +1187,14 @@ instance {α β : Type _} [TopologicalSpace α] [T2Space α] [TopologicalSpace 
   · exact separated_by_continuous continuous_isLeft <| by simp
   · exact separated_by_openEmbedding openEmbedding_inr <| ne_of_apply_ne _ h
 
-instance Pi.t2Space {α : Type _} {β : α → Type v} [∀ a, TopologicalSpace (β a)]
+instance Pi.t2Space {α : Type*} {β : α → Type v} [∀ a, TopologicalSpace (β a)]
     [∀ a, T2Space (β a)] : T2Space (∀ a, β a) :=
   ⟨fun _ _ h =>
     let ⟨i, hi⟩ := not_forall.mp (mt funext h)
     separated_by_continuous (continuous_apply i) hi⟩
 #align Pi.t2_space Pi.t2Space
 
-instance Sigma.t2Space {ι} {α : ι → Type _} [∀ i, TopologicalSpace (α i)] [∀ a, T2Space (α a)] :
+instance Sigma.t2Space {ι} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] [∀ a, T2Space (α a)] :
     T2Space (Σi, α i) := by
   constructor
   rintro ⟨i, x⟩ ⟨j, y⟩ neq
@@ -1205,7 +1205,7 @@ instance Sigma.t2Space {ι} {α : ι → Type _} [∀ i, TopologicalSpace (α i)
     exact separated_by_continuous (continuous_def.2 fun u _ => isOpen_sigma_fst_preimage u) h
 #align sigma.t2_space Sigma.t2Space
 
-variable {γ : Type _} [TopologicalSpace β] [TopologicalSpace γ]
+variable {γ : Type*} [TopologicalSpace β] [TopologicalSpace γ]
 
 theorem isClosed_eq [T2Space α] {f g : β → α} (hf : Continuous f) (hg : Continuous g) :
     IsClosed { x : β | f x = g x } :=
@@ -1315,7 +1315,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 `⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhds_of_isCompact'` where we
 don't need to assume each `V i` closed because it follows from compactness since `α` is
 assumed to be Hausdorff. -/
-theorem exists_subset_nhds_of_isCompact [T2Space α] {ι : Type _} [Nonempty ι] {V : ι → Set α}
+theorem exists_subset_nhds_of_isCompact [T2Space α] {ι : Type*} [Nonempty ι] {V : ι → Set α}
     (hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) {U : Set α}
     (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U :=
   exists_subset_nhds_of_isCompact' hV hV_cpct (fun i => (hV_cpct i).isClosed) hU
@@ -1539,7 +1539,7 @@ theorem RegularSpace.ofLift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 
   Iff.mpr ((regularSpace_TFAE α).out 0 5) h
 #align regular_space.of_lift'_closure RegularSpace.ofLift'_closure
 
-theorem RegularSpace.ofBasis {ι : α → Sort _} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set α}
+theorem RegularSpace.ofBasis {ι : α → Sort*} {p : ∀ a, ι a → Prop} {s : ∀ a, ι a → Set α}
     (h₁ : ∀ a, (𝓝 a).HasBasis (p a) (s a)) (h₂ : ∀ a i, p a i → IsClosed (s a i)) :
     RegularSpace α :=
   RegularSpace.ofLift'_closure fun a => (h₁ a).lift'_closure_eq_self (h₂ a)
@@ -1575,7 +1575,7 @@ theorem lift'_nhds_closure (a : α) : (𝓝 a).lift' closure = 𝓝 a :=
   (closed_nhds_basis a).lift'_closure_eq_self fun _ => And.right
 #align lift'_nhds_closure lift'_nhds_closure
 
-theorem Filter.HasBasis.nhds_closure {ι : Sort _} {a : α} {p : ι → Prop} {s : ι → Set α}
+theorem Filter.HasBasis.nhds_closure {ι : Sort*} {a : α} {p : ι → Prop} {s : ι → Set α}
     (h : (𝓝 a).HasBasis p s) : (𝓝 a).HasBasis p fun i => closure (s i) :=
   lift'_nhds_closure a ▸ h.lift'_closure
 #align filter.has_basis.nhds_closure Filter.HasBasis.nhds_closure
@@ -1667,7 +1667,7 @@ instance {p : α → Prop} : RegularSpace (Subtype p) :=
 instance [TopologicalSpace β] [RegularSpace β] : RegularSpace (α × β) :=
   (regularSpace_induced (@Prod.fst α β)).inf (regularSpace_induced (@Prod.snd α β))
 
-instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [∀ i, RegularSpace (π i)] :
+instance {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)] [∀ i, RegularSpace (π i)] :
     RegularSpace (∀ i, π i) :=
   regularSpace_iInf fun _ => regularSpace_induced _
 
@@ -1702,7 +1702,7 @@ instance Subtype.t3Space [T3Space α] {p : α → Prop} : T3Space (Subtype p) :=
 
 instance [TopologicalSpace β] [T3Space α] [T3Space β] : T3Space (α × β) := ⟨⟩
 
-instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [∀ i, T3Space (π i)] :
+instance {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)] [∀ i, T3Space (π i)] :
     T3Space (∀ i, π i) := ⟨⟩
 
 /-- Given two points `x ≠ y`, we can find neighbourhoods `x ∈ V₁ ⊆ U₁` and `y ∈ V₂ ⊆ U₂`,
@@ -2028,7 +2028,7 @@ end Profinite
 
 section LocallyCompact
 
-variable {H : Type _} [TopologicalSpace H] [LocallyCompactSpace H] [T2Space H]
+variable {H : Type*} [TopologicalSpace H] [LocallyCompactSpace H] [T2Space H]
 
 /-- A locally compact Hausdorff totally disconnected space has a basis with clopen elements. -/
 theorem loc_compact_Haus_tot_disc_of_zero_dim [TotallyDisconnectedSpace H] :

A step towards showing that Pmfs have countable support.

