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

@@ -368,7 +368,7 @@ def interior (s : Set α) : Set α :=
 /- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem mem_interior {s : Set α} {x : α} :
     x ∈ interior s ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t := by
-  simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc', and_left_comm]
+  simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc, and_left_comm]
 #align mem_interior mem_interiorₓ
 
 #print isOpen_interior /-
@@ -1128,7 +1128,7 @@ theorem nhds_def (a : α) : 𝓝 a = ⨅ s ∈ {s : Set α | a ∈ s ∧ IsOpen
 
 #print nhds_def' /-
 theorem nhds_def' (a : α) : 𝓝 a = ⨅ (s : Set α) (hs : IsOpen s) (ha : a ∈ s), 𝓟 s := by
-  simp only [nhds_def, mem_set_of_eq, and_comm' (a ∈ _), iInf_and]
+  simp only [nhds_def, mem_set_of_eq, and_comm (a ∈ _), iInf_and]
 #align nhds_def' nhds_def'
 -/
 
@@ -1349,7 +1349,7 @@ theorem Filter.EventuallyLE.eventuallyLE_nhds [LE β] {f g : α → β} {a : α}
 #print all_mem_nhds /-
 theorem all_mem_nhds (x : α) (P : Set α → Prop) (hP : ∀ s t, s ⊆ t → P s → P t) :
     (∀ s ∈ 𝓝 x, P s) ↔ ∀ s, IsOpen s → x ∈ s → P s :=
-  ((nhds_basis_opens x).forall_iff hP).trans <| by simp only [and_comm' (x ∈ _), and_imp]
+  ((nhds_basis_opens x).forall_iff hP).trans <| by simp only [and_comm (x ∈ _), and_imp]
 #align all_mem_nhds all_mem_nhds
 -/
 
@@ -1570,8 +1570,8 @@ theorem acc_principal_iff_cluster (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ C
 /-- `x` is an accumulation point of a set `C` iff every neighborhood
 of `x` contains a point of `C` other than `x`. -/
 theorem accPt_iff_nhds (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x := by
-  simp [acc_principal_iff_cluster, clusterPt_principal_iff, Set.Nonempty, exists_prop, and_assoc',
-    and_comm' ¬_ = x]
+  simp [acc_principal_iff_cluster, clusterPt_principal_iff, Set.Nonempty, exists_prop, and_assoc,
+    and_comm ¬_ = x]
 #align acc_pt_iff_nhds accPt_iff_nhds
 -/
 
@@ -1579,7 +1579,7 @@ theorem accPt_iff_nhds (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∀ U ∈ 
 /-- `x` is an accumulation point of a set `C` iff
 there are points near `x` in `C` and different from `x`.-/
 theorem accPt_iff_frequently (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∃ᶠ y in 𝓝 x, y ≠ x ∧ y ∈ C := by
-  simp [acc_principal_iff_cluster, clusterPt_principal_iff_frequently, and_comm']
+  simp [acc_principal_iff_cluster, clusterPt_principal_iff_frequently, and_comm]
 #align acc_pt_iff_frequently accPt_iff_frequently
 -/
 
@@ -1814,7 +1814,7 @@ theorem mem_closure_iff_nhds_basis' {a : α} {p : ι → Prop} {s : ι → Set 
 theorem mem_closure_iff_nhds_basis {a : α} {p : ι → Prop} {s : ι → Set α} (h : (𝓝 a).HasBasis p s)
     {t : Set α} : a ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i :=
   (mem_closure_iff_nhds_basis' h).trans <| by
-    simp only [Set.Nonempty, mem_inter_iff, exists_prop, and_comm']
+    simp only [Set.Nonempty, mem_inter_iff, exists_prop, and_comm]
 #align mem_closure_iff_nhds_basis mem_closure_iff_nhds_basis
 -/
 

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, Jeremy Avigad
 -/
 import Order.Filter.Ultrafilter
-import Algebra.Support
+import Algebra.Function.Support
 import Order.Filter.Lift
 
 #align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
@@ -78,8 +78,8 @@ class TopologicalSpace (α : Type u) where
 #align topological_space TopologicalSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (A «expr ⊆ » T) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (A B «expr ∈ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (A «expr ⊆ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (A B «expr ∈ » T) -/
 #print TopologicalSpace.ofClosed /-
 /-- A constructor for topologies by specifying the closed sets,
 and showing that they satisfy the appropriate conditions. -/
@@ -365,7 +365,7 @@ def interior (s : Set α) : Set α :=
 #align interior interior
 -/
 
-/- ./././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) -/
 theorem mem_interior {s : Set α} {x : α} :
     x ∈ interior s ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc', and_left_comm]
@@ -499,7 +499,7 @@ theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClose
 #align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
 -/
 
-/- ./././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 isOpen_iff_forall_mem_open /-
 theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
@@ -1168,7 +1168,7 @@ theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟
 #align nhds_le_of_le nhds_le_of_le
 -/
 
-/- ./././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) -/
 theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ a ∈ t :=
   (nhds_basis_opens a).mem_iff.trans
     ⟨fun ⟨t, ⟨hat, ht⟩, hts⟩ => ⟨t, hts, ht, hat⟩, fun ⟨t, hts, ht, hat⟩ => ⟨t, ⟨hat, ht⟩, hts⟩⟩
@@ -1962,7 +1962,7 @@ theorem mem_closure_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set
 #align mem_closure_of_tendsto mem_closure_of_tendsto
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (x «expr ∉ » s) -/
 #print tendsto_inf_principal_nhds_iff_of_forall_eq /-
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
 Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/

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

@@ -211,7 +211,7 @@ theorem Set.Finite.isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finit
 #print isOpen_iInter_of_finite /-
 theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) :
     IsOpen (⋂ i, s i) :=
-  Set.Finite.isOpen_sInter (finite_range _) (forall_range_iff.2 h)
+  Set.Finite.isOpen_sInter (finite_range _) (forall_mem_range.2 h)
 #align is_open_Inter isOpen_iInter_of_finite
 -/
 
@@ -493,7 +493,7 @@ theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClose
     by_contradiction fun hx₂ : x ∉ s =>
       have : u \ s ⊆ t := fun x ⟨h₁, h₂⟩ => Or.resolve_left (hu₂ h₁) h₂
       have : u \ s ⊆ interior t := by rwa [(IsOpen.sdiff hu₁ h₁).subset_interior_iff]
-      have : u \ s ⊆ ∅ := by rwa [h₂] at this 
+      have : u \ s ⊆ ∅ := by rwa [h₂] at this
       this ⟨hx₁, hx₂⟩
   Subset.antisymm (interior_maximal this isOpen_interior) (interior_mono <| subset_union_left _ _)
 #align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
@@ -1042,7 +1042,7 @@ theorem interior_frontier {s : Set α} (h : IsClosed s) : interior (frontier s)
   have C : interior (frontier s) ⊆ frontier s := interior_subset
   have : interior (frontier s) ⊆ interior s ∩ (s \ interior s) :=
     subset_inter B (by simpa [A] using C)
-  rwa [inter_diff_self, subset_empty_iff] at this 
+  rwa [inter_diff_self, subset_empty_iff] at this
 #align interior_frontier interior_frontier
 -/
 
@@ -1977,7 +1977,7 @@ theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : β → α} {l : Filter
   refine' ⟨fun h' => _, le_trans inf_le_left⟩
   have := sup_le h' h
   rw [sup_inf_right, sup_principal, union_compl_self, principal_univ, inf_top_eq, sup_le_iff] at
-    this 
+    this
   exact this.1
 #align tendsto_inf_principal_nhds_iff_of_forall_eq tendsto_inf_principal_nhds_iff_of_forall_eq
 -/

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

@@ -470,7 +470,11 @@ theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ inte
 #print Finset.interior_iInter /-
 @[simp]
 theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α) :
-    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by classical
+    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by
+  classical
+  refine' s.induction_on (by simp) _
+  intro i s h₁ h₂
+  simp [h₂]
 #align finset.interior_Inter Finset.interior_iInter
 -/
 
@@ -697,7 +701,11 @@ theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure
 #print Finset.closure_biUnion /-
 @[simp]
 theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α) :
-    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by classical
+    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
+  classical
+  refine' s.induction_on (by simp) _
+  intro i s h₁ h₂
+  simp [h₂]
 #align finset.closure_bUnion Finset.closure_biUnion
 -/
 

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

@@ -470,11 +470,7 @@ theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ inte
 #print Finset.interior_iInter /-
 @[simp]
 theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α) :
-    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by
-  classical
-  refine' s.induction_on (by simp) _
-  intro i s h₁ h₂
-  simp [h₂]
+    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by classical
 #align finset.interior_Inter Finset.interior_iInter
 -/
 
@@ -701,11 +697,7 @@ theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure
 #print Finset.closure_biUnion /-
 @[simp]
 theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α) :
-    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
-  classical
-  refine' s.induction_on (by simp) _
-  intro i s h₁ h₂
-  simp [h₂]
+    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by classical
 #align finset.closure_bUnion Finset.closure_biUnion
 -/
 

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

@@ -1728,10 +1728,10 @@ theorem mem_closure_iff_clusterPt {s : Set α} {a : α} : a ∈ closure s ↔ Cl
 #align mem_closure_iff_cluster_pt mem_closure_iff_clusterPt
 -/
 
-#print mem_closure_iff_nhds_neBot /-
-theorem mem_closure_iff_nhds_neBot {s : Set α} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ :=
+#print mem_closure_iff_nhds_ne_bot /-
+theorem mem_closure_iff_nhds_ne_bot {s : Set α} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ :=
   mem_closure_iff_clusterPt.trans neBot_iff
-#align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_neBot
+#align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_ne_bot
 -/
 
 #print mem_closure_iff_nhdsWithin_neBot /-
@@ -1883,7 +1883,7 @@ theorem Dense.open_subset_closure_inter {s t : Set α} (hs : Dense s) (ht : IsOp
 theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x ∈ closure (s₁ ∪ s₂))
     (h₁ : s₁ᶜ ∈ 𝓝 x) : x ∈ closure s₂ :=
   by
-  rw [mem_closure_iff_nhds_neBot] at *
+  rw [mem_closure_iff_nhds_ne_bot] at *
   rwa [←
     calc
       𝓝 x ⊓ principal (s₁ ∪ s₂) = 𝓝 x ⊓ (principal s₁ ⊔ principal s₂) := by rw [sup_principal]

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

@@ -1893,20 +1893,20 @@ theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x 
 #align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_union
 -/
 
-#print Dense.inter_of_open_left /-
+#print Dense.inter_of_isOpen_left /-
 /-- The intersection of an open dense set with a dense set is a dense set. -/
-theorem Dense.inter_of_open_left {s t : Set α} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
+theorem Dense.inter_of_isOpen_left {s t : Set α} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
     Dense (s ∩ t) := fun x =>
   closure_minimal hso.inter_closure isClosed_closure <| by simp [hs.closure_eq, ht.closure_eq]
-#align dense.inter_of_open_left Dense.inter_of_open_left
+#align dense.inter_of_open_left Dense.inter_of_isOpen_left
 -/
 
-#print Dense.inter_of_open_right /-
+#print Dense.inter_of_isOpen_right /-
 /-- The intersection of a dense set with an open dense set is a dense set. -/
-theorem Dense.inter_of_open_right {s t : Set α} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
+theorem Dense.inter_of_isOpen_right {s t : Set α} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
     Dense (s ∩ t) :=
-  inter_comm t s ▸ ht.inter_of_open_left hs hto
-#align dense.inter_of_open_right Dense.inter_of_open_right
+  inter_comm t s ▸ ht.inter_of_isOpen_left hs hto
+#align dense.inter_of_open_right Dense.inter_of_isOpen_right
 -/
 
 #print Dense.inter_nhds_nonempty /-

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

@@ -514,18 +514,18 @@ theorem interior_iInter_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ 
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-#print interior_Inter₂_subset /-
-theorem interior_Inter₂_subset (p : ι → Sort _) (s : ∀ i, p i → Set α) :
+#print interior_iInter₂_subset /-
+theorem interior_iInter₂_subset (p : ι → Sort _) (s : ∀ i, p i → Set α) :
     interior (⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), interior (s i j) :=
   (interior_iInter_subset _).trans <| iInter_mono fun i => interior_iInter_subset _
-#align interior_Inter₂_subset interior_Inter₂_subset
+#align interior_Inter₂_subset interior_iInter₂_subset
 -/
 
 #print interior_sInter_subset /-
 theorem interior_sInter_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s :=
   calc
     interior (⋂₀ S) = interior (⋂ s ∈ S, s) := by rw [sInter_eq_bInter]
-    _ ⊆ ⋂ s ∈ S, interior s := interior_Inter₂_subset _ _
+    _ ⊆ ⋂ s ∈ S, interior s := interior_iInter₂_subset _ _
 #align interior_sInter_subset interior_sInter_subset
 -/
 
@@ -709,12 +709,12 @@ theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α)
 #align finset.closure_bUnion Finset.closure_biUnion
 -/
 
-#print closure_Union_of_finite /-
+#print closure_iUnion_of_finite /-
 @[simp]
-theorem closure_Union_of_finite {ι : Type _} [Finite ι] (f : ι → Set α) :
+theorem closure_iUnion_of_finite {ι : Type _} [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by cases nonempty_fintype ι;
   convert finset.univ.closure_bUnion f <;> simp
-#align closure_Union closure_Union_of_finite
+#align closure_Union closure_iUnion_of_finite
 -/
 
 #print interior_subset_closure /-

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

@@ -3,9 +3,9 @@ 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, Jeremy Avigad
 -/
-import Mathbin.Order.Filter.Ultrafilter
-import Mathbin.Algebra.Support
-import Mathbin.Order.Filter.Lift
+import Order.Filter.Ultrafilter
+import Algebra.Support
+import Order.Filter.Lift
 
 #align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
 
@@ -78,8 +78,8 @@ class TopologicalSpace (α : Type u) where
 #align topological_space TopologicalSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A «expr ⊆ » T) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A B «expr ∈ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (A «expr ⊆ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (A B «expr ∈ » T) -/
 #print TopologicalSpace.ofClosed /-
 /-- A constructor for topologies by specifying the closed sets,
 and showing that they satisfy the appropriate conditions. -/
@@ -365,7 +365,7 @@ def interior (s : Set α) : Set α :=
 #align interior interior
 -/
 
-/- ./././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) -/
 theorem mem_interior {s : Set α} {x : α} :
     x ∈ interior s ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc', and_left_comm]
@@ -499,7 +499,7 @@ theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClose
 #align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
 -/
 
-/- ./././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 isOpen_iff_forall_mem_open /-
 theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
@@ -1168,7 +1168,7 @@ theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟
 #align nhds_le_of_le nhds_le_of_le
 -/
 
-/- ./././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) -/
 theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ a ∈ t :=
   (nhds_basis_opens a).mem_iff.trans
     ⟨fun ⟨t, ⟨hat, ht⟩, hts⟩ => ⟨t, hts, ht, hat⟩, fun ⟨t, hts, ht, hat⟩ => ⟨t, ⟨hat, ht⟩, hts⟩⟩
@@ -1962,7 +1962,7 @@ theorem mem_closure_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set
 #align mem_closure_of_tendsto mem_closure_of_tendsto
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » s) -/
 #print tendsto_inf_principal_nhds_iff_of_forall_eq /-
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
 Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/

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

@@ -188,35 +188,37 @@ theorem isOpen_empty : IsOpen (∅ : Set α) := by
 #align is_open_empty isOpen_empty
 -/
 
-#print isOpen_sInter /-
-theorem isOpen_sInter {s : Set (Set α)} (hs : s.Finite) : (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
+#print Set.Finite.isOpen_sInter /-
+theorem Set.Finite.isOpen_sInter {s : Set (Set α)} (hs : s.Finite) :
+    (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
   Finite.induction_on hs (fun _ => by rw [sInter_empty] <;> exact isOpen_univ)
     fun a s has hs ih h => by
     rw [sInter_insert] <;>
       exact IsOpen.inter (h _ <| mem_insert _ _) (ih fun t => h t ∘ mem_insert_of_mem _)
-#align is_open_sInter isOpen_sInter
+#align is_open_sInter Set.Finite.isOpen_sInter
 -/
 
-#print isOpen_biInter /-
-theorem isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finite) :
+#print Set.Finite.isOpen_biInter /-
+theorem Set.Finite.isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finite) :
     (∀ i ∈ s, IsOpen (f i)) → IsOpen (⋂ i ∈ s, f i) :=
   Finite.induction_on hs (fun _ => by rw [bInter_empty] <;> exact isOpen_univ)
     fun a s has hs ih h => by
     rw [bInter_insert] <;>
       exact IsOpen.inter (h a (mem_insert _ _)) (ih fun i hi => h i (mem_insert_of_mem _ hi))
-#align is_open_bInter isOpen_biInter
+#align is_open_bInter Set.Finite.isOpen_biInter
 -/
 
-#print isOpen_iInter /-
-theorem isOpen_iInter [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) : IsOpen (⋂ i, s i) :=
-  isOpen_sInter (finite_range _) (forall_range_iff.2 h)
-#align is_open_Inter isOpen_iInter
+#print isOpen_iInter_of_finite /-
+theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) :
+    IsOpen (⋂ i, s i) :=
+  Set.Finite.isOpen_sInter (finite_range _) (forall_range_iff.2 h)
+#align is_open_Inter isOpen_iInter_of_finite
 -/
 
 #print isOpen_biInter_finset /-
 theorem isOpen_biInter_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
-  isOpen_biInter (toFinite _) h
+  Set.Finite.isOpen_biInter (toFinite _) h
 #align is_open_bInter_finset isOpen_biInter_finset
 -/
 
@@ -319,21 +321,21 @@ theorem IsClosed.sdiff {s t : Set α} (h₁ : IsClosed s) (h₂ : IsOpen t) : Is
 #align is_closed.sdiff IsClosed.sdiff
 -/
 
-#print isClosed_biUnion /-
-theorem isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite) :
+#print Set.Finite.isClosed_biUnion /-
+theorem Set.Finite.isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite) :
     (∀ i ∈ s, IsClosed (f i)) → IsClosed (⋃ i ∈ s, f i) :=
   Finite.induction_on hs (fun _ => by rw [bUnion_empty] <;> exact isClosed_empty)
     fun a s has hs ih h => by
     rw [bUnion_insert] <;>
       exact IsClosed.union (h a (mem_insert _ _)) (ih fun i hi => h i (mem_insert_of_mem _ hi))
-#align is_closed_bUnion isClosed_biUnion
+#align is_closed_bUnion Set.Finite.isClosed_biUnion
 -/
 
-#print isClosed_iUnion /-
-theorem isClosed_iUnion [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
+#print isClosed_iUnion_of_finite /-
+theorem isClosed_iUnion_of_finite [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
     IsClosed (⋃ i, s i) := by simp only [← isOpen_compl_iff, compl_Union] at *;
-  exact isOpen_iInter h
-#align is_closed_Union isClosed_iUnion
+  exact isOpen_iInter_of_finite h
+#align is_closed_Union isClosed_iUnion_of_finite
 -/
 
 #print isClosed_imp /-
@@ -476,12 +478,12 @@ theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α)
 #align finset.interior_Inter Finset.interior_iInter
 -/
 
-#print interior_iInter /-
+#print interior_iInter_of_finite /-
 @[simp]
-theorem interior_iInter {ι : Type _} [Finite ι] (f : ι → Set α) :
+theorem interior_iInter_of_finite {ι : Type _} [Finite ι] (f : ι → Set α) :
     interior (⋂ i, f i) = ⋂ i, interior (f i) := by cases nonempty_fintype ι;
   convert finset.univ.interior_Inter f <;> simp
-#align interior_Inter interior_iInter
+#align interior_Inter interior_iInter_of_finite
 -/
 
 #print interior_union_isClosed_of_interior_empty /-
@@ -707,12 +709,12 @@ theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α)
 #align finset.closure_bUnion Finset.closure_biUnion
 -/
 
-#print closure_iUnion /-
+#print closure_Union_of_finite /-
 @[simp]
-theorem closure_iUnion {ι : Type _} [Finite ι] (f : ι → Set α) :
+theorem closure_Union_of_finite {ι : Type _} [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by cases nonempty_fintype ι;
   convert finset.univ.closure_bUnion f <;> simp
-#align closure_Union closure_iUnion
+#align closure_Union closure_Union_of_finite
 -/
 
 #print interior_subset_closure /-

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

@@ -115,11 +115,11 @@ theorem isOpen_mk {p h₁ h₂ h₃} {s : Set α} : is_open[⟨p, h₁, h₂, h
 #align is_open_mk isOpen_mk
 -/
 
-#print topologicalSpace_eq /-
+#print TopologicalSpace.ext /-
 @[ext]
-theorem topologicalSpace_eq {f g : TopologicalSpace α} (h : is_open[f] = is_open[g]) : f = g := by
+theorem TopologicalSpace.ext {f g : TopologicalSpace α} (h : is_open[f] = is_open[g]) : f = g := by
   cases f; cases g; congr; exact h
-#align topological_space_eq topologicalSpace_eq
+#align topological_space_eq TopologicalSpace.ext
 -/
 
 section
@@ -147,11 +147,11 @@ theorem isOpen_sUnion {s : Set (Set α)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (
 
 end
 
-#print topologicalSpace_eq_iff /-
-theorem topologicalSpace_eq_iff {t t' : TopologicalSpace α} :
+#print TopologicalSpace.ext_iff /-
+theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace α} :
     t = t' ↔ ∀ s, is_open[t] s ↔ is_open[t'] s :=
   ⟨fun h s => h ▸ Iff.rfl, fun h => by ext; exact h _⟩
-#align topological_space_eq_iff topologicalSpace_eq_iff
+#align topological_space_eq_iff TopologicalSpace.ext_iff
 -/
 
 #print isOpen_fold /-
@@ -668,7 +668,7 @@ theorem closure_nonempty_iff {s : Set α} : (closure s).Nonempty ↔ s.Nonempty
 #align closure_nonempty_iff closure_nonempty_iff
 -/
 
-alias closure_nonempty_iff ↔ Set.Nonempty.of_closure Set.Nonempty.closure
+alias ⟨Set.Nonempty.of_closure, Set.Nonempty.closure⟩ := closure_nonempty_iff
 #align set.nonempty.of_closure Set.Nonempty.of_closure
 #align set.nonempty.closure Set.Nonempty.closure
 
@@ -833,7 +833,7 @@ theorem dense_closure {s : Set α} : Dense (closure s) ↔ Dense s := by
 #align dense_closure dense_closure
 -/
 
-alias dense_closure ↔ Dense.of_closure Dense.closure
+alias ⟨Dense.of_closure, Dense.closure⟩ := dense_closure
 #align dense.of_closure Dense.of_closure
 #align dense.closure Dense.closure
 
@@ -858,7 +858,7 @@ theorem dense_iff_inter_open {s : Set α} :
 #align dense_iff_inter_open dense_iff_inter_open
 -/
 
-alias dense_iff_inter_open ↔ Dense.inter_open_nonempty _
+alias ⟨Dense.inter_open_nonempty, _⟩ := dense_iff_inter_open
 #align dense.inter_open_nonempty Dense.inter_open_nonempty
 
 #print Dense.exists_mem_open /-
@@ -1695,7 +1695,7 @@ theorem mem_closure_iff_frequently {s : Set α} {a : α} : a ∈ closure s ↔ 
 #align mem_closure_iff_frequently mem_closure_iff_frequently
 -/
 
-alias mem_closure_iff_frequently ↔ _ Filter.Frequently.mem_closure
+alias ⟨_, Filter.Frequently.mem_closure⟩ := mem_closure_iff_frequently
 #align filter.frequently.mem_closure Filter.Frequently.mem_closure
 
 #print isClosed_iff_frequently /-

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

@@ -2,16 +2,13 @@
 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, Jeremy Avigad
-
-! This file was ported from Lean 3 source module topology.basic
-! leanprover-community/mathlib commit e354e865255654389cc46e6032160238df2e0f40
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Order.Filter.Ultrafilter
 import Mathbin.Algebra.Support
 import Mathbin.Order.Filter.Lift
 
+#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
+
 /-!
 # Basic theory of topological spaces.
 
@@ -81,8 +78,8 @@ class TopologicalSpace (α : Type u) where
 #align topological_space TopologicalSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (A «expr ⊆ » T) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (A B «expr ∈ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A «expr ⊆ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A B «expr ∈ » T) -/
 #print TopologicalSpace.ofClosed /-
 /-- A constructor for topologies by specifying the closed sets,
 and showing that they satisfy the appropriate conditions. -/
@@ -366,7 +363,7 @@ def interior (s : Set α) : Set α :=
 #align interior interior
 -/
 
-/- ./././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) -/
 theorem mem_interior {s : Set α} {x : α} :
     x ∈ interior s ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc', and_left_comm]
@@ -500,7 +497,7 @@ theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClose
 #align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
 -/
 
-/- ./././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 isOpen_iff_forall_mem_open /-
 theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
@@ -1169,7 +1166,7 @@ theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟
 #align nhds_le_of_le nhds_le_of_le
 -/
 
-/- ./././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) -/
 theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ a ∈ t :=
   (nhds_basis_opens a).mem_iff.trans
     ⟨fun ⟨t, ⟨hat, ht⟩, hts⟩ => ⟨t, hts, ht, hat⟩, fun ⟨t, hts, ht, hat⟩ => ⟨t, ⟨hat, ht⟩, hts⟩⟩
@@ -1963,7 +1960,7 @@ theorem mem_closure_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set
 #align mem_closure_of_tendsto mem_closure_of_tendsto
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
 #print tendsto_inf_principal_nhds_iff_of_forall_eq /-
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
 Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/

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

@@ -83,6 +83,7 @@ class TopologicalSpace (α : Type u) where
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (A «expr ⊆ » T) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (A B «expr ∈ » T) -/
+#print TopologicalSpace.ofClosed /-
 /-- A constructor for topologies by specifying the closed sets,
 and showing that they satisfy the appropriate conditions. -/
 def TopologicalSpace.ofClosed {α : Type u} (T : Set (Set α)) (empty_mem : ∅ ∈ T)
@@ -96,6 +97,7 @@ def TopologicalSpace.ofClosed {α : Type u} (T : Set (Set α)) (empty_mem : ∅
     rw [Set.compl_sUnion] <;>
       exact sInter_mem (compl '' s) fun z ⟨y, hy, hz⟩ => by simpa [hz.symm] using hs y hy
 #align topological_space.of_closed TopologicalSpace.ofClosed
+-/
 
 section TopologicalSpace
 
@@ -108,12 +110,13 @@ def IsOpen [TopologicalSpace α] (s : Set α) : Prop :=
 #align is_open IsOpen
 -/
 
--- mathport name: is_open_of
 scoped[Topology] notation "is_open[" t "]" => @IsOpen hole! t
 
+#print isOpen_mk /-
 theorem isOpen_mk {p h₁ h₂ h₃} {s : Set α} : is_open[⟨p, h₁, h₂, h₃⟩] s ↔ p s :=
   Iff.rfl
 #align is_open_mk isOpen_mk
+-/
 
 #print topologicalSpace_eq /-
 @[ext]
@@ -133,9 +136,11 @@ theorem isOpen_univ : IsOpen (univ : Set α) :=
 #align is_open_univ isOpen_univ
 -/
 
+#print IsOpen.inter /-
 theorem IsOpen.inter (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∩ s₂) :=
   TopologicalSpace.isOpen_inter s₁ s₂ h₁ h₂
 #align is_open.inter IsOpen.inter
+-/
 
 #print isOpen_sUnion /-
 theorem isOpen_sUnion {s : Set (Set α)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (⋃₀ s) :=
@@ -173,9 +178,11 @@ theorem isOpen_biUnion {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen
 #align is_open_bUnion isOpen_biUnion
 -/
 
+#print IsOpen.union /-
 theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∪ s₂) := by
   rw [union_eq_Union] <;> exact isOpen_iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align is_open.union IsOpen.union
+-/
 
 #print isOpen_empty /-
 @[simp]
@@ -236,13 +243,14 @@ class IsClosed (s : Set α) : Prop where
 #align is_closed IsClosed
 -/
 
--- mathport name: is_closed_of
 scoped[Topology] notation "is_closed[" t "]" => @IsClosed hole! t
 
+#print isOpen_compl_iff /-
 @[simp]
 theorem isOpen_compl_iff {s : Set α} : IsOpen (sᶜ) ↔ IsClosed s :=
   ⟨fun h => ⟨h⟩, fun h => h.isOpen_compl⟩
 #align is_open_compl_iff isOpen_compl_iff
+-/
 
 #print isClosed_empty /-
 @[simp]
@@ -258,9 +266,11 @@ theorem isClosed_univ : IsClosed (univ : Set α) := by rw [← isOpen_compl_iff,
 #align is_closed_univ isClosed_univ
 -/
 
+#print IsClosed.union /-
 theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂) := fun h₁ h₂ => by
   rw [← isOpen_compl_iff] at *; rw [compl_union]; exact IsOpen.inter h₁ h₂
 #align is_closed.union IsClosed.union
+-/
 
 #print isClosed_sInter /-
 theorem isClosed_sInter {s : Set (Set α)} : (∀ t ∈ s, IsClosed t) → IsClosed (⋂₀ s) := by
@@ -281,26 +291,36 @@ theorem isClosed_biInter {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsCl
 #align is_closed_bInter isClosed_biInter
 -/
 
+#print isClosed_compl_iff /-
 @[simp]
 theorem isClosed_compl_iff {s : Set α} : IsClosed (sᶜ) ↔ IsOpen s := by
   rw [← isOpen_compl_iff, compl_compl]
 #align is_closed_compl_iff isClosed_compl_iff
+-/
 
+#print IsOpen.isClosed_compl /-
 theorem IsOpen.isClosed_compl {s : Set α} (hs : IsOpen s) : IsClosed (sᶜ) :=
   isClosed_compl_iff.2 hs
 #align is_open.is_closed_compl IsOpen.isClosed_compl
+-/
 
+#print IsOpen.sdiff /-
 theorem IsOpen.sdiff {s t : Set α} (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) :=
   IsOpen.inter h₁ <| isOpen_compl_iff.mpr h₂
 #align is_open.sdiff IsOpen.sdiff
+-/
 
+#print IsClosed.inter /-
 theorem IsClosed.inter (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) : IsClosed (s₁ ∩ s₂) := by
   rw [← isOpen_compl_iff] at *; rw [compl_inter]; exact IsOpen.union h₁ h₂
 #align is_closed.inter IsClosed.inter
+-/
 
+#print IsClosed.sdiff /-
 theorem IsClosed.sdiff {s t : Set α} (h₁ : IsClosed s) (h₂ : IsOpen t) : IsClosed (s \ t) :=
   IsClosed.inter h₁ (isClosed_compl_iff.mpr h₂)
 #align is_closed.sdiff IsClosed.sdiff
+-/
 
 #print isClosed_biUnion /-
 theorem isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite) :
@@ -437,6 +457,7 @@ theorem interior_interior {s : Set α} : interior (interior s) = interior s :=
 #align interior_interior interior_interior
 -/
 