Thanks Eric Rodriguez and Kevin Buzzard for helping on zulip.

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -671,13 +671,13 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
   exact mem_of_superset self_mem_nhdsWithin (subset_insert x s)
 #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert
 
+@[simp]
+theorem sInter_sets_nhds [T1Space α] (x : α) : ⋂₀ (𝓝 x).sets = {x} := by
+  simp [sInter_nhds_sets_eq_specializes]
+
 theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι → Set α} {x : α}
     (h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} := by
-  simp only [eq_singleton_iff_unique_mem, mem_iInter]
-  refine' ⟨fun i hi => mem_of_mem_nhds <| h.mem_of_mem hi, fun y hy => _⟩
-  contrapose! hy
-  rcases h.mem_iff.1 (compl_singleton_mem_nhds hy.symm) with ⟨i, hi, hsub⟩
-  exact ⟨i, hi, fun h => hsub h rfl⟩
+  rw [← h.sInter_sets, sInter_sets_nhds]
 #align bInter_basis_nhds biInter_basis_nhds
 
 @[simp]

@@ -1450,7 +1450,7 @@ theorem exists_open_superset_and_isCompact_closure [LocallyCompactSpace α] [T2S
     ⟨interior K', isOpen_interior, hKK', isCompact_closure_of_subset_compact hK' interior_subset⟩
 #align exists_open_superset_and_is_compact_closure exists_open_superset_and_isCompact_closure
 
-/-- In a locally compact T₂ space, given a compact set `K` inside an open set `U`, we can find a
+/-- In a locally compact T₂ space, given a compact set `K` inside an open set `U`, we can find an
 open set `V` between these sets with compact closure: `K ⊆ V` and the closure of `V` is inside `U`.
 -/
 theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace α] [T2Space α] {K U : Set α}

@@ -824,6 +824,16 @@ theorem ConnectedSpace.infinite [ConnectedSpace α] [Nontrivial α] [T1Space α]
   infinite_univ_iff.mp <| isPreconnected_univ.infinite_of_nontrivial nontrivial_univ
 #align connected_space.infinite ConnectedSpace.infinite
 
+/-- A non-trivial connected T1 space has no isolated points. -/
+instance ConnectedSpace.neBot_nhdsWithin_compl_of_nontrivial_of_t1space
+    [ConnectedSpace α] [Nontrivial α] [T1Space α] (x : α) :
+    NeBot (𝓝[≠] x) := by
+  by_contra contra
+  rw [not_neBot, ← isOpen_singleton_iff_punctured_nhds] at contra
+  replace contra := nonempty_inter isOpen_compl_singleton
+    contra (compl_union_self _) (Set.nonempty_compl_of_nontrivial _) (singleton_nonempty _)
+  simp [compl_inter_self {x}] at contra
+
 theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set α} [DiscreteTopology s] {x : α}
     (hx : x ∈ s) : {x} ∈ 𝓝[s] x := by
   have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete]

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,17 +2,14 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
-
-! This file was ported from Lean 3 source module topology.separation
-! leanprover-community/mathlib commit d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.SubsetProperties
 import Mathlib.Topology.Connected
 import Mathlib.Topology.NhdsSet
 import Mathlib.Topology.Inseparable
 
+#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
+
 /-!
 # Separation properties of topological spaces.
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -728,7 +728,7 @@ obtains a dense set. -/
 theorem Dense.diff_finset [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Set α} (hs : Dense s)
     (t : Finset α) : Dense (s \ t) := by
   induction t using Finset.induction_on with
-  | empty =>  simpa using hs
+  | empty => simpa using hs
   | insert _ ih =>
     rw [Finset.coe_insert, ← union_singleton, ← diff_diff]
     exact ih.diff_singleton _
@@ -1986,7 +1986,7 @@ theorem nhds_basis_clopen (x : α) : (𝓝 x).HasBasis (fun s : Set α => x ∈
       intro hU
       let N := { Z // IsClopen Z ∧ x ∈ Z }
       suffices : ∃ Z : N, Z.val ⊆ U
-      · rcases this with  ⟨⟨s, hs, hs'⟩, hs''⟩; exact ⟨s, ⟨hs', hs⟩, hs''⟩
+      · rcases this with ⟨⟨s, hs, hs'⟩, hs''⟩; exact ⟨s, ⟨hs', hs⟩, hs''⟩
       haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩
       have hNcl : ∀ Z : N, IsClosed Z.val := fun Z => Z.property.1.2
       have hdir : Directed Superset fun Z : N => Z.val := by

• depends on: https://github.com/leanprover/std4/pull/163
• depends on: #5584
• depends on: #5715
• depends on: #5729
• depends on: https://github.com/leanprover/std4/pull/173

Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -2079,9 +2079,9 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
   -- Now we show that this can be reduced to some clopen containing `↑b` being disjoint to `↑a`
   obtain ⟨U, V, hU, ha, hb, rfl⟩ : ∃ (U : Set α) (V : Set (ConnectedComponents α)),
       IsClopen U ∧ connectedComponent a ∩ U = ∅ ∧ connectedComponent b ⊆ U ∧ (↑) ⁻¹' V = U := by
-    have h := by
-      refine isClosed_connectedComponent.isCompact.elim_finite_subfamily_closed ?_ _ h
-      exact fun Z => Z.2.1.2
+    have h :=
+      (isClosed_connectedComponent (α := α)).isCompact.elim_finite_subfamily_closed
+        _ (fun Z : { Z : Set α // IsClopen Z ∧ b ∈ Z } => Z.2.1.2) h
     cases' h with fin_a ha
     -- This clopen and its complement will separate the connected components of `a` and `b`
     set U : Set α := ⋂ (i : { Z // IsClopen Z ∧ b ∈ Z }) (_ : i ∈ fin_a), i

Also add TopologicalSpace.IsTopologicalBasis.eq_iff for a T₀ space.

@@ -232,6 +232,17 @@ theorem inseparable_eq_eq [T0Space α] : Inseparable = @Eq α :=
   funext₂ fun _ _ => propext inseparable_iff_eq
 #align inseparable_eq_eq inseparable_eq_eq
 
+theorem TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set α)}
+    (hb : IsTopologicalBasis b) {x y : α} : Inseparable x y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
+  ⟨fun h s hs ↦ inseparable_iff_forall_open.1 h _ (hb.isOpen hs),
+    fun h ↦ hb.nhds_hasBasis.eq_of_same_basis $ by
+      convert hb.nhds_hasBasis using 2
+      exact and_congr_right (h _)⟩
+
+theorem TopologicalSpace.IsTopologicalBasis.eq_iff [T0Space α] {b : Set (Set α)}
+    (hb : IsTopologicalBasis b) {x y : α} : x = y ↔ ∀ s ∈ b, (x ∈ s ↔ y ∈ s) :=
+  inseparable_iff_eq.symm.trans hb.inseparable_iff
+
 theorem t0Space_iff_exists_isOpen_xor'_mem (α : Type u) [TopologicalSpace α] :
     T0Space α ↔ ∀ x y, x ≠ y → ∃ U : Set α, IsOpen U ∧ Xor' (x ∈ U) (y ∈ U) := by
   simp only [t0Space_iff_not_inseparable, xor_iff_not_iff, not_forall, exists_prop,

We had continuity of fun f : C(X, Y) ↦ f a in two cases:

• X is a locally compact space;
• X is a compact space and Y is a metric space.

In fact, it is true in general topological spaces.

@@ -1147,9 +1147,17 @@ instance Prod.t2Space {α : Type _} {β : Type _} [TopologicalSpace α] [T2Space
     (fun h₁ => separated_by_continuous continuous_fst h₁) fun h₂ =>
     separated_by_continuous continuous_snd h₂⟩
 
+/-- If the codomain of an injective continuous function is a Hausdorff space, then so is its
+domain. -/
+theorem T2Space.of_injective_continuous [TopologicalSpace β] [T2Space β] {f : α → β}
+    (hinj : Injective f) (hc : Continuous f) : T2Space α :=
+  ⟨fun _ _ h => separated_by_continuous hc (hinj.ne h)⟩
+
+/-- If the codomain of a topological embedding is a Hausdorff space, then so is its domain.
+See also `T2Space.of_continuous_injective`. -/
 theorem Embedding.t2Space [TopologicalSpace β] [T2Space β] {f : α → β} (hf : Embedding f) :
     T2Space α :=
-  ⟨fun _ _ h => separated_by_continuous hf.continuous (hf.inj.ne h)⟩
+  .of_injective_continuous hf.inj hf.continuous
 #align embedding.t2_space Embedding.t2Space
 
 instance {α β : Type _} [TopologicalSpace α] [T2Space α] [TopologicalSpace β] [T2Space β] :

@@ -664,7 +664,7 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
 #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert
 
 theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι → Set α} {x : α}
-    (h : (𝓝 x).HasBasis p s) : (⋂ (i) (_ : p i), s i) = {x} := by
+    (h : (𝓝 x).HasBasis p s) : ⋂ (i) (_ : p i), s i = {x} := by
   simp only [eq_singleton_iff_unique_mem, mem_iInter]
   refine' ⟨fun i hi => mem_of_mem_nhds <| h.mem_of_mem hi, fun y hy => _⟩
   contrapose! hy
@@ -1912,7 +1912,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
     have H2 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hv hu huv.symm
     by_cases hxu : x ∈ u <;> [left; right]
     -- The x ∈ u case.
-    · suffices (⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, ↑Z) ⊆ u
+    · suffices ⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, ↑Z ⊆ u
         from Disjoint.left_le_of_le_sup_right hab (huv.mono this hbv)
       · apply Subset.trans _ (inter_subset_right Z u)
         exact iInter_subset (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z.1)
@@ -1920,7 +1920,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
     -- If x ∉ u, we get x ∈ v since x ∈ u ∪ v. The rest is then like the x ∈ u case.
     · have h1 : x ∈ v :=
         (hab.trans (union_subset_union hau hbv) (mem_iInter.2 fun i => i.2.2)).resolve_left hxu
-      suffices (⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, ↑Z) ⊆ v
+      suffices ⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, ↑Z ⊆ v
         from (huv.symm.mono this hau).left_le_of_le_sup_left hab
       · refine Subset.trans ?_ (inter_subset_right Z v)
         exact iInter_subset (fun Z : { Z : Set α // IsClopen Z ∧ x ∈ Z } => Z.1)