+#print interior_inter /-
 @[simp]
 theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ interior t :=
   Subset.antisymm
@@ -445,7 +466,9 @@ theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ inte
     (interior_maximal (inter_subset_inter interior_subset interior_subset) <|
       IsOpen.inter isOpen_interior isOpen_interior)
 #align interior_inter interior_inter
+-/
 
+#print Finset.interior_iInter /-
 @[simp]
 theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α) :
     interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by
@@ -454,13 +477,17 @@ theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α)
   intro i s h₁ h₂
   simp [h₂]
 #align finset.interior_Inter Finset.interior_iInter
+-/
 
+#print interior_iInter /-
 @[simp]
 theorem interior_iInter {ι : Type _} [Finite ι] (f : ι → Set α) :
     interior (⋂ i, f i) = ⋂ i, interior (f i) := by cases nonempty_fintype ι;
   convert finset.univ.interior_Inter f <;> simp
 #align interior_Inter interior_iInter
+-/
 
+#print interior_union_isClosed_of_interior_empty /-
 theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClosed s)
     (h₂ : interior t = ∅) : interior (s ∪ t) = interior s :=
   have : interior (s ∪ t) ⊆ s := fun x ⟨u, ⟨(hu₁ : IsOpen u), (hu₂ : u ⊆ s ∪ t)⟩, (hx₁ : x ∈ u)⟩ =>
@@ -471,11 +498,14 @@ theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClose
       this ⟨hx₁, hx₂⟩
   Subset.antisymm (interior_maximal this isOpen_interior) (interior_mono <| subset_union_left _ _)
 #align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+#print isOpen_iff_forall_mem_open /-
 theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
 #align is_open_iff_forall_mem_open isOpen_iff_forall_mem_open
+-/
 
 #print interior_iInter_subset /-
 theorem interior_iInter_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ ⋂ i, interior (s i) :=
@@ -485,10 +515,12 @@ theorem interior_iInter_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ 
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
+#print interior_Inter₂_subset /-
 theorem interior_Inter₂_subset (p : ι → Sort _) (s : ∀ i, p i → Set α) :
     interior (⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), interior (s i j) :=
   (interior_iInter_subset _).trans <| iInter_mono fun i => interior_iInter_subset _
 #align interior_Inter₂_subset interior_Inter₂_subset
+-/
 
 #print interior_sInter_subset /-
 theorem interior_sInter_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s :=
@@ -535,15 +567,19 @@ theorem closure_minimal {s t : Set α} (h₁ : s ⊆ t) (h₂ : IsClosed t) : cl
 #align closure_minimal closure_minimal
 -/
 
+#print Disjoint.closure_left /-
 theorem Disjoint.closure_left {s t : Set α} (hd : Disjoint s t) (ht : IsOpen t) :
     Disjoint (closure s) t :=
   disjoint_compl_left.mono_left <| closure_minimal hd.subset_compl_right ht.isClosed_compl
 #align disjoint.closure_left Disjoint.closure_left
+-/
 
+#print Disjoint.closure_right /-
 theorem Disjoint.closure_right {s t : Set α} (hd : Disjoint s t) (hs : IsOpen s) :
     Disjoint s (closure t) :=
   (hd.symm.closure_left hs).symm
 #align disjoint.closure_right Disjoint.closure_right
+-/
 
 #print IsClosed.closure_eq /-
 theorem IsClosed.closure_eq {s : Set α} (h : IsClosed s) : closure s = s :=
@@ -577,18 +613,24 @@ theorem closure_mono {s t : Set α} (h : s ⊆ t) : closure s ⊆ closure t :=
 #align closure_mono closure_mono
 -/
 
+#print monotone_closure /-
 theorem monotone_closure (α : Type _) [TopologicalSpace α] : Monotone (@closure α _) := fun _ _ =>
   closure_mono
 #align monotone_closure monotone_closure
+-/
 
+#print diff_subset_closure_iff /-
 theorem diff_subset_closure_iff {s t : Set α} : s \ t ⊆ closure t ↔ s ⊆ closure t := by
   rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]
 #align diff_subset_closure_iff diff_subset_closure_iff
+-/
 
+#print closure_inter_subset_inter_closure /-
 theorem closure_inter_subset_inter_closure (s t : Set α) :
     closure (s ∩ t) ⊆ closure s ∩ closure t :=
   (monotone_closure α).map_inf_le s t
 #align closure_inter_subset_inter_closure closure_inter_subset_inter_closure
+-/
 
 #print isClosed_of_closure_subset /-
 theorem isClosed_of_closure_subset {s : Set α} (h : closure s ⊆ s) : IsClosed s := by
@@ -647,6 +689,7 @@ theorem closure_closure {s : Set α} : closure (closure s) = closure s :=
 #align closure_closure closure_closure
 -/
 
+#print closure_union /-
 @[simp]
 theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure t :=
   Subset.antisymm
@@ -654,7 +697,9 @@ theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure
       IsClosed.union isClosed_closure isClosed_closure)
     ((monotone_closure α).le_map_sup s t)
 #align closure_union closure_union
+-/
 
+#print Finset.closure_biUnion /-
 @[simp]
 theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α) :
     closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
@@ -663,12 +708,15 @@ theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α)
   intro i s h₁ h₂
   simp [h₂]
 #align finset.closure_bUnion Finset.closure_biUnion
+-/
 
+#print closure_iUnion /-
 @[simp]
 theorem closure_iUnion {ι : Type _} [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by cases nonempty_fintype ι;
   convert finset.univ.closure_bUnion f <;> simp
 #align closure_Union closure_iUnion
+-/
 
 #print interior_subset_closure /-
 theorem interior_subset_closure {s : Set α} : interior s ⊆ closure s :=
@@ -676,22 +724,29 @@ theorem interior_subset_closure {s : Set α} : interior s ⊆ closure s :=
 #align interior_subset_closure interior_subset_closure
 -/
 
+#print closure_eq_compl_interior_compl /-
 theorem closure_eq_compl_interior_compl {s : Set α} : closure s = interior (sᶜ)ᶜ :=
   by
   rw [interior, closure, compl_sUnion, compl_image_set_of]
   simp only [compl_subset_compl, isOpen_compl_iff]
 #align closure_eq_compl_interior_compl closure_eq_compl_interior_compl
+-/
 
+#print interior_compl /-
 @[simp]
 theorem interior_compl {s : Set α} : interior (sᶜ) = closure sᶜ := by
   simp [closure_eq_compl_interior_compl]
 #align interior_compl interior_compl
+-/
 
+#print closure_compl /-
 @[simp]
 theorem closure_compl {s : Set α} : closure (sᶜ) = interior sᶜ := by
   simp [closure_eq_compl_interior_compl]
 #align closure_compl closure_compl
+-/
 
+#print mem_closure_iff /-
 theorem mem_closure_iff {s : Set α} {a : α} :
     a ∈ closure s ↔ ∀ o, IsOpen o → a ∈ o → (o ∩ s).Nonempty :=
   ⟨fun h o oo ao =>
@@ -703,16 +758,21 @@ theorem mem_closure_iff {s : Set α} {a : α} :
       let ⟨x, hc, hs⟩ := H _ h₁.isOpen_compl nc
       hc (h₂ hs)⟩
 #align mem_closure_iff mem_closure_iff
+-/
 
+#print closure_inter_open_nonempty_iff /-
 theorem closure_inter_open_nonempty_iff {s t : Set α} (h : IsOpen t) :
     (closure s ∩ t).Nonempty ↔ (s ∩ t).Nonempty :=
   ⟨fun ⟨x, hxcs, hxt⟩ => inter_comm t s ▸ mem_closure_iff.1 hxcs t h hxt, fun h =>
     h.mono <| inf_le_inf_right t subset_closure⟩
 #align closure_inter_open_nonempty_iff closure_inter_open_nonempty_iff
+-/
 
+#print Filter.le_lift'_closure /-
 theorem Filter.le_lift'_closure (l : Filter α) : l ≤ l.lift' closure :=
   le_lift'.2 fun s hs => mem_of_superset hs subset_closure
 #align filter.le_lift'_closure Filter.le_lift'_closure
+-/
 
 #print Filter.HasBasis.lift'_closure /-
 theorem Filter.HasBasis.lift'_closure {l : Filter α} {p : ι → Prop} {s : ι → Set α}
@@ -729,11 +789,13 @@ theorem Filter.HasBasis.lift'_closure_eq_self {l : Filter α} {p : ι → Prop}
 #align filter.has_basis.lift'_closure_eq_self Filter.HasBasis.lift'_closure_eq_self
 -/
 
+#print Filter.lift'_closure_eq_bot /-
 @[simp]
 theorem Filter.lift'_closure_eq_bot {l : Filter α} : l.lift' closure = ⊥ ↔ l = ⊥ :=
   ⟨fun h => bot_unique <| h ▸ l.le_lift'_closure, fun h =>
     h.symm ▸ by rw [lift'_bot (monotone_closure _), closure_empty, principal_empty]⟩
 #align filter.lift'_closure_eq_bot Filter.lift'_closure_eq_bot
+-/
 
 #print Dense /-
 /-- A set is dense in a topological space if every point belongs to its closure. -/
@@ -754,13 +816,17 @@ theorem Dense.closure_eq {s : Set α} (h : Dense s) : closure s = univ :=
 #align dense.closure_eq Dense.closure_eq
 -/
 
+#print interior_eq_empty_iff_dense_compl /-
 theorem interior_eq_empty_iff_dense_compl {s : Set α} : interior s = ∅ ↔ Dense (sᶜ) := by
   rw [dense_iff_closure_eq, closure_compl, compl_univ_iff]
 #align interior_eq_empty_iff_dense_compl interior_eq_empty_iff_dense_compl
+-/
 
+#print Dense.interior_compl /-
 theorem Dense.interior_compl {s : Set α} (h : Dense s) : interior (sᶜ) = ∅ :=
   interior_eq_empty_iff_dense_compl.2 <| by rwa [compl_compl]
 #align dense.interior_compl Dense.interior_compl
+-/
 
 #print dense_closure /-
 /-- The closure of a set `s` is dense if and only if `s` is dense. -/
@@ -780,6 +846,7 @@ theorem dense_univ : Dense (univ : Set α) := fun x => subset_closure trivial
 #align dense_univ dense_univ
 -/
 
+#print dense_iff_inter_open /-
 /-- A set is dense if and only if it has a nonempty intersection with each nonempty open set. -/
 theorem dense_iff_inter_open {s : Set α} :
     Dense s ↔ ∀ U, IsOpen U → U.Nonempty → (U ∩ s).Nonempty :=
@@ -792,15 +859,18 @@ theorem dense_iff_inter_open {s : Set α} :
     intro U U_op x_in
     exact h U U_op ⟨_, x_in⟩
 #align dense_iff_inter_open dense_iff_inter_open
+-/
 
 alias dense_iff_inter_open ↔ Dense.inter_open_nonempty _
 #align dense.inter_open_nonempty Dense.inter_open_nonempty
 
+#print Dense.exists_mem_open /-
 theorem Dense.exists_mem_open {s : Set α} (hs : Dense s) {U : Set α} (ho : IsOpen U)
     (hne : U.Nonempty) : ∃ x ∈ s, x ∈ U :=
   let ⟨x, hx⟩ := hs.inter_open_nonempty U ho hne
   ⟨x, hx.2, hx.1⟩
 #align dense.exists_mem_open Dense.exists_mem_open
+-/
 
 #print Dense.nonempty_iff /-
 theorem Dense.nonempty_iff {s : Set α} (hs : Dense s) : s.Nonempty ↔ Nonempty α :=
@@ -823,6 +893,7 @@ theorem Dense.mono {s₁ s₂ : Set α} (h : s₁ ⊆ s₂) (hd : Dense s₁) :
 #align dense.mono Dense.mono
 -/
 
+#print dense_compl_singleton_iff_not_open /-
 /-- Complement to a singleton is dense if and only if the singleton is not an open set. -/
 theorem dense_compl_singleton_iff_not_open {x : α} : Dense ({x}ᶜ : Set α) ↔ ¬IsOpen ({x} : Set α) :=
   by
@@ -833,6 +904,7 @@ theorem dense_compl_singleton_iff_not_open {x : α} : Dense ({x}ᶜ : Set α) 
     obtain rfl : U = {x}; exact eq_singleton_iff_nonempty_unique_mem.2 ⟨hne, hUx⟩
     exact ho hU
 #align dense_compl_singleton_iff_not_open dense_compl_singleton_iff_not_open
+-/
 
 /-!
 ### Frontier of a set
@@ -846,25 +918,33 @@ def frontier (s : Set α) : Set α :=
 #align frontier frontier
 -/
 
+#print closure_diff_interior /-
 @[simp]
 theorem closure_diff_interior (s : Set α) : closure s \ interior s = frontier s :=
   rfl
 #align closure_diff_interior closure_diff_interior
+-/
 
+#print closure_diff_frontier /-
 @[simp]
 theorem closure_diff_frontier (s : Set α) : closure s \ frontier s = interior s := by
   rw [frontier, diff_diff_right_self, inter_eq_self_of_subset_right interior_subset_closure]
 #align closure_diff_frontier closure_diff_frontier
+-/
 
+#print self_diff_frontier /-
 @[simp]
 theorem self_diff_frontier (s : Set α) : s \ frontier s = interior s := by
   rw [frontier, diff_diff_right, diff_eq_empty.2 subset_closure,
     inter_eq_self_of_subset_right interior_subset, empty_union]
 #align self_diff_frontier self_diff_frontier
+-/
 
+#print frontier_eq_closure_inter_closure /-
 theorem frontier_eq_closure_inter_closure {s : Set α} : frontier s = closure s ∩ closure (sᶜ) := by
   rw [closure_compl, frontier, diff_eq]
 #align frontier_eq_closure_inter_closure frontier_eq_closure_inter_closure
+-/
 
 #print frontier_subset_closure /-
 theorem frontier_subset_closure {s : Set α} : frontier s ⊆ closure s :=
@@ -890,11 +970,13 @@ theorem frontier_interior_subset {s : Set α} : frontier (interior s) ⊆ fronti
 #align frontier_interior_subset frontier_interior_subset
 -/
 
+#print frontier_compl /-
 /-- The complement of a set has the same frontier as the original set. -/
 @[simp]
 theorem frontier_compl (s : Set α) : frontier (sᶜ) = frontier s := by
   simp only [frontier_eq_closure_inter_closure, compl_compl, inter_comm]
 #align frontier_compl frontier_compl
+-/
 
 #print frontier_univ /-
 @[simp]
@@ -908,6 +990,7 @@ theorem frontier_empty : frontier (∅ : Set α) = ∅ := by simp [frontier]
 #align frontier_empty frontier_empty
 -/
 
+#print frontier_inter_subset /-
 theorem frontier_inter_subset (s t : Set α) :
     frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t :=
   by
@@ -917,23 +1000,32 @@ theorem frontier_inter_subset (s t : Set α) :
   congr 2
   apply inter_comm
 #align frontier_inter_subset frontier_inter_subset
+-/
 
+#print frontier_union_subset /-
 theorem frontier_union_subset (s t : Set α) :
     frontier (s ∪ t) ⊆ frontier s ∩ closure (tᶜ) ∪ closure (sᶜ) ∩ frontier t := by
   simpa only [frontier_compl, ← compl_union] using frontier_inter_subset (sᶜ) (tᶜ)
 #align frontier_union_subset frontier_union_subset
+-/
 
+#print IsClosed.frontier_eq /-
 theorem IsClosed.frontier_eq {s : Set α} (hs : IsClosed s) : frontier s = s \ interior s := by
   rw [frontier, hs.closure_eq]
 #align is_closed.frontier_eq IsClosed.frontier_eq
+-/
 
+#print IsOpen.frontier_eq /-
 theorem IsOpen.frontier_eq {s : Set α} (hs : IsOpen s) : frontier s = closure s \ s := by
   rw [frontier, hs.interior_eq]
 #align is_open.frontier_eq IsOpen.frontier_eq
+-/
 
+#print IsOpen.inter_frontier_eq /-
 theorem IsOpen.inter_frontier_eq {s : Set α} (hs : IsOpen s) : s ∩ frontier s = ∅ := by
   rw [hs.frontier_eq, inter_diff_self]
 #align is_open.inter_frontier_eq IsOpen.inter_frontier_eq
+-/
 
 #print isClosed_frontier /-
 /-- The frontier of a set is closed. -/
@@ -955,32 +1047,44 @@ theorem interior_frontier {s : Set α} (h : IsClosed s) : interior (frontier s)
 #align interior_frontier interior_frontier
 -/
 
+#print closure_eq_interior_union_frontier /-
 theorem closure_eq_interior_union_frontier (s : Set α) : closure s = interior s ∪ frontier s :=
   (union_diff_cancel interior_subset_closure).symm
 #align closure_eq_interior_union_frontier closure_eq_interior_union_frontier
+-/
 
+#print closure_eq_self_union_frontier /-
 theorem closure_eq_self_union_frontier (s : Set α) : closure s = s ∪ frontier s :=
   (union_diff_cancel' interior_subset subset_closure).symm
 #align closure_eq_self_union_frontier closure_eq_self_union_frontier
+-/
 
+#print Disjoint.frontier_left /-
 theorem Disjoint.frontier_left (ht : IsOpen t) (hd : Disjoint s t) : Disjoint (frontier s) t :=
   subset_compl_iff_disjoint_right.1 <|
     frontier_subset_closure.trans <| closure_minimal (disjoint_left.1 hd) <| isClosed_compl_iff.2 ht
 #align disjoint.frontier_left Disjoint.frontier_left
+-/
 
+#print Disjoint.frontier_right /-
 theorem Disjoint.frontier_right (hs : IsOpen s) (hd : Disjoint s t) : Disjoint s (frontier t) :=
   (hd.symm.frontier_left hs).symm
 #align disjoint.frontier_right Disjoint.frontier_right
+-/
 
+#print frontier_eq_inter_compl_interior /-
 theorem frontier_eq_inter_compl_interior {s : Set α} : frontier s = interior sᶜ ∩ interior (sᶜ)ᶜ :=
   by rw [← frontier_compl, ← closure_compl]; rfl
 #align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interior
+-/
 
+#print compl_frontier_eq_union_interior /-
 theorem compl_frontier_eq_union_interior {s : Set α} : frontier sᶜ = interior s ∪ interior (sᶜ) :=
   by
   rw [frontier_eq_inter_compl_interior]
   simp only [compl_inter, compl_compl]
 #align compl_frontier_eq_union_interior compl_frontier_eq_union_interior
+-/
 
 /-!
 ### Neighborhoods
@@ -996,7 +1100,6 @@ irreducible_def nhds (a : α) : Filter α :=
 #align nhds nhds
 -/
 
--- mathport name: nhds
 scoped[Topology] notation "𝓝" => nhds
 
 #print nhdsWithin /-
@@ -1007,30 +1110,28 @@ def nhdsWithin (a : α) (s : Set α) : Filter α :=
 #align nhds_within nhdsWithin
 -/
 
--- mathport name: nhds_within
 scoped[Topology] notation "𝓝[" s "] " x:100 => nhdsWithin x s
 
--- mathport name: nhds_within.ne
 scoped[Topology] notation "𝓝[≠] " x:100 => nhdsWithin x ({x}ᶜ)
 
--- mathport name: nhds_within.ge
 scoped[Topology] notation "𝓝[≥] " x:100 => nhdsWithin x (Set.Ici x)
 
--- mathport name: nhds_within.le
 scoped[Topology] notation "𝓝[≤] " x:100 => nhdsWithin x (Set.Iic x)
 
--- mathport name: nhds_within.gt
 scoped[Topology] notation "𝓝[>] " x:100 => nhdsWithin x (Set.Ioi x)
 
--- mathport name: nhds_within.lt
 scoped[Topology] notation "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
 
+#print nhds_def /-
 theorem nhds_def (a : α) : 𝓝 a = ⨅ s ∈ {s : Set α | a ∈ s ∧ IsOpen s}, 𝓟 s := by rw [nhds]
 #align nhds_def nhds_def
+-/
 
+#print nhds_def' /-
 theorem nhds_def' (a : α) : 𝓝 a = ⨅ (s : Set α) (hs : IsOpen s) (ha : a ∈ s), 𝓟 s := by
   simp only [nhds_def, mem_set_of_eq, and_comm' (a ∈ _), iInf_and]
 #align nhds_def' nhds_def'
+-/
 
 #print nhds_basis_opens /-
 /-- The open sets containing `a` are a basis for the neighborhood filter. See `nhds_basis_opens'`
@@ -1046,21 +1147,27 @@ theorem nhds_basis_opens (a : α) : (𝓝 a).HasBasis (fun s : Set α => a ∈ s
 #align nhds_basis_opens nhds_basis_opens
 -/
 
+#print nhds_basis_closeds /-
 theorem nhds_basis_closeds (a : α) : (𝓝 a).HasBasis (fun s : Set α => a ∉ s ∧ IsClosed s) compl :=
   ⟨fun t =>
     (nhds_basis_opens a).mem_iff.trans <|
       compl_surjective.exists.trans <| by simp only [isOpen_compl_iff, mem_compl_iff]⟩
 #align nhds_basis_closeds nhds_basis_closeds
+-/
 
+#print le_nhds_iff /-
 /-- A filter lies below the neighborhood filter at `a` iff it contains every open set around `a`. -/
 theorem le_nhds_iff {f a} : f ≤ 𝓝 a ↔ ∀ s : Set α, a ∈ s → IsOpen s → s ∈ f := by simp [nhds_def]
 #align le_nhds_iff le_nhds_iff
+-/
 
+#print nhds_le_of_le /-
 /-- To show a filter is above the neighborhood filter at `a`, it suffices to show that it is above
 the principal filter of some open set `s` containing `a`. -/
 theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 a ≤ f := by
   rw [nhds_def] <;> exact iInf_le_of_le s (iInf_le_of_le ⟨h, o⟩ sf)
 #align nhds_le_of_le nhds_le_of_le
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ a ∈ t :=
@@ -1077,10 +1184,12 @@ theorem eventually_nhds_iff {a : α} {p : α → Prop} :
 #align eventually_nhds_iff eventually_nhds_iff
 -/
 
+#print map_nhds /-
 theorem map_nhds {a : α} {f : α → β} :
     map f (𝓝 a) = ⨅ s ∈ {s : Set α | a ∈ s ∧ IsOpen s}, 𝓟 (image f s) :=
   ((nhds_basis_opens a).map f).eq_biInf
 #align map_nhds map_nhds
+-/
 
 #print mem_of_mem_nhds /-
 theorem mem_of_mem_nhds {a : α} {s : Set α} : s ∈ 𝓝 a → a ∈ s := fun H =>
@@ -1108,9 +1217,11 @@ theorem IsOpen.mem_nhds_iff {a : α} {s : Set α} (hs : IsOpen s) : s ∈ 𝓝 a
 #align is_open.mem_nhds_iff IsOpen.mem_nhds_iff
 -/
 
+#print IsClosed.compl_mem_nhds /-
 theorem IsClosed.compl_mem_nhds {a : α} {s : Set α} (hs : IsClosed s) (ha : a ∉ s) : sᶜ ∈ 𝓝 a :=
   hs.isOpen_compl.mem_nhds (mem_compl ha)
 #align is_closed.compl_mem_nhds IsClosed.compl_mem_nhds
+-/
 
 #print IsOpen.eventually_mem /-
 theorem IsOpen.eventually_mem {a : α} {s : Set α} (hs : IsOpen s) (ha : a ∈ s) :
@@ -1144,12 +1255,14 @@ theorem exists_open_set_nhds {s U : Set α} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
 #align exists_open_set_nhds exists_open_set_nhds
 -/
 
+#print exists_open_set_nhds' /-
 /-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of s:
 it contains an open set containing `s`. -/
 theorem exists_open_set_nhds' {s U : Set α} (h : U ∈ ⨆ x ∈ s, 𝓝 x) :
     ∃ V : Set α, s ⊆ V ∧ IsOpen V ∧ V ⊆ U :=
   exists_open_set_nhds (by simpa using h)
 #align exists_open_set_nhds' exists_open_set_nhds'
+-/
 
 #print Filter.Eventually.eventually_nhds /-
 /-- If a predicate is true in a neighbourhood of `a`, then for `y` sufficiently close
@@ -1269,28 +1382,38 @@ theorem tendsto_const_nhds {a : α} {f : Filter β} : Tendsto (fun b : β => a)
 #align tendsto_const_nhds tendsto_const_nhds
 -/
 
+#print tendsto_atTop_of_eventually_const /-
 theorem tendsto_atTop_of_eventually_const {ι : Type _} [SemilatticeSup ι] [Nonempty ι] {x : α}
     {u : ι → α} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : Tendsto u atTop (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atTop.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_top_of_eventually_const tendsto_atTop_of_eventually_const
+-/
 
+#print tendsto_atBot_of_eventually_const /-
 theorem tendsto_atBot_of_eventually_const {ι : Type _} [SemilatticeInf ι] [Nonempty ι] {x : α}
     {u : ι → α} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : Tendsto u atBot (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atBot.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_bot_of_eventually_const tendsto_atBot_of_eventually_const
+-/
 
+#print pure_le_nhds /-
 theorem pure_le_nhds : pure ≤ (𝓝 : α → Filter α) := fun a s hs => mem_pure.2 <| mem_of_mem_nhds hs
 #align pure_le_nhds pure_le_nhds
+-/
 
+#print tendsto_pure_nhds /-
 theorem tendsto_pure_nhds {α : Type _} [TopologicalSpace β] (f : α → β) (a : α) :
     Tendsto f (pure a) (𝓝 (f a)) :=
   (tendsto_pure_pure f a).mono_right (pure_le_nhds _)
 #align tendsto_pure_nhds tendsto_pure_nhds
+-/
 
+#print OrderTop.tendsto_atTop_nhds /-
 theorem OrderTop.tendsto_atTop_nhds {α : Type _} [PartialOrder α] [OrderTop α] [TopologicalSpace β]
     (f : α → β) : Tendsto f atTop (𝓝 <| f ⊤) :=
   (tendsto_atTop_pure f).mono_right (pure_le_nhds _)
 #align order_top.tendsto_at_top_nhds OrderTop.tendsto_atTop_nhds
+-/
 
 #print nhds_neBot /-
 @[simp]
@@ -1316,27 +1439,35 @@ def ClusterPt (x : α) (F : Filter α) : Prop :=
 #align cluster_pt ClusterPt
 -/
 
+#print ClusterPt.neBot /-
 theorem ClusterPt.neBot {x : α} {F : Filter α} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) :=
   h
 #align cluster_pt.ne_bot ClusterPt.neBot
+-/
 
+#print Filter.HasBasis.clusterPt_iff /-
 theorem Filter.HasBasis.clusterPt_iff {ιa ιF} {pa : ιa → Prop} {sa : ιa → Set α} {pF : ιF → Prop}
     {sF : ιF → Set α} {F : Filter α} (ha : (𝓝 a).HasBasis pa sa) (hF : F.HasBasis pF sF) :
     ClusterPt a F ↔ ∀ ⦃i⦄ (hi : pa i) ⦃j⦄ (hj : pF j), (sa i ∩ sF j).Nonempty :=
   ha.inf_basis_neBot_iff hF
 #align filter.has_basis.cluster_pt_iff Filter.HasBasis.clusterPt_iff
+-/
 
+#print clusterPt_iff /-
 theorem clusterPt_iff {x : α} {F : Filter α} :
     ClusterPt x F ↔ ∀ ⦃U : Set α⦄ (hU : U ∈ 𝓝 x) ⦃V⦄ (hV : V ∈ F), (U ∩ V).Nonempty :=
   inf_neBot_iff
 #align cluster_pt_iff clusterPt_iff
+-/
 
+#print clusterPt_principal_iff /-
 /-- `x` is a cluster point of a set `s` if every neighbourhood of `x` meets `s` on a nonempty
 set. See also `mem_closure_iff_cluster_pt`. -/
 theorem clusterPt_principal_iff {x : α} {s : Set α} :
     ClusterPt x (𝓟 s) ↔ ∀ U ∈ 𝓝 x, (U ∩ s).Nonempty :=
   inf_principal_neBot_iff
 #align cluster_pt_principal_iff clusterPt_principal_iff
+-/
 
 #print clusterPt_principal_iff_frequently /-
 theorem clusterPt_principal_iff_frequently {x : α} {s : Set α} :
@@ -1345,34 +1476,48 @@ theorem clusterPt_principal_iff_frequently {x : α} {s : Set α} :
 #align cluster_pt_principal_iff_frequently clusterPt_principal_iff_frequently
 -/
 
+#print ClusterPt.of_le_nhds /-
 theorem ClusterPt.of_le_nhds {x : α} {f : Filter α} (H : f ≤ 𝓝 x) [NeBot f] : ClusterPt x f := by
   rwa [ClusterPt, inf_eq_right.mpr H]
 #align cluster_pt.of_le_nhds ClusterPt.of_le_nhds
+-/
 
+#print ClusterPt.of_le_nhds' /-
 theorem ClusterPt.of_le_nhds' {x : α} {f : Filter α} (H : f ≤ 𝓝 x) (hf : NeBot f) : ClusterPt x f :=
   ClusterPt.of_le_nhds H
 #align cluster_pt.of_le_nhds' ClusterPt.of_le_nhds'
+-/
 
+#print ClusterPt.of_nhds_le /-
 theorem ClusterPt.of_nhds_le {x : α} {f : Filter α} (H : 𝓝 x ≤ f) : ClusterPt x f := by
   simp only [ClusterPt, inf_eq_left.mpr H, nhds_neBot]
 #align cluster_pt.of_nhds_le ClusterPt.of_nhds_le
+-/
 
+#print ClusterPt.mono /-
 theorem ClusterPt.mono {x : α} {f g : Filter α} (H : ClusterPt x f) (h : f ≤ g) : ClusterPt x g :=
   ⟨ne_bot_of_le_ne_bot H.Ne <| inf_le_inf_left _ h⟩
 #align cluster_pt.mono ClusterPt.mono
+-/
 
+#print ClusterPt.of_inf_left /-
 theorem ClusterPt.of_inf_left {x : α} {f g : Filter α} (H : ClusterPt x <| f ⊓ g) : ClusterPt x f :=
   H.mono inf_le_left
 #align cluster_pt.of_inf_left ClusterPt.of_inf_left
+-/
 
+#print ClusterPt.of_inf_right /-
 theorem ClusterPt.of_inf_right {x : α} {f g : Filter α} (H : ClusterPt x <| f ⊓ g) :
     ClusterPt x g :=
   H.mono inf_le_right
 #align cluster_pt.of_inf_right ClusterPt.of_inf_right
+-/
 
+#print Ultrafilter.clusterPt_iff /-
 theorem Ultrafilter.clusterPt_iff {x : α} {f : Ultrafilter α} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
   ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩
 #align ultrafilter.cluster_pt_iff Ultrafilter.clusterPt_iff
+-/
 
 #print MapClusterPt /-
 /-- A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point
@@ -1382,11 +1527,14 @@ def MapClusterPt {ι : Type _} (x : α) (F : Filter ι) (u : ι → α) : Prop :
 #align map_cluster_pt MapClusterPt
 -/
 
+#print mapClusterPt_iff /-
 theorem mapClusterPt_iff {ι : Type _} (x : α) (F : Filter ι) (u : ι → α) :
     MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s := by
   simp_rw [MapClusterPt, ClusterPt, inf_ne_bot_iff_frequently_left, frequently_map]; rfl
 #align map_cluster_pt_iff mapClusterPt_iff
+-/
 
+#print mapClusterPt_of_comp /-
 theorem mapClusterPt_of_comp {ι δ : Type _} {F : Filter ι} {φ : δ → ι} {p : Filter δ} {x : α}
     {u : ι → α} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) : MapClusterPt x F u :=
   by
@@ -1397,6 +1545,7 @@ theorem mapClusterPt_of_comp {ι δ : Type _} {F : Filter ι} {φ : δ → ι} {
   have : map (u ∘ φ) p ≤ 𝓝 x ⊓ map u F := le_inf H this
   exact ne_bot_of_le this
 #align map_cluster_pt_of_comp mapClusterPt_of_comp
+-/
 
 #print AccPt /-
 /-- A point `x` is an accumulation point of a filter `F` if `𝓝[≠] x ⊓ F ≠ ⊥`.-/
@@ -1405,21 +1554,27 @@ def AccPt (x : α) (F : Filter α) : Prop :=
 #align acc_pt AccPt
 -/
 
+#print acc_iff_cluster /-
 theorem acc_iff_cluster (x : α) (F : Filter α) : AccPt x F ↔ ClusterPt x (𝓟 ({x}ᶜ) ⊓ F) := by
   rw [AccPt, nhdsWithin, ClusterPt, inf_assoc]
 #align acc_iff_cluster acc_iff_cluster
+-/
 
+#print acc_principal_iff_cluster /-
 /-- `x` is an accumulation point of a set `C` iff it is a cluster point of `C ∖ {x}`.-/
 theorem acc_principal_iff_cluster (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x})) :=
   by rw [acc_iff_cluster, inf_principal, inter_comm] <;> rfl
 #align acc_principal_iff_cluster acc_principal_iff_cluster
+-/
 
+#print accPt_iff_nhds /-
 /-- `x` is an accumulation point of a set `C` iff every neighborhood
 of `x` contains a point of `C` other than `x`. -/
 theorem accPt_iff_nhds (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x := by
   simp [acc_principal_iff_cluster, clusterPt_principal_iff, Set.Nonempty, exists_prop, and_assoc',
     and_comm' ¬_ = x]
 #align acc_pt_iff_nhds accPt_iff_nhds
+-/
 
 #print accPt_iff_frequently /-
 /-- `x` is an accumulation point of a set `C` iff
@@ -1429,11 +1584,13 @@ theorem accPt_iff_frequently (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∃ᶠ
 #align acc_pt_iff_frequently accPt_iff_frequently
 -/
 
+#print AccPt.mono /-
 /-- If `x` is an accumulation point of `F` and `F ≤ G`, then
 `x` is an accumulation point of `D. -/
 theorem AccPt.mono {x : α} {F G : Filter α} (h : AccPt x F) (hFG : F ≤ G) : AccPt x G :=
   ⟨ne_bot_of_le_ne_bot h.Ne (inf_le_inf_left _ hFG)⟩
 #align acc_pt.mono AccPt.mono
+-/
 
 /-!
 ### Interior, closure and frontier in terms of neighborhoods
@@ -1446,9 +1603,11 @@ theorem interior_eq_nhds' {s : Set α} : interior s = {a | s ∈ 𝓝 a} :=
 #align interior_eq_nhds' interior_eq_nhds'
 -/
 
+#print interior_eq_nhds /-
 theorem interior_eq_nhds {s : Set α} : interior s = {a | 𝓝 a ≤ 𝓟 s} :=
   interior_eq_nhds'.trans <| by simp only [le_principal_iff]
 #align interior_eq_nhds interior_eq_nhds
+-/
 
 #print mem_interior_iff_mem_nhds /-
 theorem mem_interior_iff_mem_nhds {s : Set α} {a : α} : a ∈ interior s ↔ s ∈ 𝓝 a := by
@@ -1482,11 +1641,13 @@ theorem subset_interior_iff_nhds {s V : Set α} : s ⊆ interior V ↔ ∀ x ∈
 #align subset_interior_iff_nhds subset_interior_iff_nhds
 -/
 
+#print isOpen_iff_nhds /-
 theorem isOpen_iff_nhds {s : Set α} : IsOpen s ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s :=
   calc
     IsOpen s ↔ s ⊆ interior s := subset_interior_iff_isOpen.symm
     _ ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s := by rw [interior_eq_nhds] <;> rfl
 #align is_open_iff_nhds isOpen_iff_nhds
+-/
 
 #print isOpen_iff_mem_nhds /-
 theorem isOpen_iff_mem_nhds {s : Set α} : IsOpen s ↔ ∀ a ∈ s, s ∈ 𝓝 a :=
@@ -1501,10 +1662,12 @@ theorem isOpen_iff_eventually {s : Set α} : IsOpen s ↔ ∀ x, x ∈ s → ∀
 #align is_open_iff_eventually isOpen_iff_eventually
 -/
 
+#print isOpen_iff_ultrafilter /-
 theorem isOpen_iff_ultrafilter {s : Set α} :
     IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter α), ↑l ≤ 𝓝 x → s ∈ l := by
   simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]
 #align is_open_iff_ultrafilter isOpen_iff_ultrafilter
+-/
 
 #print isOpen_singleton_iff_nhds_eq_pure /-
 theorem isOpen_singleton_iff_nhds_eq_pure (a : α) : IsOpen ({a} : Set α) ↔ 𝓝 a = pure a :=
@@ -1519,11 +1682,13 @@ theorem isOpen_singleton_iff_nhds_eq_pure (a : α) : IsOpen ({a} : Set α) ↔ 
 #align is_open_singleton_iff_nhds_eq_pure isOpen_singleton_iff_nhds_eq_pure
 -/
 
+#print isOpen_singleton_iff_punctured_nhds /-
 theorem isOpen_singleton_iff_punctured_nhds {α : Type _} [TopologicalSpace α] (a : α) :
     IsOpen ({a} : Set α) ↔ 𝓝[≠] a = ⊥ := by
   rw [isOpen_singleton_iff_nhds_eq_pure, nhdsWithin, ← mem_iff_inf_principal_compl, ← le_pure_iff,
     nhds_ne_bot.le_pure_iff]
 #align is_open_singleton_iff_punctured_nhds isOpen_singleton_iff_punctured_nhds
+-/
 
 #print mem_closure_iff_frequently /-
 theorem mem_closure_iff_frequently {s : Set α} {a : α} : a ∈ closure s ↔ ∃ᶠ x in 𝓝 a, x ∈ s := by
@@ -1564,9 +1729,11 @@ theorem mem_closure_iff_clusterPt {s : Set α} {a : α} : a ∈ closure s ↔ Cl
 #align mem_closure_iff_cluster_pt mem_closure_iff_clusterPt
 -/
 
+#print mem_closure_iff_nhds_neBot /-
 theorem mem_closure_iff_nhds_neBot {s : Set α} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ :=
   mem_closure_iff_clusterPt.trans neBot_iff
 #align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_neBot
+-/
 
 #print mem_closure_iff_nhdsWithin_neBot /-
 theorem mem_closure_iff_nhdsWithin_neBot {s : Set α} {x : α} : x ∈ closure s ↔ NeBot (𝓝[s] x) :=
@@ -1574,6 +1741,7 @@ theorem mem_closure_iff_nhdsWithin_neBot {s : Set α} {x : α} : x ∈ closure s
 #align mem_closure_iff_nhds_within_ne_bot mem_closure_iff_nhdsWithin_neBot
 -/
 
+#print dense_compl_singleton /-
 /-- If `x` is not an isolated point of a topological space, then `{x}ᶜ` is dense in the whole
 space. -/
 theorem dense_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : Set α) :=
@@ -1583,23 +1751,30 @@ theorem dense_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : S
   · rwa [mem_closure_iff_nhdsWithin_neBot]
   · exact subset_closure hne
 #align dense_compl_singleton dense_compl_singleton
+-/
 
+#print closure_compl_singleton /-
 /-- If `x` is not an isolated point of a topological space, then the closure of `{x}ᶜ` is the whole
 space. -/
 @[simp]
 theorem closure_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : closure ({x}ᶜ) = (univ : Set α) :=
   (dense_compl_singleton x).closure_eq
 #align closure_compl_singleton closure_compl_singleton
+-/
 
+#print interior_singleton /-
 /-- If `x` is not an isolated point of a topological space, then the interior of `{x}` is empty. -/
 @[simp]
 theorem interior_singleton (x : α) [NeBot (𝓝[≠] x)] : interior {x} = (∅ : Set α) :=
   interior_eq_empty_iff_dense_compl.2 (dense_compl_singleton x)
 #align interior_singleton interior_singleton
+-/
 
+#print not_isOpen_singleton /-
 theorem not_isOpen_singleton (x : α) [NeBot (𝓝[≠] x)] : ¬IsOpen ({x} : Set α) :=
   dense_compl_singleton_iff_not_open.1 (dense_compl_singleton x)
 #align not_is_open_singleton not_isOpen_singleton
+-/
 
 #print closure_eq_cluster_pts /-
 theorem closure_eq_cluster_pts {s : Set α} : closure s = {a | ClusterPt a (𝓟 s)} :=
@@ -1607,9 +1782,11 @@ theorem closure_eq_cluster_pts {s : Set α} : closure s = {a | ClusterPt a (𝓟
 #align closure_eq_cluster_pts closure_eq_cluster_pts
 -/
 
+#print mem_closure_iff_nhds /-
 theorem mem_closure_iff_nhds {s : Set α} {a : α} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, (t ∩ s).Nonempty :=
   mem_closure_iff_clusterPt.trans clusterPt_principal_iff
 #align mem_closure_iff_nhds mem_closure_iff_nhds
+-/
 
 #print mem_closure_iff_nhds' /-
 theorem mem_closure_iff_nhds' {s : Set α} {a : α} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, ∃ y : s, ↑y ∈ t := by
@@ -1626,24 +1803,30 @@ theorem mem_closure_iff_comap_neBot {A : Set α} {x : α} :
 #align mem_closure_iff_comap_ne_bot mem_closure_iff_comap_neBot
 -/
 
+#print mem_closure_iff_nhds_basis' /-
 theorem mem_closure_iff_nhds_basis' {a : α} {p : ι → Prop} {s : ι → Set α} (h : (𝓝 a).HasBasis p s)
     {t : Set α} : a ∈ closure t ↔ ∀ i, p i → (s i ∩ t).Nonempty :=
   mem_closure_iff_clusterPt.trans <|
     (h.clusterPt_iff (hasBasis_principal _)).trans <| by simp only [exists_prop, forall_const]
 #align mem_closure_iff_nhds_basis' mem_closure_iff_nhds_basis'
+-/
 
+#print mem_closure_iff_nhds_basis /-
 theorem mem_closure_iff_nhds_basis {a : α} {p : ι → Prop} {s : ι → Set α} (h : (𝓝 a).HasBasis p s)
     {t : Set α} : a ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i :=
   (mem_closure_iff_nhds_basis' h).trans <| by
     simp only [Set.Nonempty, mem_inter_iff, exists_prop, and_comm']
 #align mem_closure_iff_nhds_basis mem_closure_iff_nhds_basis
+-/
 
+#print mem_closure_iff_ultrafilter /-
 /-- `x` belongs to the closure of `s` if and only if some ultrafilter
   supported on `s` converges to `x`. -/
 theorem mem_closure_iff_ultrafilter {s : Set α} {x : α} :
     x ∈ closure s ↔ ∃ u : Ultrafilter α, s ∈ u ∧ ↑u ≤ 𝓝 x := by
   simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]
 #align mem_closure_iff_ultrafilter mem_closure_iff_ultrafilter
+-/
 
 #print isClosed_iff_clusterPt /-
 theorem isClosed_iff_clusterPt {s : Set α} : IsClosed s ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s :=
@@ -1653,10 +1836,13 @@ theorem isClosed_iff_clusterPt {s : Set α} : IsClosed s ↔ ∀ a, ClusterPt a
 #align is_closed_iff_cluster_pt isClosed_iff_clusterPt
 -/
 
+#print isClosed_iff_nhds /-
 theorem isClosed_iff_nhds {s : Set α} : IsClosed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s :=
   by simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_ne_bot_iff]
 #align is_closed_iff_nhds isClosed_iff_nhds
+-/
 
+#print IsClosed.interior_union_left /-
 theorem IsClosed.interior_union_left {s t : Set α} (h : IsClosed s) :
     interior (s ∪ t) ⊆ s ∪ interior t := fun a ⟨u, ⟨⟨hu₁, hu₂⟩, ha⟩⟩ =>
   (Classical.em (a ∈ s)).imp_right fun h =>
@@ -1664,27 +1850,37 @@ theorem IsClosed.interior_union_left {s t : Set α} (h : IsClosed s) :
       ⟨u ∩ sᶜ, fun x hx => (hu₂ hx.1).resolve_left hx.2, IsOpen.inter hu₁ IsClosed.isOpen_compl,
         ⟨ha, h⟩⟩
 #align is_closed.interior_union_left IsClosed.interior_union_left
+-/
 
+#print IsClosed.interior_union_right /-
 theorem IsClosed.interior_union_right {s t : Set α} (h : IsClosed t) :
     interior (s ∪ t) ⊆ interior s ∪ t := by simpa only [union_comm] using h.interior_union_left
 #align is_closed.interior_union_right IsClosed.interior_union_right
+-/
 
+#print IsOpen.inter_closure /-
 theorem IsOpen.inter_closure {s t : Set α} (h : IsOpen s) : s ∩ closure t ⊆ closure (s ∩ t) :=
   compl_subset_compl.mp <| by
     simpa only [← interior_compl, compl_inter] using IsClosed.interior_union_left h.is_closed_compl
 #align is_open.inter_closure IsOpen.inter_closure
+-/
 
+#print IsOpen.closure_inter /-
 theorem IsOpen.closure_inter {s t : Set α} (h : IsOpen t) : closure s ∩ t ⊆ closure (s ∩ t) := by
   simpa only [inter_comm] using h.inter_closure
 #align is_open.closure_inter IsOpen.closure_inter
+-/
 
+#print Dense.open_subset_closure_inter /-
 theorem Dense.open_subset_closure_inter {s t : Set α} (hs : Dense s) (ht : IsOpen t) :
     t ⊆ closure (t ∩ s) :=
   calc
     t = t ∩ closure s := by rw [hs.closure_eq, inter_univ]
     _ ⊆ closure (t ∩ s) := ht.inter_closure
 #align dense.open_subset_closure_inter Dense.open_subset_closure_inter
+-/
 
+#print mem_closure_of_mem_closure_union /-
 theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x ∈ closure (s₁ ∪ s₂))
     (h₁ : s₁ᶜ ∈ 𝓝 x) : x ∈ closure s₂ :=
   by
@@ -1696,25 +1892,33 @@ theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x 
       _ = ⊥ ⊔ 𝓝 x ⊓ principal s₂ := by rw [inf_principal_eq_bot.mpr h₁]
       _ = 𝓝 x ⊓ principal s₂ := bot_sup_eq]
 #align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_union
+-/
 
+#print Dense.inter_of_open_left /-
 /-- The intersection of an open dense set with a dense set is a dense set. -/
 theorem Dense.inter_of_open_left {s t : Set α} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
     Dense (s ∩ t) := fun x =>
   closure_minimal hso.inter_closure isClosed_closure <| by simp [hs.closure_eq, ht.closure_eq]
 #align dense.inter_of_open_left Dense.inter_of_open_left
+-/
 
+#print Dense.inter_of_open_right /-
 /-- The intersection of a dense set with an open dense set is a dense set. -/
 theorem Dense.inter_of_open_right {s t : Set α} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
     Dense (s ∩ t) :=
   inter_comm t s ▸ ht.inter_of_open_left hs hto
 #align dense.inter_of_open_right Dense.inter_of_open_right
+-/
 
+#print Dense.inter_nhds_nonempty /-
 theorem Dense.inter_nhds_nonempty {s t : Set α} (hs : Dense s) {x : α} (ht : t ∈ 𝓝 x) :
     (s ∩ t).Nonempty :=
   let ⟨U, hsub, ho, hx⟩ := mem_nhds_iff.1 ht
   (hs.inter_open_nonempty U ho ⟨x, hx⟩).mono fun y hy => ⟨hy.2, hsub hy.1⟩
 #align dense.inter_nhds_nonempty Dense.inter_nhds_nonempty
+-/
 
+#print closure_diff /-
 theorem closure_diff {s t : Set α} : closure s \ closure t ⊆ closure (s \ t) :=
   calc
     closure s \ closure t = closure tᶜ ∩ closure s := by simp only [diff_eq, inter_comm]
@@ -1722,6 +1926,7 @@ theorem closure_diff {s t : Set α} : closure s \ closure t ⊆ closure (s \ t)
     _ = closure (s \ closure t) := by simp only [diff_eq, inter_comm]
     _ ⊆ closure (s \ t) := closure_mono <| diff_subset_diff (Subset.refl s) subset_closure
 #align closure_diff closure_diff
+-/
 
 #print Filter.Frequently.mem_of_closed /-
 theorem Filter.Frequently.mem_of_closed {a : α} {s : Set α} (h : ∃ᶠ x in 𝓝 a, x ∈ s)
@@ -1759,6 +1964,7 @@ theorem mem_closure_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
+#print tendsto_inf_principal_nhds_iff_of_forall_eq /-
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
 Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/
 theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : β → α} {l : Filter β} {s : Set β} {a : α}
@@ -1775,6 +1981,7 @@ theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : β → α} {l : Filter
     this 
   exact this.1
 #align tendsto_inf_principal_nhds_iff_of_forall_eq tendsto_inf_principal_nhds_iff_of_forall_eq
+-/
 
 /-!
 ### Limits of filters in topological spaces
@@ -1813,6 +2020,7 @@ noncomputable def limUnder [Nonempty α] (f : Filter β) (g : β → α) : α :=
 #align lim limUnder
 -/
 
+#print le_nhds_lim /-
 /-- If a filter `f` is majorated by some `𝓝 a`, then it is majorated by `𝓝 (Lim f)`. We formulate
 this lemma with a `[nonempty α]` argument of `Lim` derived from `h` to make it useful for types
 without a `[nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify
@@ -1820,6 +2028,7 @@ this instance with any other instance. -/
 theorem le_nhds_lim {f : Filter α} (h : ∃ a, f ≤ 𝓝 a) : f ≤ 𝓝 (@lim _ _ (nonempty_of_exists h) f) :=
   epsilon_spec h
 #align le_nhds_Lim le_nhds_lim
+-/
 
 #print tendsto_nhds_limUnder /-
 /-- If `g` tends to some `𝓝 a` along `f`, then it tends to `𝓝 (lim f g)`. We formulate
@@ -1857,18 +2066,24 @@ structure Continuous (f : α → β) : Prop where
 #align continuous Continuous
 -/
 
+#print continuous_def /-
 theorem continuous_def {f : α → β} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) :=
   ⟨fun hf s hs => hf.isOpen_preimage s hs, fun h => ⟨h⟩⟩
 #align continuous_def continuous_def
+-/
 
+#print IsOpen.preimage /-
 theorem IsOpen.preimage {f : α → β} (hf : Continuous f) {s : Set β} (h : IsOpen s) :
     IsOpen (f ⁻¹' s) :=
   hf.isOpen_preimage s h
 #align is_open.preimage IsOpen.preimage
+-/
 
+#print Continuous.congr /-
 theorem Continuous.congr {f g : α → β} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g := by
   convert h; ext; rw [h']
 #align continuous.congr Continuous.congr
+-/
 
 #print ContinuousAt /-
 /-- A function between topological spaces is continuous at a point `x₀`
@@ -1878,24 +2093,32 @@ def ContinuousAt (f : α → β) (x : α) :=
 #align continuous_at ContinuousAt
 -/
 
+#print ContinuousAt.tendsto /-
 theorem ContinuousAt.tendsto {f : α → β} {x : α} (h : ContinuousAt f x) :
     Tendsto f (𝓝 x) (𝓝 (f x)) :=
   h
 #align continuous_at.tendsto ContinuousAt.tendsto
+-/
 
+#print continuousAt_def /-
 theorem continuousAt_def {f : α → β} {x : α} : ContinuousAt f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x :=
   Iff.rfl
 #align continuous_at_def continuousAt_def
+-/
 
+#print continuousAt_congr /-
 theorem continuousAt_congr {f g : α → β} {x : α} (h : f =ᶠ[𝓝 x] g) :
     ContinuousAt f x ↔ ContinuousAt g x := by
   simp only [ContinuousAt, tendsto_congr' h, h.eq_of_nhds]
 #align continuous_at_congr continuousAt_congr
+-/
 
+#print ContinuousAt.congr /-
 theorem ContinuousAt.congr {f g : α → β} {x : α} (hf : ContinuousAt f x) (h : f =ᶠ[𝓝 x] g) :
     ContinuousAt g x :=
   (continuousAt_congr h).1 hf
 #align continuous_at.congr ContinuousAt.congr
+-/
 
 #print ContinuousAt.preimage_mem_nhds /-
 theorem ContinuousAt.preimage_mem_nhds {f : α → β} {x : α} {t : Set β} (h : ContinuousAt f x)
@@ -1911,10 +2134,12 @@ theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {a : α} {f : α → M₀} :
 #align eventually_eq_zero_nhds eventuallyEq_zero_nhds
 -/
 
+#print ClusterPt.map /-
 theorem ClusterPt.map {x : α} {la : Filter α} {lb : Filter β} (H : ClusterPt x la) {f : α → β}
     (hfc : ContinuousAt f x) (hf : Tendsto f la lb) : ClusterPt (f x) lb :=
   ⟨ne_bot_of_le_ne_bot ((map_neBot_iff f).2 H).Ne <| hfc.Tendsto.inf hf⟩
 #align cluster_pt.map ClusterPt.map
+-/
 
 #print preimage_interior_subset_interior_preimage /-
 /-- See also `interior_preimage_subset_preimage_interior`. -/
@@ -1930,10 +2155,12 @@ theorem continuous_id : Continuous (id : α → α) :=
 #align continuous_id continuous_id
 -/
 
+#print Continuous.comp /-
 theorem Continuous.comp {g : β → γ} {f : α → β} (hg : Continuous g) (hf : Continuous f) :
     Continuous (g ∘ f) :=
   continuous_def.2 fun s h => (h.Preimage hg).Preimage hf
 #align continuous.comp Continuous.comp
+-/
 
 #print Continuous.iterate /-
 theorem Continuous.iterate {f : α → α} (h : Continuous f) (n : ℕ) : Continuous (f^[n]) :=
@@ -1941,32 +2168,43 @@ theorem Continuous.iterate {f : α → α} (h : Continuous f) (n : ℕ) : Contin
 #align continuous.iterate Continuous.iterate
 -/
 
+#print ContinuousAt.comp /-
 theorem ContinuousAt.comp {g : β → γ} {f : α → β} {x : α} (hg : ContinuousAt g (f x))
     (hf : ContinuousAt f x) : ContinuousAt (g ∘ f) x :=
   hg.comp hf
 #align continuous_at.comp ContinuousAt.comp
+-/
 
+#print ContinuousAt.comp_of_eq /-
 /-- See note [comp_of_eq lemmas] -/
 theorem ContinuousAt.comp_of_eq {g : β → γ} {f : α → β} {x : α} {y : β} (hg : ContinuousAt g y)
     (hf : ContinuousAt f x) (hy : f x = y) : ContinuousAt (g ∘ f) x := by subst hy; exact hg.comp hf
 #align continuous_at.comp_of_eq ContinuousAt.comp_of_eq
+-/
 
+#print Continuous.tendsto /-
 theorem Continuous.tendsto {f : α → β} (hf : Continuous f) (x) : Tendsto f (𝓝 x) (𝓝 (f x)) :=
   ((nhds_basis_opens x).tendsto_iffₓ <| nhds_basis_opens <| f x).2 fun t ⟨hxt, ht⟩ =>
     ⟨f ⁻¹' t, ⟨hxt, ht.Preimage hf⟩, Subset.refl _⟩
 #align continuous.tendsto Continuous.tendsto
+-/
 
+#print Continuous.tendsto' /-
 /-- A version of `continuous.tendsto` that allows one to specify a simpler form of the limit.
 E.g., one can write `continuous_exp.tendsto' 0 1 exp_zero`. -/
 theorem Continuous.tendsto' {f : α → β} (hf : Continuous f) (x : α) (y : β) (h : f x = y) :
     Tendsto f (𝓝 x) (𝓝 y) :=
   h ▸ hf.Tendsto x
 #align continuous.tendsto' Continuous.tendsto'
+-/
 
+#print Continuous.continuousAt /-
 theorem Continuous.continuousAt {f : α → β} {x : α} (h : Continuous f) : ContinuousAt f x :=
   h.Tendsto x
 #align continuous.continuous_at Continuous.continuousAt
+-/
 
+#print continuous_iff_continuousAt /-
 theorem continuous_iff_continuousAt {f : α → β} : Continuous f ↔ ∀ x, ContinuousAt f x :=
   ⟨Continuous.tendsto, fun hf : ∀ x, Tendsto f (𝓝 x) (𝓝 (f x)) =>
     continuous_def.2 fun s => fun hs : IsOpen s =>
@@ -1974,19 +2212,26 @@ theorem continuous_iff_continuousAt {f : α → β} : Continuous f ↔ ∀ x, Co
       show IsOpen (f ⁻¹' s) from
         isOpen_iff_nhds.2 fun a ha => le_principal_iff.2 <| hf _ (this a ha)⟩
 #align continuous_iff_continuous_at continuous_iff_continuousAt
+-/
 
+#print continuousAt_const /-
 theorem continuousAt_const {x : α} {b : β} : ContinuousAt (fun a : α => b) x :=
   tendsto_const_nhds
 #align continuous_at_const continuousAt_const
+-/
 
+#print continuous_const /-
 theorem continuous_const {b : β} : Continuous fun a : α => b :=
   continuous_iff_continuousAt.mpr fun a => continuousAt_const
 #align continuous_const continuous_const
+-/
 
+#print Filter.EventuallyEq.continuousAt /-
 theorem Filter.EventuallyEq.continuousAt {x : α} {f : α → β} {y : β} (h : f =ᶠ[𝓝 x] fun _ => y) :
     ContinuousAt f x :=
   (continuousAt_congr h).2 tendsto_const_nhds
 #align filter.eventually_eq.continuous_at Filter.EventuallyEq.continuousAt
+-/
 
 #print continuous_of_const /-
 theorem continuous_of_const {f : α → β} (h : ∀ x y, f x = f y) : Continuous f :=
@@ -2009,45 +2254,60 @@ theorem ContinuousAt.iterate {f : α → α} {x : α} (hf : ContinuousAt f x) (h
 #align continuous_at.iterate ContinuousAt.iterate
 -/
 
+#print continuous_iff_isClosed /-
 theorem continuous_iff_isClosed {f : α → β} : Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s) :=
   ⟨fun hf s hs => by simpa using (continuous_def.1 hf (sᶜ) hs.is_open_compl).isClosed_compl,
     fun hf =>
     continuous_def.2 fun s => by rw [← isClosed_compl_iff, ← isClosed_compl_iff] <;> exact hf _⟩
 #align continuous_iff_is_closed continuous_iff_isClosed
+-/
 
+#print IsClosed.preimage /-
 theorem IsClosed.preimage {f : α → β} (hf : Continuous f) {s : Set β} (h : IsClosed s) :
     IsClosed (f ⁻¹' s) :=
   continuous_iff_isClosed.mp hf s h
 #align is_closed.preimage IsClosed.preimage
+-/
 
+#print mem_closure_image /-
 theorem mem_closure_image {f : α → β} {x : α} {s : Set α} (hf : ContinuousAt f x)
     (hx : x ∈ closure s) : f x ∈ closure (f '' s) :=
   mem_closure_of_frequently_of_tendsto
     ((mem_closure_iff_frequently.1 hx).mono fun x => mem_image_of_mem _) hf
 #align mem_closure_image mem_closure_image
+-/
 
+#print continuousAt_iff_ultrafilter /-
 theorem continuousAt_iff_ultrafilter {f : α → β} {x} :
     ContinuousAt f x ↔ ∀ g : Ultrafilter α, ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) :=
   tendsto_iff_ultrafilter f (𝓝 x) (𝓝 (f x))
 #align continuous_at_iff_ultrafilter continuousAt_iff_ultrafilter
+-/
 
+#print continuous_iff_ultrafilter /-
 theorem continuous_iff_ultrafilter {f : α → β} :
     Continuous f ↔ ∀ (x) (g : Ultrafilter α), ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) := by
   simp only [continuous_iff_continuousAt, continuousAt_iff_ultrafilter]
 #align continuous_iff_ultrafilter continuous_iff_ultrafilter
+-/
 
+#print Continuous.closure_preimage_subset /-
 theorem Continuous.closure_preimage_subset {f : α → β} (hf : Continuous f) (t : Set β) :
     closure (f ⁻¹' t) ⊆ f ⁻¹' closure t :=
   by
   rw [← (is_closed_closure.preimage hf).closure_eq]
   exact closure_mono (preimage_mono subset_closure)
 #align continuous.closure_preimage_subset Continuous.closure_preimage_subset
+-/
 
+#print Continuous.frontier_preimage_subset /-
 theorem Continuous.frontier_preimage_subset {f : α → β} (hf : Continuous f) (t : Set β) :
     frontier (f ⁻¹' t) ⊆ f ⁻¹' frontier t :=
   diff_subset_diff (hf.closure_preimage_subset t) (preimage_interior_subset_interior_preimage hf)
 #align continuous.frontier_preimage_subset Continuous.frontier_preimage_subset
+-/
 
+#print Set.MapsTo.closure /-
 /-- If a continuous map `f` maps `s` to `t`, then it maps `closure s` to `closure t`. -/
 theorem Set.MapsTo.closure {s : Set α} {t : Set β} {f : α → β} (h : MapsTo f s t)
     (hc : Continuous f) : MapsTo f (closure s) (closure t) :=
@@ -2055,27 +2315,36 @@ theorem Set.MapsTo.closure {s : Set α} {t : Set β} {f : α → β} (h : MapsTo
   simp only [maps_to, mem_closure_iff_clusterPt]
   exact fun x hx => hx.map hc.continuous_at (tendsto_principal_principal.2 h)
 #align set.maps_to.closure Set.MapsTo.closure
+-/
 
+#print image_closure_subset_closure_image /-
 theorem image_closure_subset_closure_image {f : α → β} {s : Set α} (h : Continuous f) :
     f '' closure s ⊆ closure (f '' s) :=
   ((mapsTo_image f s).closure h).image_subset
 #align image_closure_subset_closure_image image_closure_subset_closure_image
+-/
 
+#print closure_subset_preimage_closure_image /-
 theorem closure_subset_preimage_closure_image {f : α → β} {s : Set α} (h : Continuous f) :
     closure s ⊆ f ⁻¹' closure (f '' s) := by rw [← Set.image_subset_iff];
   exact image_closure_subset_closure_image h
 #align closure_subset_preimage_closure_image closure_subset_preimage_closure_image
+-/
 