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -681,7 +681,7 @@ theorem compl_singleton_mem_nhdsSet_iff [T1Space α] {x : α} {s : Set α} : {x}
 theorem nhdsSet_le_iff [T1Space α] {s t : Set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t := by
   refine' ⟨_, fun h => monotone_nhdsSet h⟩
   simp_rw [Filter.le_def]; intro h x hx
-  specialize h ({x}ᶜ)
+  specialize h {x}ᶜ
   simp_rw [compl_singleton_mem_nhdsSet_iff] at h
   by_contra hxt
   exact h hxt hx
@@ -1720,7 +1720,7 @@ theorem normal_separation [NormalSpace α] {s t : Set α} (H1 : IsClosed s) (H2
 
 theorem normal_exists_closure_subset [NormalSpace α] {s t : Set α} (hs : IsClosed s) (ht : IsOpen t)
     (hst : s ⊆ t) : ∃ u, IsOpen u ∧ s ⊆ u ∧ closure u ⊆ t := by
-  have : Disjoint s (tᶜ) := Set.disjoint_left.mpr fun x hxs hxt => hxt (hst hxs)
+  have : Disjoint s tᶜ := Set.disjoint_left.mpr fun x hxs hxt => hxt (hst hxs)
   rcases normal_separation hs (isClosed_compl_iff.2 ht) this with
     ⟨s', t', hs', ht', hss', htt', hs't'⟩
   refine ⟨s', hs', hss', Subset.trans (closure_minimal ?_ (isClosed_compl_iff.2 ht'))
@@ -1952,7 +1952,7 @@ theorem compact_t2_tot_disc_iff_tot_sep : TotallyDisconnectedSpace α ↔ Totall
   rw [connectedComponent_eq_iInter_clopen, mem_iInter]
   rintro ⟨w : Set α, hw : IsClopen w, hy : y ∈ w⟩
   by_contra hx
-  exact hyp (wᶜ) w hw.2.isOpen_compl hw.1 hx hy (@isCompl_compl _ w _).symm.codisjoint.top_le
+  exact hyp wᶜ w hw.2.isOpen_compl hw.1 hx hy (@isCompl_compl _ w _).symm.codisjoint.top_le
     disjoint_compl_left
 #align compact_t2_tot_disc_iff_tot_sep compact_t2_tot_disc_iff_tot_sep
 

Fix the set tactic to not time out when dealing with slow to elaborate terms and many local hypotheses.

The root cause of this is that the rewrite [blah] at * tactic causes blah to be elaborated again and again for each local hypothesis, this is possibly a core issue that should be fixed separately, but in set we have the elaborated term already so we can just use it.

We also add some functionality to simply test / demonstrate failures when elaboration takes too long, namely sleepAtLeastHeartbeats and a sleep_heartbeats tactic.

@urkud was facing some slow running set's in https://github.com/leanprover-community/mathlib4/pull/4912, see https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Timeout.20in.20.60set.20.2E.2E.20with.60/near/367958828 that this issue was minimized from and should fix.

Some other linter failures show up due to changes to the set internals so fix these too.

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -2065,7 +2065,7 @@ instance ConnectedComponents.t2 [T2Space α] [CompactSpace α] : T2Space (Connec
       exact fun Z => Z.2.1.2
     cases' h with fin_a ha
     -- This clopen and its complement will separate the connected components of `a` and `b`
-    set U : Set α := ⋂ (i : { Z // IsClopen Z ∧ b ∈ Z }) (H : i ∈ fin_a), i
+    set U : Set α := ⋂ (i : { Z // IsClopen Z ∧ b ∈ Z }) (_ : i ∈ fin_a), i
     have hU : IsClopen U := isClopen_biInter_finset fun i _ => i.2.1
     exact ⟨U, (↑) '' U, hU, ha, subset_iInter₂ fun Z _ => Z.2.1.connectedComponent_subset Z.2.2,
       (connectedComponents_preimage_image U).symm ▸ hU.biUnion_connectedComponent_eq⟩

@@ -751,6 +751,13 @@ theorem ContinuousAt.eventually_ne [TopologicalSpace β] [T1Space β] {g : α 
   hg1.tendsto.eventually_ne hg2
 #align continuous_at.eventually_ne ContinuousAt.eventually_ne
 
+theorem eventually_ne_nhds [T1Space α] {a b : α} (h : a ≠ b) : ∀ᶠ x in 𝓝 a, x ≠ b :=
+  IsOpen.eventually_mem isOpen_ne h
+
+theorem eventually_ne_nhdsWithin [T1Space α] {a b : α} {s : Set α} (h : a ≠ b) :
+    ∀ᶠ x in 𝓝[s] a, x ≠ b :=
+  Filter.Eventually.filter_mono nhdsWithin_le_nhds <| eventually_ne_nhds h
+
 /-- To prove a function to a `T1Space` is continuous at some point `a`, it suffices to prove that
 `f` admits *some* limit at `a`. -/
 theorem continuousAt_of_tendsto_nhds [TopologicalSpace β] [T1Space β] {f : α → β} {a : α} {b : β}