+#print map_mem_closure /-
 theorem map_mem_closure {s : Set α} {t : Set β} {f : α → β} {a : α} (hf : Continuous f)
     (ha : a ∈ closure s) (ht : MapsTo f s t) : f a ∈ closure t :=
   ht.closure hf ha
 #align map_mem_closure map_mem_closure
+-/
 
+#print Set.MapsTo.closure_left /-
 /-- If a continuous map `f` maps `s` to a closed set `t`, then it maps `closure s` to `t`. -/
 theorem Set.MapsTo.closure_left {s : Set α} {t : Set β} {f : α → β} (h : MapsTo f s t)
     (hc : Continuous f) (ht : IsClosed t) : MapsTo f (closure s) t :=
   ht.closure_eq ▸ h.closure hc
 #align set.maps_to.closure_left Set.MapsTo.closure_left
+-/
 
 /-!
 ### Function with dense range
@@ -2108,13 +2377,17 @@ theorem denseRange_id : DenseRange (id : α → α) :=
 #align dense_range_id denseRange_id
 -/
 
+#print denseRange_iff_closure_range /-
 theorem denseRange_iff_closure_range : DenseRange f ↔ closure (range f) = univ :=
   dense_iff_closure_eq
 #align dense_range_iff_closure_range denseRange_iff_closure_range
+-/
 
+#print DenseRange.closure_range /-
 theorem DenseRange.closure_range (h : DenseRange f) : closure (range f) = univ :=
   h.closure_eq
 #align dense_range.closure_range DenseRange.closure_range
+-/
 
 #print Dense.denseRange_val /-
 theorem Dense.denseRange_val {s : Set α} (h : Dense s) : DenseRange (coe : s → α) := by
@@ -2122,10 +2395,12 @@ theorem Dense.denseRange_val {s : Set α} (h : Dense s) : DenseRange (coe : s 
 #align dense.dense_range_coe Dense.denseRange_val
 -/
 
+#print Continuous.range_subset_closure_image_dense /-
 theorem Continuous.range_subset_closure_image_dense {f : α → β} (hf : Continuous f) {s : Set α}
     (hs : Dense s) : range f ⊆ closure (f '' s) := by rw [← image_univ, ← hs.closure_eq];
   exact image_closure_subset_closure_image hf
 #align continuous.range_subset_closure_image_dense Continuous.range_subset_closure_image_dense
+-/
 
 #print DenseRange.dense_image /-
 /-- The image of a dense set under a continuous map with dense range is a dense set. -/
@@ -2135,12 +2410,14 @@ theorem DenseRange.dense_image {f : α → β} (hf' : DenseRange f) (hf : Contin
 #align dense_range.dense_image DenseRange.dense_image
 -/
 
+#print DenseRange.subset_closure_image_preimage_of_isOpen /-
 /-- If `f` has dense range and `s` is an open set in the codomain of `f`, then the image of the
 preimage of `s` under `f` is dense in `s`. -/
 theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) {s : Set β}
     (hs : IsOpen s) : s ⊆ closure (f '' (f ⁻¹' s)) := by rw [image_preimage_eq_inter_range];
   exact hf.open_subset_closure_inter hs
 #align dense_range.subset_closure_image_preimage_of_is_open DenseRange.subset_closure_image_preimage_of_isOpen
+-/
 
 #print DenseRange.dense_of_mapsTo /-
 /-- If a continuous map with dense range maps a dense set to a subset of `t`, then `t` is a dense
@@ -2151,20 +2428,26 @@ theorem DenseRange.dense_of_mapsTo {f : α → β} (hf' : DenseRange f) (hf : Co
 #align dense_range.dense_of_maps_to DenseRange.dense_of_mapsTo
 -/
 
+#print DenseRange.comp /-
 /-- Composition of a continuous map with dense range and a function with dense range has dense
 range. -/
 theorem DenseRange.comp {g : β → γ} {f : κ → β} (hg : DenseRange g) (hf : DenseRange f)
     (cg : Continuous g) : DenseRange (g ∘ f) := by rw [DenseRange, range_comp];
   exact hg.dense_image cg hf
 #align dense_range.comp DenseRange.comp
+-/
 
+#print DenseRange.nonempty_iff /-
 theorem DenseRange.nonempty_iff (hf : DenseRange f) : Nonempty κ ↔ Nonempty β :=
   range_nonempty_iff_nonempty.symm.trans hf.nonempty_iff
 #align dense_range.nonempty_iff DenseRange.nonempty_iff
+-/
 
+#print DenseRange.nonempty /-
 theorem DenseRange.nonempty [h : Nonempty β] (hf : DenseRange f) : Nonempty κ :=
   hf.nonempty_iff.mpr h
 #align dense_range.nonempty DenseRange.nonempty
+-/
 
 #print DenseRange.some /-
 /-- Given a function `f : α → β` with dense range and `b : β`, returns some `a : α`. -/
@@ -2173,16 +2456,20 @@ def DenseRange.some (hf : DenseRange f) (b : β) : κ :=
 #align dense_range.some DenseRange.some
 -/
 
+#print DenseRange.exists_mem_open /-
 theorem DenseRange.exists_mem_open (hf : DenseRange f) {s : Set β} (ho : IsOpen s)
     (hs : s.Nonempty) : ∃ a, f a ∈ s :=
   exists_range_iff.1 <| hf.exists_mem_open ho hs
 #align dense_range.exists_mem_open DenseRange.exists_mem_open
+-/
 
+#print DenseRange.mem_nhds /-
 theorem DenseRange.mem_nhds {f : κ → β} (h : DenseRange f) {b : β} {U : Set β} (U_in : U ∈ 𝓝 b) :
     ∃ a, f a ∈ U :=
   let ⟨a, ha⟩ := h.exists_mem_open isOpen_interior ⟨b, mem_interior_iff_mem_nhds.2 U_in⟩
   ⟨a, interior_subset ha⟩
 #align dense_range.mem_nhds DenseRange.mem_nhds
+-/
 
 end DenseRange
 

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

@@ -495,7 +495,6 @@ theorem interior_sInter_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂
   calc
     interior (⋂₀ S) = interior (⋂ s ∈ S, s) := by rw [sInter_eq_bInter]
     _ ⊆ ⋂ s ∈ S, interior s := interior_Inter₂_subset _ _
-    
 #align interior_sInter_subset interior_sInter_subset
 -/
 
@@ -1395,7 +1394,6 @@ theorem mapClusterPt_of_comp {ι δ : Type _} {F : Filter ι} {φ : δ → ι} {
     calc
       map (u ∘ φ) p = map u (map φ p) := map_map
       _ ≤ map u F := map_mono h
-      
   have : map (u ∘ φ) p ≤ 𝓝 x ⊓ map u F := le_inf H this
   exact ne_bot_of_le this
 #align map_cluster_pt_of_comp mapClusterPt_of_comp
@@ -1488,7 +1486,6 @@ theorem isOpen_iff_nhds {s : Set α} : IsOpen s ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟
   calc
     IsOpen s ↔ s ⊆ interior s := subset_interior_iff_isOpen.symm
     _ ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s := by rw [interior_eq_nhds] <;> rfl
-    
 #align is_open_iff_nhds isOpen_iff_nhds
 
 #print isOpen_iff_mem_nhds /-
@@ -1653,7 +1650,6 @@ theorem isClosed_iff_clusterPt {s : Set α} : IsClosed s ↔ ∀ a, ClusterPt a
   calc
     IsClosed s ↔ closure s ⊆ s := closure_subset_iff_isClosed.symm
     _ ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s := by simp only [subset_def, mem_closure_iff_clusterPt]
-    
 #align is_closed_iff_cluster_pt isClosed_iff_clusterPt
 -/
 
@@ -1687,7 +1683,6 @@ theorem Dense.open_subset_closure_inter {s t : Set α} (hs : Dense s) (ht : IsOp
   calc
     t = t ∩ closure s := by rw [hs.closure_eq, inter_univ]
     _ ⊆ closure (t ∩ s) := ht.inter_closure
-    
 #align dense.open_subset_closure_inter Dense.open_subset_closure_inter
 
 theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x ∈ closure (s₁ ∪ s₂))
@@ -1699,8 +1694,7 @@ theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x 
       𝓝 x ⊓ principal (s₁ ∪ s₂) = 𝓝 x ⊓ (principal s₁ ⊔ principal s₂) := by rw [sup_principal]
       _ = 𝓝 x ⊓ principal s₁ ⊔ 𝓝 x ⊓ principal s₂ := inf_sup_left
       _ = ⊥ ⊔ 𝓝 x ⊓ principal s₂ := by rw [inf_principal_eq_bot.mpr h₁]
-      _ = 𝓝 x ⊓ principal s₂ := bot_sup_eq
-      ]
+      _ = 𝓝 x ⊓ principal s₂ := bot_sup_eq]
 #align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_union
 
 /-- The intersection of an open dense set with a dense set is a dense set. -/
@@ -1727,7 +1721,6 @@ theorem closure_diff {s t : Set α} : closure s \ closure t ⊆ closure (s \ t)
     _ ⊆ closure (closure tᶜ ∩ s) := (isOpen_compl_iff.mpr <| isClosed_closure).inter_closure
     _ = closure (s \ closure t) := by simp only [diff_eq, inter_comm]
     _ ⊆ closure (s \ t) := closure_mono <| diff_subset_diff (Subset.refl s) subset_closure
-    
 #align closure_diff closure_diff
 
 #print Filter.Frequently.mem_of_closed /-

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

@@ -81,8 +81,8 @@ class TopologicalSpace (α : Type u) where
 #align topological_space TopologicalSpace
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A «expr ⊆ » T) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A B «expr ∈ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (A «expr ⊆ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (A B «expr ∈ » T) -/
 /-- A constructor for topologies by specifying the closed sets,
 and showing that they satisfy the appropriate conditions. -/
 def TopologicalSpace.ofClosed {α : Type u} (T : Set (Set α)) (empty_mem : ∅ ∈ T)
@@ -346,7 +346,7 @@ def interior (s : Set α) : Set α :=
 #align interior interior
 -/
 
-/- ./././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) -/
 theorem mem_interior {s : Set α} {x : α} :
     x ∈ interior s ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc', and_left_comm]
@@ -472,7 +472,7 @@ theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClose
   Subset.antisymm (interior_maximal this isOpen_interior) (interior_mono <| subset_union_left _ _)
 #align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
 
-/- ./././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) -/
 theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
 #align is_open_iff_forall_mem_open isOpen_iff_forall_mem_open
@@ -1063,7 +1063,7 @@ theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟
   rw [nhds_def] <;> exact iInf_le_of_le s (iInf_le_of_le ⟨h, o⟩ sf)
 #align nhds_le_of_le nhds_le_of_le
 
-/- ./././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) -/
 theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ a ∈ t :=
   (nhds_basis_opens a).mem_iff.trans
     ⟨fun ⟨t, ⟨hat, ht⟩, hts⟩ => ⟨t, hts, ht, hat⟩, fun ⟨t, hts, ht, hat⟩ => ⟨t, ⟨hat, ht⟩, hts⟩⟩
@@ -1765,7 +1765,7 @@ theorem mem_closure_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set
 #align mem_closure_of_tendsto mem_closure_of_tendsto
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (x «expr ∉ » s) -/
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
 Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/
 theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : β → α} {l : Filter β} {s : Set β} {a : α}

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
 
 ! This file was ported from Lean 3 source module topology.basic
-! leanprover-community/mathlib commit e8da5f215e815d9ed3455f0216ef52b53e05438a
+! leanprover-community/mathlib commit e354e865255654389cc46e6032160238df2e0f40
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -1953,6 +1953,11 @@ theorem ContinuousAt.comp {g : β → γ} {f : α → β} {x : α} (hg : Continu
   hg.comp hf
 #align continuous_at.comp ContinuousAt.comp
 
+/-- See note [comp_of_eq lemmas] -/
+theorem ContinuousAt.comp_of_eq {g : β → γ} {f : α → β} {x : α} {y : β} (hg : ContinuousAt g y)
+    (hf : ContinuousAt f x) (hy : f x = y) : ContinuousAt (g ∘ f) x := by subst hy; exact hg.comp hf
+#align continuous_at.comp_of_eq ContinuousAt.comp_of_eq
+
 theorem Continuous.tendsto {f : α → β} (hf : Continuous f) (x) : Tendsto f (𝓝 x) (𝓝 (f x)) :=
   ((nhds_basis_opens x).tendsto_iffₓ <| nhds_basis_opens <| f x).2 fun t ⟨hxt, ht⟩ =>
     ⟨f ⁻¹' t, ⟨hxt, ht.Preimage hf⟩, Subset.refl _⟩
@@ -2286,3 +2291,31 @@ see e.g. `continuous_at.comp_div_cases`.
 -/
 
 
+library_note "comp_of_eq lemmas"/--
+Lean's elaborator has trouble elaborating applications of lemmas that state that the composition of
+two functions satisfy some property at a point, like `continuous_at.comp` / `cont_diff_at.comp` and
+`cont_mdiff_within_at.comp`. The reason is that a lemma like this looks like
+`continuous_at g (f x) → continuous_at f x → continuous_at (g ∘ f) x`.
+Since Lean's elaborator elaborates the arguments from left-to-right, when you write `hg.comp hf`,
+the elaborator will try to figure out *both* `f` and `g` from the type of `hg`. It tries to figure
+out `f` just from the point where `g` is continuous. For example, if `hg : continuous_at g (a, x)`
+then the elaborator will assign `f` to the function `prod.mk a`, since in that case `f x = (a, x)`.
+This is undesirable in most cases where `f` is not a variable. There are some ways to work around
+this, for example by giving `f` explicitly, or to force Lean to elaborate `hf` before elaborating
+`hg`, but this is annoying.
+Another better solution is to reformulate composition lemmas to have the following shape
+`continuous_at g y → continuous_at f x → f x = y → continuous_at (g ∘ f) x`.
+This is even useful if the proof of `f x = y` is `rfl`.
+The reason that this works better is because the type of `hg` doesn't mention `f`.
+Only after elaborating the two `continuous_at` arguments, Lean will try to unify `f x` with `y`,
+which is often easy after having chosen the correct functions for `f` and `g`.
+Here is an example that shows the difference:
+```
+example {x₀ : α} (f : α → α → β) (hf : continuous_at (function.uncurry f) (x₀, x₀)) :
+  continuous_at (λ x => f x x) x₀ :=
+-- hf.comp x (continuous_at_id.prod continuous_at_id) -- type mismatch
+-- hf.comp_of_eq (continuous_at_id.prod continuous_at_id) rfl -- works
+```
+-/
+
+

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

@@ -217,14 +217,14 @@ theorem isOpen_biInter_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈
 -/
 
 #print isOpen_const /-
-theorem isOpen_const {p : Prop} : IsOpen { a : α | p } :=
+theorem isOpen_const {p : Prop} : IsOpen {a : α | p} :=
   by_cases (fun this : p => by simp only [this] <;> exact isOpen_univ) fun this : ¬p => by
     simp only [this] <;> exact isOpen_empty
 #align is_open_const isOpen_const
 -/
 
 #print IsOpen.and /-
-theorem IsOpen.and : IsOpen { a | p₁ a } → IsOpen { a | p₂ a } → IsOpen { a | p₁ a ∧ p₂ a } :=
+theorem IsOpen.and : IsOpen {a | p₁ a} → IsOpen {a | p₂ a} → IsOpen {a | p₁ a ∧ p₂ a} :=
   IsOpen.inter
 #align is_open.and IsOpen.and
 -/
@@ -320,16 +320,16 @@ theorem isClosed_iUnion [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s
 -/
 
 #print isClosed_imp /-
-theorem isClosed_imp {p q : α → Prop} (hp : IsOpen { x | p x }) (hq : IsClosed { x | q x }) :
-    IsClosed { x | p x → q x } :=
+theorem isClosed_imp {p q : α → Prop} (hp : IsOpen {x | p x}) (hq : IsClosed {x | q x}) :
+    IsClosed {x | p x → q x} :=
   by
-  have : { x | p x → q x } = { x | p x }ᶜ ∪ { x | q x } := Set.ext fun x => imp_iff_not_or
+  have : {x | p x → q x} = {x | p x}ᶜ ∪ {x | q x} := Set.ext fun x => imp_iff_not_or
   rw [this] <;> exact IsClosed.union (is_closed_compl_iff.mpr hp) hq
 #align is_closed_imp isClosed_imp
 -/
 
 #print IsClosed.not /-
-theorem IsClosed.not : IsClosed { a | p a } → IsOpen { a | ¬p a } :=
+theorem IsClosed.not : IsClosed {a | p a} → IsOpen {a | ¬p a} :=
   isOpen_compl_iff.mpr
 #align is_closed.not IsClosed.not
 -/
@@ -342,7 +342,7 @@ theorem IsClosed.not : IsClosed { a | p a } → IsOpen { a | ¬p a } :=
 #print interior /-
 /-- The interior of a set `s` is the largest open subset of `s`. -/
 def interior (s : Set α) : Set α :=
-  ⋃₀ { t | IsOpen t ∧ t ⊆ s }
+  ⋃₀ {t | IsOpen t ∧ t ⊆ s}
 #align interior interior
 -/
 
@@ -450,9 +450,9 @@ theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ inte
 theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α) :
     interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by
   classical
-    refine' s.induction_on (by simp) _
-    intro i s h₁ h₂
-    simp [h₂]
+  refine' s.induction_on (by simp) _
+  intro i s h₁ h₂
+  simp [h₂]
 #align finset.interior_Inter Finset.interior_iInter
 
 @[simp]
@@ -507,7 +507,7 @@ theorem interior_sInter_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂
 #print closure /-
 /-- The closure of `s` is the smallest closed set containing `s`. -/
 def closure (s : Set α) : Set α :=
-  ⋂₀ { t | IsClosed t ∧ s ⊆ t }
+  ⋂₀ {t | IsClosed t ∧ s ⊆ t}
 #align closure closure
 -/
 
@@ -660,9 +660,9 @@ theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure
 theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α) :
     closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
   classical
-    refine' s.induction_on (by simp) _
-    intro i s h₁ h₂
-    simp [h₂]
+  refine' s.induction_on (by simp) _
+  intro i s h₁ h₂
+  simp [h₂]
 #align finset.closure_bUnion Finset.closure_biUnion
 
 @[simp]
@@ -993,7 +993,7 @@ theorem compl_frontier_eq_union_interior {s : Set α} : frontier sᶜ = interior
 neighborhoods of `a` forms a filter, the neighborhood filter at `a`, is here defined as the
 infimum over the principal filters of all open sets containing `a`. -/
 irreducible_def nhds (a : α) : Filter α :=
-  ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 s
+  ⨅ s ∈ {s : Set α | a ∈ s ∧ IsOpen s}, 𝓟 s
 #align nhds nhds
 -/
 
@@ -1026,7 +1026,7 @@ scoped[Topology] notation "𝓝[>] " x:100 => nhdsWithin x (Set.Ioi x)
 -- mathport name: nhds_within.lt
 scoped[Topology] notation "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
 
-theorem nhds_def (a : α) : 𝓝 a = ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 s := by rw [nhds]
+theorem nhds_def (a : α) : 𝓝 a = ⨅ s ∈ {s : Set α | a ∈ s ∧ IsOpen s}, 𝓟 s := by rw [nhds]
 #align nhds_def nhds_def
 
 theorem nhds_def' (a : α) : 𝓝 a = ⨅ (s : Set α) (hs : IsOpen s) (ha : a ∈ s), 𝓟 s := by
@@ -1079,7 +1079,7 @@ theorem eventually_nhds_iff {a : α} {p : α → Prop} :
 -/
 
 theorem map_nhds {a : α} {f : α → β} :
-    map f (𝓝 a) = ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 (image f s) :=
+    map f (𝓝 a) = ⨅ s ∈ {s : Set α | a ∈ s ∧ IsOpen s}, 𝓟 (image f s) :=
   ((nhds_basis_opens a).map f).eq_biInf
 #align map_nhds map_nhds
 
@@ -1443,12 +1443,12 @@ theorem AccPt.mono {x : α} {F G : Filter α} (h : AccPt x F) (hFG : F ≤ G) :
 
 
 #print interior_eq_nhds' /-
-theorem interior_eq_nhds' {s : Set α} : interior s = { a | s ∈ 𝓝 a } :=
+theorem interior_eq_nhds' {s : Set α} : interior s = {a | s ∈ 𝓝 a} :=
   Set.ext fun x => by simp only [mem_interior, mem_nhds_iff, mem_set_of_eq]
 #align interior_eq_nhds' interior_eq_nhds'
 -/
 
-theorem interior_eq_nhds {s : Set α} : interior s = { a | 𝓝 a ≤ 𝓟 s } :=
+theorem interior_eq_nhds {s : Set α} : interior s = {a | 𝓝 a ≤ 𝓟 s} :=
   interior_eq_nhds'.trans <| by simp only [le_principal_iff]
 #align interior_eq_nhds interior_eq_nhds
 
@@ -1467,13 +1467,13 @@ theorem interior_mem_nhds {s : Set α} {a : α} : interior s ∈ 𝓝 a ↔ s 
 -/
 
 #print interior_setOf_eq /-
-theorem interior_setOf_eq {p : α → Prop} : interior { x | p x } = { x | ∀ᶠ y in 𝓝 x, p y } :=
+theorem interior_setOf_eq {p : α → Prop} : interior {x | p x} = {x | ∀ᶠ y in 𝓝 x, p y} :=
   interior_eq_nhds'
 #align interior_set_of_eq interior_setOf_eq
 -/
 
 #print isOpen_setOf_eventually_nhds /-
-theorem isOpen_setOf_eventually_nhds {p : α → Prop} : IsOpen { x | ∀ᶠ y in 𝓝 x, p y } := by
+theorem isOpen_setOf_eventually_nhds {p : α → Prop} : IsOpen {x | ∀ᶠ y in 𝓝 x, p y} := by
   simp only [← interior_setOf_eq, isOpen_interior]
 #align is_open_set_of_eventually_nhds isOpen_setOf_eventually_nhds
 -/
@@ -1553,7 +1553,7 @@ theorem isClosed_iff_frequently {s : Set α} : IsClosed s ↔ ∀ x, (∃ᶠ y i
 #print isClosed_setOf_clusterPt /-
 /-- The set of cluster points of a filter is closed. In particular, the set of limit points
 of a sequence is closed. -/
-theorem isClosed_setOf_clusterPt {f : Filter α} : IsClosed { x | ClusterPt x f } :=
+theorem isClosed_setOf_clusterPt {f : Filter α} : IsClosed {x | ClusterPt x f} :=
   by
   simp only [ClusterPt, inf_ne_bot_iff_frequently_left, set_of_forall, imp_iff_not_or]
   refine' isClosed_iInter fun p => IsClosed.union _ _ <;> apply isClosed_compl_iff.2
@@ -1605,7 +1605,7 @@ theorem not_isOpen_singleton (x : α) [NeBot (𝓝[≠] x)] : ¬IsOpen ({x} : Se
 #align not_is_open_singleton not_isOpen_singleton
 
 #print closure_eq_cluster_pts /-
-theorem closure_eq_cluster_pts {s : Set α} : closure s = { a | ClusterPt a (𝓟 s) } :=
+theorem closure_eq_cluster_pts {s : Set α} : closure s = {a | ClusterPt a (𝓟 s)} :=
   Set.ext fun x => mem_closure_iff_clusterPt
 #align closure_eq_cluster_pts closure_eq_cluster_pts
 -/

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

@@ -118,7 +118,7 @@ theorem isOpen_mk {p h₁ h₂ h₃} {s : Set α} : is_open[⟨p, h₁, h₂, h
 #print topologicalSpace_eq /-
 @[ext]
 theorem topologicalSpace_eq {f g : TopologicalSpace α} (h : is_open[f] = is_open[g]) : f = g := by
-  cases f; cases g; congr ; exact h
+  cases f; cases g; congr; exact h
 #align topological_space_eq topologicalSpace_eq
 -/
 
@@ -348,7 +348,7 @@ def interior (s : Set α) : Set α :=
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem mem_interior {s : Set α} {x : α} :
-    x ∈ interior s ↔ ∃ (t : _)(_ : t ⊆ s), IsOpen t ∧ x ∈ t := by
+    x ∈ interior s ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc', and_left_comm]
 #align mem_interior mem_interiorₓ
 
@@ -467,13 +467,13 @@ theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClose
     by_contradiction fun hx₂ : x ∉ s =>
       have : u \ s ⊆ t := fun x ⟨h₁, h₂⟩ => Or.resolve_left (hu₂ h₁) h₂
       have : u \ s ⊆ interior t := by rwa [(IsOpen.sdiff hu₁ h₁).subset_interior_iff]
-      have : u \ s ⊆ ∅ := by rwa [h₂] at this
+      have : u \ s ⊆ ∅ := by rwa [h₂] at this 
       this ⟨hx₁, hx₂⟩
   Subset.antisymm (interior_maximal this isOpen_interior) (interior_mono <| subset_union_left _ _)
 #align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
-theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _)(_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
+theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
 #align is_open_iff_forall_mem_open isOpen_iff_forall_mem_open
 
@@ -952,7 +952,7 @@ theorem interior_frontier {s : Set α} (h : IsClosed s) : interior (frontier s)
   have C : interior (frontier s) ⊆ frontier s := interior_subset
   have : interior (frontier s) ⊆ interior s ∩ (s \ interior s) :=
     subset_inter B (by simpa [A] using C)
-  rwa [inter_diff_self, subset_empty_iff] at this
+  rwa [inter_diff_self, subset_empty_iff] at this 
 #align interior_frontier interior_frontier
 -/
 
@@ -1064,7 +1064,7 @@ theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟
 #align nhds_le_of_le nhds_le_of_le
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
-theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ (t : _)(_ : t ⊆ s), IsOpen t ∧ a ∈ t :=
+theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ (t : _) (_ : t ⊆ s), IsOpen t ∧ a ∈ t :=
   (nhds_basis_opens a).mem_iff.trans
     ⟨fun ⟨t, ⟨hat, ht⟩, hts⟩ => ⟨t, hts, ht, hat⟩, fun ⟨t, hts, ht, hat⟩ => ⟨t, ⟨hat, ht⟩, hts⟩⟩
 #align mem_nhds_iff mem_nhds_iffₓ
@@ -1557,7 +1557,7 @@ theorem isClosed_setOf_clusterPt {f : Filter α} : IsClosed { x | ClusterPt x f
   by
   simp only [ClusterPt, inf_ne_bot_iff_frequently_left, set_of_forall, imp_iff_not_or]
   refine' isClosed_iInter fun p => IsClosed.union _ _ <;> apply isClosed_compl_iff.2
-  exacts[isOpen_setOf_eventually_nhds, isOpen_const]
+  exacts [isOpen_setOf_eventually_nhds, isOpen_const]
 #align is_closed_set_of_cluster_pt isClosed_setOf_clusterPt
 -/
 
@@ -1779,7 +1779,7 @@ theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : β → α} {l : Filter
   refine' ⟨fun h' => _, le_trans inf_le_left⟩
   have := sup_le h' h
   rw [sup_inf_right, sup_principal, union_compl_self, principal_univ, inf_top_eq, sup_le_iff] at
-    this
+    this 
   exact this.1
 #align tendsto_inf_principal_nhds_iff_of_forall_eq tendsto_inf_principal_nhds_iff_of_forall_eq
 

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

@@ -61,7 +61,7 @@ noncomputable section
 
 open Set Filter Classical
 
-open Classical Filter
+open scoped Classical Filter
 
 universe u v w
 
@@ -1256,11 +1256,13 @@ theorem tendsto_nhds {f : β → α} {l : Filter β} {a : α} :
 #align tendsto_nhds tendsto_nhds
 -/
 
+#print tendsto_atTop_nhds /-
 theorem tendsto_atTop_nhds [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} :
     Tendsto f atTop (𝓝 a) ↔ ∀ U : Set α, a ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U :=
   (atTop_basis.tendsto_iffₓ (nhds_basis_opens a)).trans <| by
     simp only [and_imp, exists_prop, true_and_iff, mem_Ici, ge_iff_le]
 #align tendsto_at_top_nhds tendsto_atTop_nhds
+-/
 
 #print tendsto_const_nhds /-
 theorem tendsto_const_nhds {a : α} {f : Filter β} : Tendsto (fun b : β => a) f (𝓝 a) :=
@@ -1852,7 +1854,7 @@ variable {α : Type _} {β : Type _} {γ : Type _} {δ : Type _}
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
-open Topology
+open scoped Topology
 
 #print Continuous /-
 /-- A function between topological spaces is continuous if the preimage

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

@@ -81,12 +81,6 @@ class TopologicalSpace (α : Type u) where
 #align topological_space TopologicalSpace
 -/
 
-/- warning: topological_space.of_closed -> TopologicalSpace.ofClosed is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} (T : Set.{u1} (Set.{u1} α)), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α)) T) -> (forall (A : Set.{u1} (Set.{u1} α)), (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) A T) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (Set.sInter.{u1} α A) T)) -> (forall (A : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) A T) -> (forall (B : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) B T) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) A B) T))) -> (TopologicalSpace.{u1} α)
-but is expected to have type
-  forall {α : Type.{u1}} (T : Set.{u1} (Set.{u1} α)), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α)) T) -> (forall (A : Set.{u1} (Set.{u1} α)), (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) A T) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (Set.sInter.{u1} α A) T)) -> (forall (A : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) A T) -> (forall (B : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) B T) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) A B) T))) -> (TopologicalSpace.{u1} α)
-Case conversion may be inaccurate. Consider using '#align topological_space.of_closed TopologicalSpace.ofClosedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A «expr ⊆ » T) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A B «expr ∈ » T) -/
 /-- A constructor for topologies by specifying the closed sets,
@@ -117,12 +111,6 @@ def IsOpen [TopologicalSpace α] (s : Set α) : Prop :=
 -- mathport name: is_open_of
 scoped[Topology] notation "is_open[" t "]" => @IsOpen hole! t
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open_mk isOpen_mkₓ'. -/
 theorem isOpen_mk {p h₁ h₂ h₃} {s : Set α} : is_open[⟨p, h₁, h₂, h₃⟩] s ↔ p s :=
   Iff.rfl
 #align is_open_mk isOpen_mk
@@ -145,12 +133,6 @@ theorem isOpen_univ : IsOpen (univ : Set α) :=
 #align is_open_univ isOpen_univ
 -/
 
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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_open.inter IsOpen.interₓ'. -/
 theorem IsOpen.inter (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∩ s₂) :=
   TopologicalSpace.isOpen_inter s₁ s₂ h₁ h₂
 #align is_open.inter IsOpen.inter
@@ -191,12 +173,6 @@ theorem isOpen_biUnion {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen
 #align is_open_bUnion isOpen_biUnion
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open.union IsOpen.unionₓ'. -/
 theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∪ s₂) := by
   rw [union_eq_Union] <;> exact isOpen_iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align is_open.union IsOpen.union
@@ -263,12 +239,6 @@ class IsClosed (s : Set α) : Prop where
 -- mathport name: is_closed_of
 scoped[Topology] notation "is_closed[" t "]" => @IsClosed hole! t
 