@@ -716,9 +716,9 @@ theorem Dense.diff_singleton [T1Space α] {s : Set α} (hs : Dense s) (x : α) [
 obtains a dense set. -/
 theorem Dense.diff_finset [T1Space α] [∀ x : α, NeBot (𝓝[≠] x)] {s : Set α} (hs : Dense s)
     (t : Finset α) : Dense (s \ t) := by
-  induction t using Finset.induction_on -- with x s _ ih hd
-  case empty =>  simpa using hs
-  case insert ih =>
+  induction t using Finset.induction_on with
+  | empty =>  simpa using hs
+  | insert _ ih =>
     rw [Finset.coe_insert, ← union_singleton, ← diff_diff]
     exact ih.diff_singleton _
 #align dense.diff_finset Dense.diff_finset
@@ -754,8 +754,8 @@ theorem ContinuousAt.eventually_ne [TopologicalSpace β] [T1Space β] {g : α 
 /-- To prove a function to a `T1Space` is continuous at some point `a`, it suffices to prove that
 `f` admits *some* limit at `a`. -/
 theorem continuousAt_of_tendsto_nhds [TopologicalSpace β] [T1Space β] {f : α → β} {a : α} {b : β}
-    (h : Tendsto f (𝓝 a) (𝓝 b)) : ContinuousAt f a :=
-  show Tendsto f (𝓝 a) (𝓝 <| f a) by rwa [eq_of_tendsto_nhds h]
+    (h : Tendsto f (𝓝 a) (𝓝 b)) : ContinuousAt f a := by
+  rwa [ContinuousAt, eq_of_tendsto_nhds h]
 #align continuous_at_of_tendsto_nhds continuousAt_of_tendsto_nhds
 
 @[simp]
@@ -1279,7 +1279,7 @@ theorem Bornology.relativelyCompact_eq_inCompact [T2Space α] :
 #align bornology.relatively_compact_eq_in_compact Bornology.relativelyCompact_eq_inCompact
 
 /-- If `V : ι → Set α` is a decreasing family of compact sets then any neighborhood of
-`⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhd_of_compact'` where we
+`⋂ i, V i` contains some `V i`. This is a version of `exists_subset_nhds_of_isCompact'` where we
 don't need to assume each `V i` closed because it follows from compactness since `α` is
 assumed to be Hausdorff. -/
 theorem exists_subset_nhds_of_isCompact [T2Space α] {ι : Type _} [Nonempty ι] {V : ι → Set α}

Too often tempted to change these during other PRs, so doing a mass edit here.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -485,7 +485,7 @@ theorem t1Space_TFAE (α : Type u) [ TopologicalSpace α ] :
   tfae_have 1 → 4
   · simp only [continuous_def, CofiniteTopology.isOpen_iff']
     rintro H s (rfl | hs)
-    exacts[isOpen_empty, compl_compl s ▸ (@Set.Finite.isClosed _ _ H _ hs).isOpen_compl]
+    exacts [isOpen_empty, compl_compl s ▸ (@Set.Finite.isClosed _ _ H _ hs).isOpen_compl]
   tfae_have 4 → 2
   · exact fun h x => (CofiniteTopology.isClosed_iff.2 <| Or.inr (finite_singleton _)).preimage h
   tfae_have 2 ↔ 10
@@ -1803,7 +1803,7 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
   · simp only [disjoint_left, mem_iUnion, mem_diff, not_exists, not_and, not_forall, not_not]
     rintro a ⟨u, huU, hau, haV⟩ v hvV hav
     cases' le_total (Encodable.encode u) (Encodable.encode v) with hle hle
-    exacts[⟨u, huU, hle, subset_closure hau⟩, (haV _ hvV hle <| subset_closure hav).elim]
+    exacts [⟨u, huU, hle, subset_closure hau⟩, (haV _ hvV hle <| subset_closure hav).elim]
 #align normal_space_of_t3_second_countable normalSpaceOfT3SecondCountable
 
 end Normality
@@ -1858,7 +1858,7 @@ instance [T5Space α] : T5Space (SeparationQuotient α) where
   completely_normal s t hd₁ hd₂ := by
     rw [← disjoint_comap_iff surjective_mk, comap_mk_nhdsSet, comap_mk_nhdsSet]
     apply T5Space.completely_normal <;> rw [← preimage_mk_closure]
-    exacts[hd₁.preimage mk, hd₂.preimage mk]
+    exacts [hd₁.preimage mk, hd₂.preimage mk]
 
 end CompletelyNormal
 

@@ -664,7 +664,7 @@ theorem insert_mem_nhdsWithin_of_subset_insert [T1Space α] {x y : α} {s t : Se
 #align insert_mem_nhds_within_of_subset_insert insert_mem_nhdsWithin_of_subset_insert
 