-/- warning: is_open_compl_iff -> isOpen_compl_iff is a dubious translation:
-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_open_compl_iff isOpen_compl_iffₓ'. -/
 @[simp]
 theorem isOpen_compl_iff {s : Set α} : IsOpen (sᶜ) ↔ IsClosed s :=
   ⟨fun h => ⟨h⟩, fun h => h.isOpen_compl⟩
@@ -288,12 +258,6 @@ theorem isClosed_univ : IsClosed (univ : Set α) := by rw [← isOpen_compl_iff,
 #align is_closed_univ isClosed_univ
 -/
 
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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.union IsClosed.unionₓ'. -/
 theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂) := fun h₁ h₂ => by
   rw [← isOpen_compl_iff] at *; rw [compl_union]; exact IsOpen.inter h₁ h₂
 #align is_closed.union IsClosed.union
@@ -317,53 +281,23 @@ theorem isClosed_biInter {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsCl
 #align is_closed_bInter isClosed_biInter
 -/
 
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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_compl_iff isClosed_compl_iffₓ'. -/
 @[simp]
 theorem isClosed_compl_iff {s : Set α} : IsClosed (sᶜ) ↔ IsOpen s := by
   rw [← isOpen_compl_iff, compl_compl]
 #align is_closed_compl_iff isClosed_compl_iff
 
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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_open.is_closed_compl IsOpen.isClosed_complₓ'. -/
 theorem IsOpen.isClosed_compl {s : Set α} (hs : IsOpen s) : IsClosed (sᶜ) :=
   isClosed_compl_iff.2 hs
 #align is_open.is_closed_compl IsOpen.isClosed_compl
 
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-Case conversion may be inaccurate. Consider using '#align is_open.sdiff IsOpen.sdiffₓ'. -/
 theorem IsOpen.sdiff {s t : Set α} (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) :=
   IsOpen.inter h₁ <| isOpen_compl_iff.mpr h₂
 #align is_open.sdiff IsOpen.sdiff
 
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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.inter IsClosed.interₓ'. -/
 theorem IsClosed.inter (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) : IsClosed (s₁ ∩ s₂) := by
   rw [← isOpen_compl_iff] at *; rw [compl_inter]; exact IsOpen.union h₁ h₂
 #align is_closed.inter IsClosed.inter
 
-/- warning: is_closed.sdiff -> IsClosed.sdiff is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align is_closed.sdiff IsClosed.sdiffₓ'. -/
 theorem IsClosed.sdiff {s t : Set α} (h₁ : IsClosed s) (h₂ : IsOpen t) : IsClosed (s \ t) :=
   IsClosed.inter h₁ (isClosed_compl_iff.mpr h₂)
 #align is_closed.sdiff IsClosed.sdiff
@@ -503,12 +437,6 @@ theorem interior_interior {s : Set α} : interior (interior s) = interior s :=
 #align interior_interior interior_interior
 -/
 
-/- warning: interior_inter -> interior_inter is a dubious translation:
-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 interior_inter interior_interₓ'. -/
 @[simp]
 theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ interior t :=
   Subset.antisymm
@@ -518,12 +446,6 @@ theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ inte
       IsOpen.inter isOpen_interior isOpen_interior)
 #align interior_inter interior_inter
 
-/- warning: finset.interior_Inter -> Finset.interior_iInter is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align finset.interior_Inter Finset.interior_iInterₓ'. -/
 @[simp]
 theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α) :
     interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by
@@ -533,24 +455,12 @@ theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α)
     simp [h₂]
 #align finset.interior_Inter Finset.interior_iInter
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align interior_Inter interior_iInterₓ'. -/
 @[simp]
 theorem interior_iInter {ι : Type _} [Finite ι] (f : ι → Set α) :
     interior (⋂ i, f i) = ⋂ i, interior (f i) := by cases nonempty_fintype ι;
   convert finset.univ.interior_Inter f <;> simp
 #align interior_Inter interior_iInter
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_emptyₓ'. -/
 theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClosed s)
     (h₂ : interior t = ∅) : interior (s ∪ t) = interior s :=
   have : interior (s ∪ t) ⊆ s := fun x ⟨u, ⟨(hu₁ : IsOpen u), (hu₂ : u ⊆ s ∪ t)⟩, (hx₁ : x ∈ u)⟩ =>
@@ -562,12 +472,6 @@ theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClose
   Subset.antisymm (interior_maximal this isOpen_interior) (interior_mono <| subset_union_left _ _)
 #align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
 
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-Case conversion may be inaccurate. Consider using '#align is_open_iff_forall_mem_open isOpen_iff_forall_mem_openₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _)(_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
@@ -579,12 +483,6 @@ theorem interior_iInter_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ 
 #align interior_Inter_subset interior_iInter_subset
 -/
 
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-Case conversion may be inaccurate. Consider using '#align interior_Inter₂_subset interior_Inter₂_subsetₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 theorem interior_Inter₂_subset (p : ι → Sort _) (s : ∀ i, p i → Set α) :
@@ -638,23 +536,11 @@ theorem closure_minimal {s t : Set α} (h₁ : s ⊆ t) (h₂ : IsClosed t) : cl
 #align closure_minimal closure_minimal
 -/
 
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-Case conversion may be inaccurate. Consider using '#align disjoint.closure_left Disjoint.closure_leftₓ'. -/
 theorem Disjoint.closure_left {s t : Set α} (hd : Disjoint s t) (ht : IsOpen t) :
     Disjoint (closure s) t :=
   disjoint_compl_left.mono_left <| closure_minimal hd.subset_compl_right ht.isClosed_compl
 #align disjoint.closure_left Disjoint.closure_left
 
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 theorem Disjoint.closure_right {s t : Set α} (hd : Disjoint s t) (hs : IsOpen s) :
     Disjoint s (closure t) :=
   (hd.symm.closure_left hs).symm
@@ -692,32 +578,14 @@ theorem closure_mono {s t : Set α} (h : s ⊆ t) : closure s ⊆ closure t :=
 #align closure_mono closure_mono
 -/
 
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 theorem monotone_closure (α : Type _) [TopologicalSpace α] : Monotone (@closure α _) := fun _ _ =>
   closure_mono
 #align monotone_closure monotone_closure
 
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 theorem diff_subset_closure_iff {s t : Set α} : s \ t ⊆ closure t ↔ s ⊆ closure t := by
   rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]
 #align diff_subset_closure_iff diff_subset_closure_iff
 
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 theorem closure_inter_subset_inter_closure (s t : Set α) :
     closure (s ∩ t) ⊆ closure s ∩ closure t :=
   (monotone_closure α).map_inf_le s t
@@ -780,12 +648,6 @@ theorem closure_closure {s : Set α} : closure (closure s) = closure s :=
 #align closure_closure closure_closure
 -/
 
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 @[simp]
 theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure t :=
   Subset.antisymm
@@ -794,12 +656,6 @@ theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure
     ((monotone_closure α).le_map_sup s t)
 #align closure_union closure_union
 
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-Case conversion may be inaccurate. Consider using '#align finset.closure_bUnion Finset.closure_biUnionₓ'. -/
 @[simp]
 theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α) :
     closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
@@ -809,12 +665,6 @@ theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α)
     simp [h₂]
 #align finset.closure_bUnion Finset.closure_biUnion
 
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-Case conversion may be inaccurate. Consider using '#align closure_Union closure_iUnionₓ'. -/
 @[simp]
 theorem closure_iUnion {ι : Type _} [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by cases nonempty_fintype ι;
@@ -827,46 +677,22 @@ theorem interior_subset_closure {s : Set α} : interior s ⊆ closure s :=
 #align interior_subset_closure interior_subset_closure
 -/
 
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 theorem closure_eq_compl_interior_compl {s : Set α} : closure s = interior (sᶜ)ᶜ :=
   by
   rw [interior, closure, compl_sUnion, compl_image_set_of]
   simp only [compl_subset_compl, isOpen_compl_iff]
 #align closure_eq_compl_interior_compl closure_eq_compl_interior_compl
 
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 @[simp]
 theorem interior_compl {s : Set α} : interior (sᶜ) = closure sᶜ := by
   simp [closure_eq_compl_interior_compl]
 #align interior_compl interior_compl
 
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 @[simp]
 theorem closure_compl {s : Set α} : closure (sᶜ) = interior sᶜ := by
   simp [closure_eq_compl_interior_compl]
 #align closure_compl closure_compl
 
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 theorem mem_closure_iff {s : Set α} {a : α} :
     a ∈ closure s ↔ ∀ o, IsOpen o → a ∈ o → (o ∩ s).Nonempty :=
   ⟨fun h o oo ao =>
@@ -879,24 +705,12 @@ theorem mem_closure_iff {s : Set α} {a : α} :
       hc (h₂ hs)⟩
 #align mem_closure_iff mem_closure_iff
 
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 theorem closure_inter_open_nonempty_iff {s t : Set α} (h : IsOpen t) :
     (closure s ∩ t).Nonempty ↔ (s ∩ t).Nonempty :=
   ⟨fun ⟨x, hxcs, hxt⟩ => inter_comm t s ▸ mem_closure_iff.1 hxcs t h hxt, fun h =>
     h.mono <| inf_le_inf_right t subset_closure⟩
 #align closure_inter_open_nonempty_iff closure_inter_open_nonempty_iff
 
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 theorem Filter.le_lift'_closure (l : Filter α) : l ≤ l.lift' closure :=
   le_lift'.2 fun s hs => mem_of_superset hs subset_closure
 #align filter.le_lift'_closure Filter.le_lift'_closure
@@ -916,12 +730,6 @@ theorem Filter.HasBasis.lift'_closure_eq_self {l : Filter α} {p : ι → Prop}
 #align filter.has_basis.lift'_closure_eq_self Filter.HasBasis.lift'_closure_eq_self
 -/
 
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-Case conversion may be inaccurate. Consider using '#align filter.lift'_closure_eq_bot Filter.lift'_closure_eq_botₓ'. -/
 @[simp]
 theorem Filter.lift'_closure_eq_bot {l : Filter α} : l.lift' closure = ⊥ ↔ l = ⊥ :=
   ⟨fun h => bot_unique <| h ▸ l.le_lift'_closure, fun h =>
@@ -947,22 +755,10 @@ theorem Dense.closure_eq {s : Set α} (h : Dense s) : closure s = univ :=
 #align dense.closure_eq Dense.closure_eq
 -/
 
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 theorem interior_eq_empty_iff_dense_compl {s : Set α} : interior s = ∅ ↔ Dense (sᶜ) := by
   rw [dense_iff_closure_eq, closure_compl, compl_univ_iff]
 #align interior_eq_empty_iff_dense_compl interior_eq_empty_iff_dense_compl
 
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 theorem Dense.interior_compl {s : Set α} (h : Dense s) : interior (sᶜ) = ∅ :=
   interior_eq_empty_iff_dense_compl.2 <| by rwa [compl_compl]
 #align dense.interior_compl Dense.interior_compl
@@ -985,12 +781,6 @@ theorem dense_univ : Dense (univ : Set α) := fun x => subset_closure trivial
 #align dense_univ dense_univ
 -/
 
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-Case conversion may be inaccurate. Consider using '#align dense_iff_inter_open dense_iff_inter_openₓ'. -/
 /-- A set is dense if and only if it has a nonempty intersection with each nonempty open set. -/
 theorem dense_iff_inter_open {s : Set α} :
     Dense s ↔ ∀ U, IsOpen U → U.Nonempty → (U ∩ s).Nonempty :=
@@ -1004,21 +794,9 @@ theorem dense_iff_inter_open {s : Set α} :
     exact h U U_op ⟨_, x_in⟩
 #align dense_iff_inter_open dense_iff_inter_open
 
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-Case conversion may be inaccurate. Consider using '#align dense.inter_open_nonempty Dense.inter_open_nonemptyₓ'. -/
 alias dense_iff_inter_open ↔ Dense.inter_open_nonempty _
 #align dense.inter_open_nonempty Dense.inter_open_nonempty
 
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-Case conversion may be inaccurate. Consider using '#align dense.exists_mem_open Dense.exists_mem_openₓ'. -/
 theorem Dense.exists_mem_open {s : Set α} (hs : Dense s) {U : Set α} (ho : IsOpen U)
     (hne : U.Nonempty) : ∃ x ∈ s, x ∈ U :=
   let ⟨x, hx⟩ := hs.inter_open_nonempty U ho hne
@@ -1046,12 +824,6 @@ theorem Dense.mono {s₁ s₂ : Set α} (h : s₁ ⊆ s₂) (hd : Dense s₁) :
 #align dense.mono Dense.mono
 -/
 
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 /-- Complement to a singleton is dense if and only if the singleton is not an open set. -/
 theorem dense_compl_singleton_iff_not_open {x : α} : Dense ({x}ᶜ : Set α) ↔ ¬IsOpen ({x} : Set α) :=
   by
@@ -1075,46 +847,22 @@ def frontier (s : Set α) : Set α :=
 #align frontier frontier
 -/
 
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 @[simp]
 theorem closure_diff_interior (s : Set α) : closure s \ interior s = frontier s :=
   rfl
 #align closure_diff_interior closure_diff_interior
 
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 @[simp]
 theorem closure_diff_frontier (s : Set α) : closure s \ frontier s = interior s := by
   rw [frontier, diff_diff_right_self, inter_eq_self_of_subset_right interior_subset_closure]
 #align closure_diff_frontier closure_diff_frontier
 
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 @[simp]
 theorem self_diff_frontier (s : Set α) : s \ frontier s = interior s := by
   rw [frontier, diff_diff_right, diff_eq_empty.2 subset_closure,
     inter_eq_self_of_subset_right interior_subset, empty_union]
 #align self_diff_frontier self_diff_frontier
 
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 theorem frontier_eq_closure_inter_closure {s : Set α} : frontier s = closure s ∩ closure (sᶜ) := by
   rw [closure_compl, frontier, diff_eq]
 #align frontier_eq_closure_inter_closure frontier_eq_closure_inter_closure
@@ -1143,12 +891,6 @@ theorem frontier_interior_subset {s : Set α} : frontier (interior s) ⊆ fronti
 #align frontier_interior_subset frontier_interior_subset
 -/
 
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 /-- The complement of a set has the same frontier as the original set. -/
 @[simp]
 theorem frontier_compl (s : Set α) : frontier (sᶜ) = frontier s := by
@@ -1167,12 +909,6 @@ theorem frontier_empty : frontier (∅ : Set α) = ∅ := by simp [frontier]
 #align frontier_empty frontier_empty
 -/
 
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 theorem frontier_inter_subset (s t : Set α) :
     frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t :=
   by
@@ -1183,43 +919,19 @@ theorem frontier_inter_subset (s t : Set α) :
   apply inter_comm
 #align frontier_inter_subset frontier_inter_subset
 
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 theorem frontier_union_subset (s t : Set α) :
     frontier (s ∪ t) ⊆ frontier s ∩ closure (tᶜ) ∪ closure (sᶜ) ∩ frontier t := by
   simpa only [frontier_compl, ← compl_union] using frontier_inter_subset (sᶜ) (tᶜ)
 #align frontier_union_subset frontier_union_subset
 
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 theorem IsClosed.frontier_eq {s : Set α} (hs : IsClosed s) : frontier s = s \ interior s := by
   rw [frontier, hs.closure_eq]
 #align is_closed.frontier_eq IsClosed.frontier_eq
 
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 theorem IsOpen.frontier_eq {s : Set α} (hs : IsOpen s) : frontier s = closure s \ s := by
   rw [frontier, hs.interior_eq]
 #align is_open.frontier_eq IsOpen.frontier_eq
 
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 theorem IsOpen.inter_frontier_eq {s : Set α} (hs : IsOpen s) : s ∩ frontier s = ∅ := by
   rw [hs.frontier_eq, inter_diff_self]
 #align is_open.inter_frontier_eq IsOpen.inter_frontier_eq
@@ -1244,63 +956,27 @@ theorem interior_frontier {s : Set α} (h : IsClosed s) : interior (frontier s)
 #align interior_frontier interior_frontier
 -/
 
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 theorem closure_eq_interior_union_frontier (s : Set α) : closure s = interior s ∪ frontier s :=
   (union_diff_cancel interior_subset_closure).symm
 #align closure_eq_interior_union_frontier closure_eq_interior_union_frontier
 
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 theorem closure_eq_self_union_frontier (s : Set α) : closure s = s ∪ frontier s :=
   (union_diff_cancel' interior_subset subset_closure).symm
 #align closure_eq_self_union_frontier closure_eq_self_union_frontier
 
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 theorem Disjoint.frontier_left (ht : IsOpen t) (hd : Disjoint s t) : Disjoint (frontier s) t :=
   subset_compl_iff_disjoint_right.1 <|
     frontier_subset_closure.trans <| closure_minimal (disjoint_left.1 hd) <| isClosed_compl_iff.2 ht
 #align disjoint.frontier_left Disjoint.frontier_left
 
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-Case conversion may be inaccurate. Consider using '#align disjoint.frontier_right Disjoint.frontier_rightₓ'. -/
 theorem Disjoint.frontier_right (hs : IsOpen s) (hd : Disjoint s t) : Disjoint s (frontier t) :=
   (hd.symm.frontier_left hs).symm
 #align disjoint.frontier_right Disjoint.frontier_right
 
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-Case conversion may be inaccurate. Consider using '#align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interiorₓ'. -/
 theorem frontier_eq_inter_compl_interior {s : Set α} : frontier s = interior sᶜ ∩ interior (sᶜ)ᶜ :=
   by rw [← frontier_compl, ← closure_compl]; rfl
 #align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interior
 
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-Case conversion may be inaccurate. Consider using '#align compl_frontier_eq_union_interior compl_frontier_eq_union_interiorₓ'. -/
 theorem compl_frontier_eq_union_interior {s : Set α} : frontier sᶜ = interior s ∪ interior (sᶜ) :=
   by
   rw [frontier_eq_inter_compl_interior]
@@ -1350,21 +1026,9 @@ scoped[Topology] notation "𝓝[>] " x:100 => nhdsWithin x (Set.Ioi x)
 -- mathport name: nhds_within.lt
 scoped[Topology] notation "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
 
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 theorem nhds_def (a : α) : 𝓝 a = ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 s := by rw [nhds]
 #align nhds_def nhds_def
 
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 theorem nhds_def' (a : α) : 𝓝 a = ⨅ (s : Set α) (hs : IsOpen s) (ha : a ∈ s), 𝓟 s := by
   simp only [nhds_def, mem_set_of_eq, and_comm' (a ∈ _), iInf_and]
 #align nhds_def' nhds_def'
@@ -1383,34 +1047,16 @@ theorem nhds_basis_opens (a : α) : (𝓝 a).HasBasis (fun s : Set α => a ∈ s
 #align nhds_basis_opens nhds_basis_opens
 -/
 
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-Case conversion may be inaccurate. Consider using '#align nhds_basis_closeds nhds_basis_closedsₓ'. -/
 theorem nhds_basis_closeds (a : α) : (𝓝 a).HasBasis (fun s : Set α => a ∉ s ∧ IsClosed s) compl :=
   ⟨fun t =>
     (nhds_basis_opens a).mem_iff.trans <|
       compl_surjective.exists.trans <| by simp only [isOpen_compl_iff, mem_compl_iff]⟩
 #align nhds_basis_closeds nhds_basis_closeds
 
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 /-- A filter lies below the neighborhood filter at `a` iff it contains every open set around `a`. -/
 theorem le_nhds_iff {f a} : f ≤ 𝓝 a ↔ ∀ s : Set α, a ∈ s → IsOpen s → s ∈ f := by simp [nhds_def]
 #align le_nhds_iff le_nhds_iff
 
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 /-- To show a filter is above the neighborhood filter at `a`, it suffices to show that it is above
 the principal filter of some open set `s` containing `a`. -/
 theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 a ≤ f := by
@@ -1432,12 +1078,6 @@ theorem eventually_nhds_iff {a : α} {p : α → Prop} :
 #align eventually_nhds_iff eventually_nhds_iff
 -/
 
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 theorem map_nhds {a : α} {f : α → β} :
     map f (𝓝 a) = ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 (image f s) :=
   ((nhds_basis_opens a).map f).eq_biInf
@@ -1469,12 +1109,6 @@ theorem IsOpen.mem_nhds_iff {a : α} {s : Set α} (hs : IsOpen s) : s ∈ 𝓝 a
 #align is_open.mem_nhds_iff IsOpen.mem_nhds_iff
 -/
 
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 theorem IsClosed.compl_mem_nhds {a : α} {s : Set α} (hs : IsClosed s) (ha : a ∉ s) : sᶜ ∈ 𝓝 a :=
   hs.isOpen_compl.mem_nhds (mem_compl ha)
 #align is_closed.compl_mem_nhds IsClosed.compl_mem_nhds
@@ -1511,12 +1145,6 @@ theorem exists_open_set_nhds {s U : Set α} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
 #align exists_open_set_nhds exists_open_set_nhds
 -/
 
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 /-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of s:
 it contains an open set containing `s`. -/
 theorem exists_open_set_nhds' {s U : Set α} (h : U ∈ ⨆ x ∈ s, 𝓝 x) :
@@ -1628,12 +1256,6 @@ theorem tendsto_nhds {f : β → α} {l : Filter β} {a : α} :
 #align tendsto_nhds tendsto_nhds
 -/
 
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 theorem tendsto_atTop_nhds [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} :
     Tendsto f atTop (𝓝 a) ↔ ∀ U : Set α, a ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U :=
   (atTop_basis.tendsto_iffₓ (nhds_basis_opens a)).trans <| by
@@ -1646,54 +1268,24 @@ theorem tendsto_const_nhds {a : α} {f : Filter β} : Tendsto (fun b : β => a)
 #align tendsto_const_nhds tendsto_const_nhds
 -/
 
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 theorem tendsto_atTop_of_eventually_const {ι : Type _} [SemilatticeSup ι] [Nonempty ι] {x : α}
     {u : ι → α} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : Tendsto u atTop (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atTop.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_top_of_eventually_const tendsto_atTop_of_eventually_const
 
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 theorem tendsto_atBot_of_eventually_const {ι : Type _} [SemilatticeInf ι] [Nonempty ι] {x : α}
     {u : ι → α} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : Tendsto u atBot (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atBot.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_bot_of_eventually_const tendsto_atBot_of_eventually_const
 
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 theorem pure_le_nhds : pure ≤ (𝓝 : α → Filter α) := fun a s hs => mem_pure.2 <| mem_of_mem_nhds hs
 #align pure_le_nhds pure_le_nhds
 
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-Case conversion may be inaccurate. Consider using '#align tendsto_pure_nhds tendsto_pure_nhdsₓ'. -/
 theorem tendsto_pure_nhds {α : Type _} [TopologicalSpace β] (f : α → β) (a : α) :
     Tendsto f (pure a) (𝓝 (f a)) :=
   (tendsto_pure_pure f a).mono_right (pure_le_nhds _)
 #align tendsto_pure_nhds tendsto_pure_nhds
 
-/- warning: order_top.tendsto_at_top_nhds -> OrderTop.tendsto_atTop_nhds is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} {α : Type.{u2}} [_inst_2 : PartialOrder.{u2} α] [_inst_3 : OrderTop.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2))] [_inst_4 : TopologicalSpace.{u1} β] (f : α -> β), Filter.Tendsto.{u2, u1} α β f (Filter.atTop.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2)) (nhds.{u1} β _inst_4 (f (Top.top.{u2} α (OrderTop.toHasTop.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2)) _inst_3))))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align order_top.tendsto_at_top_nhds OrderTop.tendsto_atTop_nhdsₓ'. -/
 theorem OrderTop.tendsto_atTop_nhds {α : Type _} [PartialOrder α] [OrderTop α] [TopologicalSpace β]
     (f : α → β) : Tendsto f atTop (𝓝 <| f ⊤) :=
   (tendsto_atTop_pure f).mono_right (pure_le_nhds _)
@@ -1723,45 +1315,21 @@ def ClusterPt (x : α) (F : Filter α) : Prop :=
 #align cluster_pt ClusterPt
 -/
 
-/- warning: cluster_pt.ne_bot -> ClusterPt.neBot is a dubious translation:
-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 cluster_pt.ne_bot ClusterPt.neBotₓ'. -/
 theorem ClusterPt.neBot {x : α} {F : Filter α} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) :=
   h
 #align cluster_pt.ne_bot ClusterPt.neBot
 
-/- warning: filter.has_basis.cluster_pt_iff -> Filter.HasBasis.clusterPt_iff is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align filter.has_basis.cluster_pt_iff Filter.HasBasis.clusterPt_iffₓ'. -/
 theorem Filter.HasBasis.clusterPt_iff {ιa ιF} {pa : ιa → Prop} {sa : ιa → Set α} {pF : ιF → Prop}
     {sF : ιF → Set α} {F : Filter α} (ha : (𝓝 a).HasBasis pa sa) (hF : F.HasBasis pF sF) :
     ClusterPt a F ↔ ∀ ⦃i⦄ (hi : pa i) ⦃j⦄ (hj : pF j), (sa i ∩ sF j).Nonempty :=
   ha.inf_basis_neBot_iff hF
 #align filter.has_basis.cluster_pt_iff Filter.HasBasis.clusterPt_iff
 
-/- warning: cluster_pt_iff -> clusterPt_iff is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align cluster_pt_iff clusterPt_iffₓ'. -/
 theorem clusterPt_iff {x : α} {F : Filter α} :
     ClusterPt x F ↔ ∀ ⦃U : Set α⦄ (hU : U ∈ 𝓝 x) ⦃V⦄ (hV : V ∈ F), (U ∩ V).Nonempty :=
   inf_neBot_iff
 #align cluster_pt_iff clusterPt_iff
 
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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 cluster_pt_principal_iff clusterPt_principal_iffₓ'. -/
 /-- `x` is a cluster point of a set `s` if every neighbourhood of `x` meets `s` on a nonempty
 set. See also `mem_closure_iff_cluster_pt`. -/
 theorem clusterPt_principal_iff {x : α} {s : Set α} :
@@ -1776,73 +1344,31 @@ theorem clusterPt_principal_iff_frequently {x : α} {s : Set α} :
 #align cluster_pt_principal_iff_frequently clusterPt_principal_iff_frequently
 -/
 
-/- warning: cluster_pt.of_le_nhds -> ClusterPt.of_le_nhds is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align cluster_pt.of_le_nhds ClusterPt.of_le_nhdsₓ'. -/
 theorem ClusterPt.of_le_nhds {x : α} {f : Filter α} (H : f ≤ 𝓝 x) [NeBot f] : ClusterPt x f := by
   rwa [ClusterPt, inf_eq_right.mpr H]
 #align cluster_pt.of_le_nhds ClusterPt.of_le_nhds
 
-/- warning: cluster_pt.of_le_nhds' -> ClusterPt.of_le_nhds' is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align cluster_pt.of_le_nhds' ClusterPt.of_le_nhds'ₓ'. -/
 theorem ClusterPt.of_le_nhds' {x : α} {f : Filter α} (H : f ≤ 𝓝 x) (hf : NeBot f) : ClusterPt x f :=
   ClusterPt.of_le_nhds H
 #align cluster_pt.of_le_nhds' ClusterPt.of_le_nhds'
 
-/- warning: cluster_pt.of_nhds_le -> ClusterPt.of_nhds_le is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align cluster_pt.of_nhds_le ClusterPt.of_nhds_leₓ'. -/
 theorem ClusterPt.of_nhds_le {x : α} {f : Filter α} (H : 𝓝 x ≤ f) : ClusterPt x f := by
   simp only [ClusterPt, inf_eq_left.mpr H, nhds_neBot]
 #align cluster_pt.of_nhds_le ClusterPt.of_nhds_le
 
-/- warning: cluster_pt.mono -> ClusterPt.mono is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align cluster_pt.mono ClusterPt.monoₓ'. -/
 theorem ClusterPt.mono {x : α} {f g : Filter α} (H : ClusterPt x f) (h : f ≤ g) : ClusterPt x g :=
   ⟨ne_bot_of_le_ne_bot H.Ne <| inf_le_inf_left _ h⟩
 #align cluster_pt.mono ClusterPt.mono
 
-/- warning: cluster_pt.of_inf_left -> ClusterPt.of_inf_left is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align cluster_pt.of_inf_left ClusterPt.of_inf_leftₓ'. -/
 theorem ClusterPt.of_inf_left {x : α} {f g : Filter α} (H : ClusterPt x <| f ⊓ g) : ClusterPt x f :=
   H.mono inf_le_left
 #align cluster_pt.of_inf_left ClusterPt.of_inf_left
 
-/- warning: cluster_pt.of_inf_right -> ClusterPt.of_inf_right is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align cluster_pt.of_inf_right ClusterPt.of_inf_rightₓ'. -/
 theorem ClusterPt.of_inf_right {x : α} {f g : Filter α} (H : ClusterPt x <| f ⊓ g) :
     ClusterPt x g :=
   H.mono inf_le_right
 #align cluster_pt.of_inf_right ClusterPt.of_inf_right
 
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-Case conversion may be inaccurate. Consider using '#align ultrafilter.cluster_pt_iff Ultrafilter.clusterPt_iffₓ'. -/
 theorem Ultrafilter.clusterPt_iff {x : α} {f : Ultrafilter α} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
   ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩
 #align ultrafilter.cluster_pt_iff Ultrafilter.clusterPt_iff
@@ -1855,23 +1381,11 @@ def MapClusterPt {ι : Type _} (x : α) (F : Filter ι) (u : ι → α) : Prop :
 #align map_cluster_pt MapClusterPt
 -/
 
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-Case conversion may be inaccurate. Consider using '#align map_cluster_pt_iff mapClusterPt_iffₓ'. -/
 theorem mapClusterPt_iff {ι : Type _} (x : α) (F : Filter ι) (u : ι → α) :
     MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s := by
   simp_rw [MapClusterPt, ClusterPt, inf_ne_bot_iff_frequently_left, frequently_map]; rfl
 #align map_cluster_pt_iff mapClusterPt_iff
 
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-Case conversion may be inaccurate. Consider using '#align map_cluster_pt_of_comp mapClusterPt_of_compₓ'. -/
 theorem mapClusterPt_of_comp {ι δ : Type _} {F : Filter ι} {φ : δ → ι} {p : Filter δ} {x : α}
     {u : ι → α} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) : MapClusterPt x F u :=
   by
@@ -1891,33 +1405,15 @@ def AccPt (x : α) (F : Filter α) : Prop :=
 #align acc_pt AccPt
 -/
 
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-Case conversion may be inaccurate. Consider using '#align acc_iff_cluster acc_iff_clusterₓ'. -/
 theorem acc_iff_cluster (x : α) (F : Filter α) : AccPt x F ↔ ClusterPt x (𝓟 ({x}ᶜ) ⊓ F) := by
   rw [AccPt, nhdsWithin, ClusterPt, inf_assoc]
 #align acc_iff_cluster acc_iff_cluster
 
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-Case conversion may be inaccurate. Consider using '#align acc_principal_iff_cluster acc_principal_iff_clusterₓ'. -/
 /-- `x` is an accumulation point of a set `C` iff it is a cluster point of `C ∖ {x}`.-/
 theorem acc_principal_iff_cluster (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x})) :=
   by rw [acc_iff_cluster, inf_principal, inter_comm] <;> rfl
 #align acc_principal_iff_cluster acc_principal_iff_cluster
 
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-Case conversion may be inaccurate. Consider using '#align acc_pt_iff_nhds accPt_iff_nhdsₓ'. -/
 /-- `x` is an accumulation point of a set `C` iff every neighborhood
 of `x` contains a point of `C` other than `x`. -/
 theorem accPt_iff_nhds (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x := by
@@ -1933,12 +1429,6 @@ theorem accPt_iff_frequently (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∃ᶠ
 #align acc_pt_iff_frequently accPt_iff_frequently
 -/
 
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-Case conversion may be inaccurate. Consider using '#align acc_pt.mono AccPt.monoₓ'. -/
 /-- If `x` is an accumulation point of `F` and `F ≤ G`, then
 `x` is an accumulation point of `D. -/
 theorem AccPt.mono {x : α} {F G : Filter α} (h : AccPt x F) (hFG : F ≤ G) : AccPt x G :=
@@ -1956,12 +1446,6 @@ theorem interior_eq_nhds' {s : Set α} : interior s = { a | s ∈ 𝓝 a } :=
 #align interior_eq_nhds' interior_eq_nhds'
 -/
 
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 theorem interior_eq_nhds {s : Set α} : interior s = { a | 𝓝 a ≤ 𝓟 s } :=
   interior_eq_nhds'.trans <| by simp only [le_principal_iff]
 #align interior_eq_nhds interior_eq_nhds
@@ -1998,12 +1482,6 @@ theorem subset_interior_iff_nhds {s V : Set α} : s ⊆ interior V ↔ ∀ x ∈
 #align subset_interior_iff_nhds subset_interior_iff_nhds
 -/
 
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 theorem isOpen_iff_nhds {s : Set α} : IsOpen s ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s :=
   calc
     IsOpen s ↔ s ⊆ interior s := subset_interior_iff_isOpen.symm
@@ -2024,12 +1502,6 @@ theorem isOpen_iff_eventually {s : Set α} : IsOpen s ↔ ∀ x, x ∈ s → ∀
 #align is_open_iff_eventually isOpen_iff_eventually
 -/
 
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 theorem isOpen_iff_ultrafilter {s : Set α} :
     IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter α), ↑l ≤ 𝓝 x → s ∈ l := by
   simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]
@@ -2048,12 +1520,6 @@ theorem isOpen_singleton_iff_nhds_eq_pure (a : α) : IsOpen ({a} : Set α) ↔ 
 #align is_open_singleton_iff_nhds_eq_pure isOpen_singleton_iff_nhds_eq_pure
 -/
 
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 theorem isOpen_singleton_iff_punctured_nhds {α : Type _} [TopologicalSpace α] (a : α) :
     IsOpen ({a} : Set α) ↔ 𝓝[≠] a = ⊥ := by
   rw [isOpen_singleton_iff_nhds_eq_pure, nhdsWithin, ← mem_iff_inf_principal_compl, ← le_pure_iff,
@@ -2099,12 +1565,6 @@ theorem mem_closure_iff_clusterPt {s : Set α} {a : α} : a ∈ closure s ↔ Cl
 #align mem_closure_iff_cluster_pt mem_closure_iff_clusterPt
 -/
 
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 theorem mem_closure_iff_nhds_neBot {s : Set α} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ :=
   mem_closure_iff_clusterPt.trans neBot_iff
 #align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_neBot
@@ -2115,12 +1575,6 @@ theorem mem_closure_iff_nhdsWithin_neBot {s : Set α} {x : α} : x ∈ closure s
 #align mem_closure_iff_nhds_within_ne_bot mem_closure_iff_nhdsWithin_neBot
 -/
 
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-Case conversion may be inaccurate. Consider using '#align dense_compl_singleton dense_compl_singletonₓ'. -/
 /-- If `x` is not an isolated point of a topological space, then `{x}ᶜ` is dense in the whole
 space. -/
 theorem dense_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : Set α) :=
@@ -2131,12 +1585,6 @@ theorem dense_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : S
   · exact subset_closure hne
 #align dense_compl_singleton dense_compl_singleton
 
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 /-- If `x` is not an isolated point of a topological space, then the closure of `{x}ᶜ` is the whole
 space. -/
 @[simp]
@@ -2144,24 +1592,12 @@ theorem closure_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : closure ({x}ᶜ
   (dense_compl_singleton x).closure_eq
 #align closure_compl_singleton closure_compl_singleton
 
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 /-- If `x` is not an isolated point of a topological space, then the interior of `{x}` is empty. -/
 @[simp]
 theorem interior_singleton (x : α) [NeBot (𝓝[≠] x)] : interior {x} = (∅ : Set α) :=
   interior_eq_empty_iff_dense_compl.2 (dense_compl_singleton x)
 #align interior_singleton interior_singleton
 
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 theorem not_isOpen_singleton (x : α) [NeBot (𝓝[≠] x)] : ¬IsOpen ({x} : Set α) :=
   dense_compl_singleton_iff_not_open.1 (dense_compl_singleton x)
 #align not_is_open_singleton not_isOpen_singleton
@@ -2172,12 +1608,6 @@ theorem closure_eq_cluster_pts {s : Set α} : closure s = { a | ClusterPt a (
 #align closure_eq_cluster_pts closure_eq_cluster_pts
 -/
 
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-Case conversion may be inaccurate. Consider using '#align mem_closure_iff_nhds mem_closure_iff_nhdsₓ'. -/
 theorem mem_closure_iff_nhds {s : Set α} {a : α} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, (t ∩ s).Nonempty :=
   mem_closure_iff_clusterPt.trans clusterPt_principal_iff
 #align mem_closure_iff_nhds mem_closure_iff_nhds
@@ -2197,36 +1627,18 @@ theorem mem_closure_iff_comap_neBot {A : Set α} {x : α} :
 #align mem_closure_iff_comap_ne_bot mem_closure_iff_comap_neBot
 -/
 
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 theorem mem_closure_iff_nhds_basis' {a : α} {p : ι → Prop} {s : ι → Set α} (h : (𝓝 a).HasBasis p s)
     {t : Set α} : a ∈ closure t ↔ ∀ i, p i → (s i ∩ t).Nonempty :=
   mem_closure_iff_clusterPt.trans <|
     (h.clusterPt_iff (hasBasis_principal _)).trans <| by simp only [exists_prop, forall_const]
 #align mem_closure_iff_nhds_basis' mem_closure_iff_nhds_basis'
 
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 theorem mem_closure_iff_nhds_basis {a : α} {p : ι → Prop} {s : ι → Set α} (h : (𝓝 a).HasBasis p s)
     {t : Set α} : a ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i :=
   (mem_closure_iff_nhds_basis' h).trans <| by
     simp only [Set.Nonempty, mem_inter_iff, exists_prop, and_comm']
 #align mem_closure_iff_nhds_basis mem_closure_iff_nhds_basis
 
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 /-- `x` belongs to the closure of `s` if and only if some ultrafilter
   supported on `s` converges to `x`. -/
 theorem mem_closure_iff_ultrafilter {s : Set α} {x : α} :
@@ -2243,22 +1655,10 @@ theorem isClosed_iff_clusterPt {s : Set α} : IsClosed s ↔ ∀ a, ClusterPt a
 #align is_closed_iff_cluster_pt isClosed_iff_clusterPt
 -/
 
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 theorem isClosed_iff_nhds {s : Set α} : IsClosed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s :=
   by simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_ne_bot_iff]
 #align is_closed_iff_nhds isClosed_iff_nhds
 
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 theorem IsClosed.interior_union_left {s t : Set α} (h : IsClosed s) :
     interior (s ∪ t) ⊆ s ∪ interior t := fun a ⟨u, ⟨⟨hu₁, hu₂⟩, ha⟩⟩ =>
   (Classical.em (a ∈ s)).imp_right fun h =>
@@ -2267,43 +1667,19 @@ theorem IsClosed.interior_union_left {s t : Set α} (h : IsClosed s) :
         ⟨ha, h⟩⟩
 #align is_closed.interior_union_left IsClosed.interior_union_left
 
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 theorem IsClosed.interior_union_right {s t : Set α} (h : IsClosed t) :
     interior (s ∪ t) ⊆ interior s ∪ t := by simpa only [union_comm] using h.interior_union_left
 #align is_closed.interior_union_right IsClosed.interior_union_right
 
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-Case conversion may be inaccurate. Consider using '#align is_open.inter_closure IsOpen.inter_closureₓ'. -/
 theorem IsOpen.inter_closure {s t : Set α} (h : IsOpen s) : s ∩ closure t ⊆ closure (s ∩ t) :=
   compl_subset_compl.mp <| by
     simpa only [← interior_compl, compl_inter] using IsClosed.interior_union_left h.is_closed_compl
 #align is_open.inter_closure IsOpen.inter_closure
 
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 theorem IsOpen.closure_inter {s t : Set α} (h : IsOpen t) : closure s ∩ t ⊆ closure (s ∩ t) := by
   simpa only [inter_comm] using h.inter_closure
 #align is_open.closure_inter IsOpen.closure_inter
 
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-Case conversion may be inaccurate. Consider using '#align dense.open_subset_closure_inter Dense.open_subset_closure_interₓ'. -/
 theorem Dense.open_subset_closure_inter {s t : Set α} (hs : Dense s) (ht : IsOpen t) :
     t ⊆ closure (t ∩ s) :=
   calc
@@ -2312,12 +1688,6 @@ theorem Dense.open_subset_closure_inter {s t : Set α} (hs : Dense s) (ht : IsOp
     
 #align dense.open_subset_closure_inter Dense.open_subset_closure_inter
 
-/- warning: mem_closure_of_mem_closure_union -> mem_closure_of_mem_closure_union is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_unionₓ'. -/
 theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x ∈ closure (s₁ ∪ s₂))
     (h₁ : s₁ᶜ ∈ 𝓝 x) : x ∈ closure s₂ :=
   by
@@ -2331,48 +1701,24 @@ theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x 
       ]
 #align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_union
 
-/- warning: dense.inter_of_open_left -> Dense.inter_of_open_left is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align dense.inter_of_open_left Dense.inter_of_open_leftₓ'. -/
 /-- The intersection of an open dense set with a dense set is a dense set. -/
 theorem Dense.inter_of_open_left {s t : Set α} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
     Dense (s ∩ t) := fun x =>
   closure_minimal hso.inter_closure isClosed_closure <| by simp [hs.closure_eq, ht.closure_eq]
 #align dense.inter_of_open_left Dense.inter_of_open_left
 
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-Case conversion may be inaccurate. Consider using '#align dense.inter_of_open_right Dense.inter_of_open_rightₓ'. -/
 /-- The intersection of a dense set with an open dense set is a dense set. -/
 theorem Dense.inter_of_open_right {s t : Set α} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
     Dense (s ∩ t) :=
   inter_comm t s ▸ ht.inter_of_open_left hs hto
 #align dense.inter_of_open_right Dense.inter_of_open_right
 
-/- warning: dense.inter_nhds_nonempty -> Dense.inter_nhds_nonempty is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align dense.inter_nhds_nonempty Dense.inter_nhds_nonemptyₓ'. -/
 theorem Dense.inter_nhds_nonempty {s t : Set α} (hs : Dense s) {x : α} (ht : t ∈ 𝓝 x) :
     (s ∩ t).Nonempty :=
   let ⟨U, hsub, ho, hx⟩ := mem_nhds_iff.1 ht
   (hs.inter_open_nonempty U ho ⟨x, hx⟩).mono fun y hy => ⟨hy.2, hsub hy.1⟩
 #align dense.inter_nhds_nonempty Dense.inter_nhds_nonempty
 
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-Case conversion may be inaccurate. Consider using '#align closure_diff closure_diffₓ'. -/
 theorem closure_diff {s t : Set α} : closure s \ closure t ⊆ closure (s \ t) :=
   calc
     closure s \ closure t = closure tᶜ ∩ closure s := by simp only [diff_eq, inter_comm]
@@ -2417,12 +1763,6 @@ theorem mem_closure_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set
 #align mem_closure_of_tendsto mem_closure_of_tendsto
 -/
 
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-Case conversion may be inaccurate. Consider using '#align tendsto_inf_principal_nhds_iff_of_forall_eq tendsto_inf_principal_nhds_iff_of_forall_eqₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
 Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/
@@ -2478,12 +1818,6 @@ noncomputable def limUnder [Nonempty α] (f : Filter β) (g : β → α) : α :=
 #align lim limUnder
 -/
 
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-Case conversion may be inaccurate. Consider using '#align le_nhds_Lim le_nhds_limₓ'. -/
 /-- If a filter `f` is majorated by some `𝓝 a`, then it is majorated by `𝓝 (Lim f)`. We formulate
 this lemma with a `[nonempty α]` argument of `Lim` derived from `h` to make it useful for types
 without a `[nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify
@@ -2528,33 +1862,15 @@ structure Continuous (f : α → β) : Prop where
 #align continuous Continuous
 -/
 
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-Case conversion may be inaccurate. Consider using '#align continuous_def continuous_defₓ'. -/
 theorem continuous_def {f : α → β} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) :=
   ⟨fun hf s hs => hf.isOpen_preimage s hs, fun h => ⟨h⟩⟩
 #align continuous_def continuous_def
 
-/- warning: is_open.preimage -> IsOpen.preimage is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align is_open.preimage IsOpen.preimageₓ'. -/
 theorem IsOpen.preimage {f : α → β} (hf : Continuous f) {s : Set β} (h : IsOpen s) :
     IsOpen (f ⁻¹' s) :=
   hf.isOpen_preimage s h
 #align is_open.preimage IsOpen.preimage
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous.congr Continuous.congrₓ'. -/
 theorem Continuous.congr {f g : α → β} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g := by
   convert h; ext; rw [h']
 #align continuous.congr Continuous.congr
@@ -2567,44 +1883,20 @@ def ContinuousAt (f : α → β) (x : α) :=
 #align continuous_at ContinuousAt
 -/
 
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-Case conversion may be inaccurate. Consider using '#align continuous_at.tendsto ContinuousAt.tendstoₓ'. -/
 theorem ContinuousAt.tendsto {f : α → β} {x : α} (h : ContinuousAt f x) :
     Tendsto f (𝓝 x) (𝓝 (f x)) :=
   h
 #align continuous_at.tendsto ContinuousAt.tendsto
 
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-Case conversion may be inaccurate. Consider using '#align continuous_at_def continuousAt_defₓ'. -/
 theorem continuousAt_def {f : α → β} {x : α} : ContinuousAt f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x :=
   Iff.rfl
 #align continuous_at_def continuousAt_def
 
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-Case conversion may be inaccurate. Consider using '#align continuous_at_congr continuousAt_congrₓ'. -/
 theorem continuousAt_congr {f g : α → β} {x : α} (h : f =ᶠ[𝓝 x] g) :
     ContinuousAt f x ↔ ContinuousAt g x := by
   simp only [ContinuousAt, tendsto_congr' h, h.eq_of_nhds]
 #align continuous_at_congr continuousAt_congr
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous_at.congr ContinuousAt.congrₓ'. -/
 theorem ContinuousAt.congr {f g : α → β} {x : α} (hf : ContinuousAt f x) (h : f =ᶠ[𝓝 x] g) :
     ContinuousAt g x :=
   (continuousAt_congr h).1 hf
@@ -2624,12 +1916,6 @@ theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {a : α} {f : α → M₀} :
 #align eventually_eq_zero_nhds eventuallyEq_zero_nhds
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align cluster_pt.map ClusterPt.mapₓ'. -/
 theorem ClusterPt.map {x : α} {la : Filter α} {lb : Filter β} (H : ClusterPt x la) {f : α → β}
     (hfc : ContinuousAt f x) (hf : Tendsto f la lb) : ClusterPt (f x) lb :=
   ⟨ne_bot_of_le_ne_bot ((map_neBot_iff f).2 H).Ne <| hfc.Tendsto.inf hf⟩
@@ -2649,12 +1935,6 @@ theorem continuous_id : Continuous (id : α → α) :=
 #align continuous_id continuous_id
 -/
 
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-Case conversion may be inaccurate. Consider using '#align continuous.comp Continuous.compₓ'. -/
 theorem Continuous.comp {g : β → γ} {f : α → β} (hg : Continuous g) (hf : Continuous f) :
     Continuous (g ∘ f) :=
   continuous_def.2 fun s h => (h.Preimage hg).Preimage hf
@@ -2666,34 +1946,16 @@ theorem Continuous.iterate {f : α → α} (h : Continuous f) (n : ℕ) : Contin
 #align continuous.iterate Continuous.iterate
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous_at.comp ContinuousAt.compₓ'. -/
 theorem ContinuousAt.comp {g : β → γ} {f : α → β} {x : α} (hg : ContinuousAt g (f x))
     (hf : ContinuousAt f x) : ContinuousAt (g ∘ f) x :=
   hg.comp hf
 #align continuous_at.comp ContinuousAt.comp
 
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-Case conversion may be inaccurate. Consider using '#align continuous.tendsto Continuous.tendstoₓ'. -/
 theorem Continuous.tendsto {f : α → β} (hf : Continuous f) (x) : Tendsto f (𝓝 x) (𝓝 (f x)) :=
   ((nhds_basis_opens x).tendsto_iffₓ <| nhds_basis_opens <| f x).2 fun t ⟨hxt, ht⟩ =>
     ⟨f ⁻¹' t, ⟨hxt, ht.Preimage hf⟩, Subset.refl _⟩
 #align continuous.tendsto Continuous.tendsto
 
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-Case conversion may be inaccurate. Consider using '#align continuous.tendsto' Continuous.tendsto'ₓ'. -/
 /-- A version of `continuous.tendsto` that allows one to specify a simpler form of the limit.
 E.g., one can write `continuous_exp.tendsto' 0 1 exp_zero`. -/
 theorem Continuous.tendsto' {f : α → β} (hf : Continuous f) (x : α) (y : β) (h : f x = y) :
@@ -2701,22 +1963,10 @@ theorem Continuous.tendsto' {f : α → β} (hf : Continuous f) (x : α) (y : β
   h ▸ hf.Tendsto x
 #align continuous.tendsto' Continuous.tendsto'
 
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-Case conversion may be inaccurate. Consider using '#align continuous.continuous_at Continuous.continuousAtₓ'. -/
 theorem Continuous.continuousAt {f : α → β} {x : α} (h : Continuous f) : ContinuousAt f x :=
   h.Tendsto x
 #align continuous.continuous_at Continuous.continuousAt
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align continuous_iff_continuous_at continuous_iff_continuousAtₓ'. -/
 theorem continuous_iff_continuousAt {f : α → β} : Continuous f ↔ ∀ x, ContinuousAt f x :=
   ⟨Continuous.tendsto, fun hf : ∀ x, Tendsto f (𝓝 x) (𝓝 (f x)) =>
     continuous_def.2 fun s => fun hs : IsOpen s =>
@@ -2725,32 +1975,14 @@ theorem continuous_iff_continuousAt {f : α → β} : Continuous f ↔ ∀ x, Co
         isOpen_iff_nhds.2 fun a ha => le_principal_iff.2 <| hf _ (this a ha)⟩
 #align continuous_iff_continuous_at continuous_iff_continuousAt
 
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-Case conversion may be inaccurate. Consider using '#align continuous_at_const continuousAt_constₓ'. -/
 theorem continuousAt_const {x : α} {b : β} : ContinuousAt (fun a : α => b) x :=
   tendsto_const_nhds
 #align continuous_at_const continuousAt_const
 
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-Case conversion may be inaccurate. Consider using '#align continuous_const continuous_constₓ'. -/
 theorem continuous_const {b : β} : Continuous fun a : α => b :=
   continuous_iff_continuousAt.mpr fun a => continuousAt_const
 #align continuous_const continuous_const
 
-/- warning: filter.eventually_eq.continuous_at -> Filter.EventuallyEq.continuousAt is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align filter.eventually_eq.continuous_at Filter.EventuallyEq.continuousAtₓ'. -/
 theorem Filter.EventuallyEq.continuousAt {x : α} {f : α → β} {y : β} (h : f =ᶠ[𝓝 x] fun _ => y) :
     ContinuousAt f x :=
   (continuousAt_congr h).2 tendsto_const_nhds
@@ -2777,69 +2009,33 @@ theorem ContinuousAt.iterate {f : α → α} {x : α} (hf : ContinuousAt f x) (h
 #align continuous_at.iterate ContinuousAt.iterate
 -/
 
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 theorem continuous_iff_isClosed {f : α → β} : Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s) :=
   ⟨fun hf s hs => by simpa using (continuous_def.1 hf (sᶜ) hs.is_open_compl).isClosed_compl,
     fun hf =>
     continuous_def.2 fun s => by rw [← isClosed_compl_iff, ← isClosed_compl_iff] <;> exact hf _⟩
 #align continuous_iff_is_closed continuous_iff_isClosed
 
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 theorem IsClosed.preimage {f : α → β} (hf : Continuous f) {s : Set β} (h : IsClosed s) :
     IsClosed (f ⁻¹' s) :=
   continuous_iff_isClosed.mp hf s h
 #align is_closed.preimage IsClosed.preimage
 
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 theorem mem_closure_image {f : α → β} {x : α} {s : Set α} (hf : ContinuousAt f x)
     (hx : x ∈ closure s) : f x ∈ closure (f '' s) :=
   mem_closure_of_frequently_of_tendsto
     ((mem_closure_iff_frequently.1 hx).mono fun x => mem_image_of_mem _) hf
 #align mem_closure_image mem_closure_image
 
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 theorem continuousAt_iff_ultrafilter {f : α → β} {x} :
     ContinuousAt f x ↔ ∀ g : Ultrafilter α, ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) :=
   tendsto_iff_ultrafilter f (𝓝 x) (𝓝 (f x))
 #align continuous_at_iff_ultrafilter continuousAt_iff_ultrafilter
 
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-Case conversion may be inaccurate. Consider using '#align continuous_iff_ultrafilter continuous_iff_ultrafilterₓ'. -/
 theorem continuous_iff_ultrafilter {f : α → β} :
     Continuous f ↔ ∀ (x) (g : Ultrafilter α), ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) := by
   simp only [continuous_iff_continuousAt, continuousAt_iff_ultrafilter]
 #align continuous_iff_ultrafilter continuous_iff_ultrafilter
 
-/- warning: continuous.closure_preimage_subset -> Continuous.closure_preimage_subset 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] {f : α -> β}, (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (forall (t : Set.{u2} β), HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (closure.{u1} α _inst_1 (Set.preimage.{u1, u2} α β f t)) (Set.preimage.{u1, u2} α β f (closure.{u2} β _inst_2 t)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous.closure_preimage_subset Continuous.closure_preimage_subsetₓ'. -/
 theorem Continuous.closure_preimage_subset {f : α → β} (hf : Continuous f) (t : Set β) :
     closure (f ⁻¹' t) ⊆ f ⁻¹' closure t :=
   by
@@ -2847,23 +2043,11 @@ theorem Continuous.closure_preimage_subset {f : α → β} (hf : Continuous f) (
   exact closure_mono (preimage_mono subset_closure)
 #align continuous.closure_preimage_subset Continuous.closure_preimage_subset
 
-/- warning: continuous.frontier_preimage_subset -> Continuous.frontier_preimage_subset 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] {f : α -> β}, (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (forall (t : Set.{u2} β), HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (frontier.{u1} α _inst_1 (Set.preimage.{u1, u2} α β f t)) (Set.preimage.{u1, u2} α β f (frontier.{u2} β _inst_2 t)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align continuous.frontier_preimage_subset Continuous.frontier_preimage_subsetₓ'. -/
 theorem Continuous.frontier_preimage_subset {f : α → β} (hf : Continuous f) (t : Set β) :
     frontier (f ⁻¹' t) ⊆ f ⁻¹' frontier t :=
   diff_subset_diff (hf.closure_preimage_subset t) (preimage_interior_subset_interior_preimage hf)
 #align continuous.frontier_preimage_subset Continuous.frontier_preimage_subset
 
-/- warning: set.maps_to.closure -> Set.MapsTo.closure 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] {s : Set.{u1} α} {t : Set.{u2} β} {f : α -> β}, (Set.MapsTo.{u1, u2} α β f s t) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Set.MapsTo.{u1, u2} α β f (closure.{u1} α _inst_1 s) (closure.{u2} β _inst_2 t))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {s : Set.{u2} α} {t : Set.{u1} β} {f : α -> β}, (Set.MapsTo.{u2, u1} α β f s t) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Set.MapsTo.{u2, u1} α β f (closure.{u2} α _inst_1 s) (closure.{u1} β _inst_2 t))
-Case conversion may be inaccurate. Consider using '#align set.maps_to.closure Set.MapsTo.closureₓ'. -/
 /-- If a continuous map `f` maps `s` to `t`, then it maps `closure s` to `closure t`. -/
 theorem Set.MapsTo.closure {s : Set α} {t : Set β} {f : α → β} (h : MapsTo f s t)
     (hc : Continuous f) : MapsTo f (closure s) (closure t) :=
@@ -2872,45 +2056,21 @@ theorem Set.MapsTo.closure {s : Set α} {t : Set β} {f : α → β} (h : MapsTo
   exact fun x hx => hx.map hc.continuous_at (tendsto_principal_principal.2 h)
 #align set.maps_to.closure Set.MapsTo.closure
 
-/- warning: image_closure_subset_closure_image -> image_closure_subset_closure_image 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] {f : α -> β} {s : Set.{u1} α}, (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.hasSubset.{u2} β) (Set.image.{u1, u2} α β f (closure.{u1} α _inst_1 s)) (closure.{u2} β _inst_2 (Set.image.{u1, u2} α β f s)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align image_closure_subset_closure_image image_closure_subset_closure_imageₓ'. -/
 theorem image_closure_subset_closure_image {f : α → β} {s : Set α} (h : Continuous f) :
     f '' closure s ⊆ closure (f '' s) :=
   ((mapsTo_image f s).closure h).image_subset
 #align image_closure_subset_closure_image image_closure_subset_closure_image
 
-/- warning: closure_subset_preimage_closure_image -> closure_subset_preimage_closure_image 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] {f : α -> β} {s : Set.{u1} α}, (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (closure.{u1} α _inst_1 s) (Set.preimage.{u1, u2} α β f (closure.{u2} β _inst_2 (Set.image.{u1, u2} α β f s))))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align closure_subset_preimage_closure_image closure_subset_preimage_closure_imageₓ'. -/
 theorem closure_subset_preimage_closure_image {f : α → β} {s : Set α} (h : Continuous f) :
     closure s ⊆ f ⁻¹' closure (f '' s) := by rw [← Set.image_subset_iff];
   exact image_closure_subset_closure_image h
 #align closure_subset_preimage_closure_image closure_subset_preimage_closure_image
 
-/- warning: map_mem_closure -> map_mem_closure is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {s : Set.{u2} α} {t : Set.{u1} β} {f : α -> β} {a : α}, (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) a (closure.{u2} α _inst_1 s)) -> (Set.MapsTo.{u2, u1} α β f s t) -> (Membership.mem.{u1, u1} β (Set.{u1} β) (Set.instMembershipSet.{u1} β) (f a) (closure.{u1} β _inst_2 t))
-Case conversion may be inaccurate. Consider using '#align map_mem_closure map_mem_closureₓ'. -/
 theorem map_mem_closure {s : Set α} {t : Set β} {f : α → β} {a : α} (hf : Continuous f)
     (ha : a ∈ closure s) (ht : MapsTo f s t) : f a ∈ closure t :=
   ht.closure hf ha
 #align map_mem_closure map_mem_closure
 
-/- warning: set.maps_to.closure_left -> Set.MapsTo.closure_left 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] {s : Set.{u1} α} {t : Set.{u2} β} {f : α -> β}, (Set.MapsTo.{u1, u2} α β f s t) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (IsClosed.{u2} β _inst_2 t) -> (Set.MapsTo.{u1, u2} α β f (closure.{u1} α _inst_1 s) t)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {s : Set.{u2} α} {t : Set.{u1} β} {f : α -> β}, (Set.MapsTo.{u2, u1} α β f s t) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (IsClosed.{u1} β _inst_2 t) -> (Set.MapsTo.{u2, u1} α β f (closure.{u2} α _inst_1 s) t)
-Case conversion may be inaccurate. Consider using '#align set.maps_to.closure_left Set.MapsTo.closure_leftₓ'. -/
 /-- If a continuous map `f` maps `s` to a closed set `t`, then it maps `closure s` to `t`. -/
 theorem Set.MapsTo.closure_left {s : Set α} {t : Set β} {f : α → β} (h : MapsTo f s t)
     (hc : Continuous f) (ht : IsClosed t) : MapsTo f (closure s) t :=
@@ -2948,22 +2108,10 @@ theorem denseRange_id : DenseRange (id : α → α) :=
 #align dense_range_id denseRange_id
 -/
 
-/- warning: dense_range_iff_closure_range -> denseRange_iff_closure_range is a dubious translation:
-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 dense_range_iff_closure_range denseRange_iff_closure_rangeₓ'. -/
 theorem denseRange_iff_closure_range : DenseRange f ↔ closure (range f) = univ :=
   dense_iff_closure_eq
 #align dense_range_iff_closure_range denseRange_iff_closure_range
 
-/- warning: dense_range.closure_range -> DenseRange.closure_range is a dubious translation:
-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 dense_range.closure_range DenseRange.closure_rangeₓ'. -/
 theorem DenseRange.closure_range (h : DenseRange f) : closure (range f) = univ :=
   h.closure_eq
 #align dense_range.closure_range DenseRange.closure_range
@@ -2974,12 +2122,6 @@ theorem Dense.denseRange_val {s : Set α} (h : Dense s) : DenseRange (coe : s 
 #align dense.dense_range_coe Dense.denseRange_val
 -/
 
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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 continuous.range_subset_closure_image_dense Continuous.range_subset_closure_image_denseₓ'. -/
 theorem Continuous.range_subset_closure_image_dense {f : α → β} (hf : Continuous f) {s : Set α}
     (hs : Dense s) : range f ⊆ closure (f '' s) := by rw [← image_univ, ← hs.closure_eq];
   exact image_closure_subset_closure_image hf
@@ -2993,12 +2135,6 @@ theorem DenseRange.dense_image {f : α → β} (hf' : DenseRange f) (hf : Contin
 #align dense_range.dense_image DenseRange.dense_image
 -/
 