 theorem biInter_basis_nhds [T1Space α] {ι : Sort _} {p : ι → Prop} {s : ι → Set α} {x : α}
-    (h : (𝓝 x).HasBasis p s) : (⋂ (i) (_h : p i), s i) = {x} := by
+    (h : (𝓝 x).HasBasis p s) : (⋂ (i) (_ : p i), s i) = {x} := by
   simp only [eq_singleton_iff_unique_mem, mem_iInter]
   refine' ⟨fun i hi => mem_of_mem_nhds <| h.mem_of_mem hi, fun y hy => _⟩
   contrapose! hy
@@ -1766,7 +1766,7 @@ This lemma is not an instance to avoid a loop. -/
 theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α] : NormalSpace α := by
   have key : ∀ {s t : Set α}, IsClosed t → Disjoint s t →
     ∃ U : Set (countableBasis α), (s ⊆ ⋃ u ∈ U, ↑u) ∧ (∀ u ∈ U, Disjoint (closure ↑u) t) ∧
-      ∀ n : ℕ, IsClosed (⋃ (u ∈ U) (_h : Encodable.encode u ≤ n), closure (u : Set α)) := by
+      ∀ n : ℕ, IsClosed (⋃ (u ∈ U) (_ : Encodable.encode u ≤ n), closure (u : Set α)) := by
     intro s t hc hd
     rw [disjoint_left] at hd
     have : ∀ x ∈ s, ∃ U ∈ countableBasis α, x ∈ U ∧ Disjoint (closure U) t := by
@@ -1785,8 +1785,8 @@ theorem normalSpaceOfT3SecondCountable [SecondCountableTopology α] [T3Space α]
   refine' ⟨fun s t hs ht hd => _⟩
   rcases key ht hd with ⟨U, hsU, hUd, hUc⟩
   rcases key hs hd.symm with ⟨V, htV, hVd, hVc⟩
-  refine ⟨⋃ u ∈ U, ↑u \ ⋃ (v ∈ V) (_hv : Encodable.encode v ≤ Encodable.encode u), closure ↑v,
-    ⋃ v ∈ V, ↑v \ ⋃ (u ∈ U) (_hu : Encodable.encode u ≤ Encodable.encode v), closure ↑u,
+  refine ⟨⋃ u ∈ U, ↑u \ ⋃ (v ∈ V) (_ : Encodable.encode v ≤ Encodable.encode u), closure ↑v,
+    ⋃ v ∈ V, ↑v \ ⋃ (u ∈ U) (_ : Encodable.encode u ≤ Encodable.encode v), closure ↑u,
     isOpen_biUnion fun u _ => (isOpen_of_mem_countableBasis u.2).sdiff (hVc _),
     isOpen_biUnion fun v _ => (isOpen_of_mem_countableBasis v.2).sdiff (hUc _),
     fun x hx => ?_, fun x hx => ?_, ?_⟩

@@ -455,10 +455,8 @@ protected theorem Finset.isClosed [T1Space α] (s : Finset α) : IsClosed (s : S
   s.finite_toSet.isClosed
 #align finset.is_closed Finset.isClosed
 
--- Porting note: used `tfae` tactic
-open List in
 theorem t1Space_TFAE (α : Type u) [ TopologicalSpace α ] :
-    TFAE [T1Space α,
+    List.TFAE [T1Space α,
       ∀ x, IsClosed ({ x } : Set α),
       ∀ x, IsOpen ({ x }ᶜ : Set α),
       Continuous (@CofiniteTopology.of α),
@@ -468,22 +466,32 @@ theorem t1Space_TFAE (α : Type u) [ TopologicalSpace α ] :
       ∀ ⦃x y : α⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
       ∀ ⦃x y : α⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
       ∀ ⦃x y : α⦄, x ⤳ y → x = y] := by
-  simp only [Ne, @not_imp_comm (_ = _), disjoint_comm (b := 𝓝 _), isOpen_compl_iff, not_exists,
-    not_and, not_not, ← specializes_iff_forall_open, ← clusterPt_iff_not_disjoint,
-    ← specializes_iff_clusterPt, continuous_iff_isClosed, CofiniteTopology.isClosed_iff]
-  simp only [← Filter.mem_pure, ← Filter.le_def, ← specializes_iff_pure]
-  simp only [← mem_interior_iff_mem_nhds, interior_compl, mem_compl_iff, not_not,
-    ← specializes_iff_mem_closure]
-  apply_rules [tfae_of_cycle, Chain.cons, Chain.nil] <;> dsimp only [ilast'] <;> try { exact id }
-  · exact @T1Space.t1 _ _
-  · rintro h U (rfl | hU)
-    · exact isClosed_univ
-    · rw [← biUnion_of_singleton U]
-      exact isClosed_biUnion hU fun x _ => h x
-  · exact fun h x y hyx => (h {y} (.inr <| finite_singleton _)).closure_subset hyx.mem_closure
-  · exact fun h₁ _ _ h₂ => (h₁ h₂).symm
-  · refine' fun h => ⟨fun x => isClosed_iff_clusterPt.2 fun y hy => (h _).symm⟩
-    rwa [specializes_iff_clusterPt, ← principal_singleton]
+  tfae_have 1 ↔ 2
+  · exact ⟨fun h => h.1, fun h => ⟨h⟩⟩
+  tfae_have 2 ↔ 3
+  · simp only [isOpen_compl_iff]
+  tfae_have 5 ↔ 3
+  · refine' forall_swap.trans _
+    simp only [isOpen_iff_mem_nhds, mem_compl_iff, mem_singleton_iff]
+  tfae_have 5 ↔ 6
+  · simp only [← subset_compl_singleton_iff, exists_mem_subset_iff]
+  tfae_have 5 ↔ 7
+  · simp only [(nhds_basis_opens _).mem_iff, subset_compl_singleton_iff, exists_prop, and_assoc,
+      and_left_comm]
+  tfae_have 5 ↔ 8
+  · simp only [← principal_singleton, disjoint_principal_right]
+  tfae_have 8 ↔ 9
+  · exact forall_swap.trans (by simp only [disjoint_comm, ne_comm])
+  tfae_have 1 → 4
+  · simp only [continuous_def, CofiniteTopology.isOpen_iff']
+    rintro H s (rfl | hs)
+    exacts[isOpen_empty, compl_compl s ▸ (@Set.Finite.isClosed _ _ H _ hs).isOpen_compl]
+  tfae_have 4 → 2
+  · exact fun h x => (CofiniteTopology.isClosed_iff.2 <| Or.inr (finite_singleton _)).preimage h
+  tfae_have 2 ↔ 10
+  · simp only [← closure_subset_iff_isClosed, specializes_iff_mem_closure, subset_def,
+      mem_singleton_iff, eq_comm]
+  tfae_finish
 #align t1_space_tfae t1Space_TFAE
 
 theorem t1Space_iff_continuous_cofinite_of {α : Type _} [TopologicalSpace α] :
@@ -1460,34 +1468,38 @@ class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
   regular : ∀ {s : Set X} {a}, IsClosed s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)
 #align regular_space RegularSpace
 
--- Porting note: used `tfae` tactic
-open List in
 theorem regularSpace_TFAE (X : Type u) [ TopologicalSpace X ] :
-    TFAE [RegularSpace X,
+    List.TFAE [RegularSpace X,
       ∀ (s : Set X) a, a ∉ closure s → Disjoint (𝓝ˢ s) (𝓝 a),
       ∀ (a : X) (s : Set X), Disjoint (𝓝ˢ s) (𝓝 a) ↔ a ∉ closure s,
       ∀ (a : X) (s : Set X), s ∈ 𝓝 a → ∃ t ∈ 𝓝 a, IsClosed t ∧ t ⊆ s,
       ∀ a : X, (𝓝 a).lift' closure ≤ 𝓝 a,
       ∀ a : X , (𝓝 a).lift' closure = 𝓝 a] := by
-  apply_rules [tfae_of_cycle, Chain.cons, Chain.nil]
-  · exact fun h s a ha => (h.1 isClosed_closure ha).mono_left (nhdsSet_mono subset_closure)
-  · refine fun h a s => ⟨fun hd => ?_, h s a⟩
-    replace hd := hd.mono_left principal_le_nhdsSet
-    rwa [← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds, ← disjoint_principal_left]
-  · refine fun h a s ha => ?_
-    rw [← mem_interior_iff_mem_nhds, ← compl_compl s, interior_compl, mem_compl_iff, ← h] at ha
-    rcases (hasBasis_nhdsSet _).disjoint_iff_left.1 ha with ⟨U, ⟨hUo, hsU⟩, ha⟩
-    exact ⟨Uᶜ, ha, hUo.isClosed_compl, compl_subset_comm.2 hsU⟩
-  · refine fun h a s hs => ?_
-    rcases h a s hs with ⟨t, ha, hc, hts⟩
-    exact flip mem_of_superset hts <| hc.closure_eq ▸ mem_lift' ha
-  · exact fun h a => (h a).antisymm (le_lift'_closure _)
-  · refine fun h => ⟨fun hs ha => ?_⟩
-    rw [← hs.closure_eq, ← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds, ← h,
-      mem_lift'_sets (monotone_closure X)] at ha
-    rcases ha with ⟨t, ha, hs⟩
-    refine' disjoint_of_disjoint_of_mem disjoint_compl_left ?_ ha
-    rwa [← subset_interior_iff_mem_nhdsSet, interior_compl, subset_compl_comm]
+  tfae_have 1 ↔ 5
+  · rw [regularSpace_iff, (@compl_surjective (Set X) _).forall, forall_swap]
+    simp only [isClosed_compl_iff, mem_compl_iff, Classical.not_not, @and_comm (_ ∈ _),
+      (nhds_basis_opens _).lift'_closure.le_basis_iff (nhds_basis_opens _), and_imp,
+      (nhds_basis_opens _).disjoint_iff_right, exists_prop, ← subset_interior_iff_mem_nhdsSet,
+      interior_compl, compl_subset_compl]
+  tfae_have 5 → 6
+  · exact fun h a => (h a).antisymm (𝓝 _).le_lift'_closure
+  tfae_have 6 → 4
+  · intro H a s hs
+    rw [← H] at hs
+    rcases(𝓝 a).basis_sets.lift'_closure.mem_iff.mp hs with ⟨U, hU, hUs⟩
+    exact ⟨closure U, mem_of_superset hU subset_closure, isClosed_closure, hUs⟩
+  tfae_have 4 → 2
+  · intro H s a ha
+    have ha' : sᶜ ∈ 𝓝 a := by rwa [← mem_interior_iff_mem_nhds, interior_compl]
+    rcases H _ _ ha' with ⟨U, hU, hUc, hUs⟩
+    refine' disjoint_of_disjoint_of_mem disjoint_compl_left _ hU
+    rwa [← subset_interior_iff_mem_nhdsSet, hUc.isOpen_compl.interior_eq, subset_compl_comm]
+  tfae_have 2 → 3
+  · refine' fun H a s => ⟨fun hd has => mem_closure_iff_nhds_neBot.mp has _, H s a⟩
+    exact (hd.symm.mono_right <| @principal_le_nhdsSet _ _ s).eq_bot
+  tfae_have 3 → 1
+  · exact fun H => ⟨fun hs ha => (H _ _).mpr <| hs.closure_eq.symm ▸ ha⟩
+  tfae_finish
 #align regular_space_tfae regularSpace_TFAE
 
 theorem RegularSpace.ofLift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 𝓝 a) : RegularSpace α :=