-/- warning: dense_range.subset_closure_image_preimage_of_is_open -> DenseRange.subset_closure_image_preimage_of_isOpen is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {κ : Type.{u2}} {f : κ -> β}, (DenseRange.{u1, u2} β _inst_2 κ f) -> (forall {s : Set.{u1} β}, (IsOpen.{u1} β _inst_2 s) -> (HasSubset.Subset.{u1} (Set.{u1} β) (Set.hasSubset.{u1} β) s (closure.{u1} β _inst_2 (Set.image.{u2, u1} κ β f (Set.preimage.{u2, u1} κ β f s)))))
-but is expected to have type
-  forall {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {κ : Type.{u1}} {f : κ -> β}, (DenseRange.{u2, u1} β _inst_2 κ f) -> (forall {s : Set.{u2} β}, (IsOpen.{u2} β _inst_2 s) -> (HasSubset.Subset.{u2} (Set.{u2} β) (Set.instHasSubsetSet.{u2} β) s (closure.{u2} β _inst_2 (Set.image.{u1, u2} κ β f (Set.preimage.{u1, u2} κ β f s)))))
-Case conversion may be inaccurate. Consider using '#align dense_range.subset_closure_image_preimage_of_is_open DenseRange.subset_closure_image_preimage_of_isOpenₓ'. -/
 /-- If `f` has dense range and `s` is an open set in the codomain of `f`, then the image of the
 preimage of `s` under `f` is dense in `s`. -/
 theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) {s : Set β}
@@ -3015,12 +2151,6 @@ theorem DenseRange.dense_of_mapsTo {f : α → β} (hf' : DenseRange f) (hf : Co
 #align dense_range.dense_of_maps_to DenseRange.dense_of_mapsTo
 -/
 
-/- warning: dense_range.comp -> DenseRange.comp is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} {γ : Type.{u2}} [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : TopologicalSpace.{u2} γ] {κ : Type.{u3}} {g : β -> γ} {f : κ -> β}, (DenseRange.{u2, u1} γ _inst_3 β g) -> (DenseRange.{u1, u3} β _inst_2 κ f) -> (Continuous.{u1, u2} β γ _inst_2 _inst_3 g) -> (DenseRange.{u2, u3} γ _inst_3 κ (Function.comp.{succ u3, succ u1, succ u2} κ β γ g f))
-but is expected to have type
-  forall {β : Type.{u2}} {γ : Type.{u3}} [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {κ : Type.{u1}} {g : β -> γ} {f : κ -> β}, (DenseRange.{u3, u2} γ _inst_3 β g) -> (DenseRange.{u2, u1} β _inst_2 κ f) -> (Continuous.{u2, u3} β γ _inst_2 _inst_3 g) -> (DenseRange.{u3, u1} γ _inst_3 κ (Function.comp.{succ u1, succ u2, succ u3} κ β γ g f))
-Case conversion may be inaccurate. Consider using '#align dense_range.comp DenseRange.compₓ'. -/
 /-- Composition of a continuous map with dense range and a function with dense range has dense
 range. -/
 theorem DenseRange.comp {g : β → γ} {f : κ → β} (hg : DenseRange g) (hf : DenseRange f)
@@ -3028,22 +2158,10 @@ theorem DenseRange.comp {g : β → γ} {f : κ → β} (hg : DenseRange g) (hf
   exact hg.dense_image cg hf
 #align dense_range.comp DenseRange.comp
 
-/- warning: dense_range.nonempty_iff -> DenseRange.nonempty_iff is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {κ : Type.{u2}} {f : κ -> β}, (DenseRange.{u1, u2} β _inst_2 κ f) -> (Iff (Nonempty.{succ u2} κ) (Nonempty.{succ u1} β))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align dense_range.nonempty_iff DenseRange.nonempty_iffₓ'. -/
 theorem DenseRange.nonempty_iff (hf : DenseRange f) : Nonempty κ ↔ Nonempty β :=
   range_nonempty_iff_nonempty.symm.trans hf.nonempty_iff
 #align dense_range.nonempty_iff DenseRange.nonempty_iff
 
-/- warning: dense_range.nonempty -> DenseRange.nonempty is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {κ : Type.{u2}} {f : κ -> β} [h : Nonempty.{succ u1} β], (DenseRange.{u1, u2} β _inst_2 κ f) -> (Nonempty.{succ u2} κ)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align dense_range.nonempty DenseRange.nonemptyₓ'. -/
 theorem DenseRange.nonempty [h : Nonempty β] (hf : DenseRange f) : Nonempty κ :=
   hf.nonempty_iff.mpr h
 #align dense_range.nonempty DenseRange.nonempty
@@ -3055,23 +2173,11 @@ def DenseRange.some (hf : DenseRange f) (b : β) : κ :=
 #align dense_range.some DenseRange.some
 -/
 
-/- warning: dense_range.exists_mem_open -> DenseRange.exists_mem_open is a dubious translation:
-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 dense_range.exists_mem_open DenseRange.exists_mem_openₓ'. -/
 theorem DenseRange.exists_mem_open (hf : DenseRange f) {s : Set β} (ho : IsOpen s)
     (hs : s.Nonempty) : ∃ a, f a ∈ s :=
   exists_range_iff.1 <| hf.exists_mem_open ho hs
 #align dense_range.exists_mem_open DenseRange.exists_mem_open
 
-/- warning: dense_range.mem_nhds -> DenseRange.mem_nhds is a dubious translation:
-lean 3 declaration is
-  forall {β : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} β] {κ : Type.{u2}} {f : κ -> β}, (DenseRange.{u1, u2} β _inst_2 κ f) -> (forall {b : β} {U : Set.{u1} β}, (Membership.Mem.{u1, u1} (Set.{u1} β) (Filter.{u1} β) (Filter.hasMem.{u1} β) U (nhds.{u1} β _inst_2 b)) -> (Exists.{succ u2} κ (fun (a : κ) => Membership.Mem.{u1, u1} β (Set.{u1} β) (Set.hasMem.{u1} β) (f a) U)))
-but is expected to have type
-  forall {β : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} β] {κ : Type.{u1}} {f : κ -> β}, (DenseRange.{u2, u1} β _inst_2 κ f) -> (forall {b : β} {U : Set.{u2} β}, (Membership.mem.{u2, u2} (Set.{u2} β) (Filter.{u2} β) (instMembershipSetFilter.{u2} β) U (nhds.{u2} β _inst_2 b)) -> (Exists.{succ u1} κ (fun (a : κ) => Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) (f a) U)))
-Case conversion may be inaccurate. Consider using '#align dense_range.mem_nhds DenseRange.mem_nhdsₓ'. -/
 theorem DenseRange.mem_nhds {f : κ → β} (h : DenseRange f) {b : β} {U : Set β} (U_in : U ∈ 𝓝 b) :
     ∃ a, f a ∈ U :=
   let ⟨a, ha⟩ := h.exists_mem_open isOpen_interior ⟨b, mem_interior_iff_mem_nhds.2 U_in⟩

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

@@ -129,12 +129,8 @@ theorem isOpen_mk {p h₁ h₂ h₃} {s : Set α} : is_open[⟨p, h₁, h₂, h
 
 #print topologicalSpace_eq /-
 @[ext]
-theorem topologicalSpace_eq {f g : TopologicalSpace α} (h : is_open[f] = is_open[g]) : f = g :=
-  by
-  cases f
-  cases g
-  congr
-  exact h
+theorem topologicalSpace_eq {f g : TopologicalSpace α} (h : is_open[f] = is_open[g]) : f = g := by
+  cases f; cases g; congr ; exact h
 #align topological_space_eq topologicalSpace_eq
 -/
 
@@ -170,9 +166,7 @@ end
 #print topologicalSpace_eq_iff /-
 theorem topologicalSpace_eq_iff {t t' : TopologicalSpace α} :
     t = t' ↔ ∀ s, is_open[t] s ↔ is_open[t'] s :=
-  ⟨fun h s => h ▸ Iff.rfl, fun h => by
-    ext
-    exact h _⟩
+  ⟨fun h s => h ▸ Iff.rfl, fun h => by ext; exact h _⟩
 #align topological_space_eq_iff topologicalSpace_eq_iff
 -/
 
@@ -282,18 +276,14 @@ theorem isOpen_compl_iff {s : Set α} : IsOpen (sᶜ) ↔ IsClosed s :=
 
 #print isClosed_empty /-
 @[simp]
-theorem isClosed_empty : IsClosed (∅ : Set α) :=
-  by
-  rw [← isOpen_compl_iff, compl_empty]
+theorem isClosed_empty : IsClosed (∅ : Set α) := by rw [← isOpen_compl_iff, compl_empty];
   exact isOpen_univ
 #align is_closed_empty isClosed_empty
 -/
 
 #print isClosed_univ /-
 @[simp]
-theorem isClosed_univ : IsClosed (univ : Set α) :=
-  by
-  rw [← isOpen_compl_iff, compl_univ]
+theorem isClosed_univ : IsClosed (univ : Set α) := by rw [← isOpen_compl_iff, compl_univ];
   exact isOpen_empty
 #align is_closed_univ isClosed_univ
 -/
@@ -304,11 +294,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} {s₁ : Set.{u1} α} {s₂ : Set.{u1} α} [_inst_1 : TopologicalSpace.{u1} α], (IsClosed.{u1} α _inst_1 s₁) -> (IsClosed.{u1} α _inst_1 s₂) -> (IsClosed.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s₁ s₂))
 Case conversion may be inaccurate. Consider using '#align is_closed.union IsClosed.unionₓ'. -/
-theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂) := fun h₁ h₂ =>
-  by
-  rw [← isOpen_compl_iff] at *
-  rw [compl_union]
-  exact IsOpen.inter h₁ h₂
+theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂) := fun h₁ h₂ => by
+  rw [← isOpen_compl_iff] at *; rw [compl_union]; exact IsOpen.inter h₁ h₂
 #align is_closed.union IsClosed.union
 
 #print isClosed_sInter /-
@@ -367,11 +354,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} {s₁ : Set.{u1} α} {s₂ : Set.{u1} α} [_inst_1 : TopologicalSpace.{u1} α], (IsClosed.{u1} α _inst_1 s₁) -> (IsClosed.{u1} α _inst_1 s₂) -> (IsClosed.{u1} α _inst_1 (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s₁ s₂))
 Case conversion may be inaccurate. Consider using '#align is_closed.inter IsClosed.interₓ'. -/
-theorem IsClosed.inter (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) : IsClosed (s₁ ∩ s₂) :=
-  by
-  rw [← isOpen_compl_iff] at *
-  rw [compl_inter]
-  exact IsOpen.union h₁ h₂
+theorem IsClosed.inter (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) : IsClosed (s₁ ∩ s₂) := by
+  rw [← isOpen_compl_iff] at *; rw [compl_inter]; exact IsOpen.union h₁ h₂
 #align is_closed.inter IsClosed.inter
 
 /- warning: is_closed.sdiff -> IsClosed.sdiff is a dubious translation:
@@ -396,9 +380,7 @@ theorem isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite) :
 
 #print isClosed_iUnion /-
 theorem isClosed_iUnion [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
-    IsClosed (⋃ i, s i) :=
-  by
-  simp only [← isOpen_compl_iff, compl_Union] at *
+    IsClosed (⋃ i, s i) := by simp only [← isOpen_compl_iff, compl_Union] at *;
   exact isOpen_iInter h
 #align is_closed_Union isClosed_iUnion
 -/
@@ -559,9 +541,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align interior_Inter interior_iInterₓ'. -/
 @[simp]
 theorem interior_iInter {ι : Type _} [Finite ι] (f : ι → Set α) :
-    interior (⋂ i, f i) = ⋂ i, interior (f i) :=
-  by
-  cases nonempty_fintype ι
+    interior (⋂ i, f i) = ⋂ i, interior (f i) := by cases nonempty_fintype ι;
   convert finset.univ.interior_Inter f <;> simp
 #align interior_Inter interior_iInter
 
@@ -837,9 +817,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align closure_Union closure_iUnionₓ'. -/
 @[simp]
 theorem closure_iUnion {ι : Type _} [Finite ι] (f : ι → Set α) :
-    closure (⋃ i, f i) = ⋃ i, closure (f i) :=
-  by
-  cases nonempty_fintype ι
+    closure (⋃ i, f i) = ⋃ i, closure (f i) := by cases nonempty_fintype ι;
   convert finset.univ.closure_bUnion f <;> simp
 #align closure_Union closure_iUnion
 
@@ -1081,8 +1059,7 @@ theorem dense_compl_singleton_iff_not_open {x : α} : Dense ({x}ᶜ : Set α) 
   · intro hd ho
     exact (hd.inter_open_nonempty _ ho (singleton_nonempty _)).ne_empty (inter_compl_self _)
   · refine' fun ho => dense_iff_inter_open.2 fun U hU hne => inter_compl_nonempty_iff.2 fun hUx => _
-    obtain rfl : U = {x}
-    exact eq_singleton_iff_nonempty_unique_mem.2 ⟨hne, hUx⟩
+    obtain rfl : U = {x}; exact eq_singleton_iff_nonempty_unique_mem.2 ⟨hne, hUx⟩
     exact ho hU
 #align dense_compl_singleton_iff_not_open dense_compl_singleton_iff_not_open
 
@@ -1315,9 +1292,7 @@ but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Eq.{succ u1} (Set.{u1} α) (frontier.{u1} α _inst_1 s) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (interior.{u1} α _inst_1 s)) (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (interior.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) s))))
 Case conversion may be inaccurate. Consider using '#align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interiorₓ'. -/
 theorem frontier_eq_inter_compl_interior {s : Set α} : frontier s = interior sᶜ ∩ interior (sᶜ)ᶜ :=
-  by
-  rw [← frontier_compl, ← closure_compl]
-  rfl
+  by rw [← frontier_compl, ← closure_compl]; rfl
 #align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interior
 
 /- warning: compl_frontier_eq_union_interior -> compl_frontier_eq_union_interior is a dubious translation:
@@ -1887,10 +1862,8 @@ but is expected to have type
   forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} (x : α) (F : Filter.{u1} ι) (u : ι -> α), Iff (MapClusterPt.{u2, u1} α _inst_1 ι x F u) (forall (s : Set.{u2} α), (Membership.mem.{u2, u2} (Set.{u2} α) (Filter.{u2} α) (instMembershipSetFilter.{u2} α) s (nhds.{u2} α _inst_1 x)) -> (Filter.Frequently.{u1} ι (fun (a : ι) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) (u a) s) F))
 Case conversion may be inaccurate. Consider using '#align map_cluster_pt_iff mapClusterPt_iffₓ'. -/
 theorem mapClusterPt_iff {ι : Type _} (x : α) (F : Filter ι) (u : ι → α) :
-    MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s :=
-  by
-  simp_rw [MapClusterPt, ClusterPt, inf_ne_bot_iff_frequently_left, frequently_map]
-  rfl
+    MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s := by
+  simp_rw [MapClusterPt, ClusterPt, inf_ne_bot_iff_frequently_left, frequently_map]; rfl
 #align map_cluster_pt_iff mapClusterPt_iff
 
 /- warning: map_cluster_pt_of_comp -> mapClusterPt_of_comp is a dubious translation:
@@ -2582,11 +2555,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {g : α -> β}, (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (forall (x : α), Eq.{succ u1} β (f x) (g x)) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 g)
 Case conversion may be inaccurate. Consider using '#align continuous.congr Continuous.congrₓ'. -/
-theorem Continuous.congr {f g : α → β} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g :=
-  by
-  convert h
-  ext
-  rw [h']
+theorem Continuous.congr {f g : α → β} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g := by
+  convert h; ext; rw [h']
 #align continuous.congr Continuous.congr
 
 #print ContinuousAt /-
@@ -2920,9 +2890,7 @@ but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {s : Set.{u2} α}, (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (closure.{u2} α _inst_1 s) (Set.preimage.{u2, u1} α β f (closure.{u1} β _inst_2 (Set.image.{u2, u1} α β f s))))
 Case conversion may be inaccurate. Consider using '#align closure_subset_preimage_closure_image closure_subset_preimage_closure_imageₓ'. -/
 theorem closure_subset_preimage_closure_image {f : α → β} {s : Set α} (h : Continuous f) :
-    closure s ⊆ f ⁻¹' closure (f '' s) :=
-  by
-  rw [← Set.image_subset_iff]
+    closure s ⊆ f ⁻¹' closure (f '' s) := by rw [← Set.image_subset_iff];
   exact image_closure_subset_closure_image h
 #align closure_subset_preimage_closure_image closure_subset_preimage_closure_image
 
@@ -3013,9 +2981,7 @@ but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (forall {s : Set.{u2} α}, (Dense.{u2} α _inst_1 s) -> (HasSubset.Subset.{u1} (Set.{u1} β) (Set.instHasSubsetSet.{u1} β) (Set.range.{u1, succ u2} β α f) (closure.{u1} β _inst_2 (Set.image.{u2, u1} α β f s))))
 Case conversion may be inaccurate. Consider using '#align continuous.range_subset_closure_image_dense Continuous.range_subset_closure_image_denseₓ'. -/
 theorem Continuous.range_subset_closure_image_dense {f : α → β} (hf : Continuous f) {s : Set α}
-    (hs : Dense s) : range f ⊆ closure (f '' s) :=
-  by
-  rw [← image_univ, ← hs.closure_eq]
+    (hs : Dense s) : range f ⊆ closure (f '' s) := by rw [← image_univ, ← hs.closure_eq];
   exact image_closure_subset_closure_image hf
 #align continuous.range_subset_closure_image_dense Continuous.range_subset_closure_image_dense
 
@@ -3036,9 +3002,7 @@ Case conversion may be inaccurate. Consider using '#align dense_range.subset_clo
 /-- If `f` has dense range and `s` is an open set in the codomain of `f`, then the image of the
 preimage of `s` under `f` is dense in `s`. -/
 theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) {s : Set β}
-    (hs : IsOpen s) : s ⊆ closure (f '' (f ⁻¹' s)) :=
-  by
-  rw [image_preimage_eq_inter_range]
+    (hs : IsOpen s) : s ⊆ closure (f '' (f ⁻¹' s)) := by rw [image_preimage_eq_inter_range];
   exact hf.open_subset_closure_inter hs
 #align dense_range.subset_closure_image_preimage_of_is_open DenseRange.subset_closure_image_preimage_of_isOpen
 
@@ -3060,9 +3024,7 @@ Case conversion may be inaccurate. Consider using '#align dense_range.comp Dense
 /-- Composition of a continuous map with dense range and a function with dense range has dense
 range. -/
 theorem DenseRange.comp {g : β → γ} {f : κ → β} (hg : DenseRange g) (hf : DenseRange f)
-    (cg : Continuous g) : DenseRange (g ∘ f) :=
-  by
-  rw [DenseRange, range_comp]
+    (cg : Continuous g) : DenseRange (g ∘ f) := by rw [DenseRange, range_comp];
   exact hg.dense_image cg hf
 #align dense_range.comp DenseRange.comp
 

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

@@ -915,7 +915,7 @@ theorem closure_inter_open_nonempty_iff {s t : Set α} (h : IsOpen t) :
 
 /- warning: filter.le_lift'_closure -> Filter.le_lift'_closure is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (l : Filter.{u1} α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) l (Filter.lift'.{u1, u1} α α l (closure.{u1} α _inst_1))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (l : Filter.{u1} α), LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) l (Filter.lift'.{u1, u1} α α l (closure.{u1} α _inst_1))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (l : Filter.{u1} α), LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) l (Filter.lift'.{u1, u1} α α l (closure.{u1} α _inst_1))
 Case conversion may be inaccurate. Consider using '#align filter.le_lift'_closure Filter.le_lift'_closureₓ'. -/
@@ -1422,7 +1422,7 @@ theorem nhds_basis_closeds (a : α) : (𝓝 a).HasBasis (fun s : Set α => a ∉
 
 /- warning: le_nhds_iff -> le_nhds_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} {a : α}, Iff (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 (s : Set.{u1} α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsOpen.{u1} α _inst_1 s) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s f))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} {a : α}, Iff (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 (s : Set.{u1} α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsOpen.{u1} α _inst_1 s) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s f))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} {a : α}, Iff (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 (s : Set.{u1} α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsOpen.{u1} α _inst_1 s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s f))
 Case conversion may be inaccurate. Consider using '#align le_nhds_iff le_nhds_iffₓ'. -/
@@ -1432,7 +1432,7 @@ theorem le_nhds_iff {f a} : f ≤ 𝓝 a ↔ ∀ s : Set α, a ∈ s → IsOpen
 
 /- warning: nhds_le_of_le -> nhds_le_of_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} {a : α} {s : Set.{u1} α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsOpen.{u1} α _inst_1 s) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.principal.{u1} α s) f) -> (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) f)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} {a : α} {s : Set.{u1} α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (IsOpen.{u1} α _inst_1 s) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) (Filter.principal.{u1} α s) f) -> (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) f)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} {a : α} {s : Set.{u1} α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (IsOpen.{u1} α _inst_1 s) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) (Filter.principal.{u1} α s) f) -> (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) f)
 Case conversion may be inaccurate. Consider using '#align nhds_le_of_le nhds_le_of_leₓ'. -/
@@ -1653,13 +1653,17 @@ theorem tendsto_nhds {f : β → α} {l : Filter β} {a : α} :
 #align tendsto_nhds tendsto_nhds
 -/
 
-#print tendsto_atTop_nhds /-
+/- warning: tendsto_at_top_nhds -> tendsto_atTop_nhds is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {f : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α f (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α _inst_1 a)) (forall (U : Set.{u1} α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a U) -> (IsOpen.{u1} α _inst_1 U) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (LE.le.{u2} β (Preorder.toHasLe.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N n) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) (f n) U))))
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : Nonempty.{succ u2} β] [_inst_3 : SemilatticeSup.{u2} β] {f : β -> α} {a : α}, Iff (Filter.Tendsto.{u2, u1} β α f (Filter.atTop.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) (nhds.{u1} α _inst_1 a)) (forall (U : Set.{u1} α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a U) -> (IsOpen.{u1} α _inst_1 U) -> (Exists.{succ u2} β (fun (N : β) => forall (n : β), (LE.le.{u2} β (Preorder.toLE.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeSup.toPartialOrder.{u2} β _inst_3))) N n) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) (f n) U))))
+Case conversion may be inaccurate. Consider using '#align tendsto_at_top_nhds tendsto_atTop_nhdsₓ'. -/
 theorem tendsto_atTop_nhds [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} :
     Tendsto f atTop (𝓝 a) ↔ ∀ U : Set α, a ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U :=
   (atTop_basis.tendsto_iffₓ (nhds_basis_opens a)).trans <| by
     simp only [and_imp, exists_prop, true_and_iff, mem_Ici, ge_iff_le]
 #align tendsto_at_top_nhds tendsto_atTop_nhds
--/
 
 #print tendsto_const_nhds /-
 theorem tendsto_const_nhds {a : α} {f : Filter β} : Tendsto (fun b : β => a) f (𝓝 a) :=
@@ -1669,7 +1673,7 @@ theorem tendsto_const_nhds {a : α} {f : Filter β} : Tendsto (fun b : β => a)
 
 /- warning: tendsto_at_top_of_eventually_const -> tendsto_atTop_of_eventually_const is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_2 : SemilatticeSup.{u2} ι] [_inst_3 : Nonempty.{succ u2} ι] {x : α} {u : ι -> α} {i₀ : ι}, (forall (i : ι), (GE.ge.{u2} ι (Preorder.toLE.{u2} ι (PartialOrder.toPreorder.{u2} ι (SemilatticeSup.toPartialOrder.{u2} ι _inst_2))) i i₀) -> (Eq.{succ u1} α (u i) x)) -> (Filter.Tendsto.{u2, u1} ι α u (Filter.atTop.{u2} ι (PartialOrder.toPreorder.{u2} ι (SemilatticeSup.toPartialOrder.{u2} ι _inst_2))) (nhds.{u1} α _inst_1 x))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_2 : SemilatticeSup.{u2} ι] [_inst_3 : Nonempty.{succ u2} ι] {x : α} {u : ι -> α} {i₀ : ι}, (forall (i : ι), (GE.ge.{u2} ι (Preorder.toHasLe.{u2} ι (PartialOrder.toPreorder.{u2} ι (SemilatticeSup.toPartialOrder.{u2} ι _inst_2))) i i₀) -> (Eq.{succ u1} α (u i) x)) -> (Filter.Tendsto.{u2, u1} ι α u (Filter.atTop.{u2} ι (PartialOrder.toPreorder.{u2} ι (SemilatticeSup.toPartialOrder.{u2} ι _inst_2))) (nhds.{u1} α _inst_1 x))
 but is expected to have type
   forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_2 : SemilatticeSup.{u1} ι] [_inst_3 : Nonempty.{succ u1} ι] {x : α} {u : ι -> α} {i₀ : ι}, (forall (i : ι), (GE.ge.{u1} ι (Preorder.toLE.{u1} ι (PartialOrder.toPreorder.{u1} ι (SemilatticeSup.toPartialOrder.{u1} ι _inst_2))) i i₀) -> (Eq.{succ u2} α (u i) x)) -> (Filter.Tendsto.{u1, u2} ι α u (Filter.atTop.{u1} ι (PartialOrder.toPreorder.{u1} ι (SemilatticeSup.toPartialOrder.{u1} ι _inst_2))) (nhds.{u2} α _inst_1 x))
 Case conversion may be inaccurate. Consider using '#align tendsto_at_top_of_eventually_const tendsto_atTop_of_eventually_constₓ'. -/
@@ -1680,7 +1684,7 @@ theorem tendsto_atTop_of_eventually_const {ι : Type _} [SemilatticeSup ι] [Non
 
 /- warning: tendsto_at_bot_of_eventually_const -> tendsto_atBot_of_eventually_const is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_2 : SemilatticeInf.{u2} ι] [_inst_3 : Nonempty.{succ u2} ι] {x : α} {u : ι -> α} {i₀ : ι}, (forall (i : ι), (LE.le.{u2} ι (Preorder.toLE.{u2} ι (PartialOrder.toPreorder.{u2} ι (SemilatticeInf.toPartialOrder.{u2} ι _inst_2))) i i₀) -> (Eq.{succ u1} α (u i) x)) -> (Filter.Tendsto.{u2, u1} ι α u (Filter.atBot.{u2} ι (PartialOrder.toPreorder.{u2} ι (SemilatticeInf.toPartialOrder.{u2} ι _inst_2))) (nhds.{u1} α _inst_1 x))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_2 : SemilatticeInf.{u2} ι] [_inst_3 : Nonempty.{succ u2} ι] {x : α} {u : ι -> α} {i₀ : ι}, (forall (i : ι), (LE.le.{u2} ι (Preorder.toHasLe.{u2} ι (PartialOrder.toPreorder.{u2} ι (SemilatticeInf.toPartialOrder.{u2} ι _inst_2))) i i₀) -> (Eq.{succ u1} α (u i) x)) -> (Filter.Tendsto.{u2, u1} ι α u (Filter.atBot.{u2} ι (PartialOrder.toPreorder.{u2} ι (SemilatticeInf.toPartialOrder.{u2} ι _inst_2))) (nhds.{u1} α _inst_1 x))
 but is expected to have type
   forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_2 : SemilatticeInf.{u1} ι] [_inst_3 : Nonempty.{succ u1} ι] {x : α} {u : ι -> α} {i₀ : ι}, (forall (i : ι), (LE.le.{u1} ι (Preorder.toLE.{u1} ι (PartialOrder.toPreorder.{u1} ι (SemilatticeInf.toPartialOrder.{u1} ι _inst_2))) i i₀) -> (Eq.{succ u2} α (u i) x)) -> (Filter.Tendsto.{u1, u2} ι α u (Filter.atBot.{u1} ι (PartialOrder.toPreorder.{u1} ι (SemilatticeInf.toPartialOrder.{u1} ι _inst_2))) (nhds.{u2} α _inst_1 x))
 Case conversion may be inaccurate. Consider using '#align tendsto_at_bot_of_eventually_const tendsto_atBot_of_eventually_constₓ'. -/
@@ -1691,7 +1695,7 @@ theorem tendsto_atBot_of_eventually_const {ι : Type _} [SemilatticeInf ι] [Non
 
 /- warning: pure_le_nhds -> pure_le_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], LE.le.{u1} (α -> (Filter.{u1} α)) (Pi.hasLe.{u1, u1} α (fun (ᾰ : α) => Filter.{u1} α) (fun (i : α) => Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α)))) (Pure.pure.{u1, u1} (fun {α : Type.{u1}} => Filter.{u1} α) Filter.hasPure.{u1} α) (nhds.{u1} α _inst_1)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], LE.le.{u1} (α -> (Filter.{u1} α)) (Pi.hasLe.{u1, u1} α (fun (ᾰ : α) => Filter.{u1} α) (fun (i : α) => Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α)))) (Pure.pure.{u1, u1} (fun {α : Type.{u1}} => Filter.{u1} α) Filter.hasPure.{u1} α) (nhds.{u1} α _inst_1)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α], LE.le.{u1} (α -> (Filter.{u1} α)) (Pi.hasLe.{u1, u1} α (fun (ᾰ : α) => Filter.{u1} α) (fun (i : 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) (nhds.{u1} α _inst_1)
 Case conversion may be inaccurate. Consider using '#align pure_le_nhds pure_le_nhdsₓ'. -/
@@ -1711,7 +1715,7 @@ theorem tendsto_pure_nhds {α : Type _} [TopologicalSpace β] (f : α → β) (a
 
 /- warning: order_top.tendsto_at_top_nhds -> OrderTop.tendsto_atTop_nhds is a dubious translation:
 lean 3 declaration is
-  forall {β : Type.{u1}} {α : Type.{u2}} [_inst_2 : PartialOrder.{u2} α] [_inst_3 : OrderTop.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2))] [_inst_4 : TopologicalSpace.{u1} β] (f : α -> β), Filter.Tendsto.{u2, u1} α β f (Filter.atTop.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2)) (nhds.{u1} β _inst_4 (f (Top.top.{u2} α (OrderTop.toHasTop.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2)) _inst_3))))
+  forall {β : Type.{u1}} {α : Type.{u2}} [_inst_2 : PartialOrder.{u2} α] [_inst_3 : OrderTop.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2))] [_inst_4 : TopologicalSpace.{u1} β] (f : α -> β), Filter.Tendsto.{u2, u1} α β f (Filter.atTop.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2)) (nhds.{u1} β _inst_4 (f (Top.top.{u2} α (OrderTop.toHasTop.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α _inst_2)) _inst_3))))
 but is expected to have type
   forall {β : Type.{u2}} {α : Type.{u1}} [_inst_2 : PartialOrder.{u1} α] [_inst_3 : OrderTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2))] [_inst_4 : TopologicalSpace.{u2} β] (f : α -> β), Filter.Tendsto.{u1, u2} α β f (Filter.atTop.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) (nhds.{u2} β _inst_4 (f (Top.top.{u1} α (OrderTop.toTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) _inst_3))))
 Case conversion may be inaccurate. Consider using '#align order_top.tendsto_at_top_nhds OrderTop.tendsto_atTop_nhdsₓ'. -/
@@ -1799,7 +1803,7 @@ theorem clusterPt_principal_iff_frequently {x : α} {s : Set α} :
 
 /- warning: cluster_pt.of_le_nhds -> ClusterPt.of_le_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} 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 x)) -> (forall [_inst_2 : Filter.NeBot.{u1} α f], ClusterPt.{u1} α _inst_1 x f)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} 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 x)) -> (forall [_inst_2 : Filter.NeBot.{u1} α f], ClusterPt.{u1} α _inst_1 x f)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} 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 x)) -> (forall [_inst_2 : Filter.NeBot.{u1} α f], ClusterPt.{u1} α _inst_1 x f)
 Case conversion may be inaccurate. Consider using '#align cluster_pt.of_le_nhds ClusterPt.of_le_nhdsₓ'. -/
@@ -1809,7 +1813,7 @@ theorem ClusterPt.of_le_nhds {x : α} {f : Filter α} (H : f ≤ 𝓝 x) [NeBot
 