Forward-port of leanprover-community/mathlib#19071

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 
 ! This file was ported from Lean 3 source module topology.separation
-! leanprover-community/mathlib commit 195fcd60ff2bfe392543bceb0ec2adcdb472db4c
+! leanprover-community/mathlib commit d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1320,8 +1320,8 @@ theorem IsCompact.binary_compact_cover [T2Space α] {K U V : Set α} (hK : IsCom
 #align is_compact.binary_compact_cover IsCompact.binary_compact_cover
 
 /-- A continuous map from a compact space to a Hausdorff space is a closed map. -/
-theorem Continuous.isClosedMap [CompactSpace α] [T2Space β] {f : α → β} (h : Continuous f) :
-    IsClosedMap f := fun _s hs => (hs.isCompact.image h).isClosed
+protected theorem Continuous.isClosedMap [CompactSpace α] [T2Space β] {f : α → β}
+    (h : Continuous f) : IsClosedMap f := fun _s hs => (hs.isCompact.image h).isClosed
 #align continuous.is_closed_map Continuous.isClosedMap
 
 /-- A continuous injective map from a compact space to a Hausdorff space is a closed embedding. -/
@@ -1330,6 +1330,12 @@ theorem Continuous.closedEmbedding [CompactSpace α] [T2Space β] {f : α → β
   closedEmbedding_of_continuous_injective_closed h hf h.isClosedMap
 #align continuous.closed_embedding Continuous.closedEmbedding
 
+/-- A continuous surjective map from a compact space to a Hausdorff space is a quotient map. -/
+theorem QuotientMap.of_surjective_continuous [CompactSpace α] [T2Space β] {f : α → β}
+    (hsurj : Surjective f) (hcont : Continuous f) : QuotientMap f :=
+  hcont.isClosedMap.to_quotientMap hcont hsurj
+#align quotient_map.of_surjective_continuous QuotientMap.of_surjective_continuous
+
 section
 
 open Finset Function

Co-authored-by: Jireh Loreaux <loreaujy@gmail.com>

@@ -343,9 +343,11 @@ theorem t0Space_iff_or_not_mem_closure (α : Type u) [TopologicalSpace α] :
 instance [TopologicalSpace β] [T0Space α] [T0Space β] : T0Space (α × β) :=
   ⟨fun _ _ h => Prod.ext (h.map continuous_fst).eq (h.map continuous_snd).eq⟩
 
-instance {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)] [∀ i, T0Space (π i)] :
+instance Pi.instT0Space {ι : Type _} {π : ι → Type _} [∀ i, TopologicalSpace (π i)]
+    [∀ i, T0Space (π i)] :
     T0Space (∀ i, π i) :=
   ⟨fun _ _ h => funext fun i => (h.map (continuous_apply i)).eq⟩
+#align pi.t0_space Pi.instT0Space
 
 theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set α, x ∈ s ∧ y ∈ s ∧ T0Space s) :
     T0Space α := by

The main breaking change is that tac <;> [t1, t2] is now written tac <;> [t1; t2], to avoid clashing with tactics like cases and use that take comma-separated lists.

@@ -1883,7 +1883,7 @@ theorem connectedComponent_eq_iInter_clopen [T2Space α] [CompactSpace α] (x :
   · have H1 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hu hv huv
     rw [union_comm] at H
     have H2 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hv hu huv.symm
-    by_cases hxu : x ∈ u <;> [left, right]
+    by_cases hxu : x ∈ u <;> [left; right]
     -- The x ∈ u case.
     · suffices (⋂ Z : { Z : Set α // IsClopen Z ∧ x ∈ Z }, ↑Z) ⊆ u
         from Disjoint.left_le_of_le_sup_right hab (huv.mono this hbv)

This makes a mathlib4 version of mathlib3's tactic.basic, now called Mathlib.Tactic.Common, which imports all tactics which do not have significant theory requirements, and then is imported all across the base of the hierarchy.

This ensures that all common tactics are available nearly everywhere in the library, rather than having to be imported one-by-one as you need them.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -8,8 +8,6 @@ Authors: Johannes Hölzl, Mario Carneiro
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathlib.Tactic.RSuffices
-import Mathlib.Tactic.WLOG
 import Mathlib.Topology.SubsetProperties
 import Mathlib.Topology.Connected
 import Mathlib.Topology.NhdsSet

As discussed on Zulip

Renames

• supₛ → sSup
• infₛ → sInf
• supᵢ → iSup
• infᵢ → iInf
• bsupₛ → bsSup
• binfₛ → bsInf
• bsupᵢ → biSup
• binfᵢ → biInf
• csupₛ → csSup
• cinfₛ → csInf
• csupᵢ → ciSup
• cinfᵢ → ciInf
• unionₛ → sUnion
• interₛ → sInter
• unionᵢ → iUnion
• interᵢ → iInter
• bunionₛ → bsUnion
• binterₛ → bsInter
• bunionᵢ → biUnion
• binterᵢ → biInter