 /- warning: cluster_pt.of_le_nhds' -> ClusterPt.of_le_nhds' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} 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 x)) -> (Filter.NeBot.{u1} α f) -> (ClusterPt.{u1} α _inst_1 x f)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} 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 x)) -> (Filter.NeBot.{u1} α f) -> (ClusterPt.{u1} α _inst_1 x f)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} 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 x)) -> (Filter.NeBot.{u1} α f) -> (ClusterPt.{u1} α _inst_1 x f)
 Case conversion may be inaccurate. Consider using '#align cluster_pt.of_le_nhds' ClusterPt.of_le_nhds'ₓ'. -/
@@ -1819,7 +1823,7 @@ theorem ClusterPt.of_le_nhds' {x : α} {f : Filter α} (H : f ≤ 𝓝 x) (hf :
 
 /- warning: cluster_pt.of_nhds_le -> ClusterPt.of_nhds_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α}, (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) f) -> (ClusterPt.{u1} α _inst_1 x f)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α}, (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) f) -> (ClusterPt.{u1} α _inst_1 x f)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α}, (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) f) -> (ClusterPt.{u1} α _inst_1 x f)
 Case conversion may be inaccurate. Consider using '#align cluster_pt.of_nhds_le ClusterPt.of_nhds_leₓ'. -/
@@ -1829,7 +1833,7 @@ theorem ClusterPt.of_nhds_le {x : α} {f : Filter α} (H : 𝓝 x ≤ f) : Clust
 
 /- warning: cluster_pt.mono -> ClusterPt.mono is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x f) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) f g) -> (ClusterPt.{u1} α _inst_1 x g)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x f) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) f g) -> (ClusterPt.{u1} α _inst_1 x g)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x f) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) f g) -> (ClusterPt.{u1} α _inst_1 x g)
 Case conversion may be inaccurate. Consider using '#align cluster_pt.mono ClusterPt.monoₓ'. -/
@@ -1860,7 +1864,7 @@ theorem ClusterPt.of_inf_right {x : α} {f g : Filter α} (H : ClusterPt x <| f
 
 /- warning: ultrafilter.cluster_pt_iff -> Ultrafilter.clusterPt_iff is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Ultrafilter.{u1} α}, Iff (ClusterPt.{u1} α _inst_1 x ((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)) (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] {x : α} {f : Ultrafilter.{u1} α}, Iff (ClusterPt.{u1} α _inst_1 x ((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)) (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] {x : α} {f : Ultrafilter.{u1} α}, Iff (ClusterPt.{u1} α _inst_1 x (Ultrafilter.toFilter.{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))) (Ultrafilter.toFilter.{u1} α f) (nhds.{u1} α _inst_1 x))
 Case conversion may be inaccurate. Consider using '#align ultrafilter.cluster_pt_iff Ultrafilter.clusterPt_iffₓ'. -/
@@ -1958,7 +1962,7 @@ theorem accPt_iff_frequently (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∃ᶠ
 
 /- warning: acc_pt.mono -> AccPt.mono is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {F : Filter.{u1} α} {G : Filter.{u1} α}, (AccPt.{u1} α _inst_1 x F) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) F G) -> (AccPt.{u1} α _inst_1 x G)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {F : Filter.{u1} α} {G : Filter.{u1} α}, (AccPt.{u1} α _inst_1 x F) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toHasLe.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.partialOrder.{u1} α))) F G) -> (AccPt.{u1} α _inst_1 x G)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {F : Filter.{u1} α} {G : Filter.{u1} α}, (AccPt.{u1} α _inst_1 x F) -> (LE.le.{u1} (Filter.{u1} α) (Preorder.toLE.{u1} (Filter.{u1} α) (PartialOrder.toPreorder.{u1} (Filter.{u1} α) (Filter.instPartialOrderFilter.{u1} α))) F G) -> (AccPt.{u1} α _inst_1 x G)
 Case conversion may be inaccurate. Consider using '#align acc_pt.mono AccPt.monoₓ'. -/
@@ -1981,7 +1985,7 @@ theorem interior_eq_nhds' {s : Set α} : interior s = { a | s ∈ 𝓝 a } :=
 
 /- warning: interior_eq_nhds -> interior_eq_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Eq.{succ u1} (Set.{u1} α) (interior.{u1} α _inst_1 s) (setOf.{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))) (nhds.{u1} α _inst_1 a) (Filter.principal.{u1} α s)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Eq.{succ u1} (Set.{u1} α) (interior.{u1} α _inst_1 s) (setOf.{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))) (nhds.{u1} α _inst_1 a) (Filter.principal.{u1} α s)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Eq.{succ u1} (Set.{u1} α) (interior.{u1} α _inst_1 s) (setOf.{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))) (nhds.{u1} α _inst_1 a) (Filter.principal.{u1} α s)))
 Case conversion may be inaccurate. Consider using '#align interior_eq_nhds interior_eq_nhdsₓ'. -/
@@ -2023,7 +2027,7 @@ theorem subset_interior_iff_nhds {s V : Set α} : s ⊆ interior V ↔ ∀ x ∈
 
 /- warning: is_open_iff_nhds -> isOpen_iff_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α _inst_1 s) (forall (a : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (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) (Filter.principal.{u1} α s)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α _inst_1 s) (forall (a : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (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) (Filter.principal.{u1} α s)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α _inst_1 s) (forall (a : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (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) (Filter.principal.{u1} α s)))
 Case conversion may be inaccurate. Consider using '#align is_open_iff_nhds isOpen_iff_nhdsₓ'. -/
@@ -2049,7 +2053,7 @@ theorem isOpen_iff_eventually {s : Set α} : IsOpen s ↔ ∀ x, x ∈ s → ∀
 
 /- warning: is_open_iff_ultrafilter -> isOpen_iff_ultrafilter is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α _inst_1 s) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (l : 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))) l) (nhds.{u1} α _inst_1 x)) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Ultrafilter.{u1} α) (Ultrafilter.hasMem.{u1} α) s l)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α _inst_1 s) (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (forall (l : 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))) l) (nhds.{u1} α _inst_1 x)) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Ultrafilter.{u1} α) (Ultrafilter.hasMem.{u1} α) s l)))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (IsOpen.{u1} α _inst_1 s) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (forall (l : 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 l) (nhds.{u1} α _inst_1 x)) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Ultrafilter.{u1} α) (Ultrafilter.instMembershipSetUltrafilter.{u1} α) s l)))
 Case conversion may be inaccurate. Consider using '#align is_open_iff_ultrafilter isOpen_iff_ultrafilterₓ'. -/
@@ -2246,7 +2250,7 @@ theorem mem_closure_iff_nhds_basis {a : α} {p : ι → Prop} {s : ι → Set α
 
 /- warning: mem_closure_iff_ultrafilter -> mem_closure_iff_ultrafilter is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {x : α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (closure.{u1} α _inst_1 s)) (Exists.{succ u1} (Ultrafilter.{u1} α) (fun (u : Ultrafilter.{u1} α) => And (Membership.Mem.{u1, u1} (Set.{u1} α) (Ultrafilter.{u1} α) (Ultrafilter.hasMem.{u1} α) s u) (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))) u) (nhds.{u1} α _inst_1 x))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {x : α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (closure.{u1} α _inst_1 s)) (Exists.{succ u1} (Ultrafilter.{u1} α) (fun (u : Ultrafilter.{u1} α) => And (Membership.Mem.{u1, u1} (Set.{u1} α) (Ultrafilter.{u1} α) (Ultrafilter.hasMem.{u1} α) s u) (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))) u) (nhds.{u1} α _inst_1 x))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {x : α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (closure.{u1} α _inst_1 s)) (Exists.{succ u1} (Ultrafilter.{u1} α) (fun (u : Ultrafilter.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Ultrafilter.{u1} α) (Ultrafilter.instMembershipSetUltrafilter.{u1} α) s u) (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 u) (nhds.{u1} α _inst_1 x))))
 Case conversion may be inaccurate. Consider using '#align mem_closure_iff_ultrafilter mem_closure_iff_ultrafilterₓ'. -/
@@ -2503,7 +2507,7 @@ noncomputable def limUnder [Nonempty α] (f : Filter β) (g : β → α) : α :=
 
 /- warning: le_nhds_Lim -> le_nhds_lim is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} (h : 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))), 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 (lim.{u1} α _inst_1 (nonempty_of_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)) h) f))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} (h : 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))), 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 (lim.{u1} α _inst_1 (nonempty_of_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)) h) f))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {f : Filter.{u1} α} (h : 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))), 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 (lim.{u1} α _inst_1 (nonempty_of_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)) h) f))
 Case conversion may be inaccurate. Consider using '#align le_nhds_Lim le_nhds_limₓ'. -/
@@ -2840,7 +2844,7 @@ theorem mem_closure_image {f : α → β} {x : α} {s : Set α} (hf : Continuous
 
 /- warning: continuous_at_iff_ultrafilter -> continuousAt_iff_ultrafilter 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] {f : α -> β} {x : α}, Iff (ContinuousAt.{u1, u2} α β _inst_1 _inst_2 f x) (forall (g : 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))) g) (nhds.{u1} α _inst_1 x)) -> (Filter.Tendsto.{u1, u2} α β f ((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))) g) (nhds.{u2} β _inst_2 (f x))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {x : α}, Iff (ContinuousAt.{u1, u2} α β _inst_1 _inst_2 f x) (forall (g : 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))) g) (nhds.{u1} α _inst_1 x)) -> (Filter.Tendsto.{u1, u2} α β f ((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))) g) (nhds.{u2} β _inst_2 (f x))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {x : α}, Iff (ContinuousAt.{u2, u1} α β _inst_1 _inst_2 f x) (forall (g : Ultrafilter.{u2} α), (LE.le.{u2} (Filter.{u2} α) (Preorder.toLE.{u2} (Filter.{u2} α) (PartialOrder.toPreorder.{u2} (Filter.{u2} α) (Filter.instPartialOrderFilter.{u2} α))) (Ultrafilter.toFilter.{u2} α g) (nhds.{u2} α _inst_1 x)) -> (Filter.Tendsto.{u2, u1} α β f (Ultrafilter.toFilter.{u2} α g) (nhds.{u1} β _inst_2 (f x))))
 Case conversion may be inaccurate. Consider using '#align continuous_at_iff_ultrafilter continuousAt_iff_ultrafilterₓ'. -/
@@ -2851,7 +2855,7 @@ theorem continuousAt_iff_ultrafilter {f : α → β} {x} :
 
 /- warning: continuous_iff_ultrafilter -> continuous_iff_ultrafilter 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] {f : α -> β}, Iff (Continuous.{u1, u2} α β _inst_1 _inst_2 f) (forall (x : α) (g : 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))) g) (nhds.{u1} α _inst_1 x)) -> (Filter.Tendsto.{u1, u2} α β f ((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))) g) (nhds.{u2} β _inst_2 (f x))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, Iff (Continuous.{u1, u2} α β _inst_1 _inst_2 f) (forall (x : α) (g : 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))) g) (nhds.{u1} α _inst_1 x)) -> (Filter.Tendsto.{u1, u2} α β f ((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))) g) (nhds.{u2} β _inst_2 (f x))))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, Iff (Continuous.{u2, u1} α β _inst_1 _inst_2 f) (forall (x : α) (g : Ultrafilter.{u2} α), (LE.le.{u2} (Filter.{u2} α) (Preorder.toLE.{u2} (Filter.{u2} α) (PartialOrder.toPreorder.{u2} (Filter.{u2} α) (Filter.instPartialOrderFilter.{u2} α))) (Ultrafilter.toFilter.{u2} α g) (nhds.{u2} α _inst_1 x)) -> (Filter.Tendsto.{u2, u1} α β f (Ultrafilter.toFilter.{u2} α g) (nhds.{u1} β _inst_2 (f x))))
 Case conversion may be inaccurate. Consider using '#align continuous_iff_ultrafilter continuous_iff_ultrafilterₓ'. -/

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

@@ -77,15 +77,15 @@ class TopologicalSpace (α : Type u) where
   IsOpen : Set α → Prop
   isOpen_univ : IsOpen univ
   isOpen_inter : ∀ s t, IsOpen s → IsOpen t → IsOpen (s ∩ t)
-  isOpen_unionₛ : ∀ s, (∀ t ∈ s, IsOpen t) → IsOpen (⋃₀ s)
+  isOpen_sUnion : ∀ s, (∀ t ∈ s, IsOpen t) → IsOpen (⋃₀ s)
 #align topological_space TopologicalSpace
 -/
 
 /- warning: topological_space.of_closed -> TopologicalSpace.ofClosed is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} (T : Set.{u1} (Set.{u1} α)), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α)) T) -> (forall (A : Set.{u1} (Set.{u1} α)), (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) A T) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (Set.interₛ.{u1} α A) T)) -> (forall (A : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) A T) -> (forall (B : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) B T) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) A B) T))) -> (TopologicalSpace.{u1} α)
+  forall {α : Type.{u1}} (T : Set.{u1} (Set.{u1} α)), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α)) T) -> (forall (A : Set.{u1} (Set.{u1} α)), (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) A T) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (Set.sInter.{u1} α A) T)) -> (forall (A : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) A T) -> (forall (B : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) B T) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) A B) T))) -> (TopologicalSpace.{u1} α)
 but is expected to have type
-  forall {α : Type.{u1}} (T : Set.{u1} (Set.{u1} α)), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α)) T) -> (forall (A : Set.{u1} (Set.{u1} α)), (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) A T) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (Set.interₛ.{u1} α A) T)) -> (forall (A : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) A T) -> (forall (B : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) B T) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) A B) T))) -> (TopologicalSpace.{u1} α)
+  forall {α : Type.{u1}} (T : Set.{u1} (Set.{u1} α)), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α)) T) -> (forall (A : Set.{u1} (Set.{u1} α)), (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) A T) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (Set.sInter.{u1} α A) T)) -> (forall (A : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) A T) -> (forall (B : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) B T) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) A B) T))) -> (TopologicalSpace.{u1} α)
 Case conversion may be inaccurate. Consider using '#align topological_space.of_closed TopologicalSpace.ofClosedₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A «expr ⊆ » T) -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A B «expr ∈ » T) -/
@@ -98,8 +98,8 @@ def TopologicalSpace.ofClosed {α : Type u} (T : Set (Set α)) (empty_mem : ∅
   IsOpen X := Xᶜ ∈ T
   isOpen_univ := by simp [empty_mem]
   isOpen_inter s t hs ht := by simpa only [compl_inter] using union_mem (sᶜ) hs (tᶜ) ht
-  isOpen_unionₛ s hs := by
-    rw [Set.compl_unionₛ] <;>
+  isOpen_sUnion s hs := by
+    rw [Set.compl_sUnion] <;>
       exact sInter_mem (compl '' s) fun z ⟨y, hy, hz⟩ => by simpa [hz.symm] using hs y hy
 #align topological_space.of_closed TopologicalSpace.ofClosed
 
@@ -119,9 +119,9 @@ scoped[Topology] notation "is_open[" t "]" => @IsOpen hole! t
 
 /- warning: is_open_mk -> isOpen_mk is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {p : (Set.{u1} α) -> Prop} {h₁ : p (Set.univ.{u1} α)} {h₂ : forall (s : Set.{u1} α) (t : Set.{u1} α), (p s) -> (p t) -> (p (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t))} {h₃ : forall (s : Set.{u1} (Set.{u1} α)), (forall (t : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t s) -> (p t)) -> (p (Set.unionₛ.{u1} α s))} {s : Set.{u1} α}, Iff (IsOpen.{u1} α (TopologicalSpace.mk.{u1} α p h₁ h₂ h₃) s) (p s)
+  forall {α : Type.{u1}} {p : (Set.{u1} α) -> Prop} {h₁ : p (Set.univ.{u1} α)} {h₂ : forall (s : Set.{u1} α) (t : Set.{u1} α), (p s) -> (p t) -> (p (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t))} {h₃ : forall (s : Set.{u1} (Set.{u1} α)), (forall (t : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t s) -> (p t)) -> (p (Set.sUnion.{u1} α s))} {s : Set.{u1} α}, Iff (IsOpen.{u1} α (TopologicalSpace.mk.{u1} α p h₁ h₂ h₃) s) (p s)
 but is expected to have type
-  forall {α : Type.{u1}} {p : (Set.{u1} α) -> Prop} {h₁ : p (Set.univ.{u1} α)} {h₂ : forall (s : Set.{u1} α) (t : Set.{u1} α), (p s) -> (p t) -> (p (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t))} {h₃ : forall (s : Set.{u1} (Set.{u1} α)), (forall (t : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t s) -> (p t)) -> (p (Set.unionₛ.{u1} α s))} {s : Set.{u1} α}, Iff (IsOpen.{u1} α (TopologicalSpace.mk.{u1} α p h₁ h₂ h₃) s) (p s)
+  forall {α : Type.{u1}} {p : (Set.{u1} α) -> Prop} {h₁ : p (Set.univ.{u1} α)} {h₂ : forall (s : Set.{u1} α) (t : Set.{u1} α), (p s) -> (p t) -> (p (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t))} {h₃ : forall (s : Set.{u1} (Set.{u1} α)), (forall (t : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t s) -> (p t)) -> (p (Set.sUnion.{u1} α s))} {s : Set.{u1} α}, Iff (IsOpen.{u1} α (TopologicalSpace.mk.{u1} α p h₁ h₂ h₃) s) (p s)
 Case conversion may be inaccurate. Consider using '#align is_open_mk isOpen_mkₓ'. -/
 theorem isOpen_mk {p h₁ h₂ h₃} {s : Set α} : is_open[⟨p, h₁, h₂, h₃⟩] s ↔ p s :=
   Iff.rfl
@@ -159,10 +159,10 @@ theorem IsOpen.inter (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ 
   TopologicalSpace.isOpen_inter s₁ s₂ h₁ h₂
 #align is_open.inter IsOpen.inter
 
-#print isOpen_unionₛ /-
-theorem isOpen_unionₛ {s : Set (Set α)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (⋃₀ s) :=
-  TopologicalSpace.isOpen_unionₛ s h
-#align is_open_sUnion isOpen_unionₛ
+#print isOpen_sUnion /-
+theorem isOpen_sUnion {s : Set (Set α)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (⋃₀ s) :=
+  TopologicalSpace.isOpen_sUnion s h
+#align is_open_sUnion isOpen_sUnion
 -/
 
 end
@@ -184,17 +184,17 @@ theorem isOpen_fold {s : Set α} {t : TopologicalSpace α} : t.IsOpen s = is_ope
 
 variable [TopologicalSpace α]
 
-#print isOpen_unionᵢ /-
-theorem isOpen_unionᵢ {f : ι → Set α} (h : ∀ i, IsOpen (f i)) : IsOpen (⋃ i, f i) :=
-  isOpen_unionₛ <| by rintro _ ⟨i, rfl⟩ <;> exact h i
-#align is_open_Union isOpen_unionᵢ
+#print isOpen_iUnion /-
+theorem isOpen_iUnion {f : ι → Set α} (h : ∀ i, IsOpen (f i)) : IsOpen (⋃ i, f i) :=
+  isOpen_sUnion <| by rintro _ ⟨i, rfl⟩ <;> exact h i
+#align is_open_Union isOpen_iUnion
 -/
 
-#print isOpen_bunionᵢ /-
-theorem isOpen_bunionᵢ {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
+#print isOpen_biUnion /-
+theorem isOpen_biUnion {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋃ i ∈ s, f i) :=
-  isOpen_unionᵢ fun i => isOpen_unionᵢ fun hi => h i hi
-#align is_open_bUnion isOpen_bunionᵢ
+  isOpen_iUnion fun i => isOpen_iUnion fun hi => h i hi
+#align is_open_bUnion isOpen_biUnion
 -/
 
 /- warning: is_open.union -> IsOpen.union is a dubious translation:
@@ -204,46 +204,46 @@ but is expected to have type
   forall {α : Type.{u1}} {s₁ : Set.{u1} α} {s₂ : Set.{u1} α} [_inst_1 : TopologicalSpace.{u1} α], (IsOpen.{u1} α _inst_1 s₁) -> (IsOpen.{u1} α _inst_1 s₂) -> (IsOpen.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s₁ s₂))
 Case conversion may be inaccurate. Consider using '#align is_open.union IsOpen.unionₓ'. -/
 theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∪ s₂) := by
-  rw [union_eq_Union] <;> exact isOpen_unionᵢ (Bool.forall_bool.2 ⟨h₂, h₁⟩)
+  rw [union_eq_Union] <;> exact isOpen_iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align is_open.union IsOpen.union
 
 #print isOpen_empty /-
 @[simp]
 theorem isOpen_empty : IsOpen (∅ : Set α) := by
-  rw [← sUnion_empty] <;> exact isOpen_unionₛ fun a => False.elim
+  rw [← sUnion_empty] <;> exact isOpen_sUnion fun a => False.elim
 #align is_open_empty isOpen_empty
 -/
 
-#print isOpen_interₛ /-
-theorem isOpen_interₛ {s : Set (Set α)} (hs : s.Finite) : (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
+#print isOpen_sInter /-
+theorem isOpen_sInter {s : Set (Set α)} (hs : s.Finite) : (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
   Finite.induction_on hs (fun _ => by rw [sInter_empty] <;> exact isOpen_univ)
     fun a s has hs ih h => by
     rw [sInter_insert] <;>
       exact IsOpen.inter (h _ <| mem_insert _ _) (ih fun t => h t ∘ mem_insert_of_mem _)
-#align is_open_sInter isOpen_interₛ
+#align is_open_sInter isOpen_sInter
 -/
 
-#print isOpen_binterᵢ /-
-theorem isOpen_binterᵢ {s : Set β} {f : β → Set α} (hs : s.Finite) :
+#print isOpen_biInter /-
+theorem isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finite) :
     (∀ i ∈ s, IsOpen (f i)) → IsOpen (⋂ i ∈ s, f i) :=
   Finite.induction_on hs (fun _ => by rw [bInter_empty] <;> exact isOpen_univ)
     fun a s has hs ih h => by
     rw [bInter_insert] <;>
       exact IsOpen.inter (h a (mem_insert _ _)) (ih fun i hi => h i (mem_insert_of_mem _ hi))
-#align is_open_bInter isOpen_binterᵢ
+#align is_open_bInter isOpen_biInter
 -/
 
-#print isOpen_interᵢ /-
-theorem isOpen_interᵢ [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) : IsOpen (⋂ i, s i) :=
-  isOpen_interₛ (finite_range _) (forall_range_iff.2 h)
-#align is_open_Inter isOpen_interᵢ
+#print isOpen_iInter /-
+theorem isOpen_iInter [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) : IsOpen (⋂ i, s i) :=
+  isOpen_sInter (finite_range _) (forall_range_iff.2 h)
+#align is_open_Inter isOpen_iInter
 -/
 
-#print isOpen_binterᵢ_finset /-
-theorem isOpen_binterᵢ_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
+#print isOpen_biInter_finset /-
+theorem isOpen_biInter_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
-  isOpen_binterᵢ (toFinite _) h
-#align is_open_bInter_finset isOpen_binterᵢ_finset
+  isOpen_biInter (toFinite _) h
+#align is_open_bInter_finset isOpen_biInter_finset
 -/
 
 #print isOpen_const /-
@@ -311,23 +311,23 @@ theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪
   exact IsOpen.inter h₁ h₂
 #align is_closed.union IsClosed.union
 
-#print isClosed_interₛ /-
-theorem isClosed_interₛ {s : Set (Set α)} : (∀ t ∈ s, IsClosed t) → IsClosed (⋂₀ s) := by
-  simpa only [← isOpen_compl_iff, compl_sInter, sUnion_image] using isOpen_bunionᵢ
-#align is_closed_sInter isClosed_interₛ
+#print isClosed_sInter /-
+theorem isClosed_sInter {s : Set (Set α)} : (∀ t ∈ s, IsClosed t) → IsClosed (⋂₀ s) := by
+  simpa only [← isOpen_compl_iff, compl_sInter, sUnion_image] using isOpen_biUnion
+#align is_closed_sInter isClosed_sInter
 -/
 
-#print isClosed_interᵢ /-
-theorem isClosed_interᵢ {f : ι → Set α} (h : ∀ i, IsClosed (f i)) : IsClosed (⋂ i, f i) :=
-  isClosed_interₛ fun t ⟨i, (HEq : f i = t)⟩ => HEq ▸ h i
-#align is_closed_Inter isClosed_interᵢ
+#print isClosed_iInter /-
+theorem isClosed_iInter {f : ι → Set α} (h : ∀ i, IsClosed (f i)) : IsClosed (⋂ i, f i) :=
+  isClosed_sInter fun t ⟨i, (HEq : f i = t)⟩ => HEq ▸ h i
+#align is_closed_Inter isClosed_iInter
 -/
 
-#print isClosed_binterᵢ /-
-theorem isClosed_binterᵢ {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsClosed (f i)) :
+#print isClosed_biInter /-
+theorem isClosed_biInter {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsClosed (f i)) :
     IsClosed (⋂ i ∈ s, f i) :=
-  isClosed_interᵢ fun i => isClosed_interᵢ <| h i
-#align is_closed_bInter isClosed_binterᵢ
+  isClosed_iInter fun i => isClosed_iInter <| h i
+#align is_closed_bInter isClosed_biInter
 -/
 
 /- warning: is_closed_compl_iff -> isClosed_compl_iff is a dubious translation:
@@ -384,23 +384,23 @@ theorem IsClosed.sdiff {s t : Set α} (h₁ : IsClosed s) (h₂ : IsOpen t) : Is
   IsClosed.inter h₁ (isClosed_compl_iff.mpr h₂)
 #align is_closed.sdiff IsClosed.sdiff
 
-#print isClosed_bunionᵢ /-
-theorem isClosed_bunionᵢ {s : Set β} {f : β → Set α} (hs : s.Finite) :
+#print isClosed_biUnion /-
+theorem isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite) :
     (∀ i ∈ s, IsClosed (f i)) → IsClosed (⋃ i ∈ s, f i) :=
   Finite.induction_on hs (fun _ => by rw [bUnion_empty] <;> exact isClosed_empty)
     fun a s has hs ih h => by
     rw [bUnion_insert] <;>
       exact IsClosed.union (h a (mem_insert _ _)) (ih fun i hi => h i (mem_insert_of_mem _ hi))
-#align is_closed_bUnion isClosed_bunionᵢ
+#align is_closed_bUnion isClosed_biUnion
 -/
 
-#print isClosed_unionᵢ /-
-theorem isClosed_unionᵢ [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
+#print isClosed_iUnion /-
+theorem isClosed_iUnion [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
     IsClosed (⋃ i, s i) :=
   by
   simp only [← isOpen_compl_iff, compl_Union] at *
-  exact isOpen_interᵢ h
-#align is_closed_Union isClosed_unionᵢ
+  exact isOpen_iInter h
+#align is_closed_Union isClosed_iUnion
 -/
 
 #print isClosed_imp /-
@@ -439,19 +439,19 @@ theorem mem_interior {s : Set α} {x : α} :
 #print isOpen_interior /-
 @[simp]
 theorem isOpen_interior {s : Set α} : IsOpen (interior s) :=
-  isOpen_unionₛ fun t ⟨h₁, h₂⟩ => h₁
+  isOpen_sUnion fun t ⟨h₁, h₂⟩ => h₁
 #align is_open_interior isOpen_interior
 -/
 
 #print interior_subset /-
 theorem interior_subset {s : Set α} : interior s ⊆ s :=
-  unionₛ_subset fun t ⟨h₁, h₂⟩ => h₂
+  sUnion_subset fun t ⟨h₁, h₂⟩ => h₂
 #align interior_subset interior_subset
 -/
 
 #print interior_maximal /-
 theorem interior_maximal {s t : Set α} (h₁ : t ⊆ s) (h₂ : IsOpen t) : t ⊆ interior s :=
-  subset_unionₛ_of_mem ⟨h₂, h₁⟩
+  subset_sUnion_of_mem ⟨h₂, h₁⟩
 #align interior_maximal interior_maximal
 -/
 
@@ -536,34 +536,34 @@ theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ inte
       IsOpen.inter isOpen_interior isOpen_interior)
 #align interior_inter interior_inter
 
-/- warning: finset.interior_Inter -> Finset.interior_interᵢ is a dubious translation:
+/- warning: finset.interior_Inter -> Finset.interior_iInter is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} (s : Finset.{u2} ι) (f : ι -> (Set.{u1} α)), Eq.{succ u1} (Set.{u1} α) (interior.{u1} α _inst_1 (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.interᵢ.{u1, 0} α (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i)))) (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.interᵢ.{u1, 0} α (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => interior.{u1} α _inst_1 (f i))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} (s : Finset.{u2} ι) (f : ι -> (Set.{u1} α)), Eq.{succ u1} (Set.{u1} α) (interior.{u1} α _inst_1 (Set.iInter.{u1, succ u2} α ι (fun (i : ι) => Set.iInter.{u1, 0} α (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i)))) (Set.iInter.{u1, succ u2} α ι (fun (i : ι) => Set.iInter.{u1, 0} α (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => interior.{u1} α _inst_1 (f i))))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} (s : Finset.{u1} ι) (f : ι -> (Set.{u2} α)), Eq.{succ u2} (Set.{u2} α) (interior.{u2} α _inst_1 (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => Set.interᵢ.{u2, 0} α (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => f i)))) (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => Set.interᵢ.{u2, 0} α (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => interior.{u2} α _inst_1 (f i))))
-Case conversion may be inaccurate. Consider using '#align finset.interior_Inter Finset.interior_interᵢₓ'. -/
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} (s : Finset.{u1} ι) (f : ι -> (Set.{u2} α)), Eq.{succ u2} (Set.{u2} α) (interior.{u2} α _inst_1 (Set.iInter.{u2, succ u1} α ι (fun (i : ι) => Set.iInter.{u2, 0} α (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => f i)))) (Set.iInter.{u2, succ u1} α ι (fun (i : ι) => Set.iInter.{u2, 0} α (Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => interior.{u2} α _inst_1 (f i))))
+Case conversion may be inaccurate. Consider using '#align finset.interior_Inter Finset.interior_iInterₓ'. -/
 @[simp]
-theorem Finset.interior_interᵢ {ι : Type _} (s : Finset ι) (f : ι → Set α) :
+theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α) :
     interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by
   classical
     refine' s.induction_on (by simp) _
     intro i s h₁ h₂
     simp [h₂]
-#align finset.interior_Inter Finset.interior_interᵢ
+#align finset.interior_Inter Finset.interior_iInter
 
-/- warning: interior_Inter -> interior_interᵢ is a dubious translation:
+/- warning: interior_Inter -> interior_iInter is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_2 : Finite.{succ u2} ι] (f : ι -> (Set.{u1} α)), Eq.{succ u1} (Set.{u1} α) (interior.{u1} α _inst_1 (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => f i))) (Set.interᵢ.{u1, succ u2} α ι (fun (i : ι) => interior.{u1} α _inst_1 (f i)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_2 : Finite.{succ u2} ι] (f : ι -> (Set.{u1} α)), Eq.{succ u1} (Set.{u1} α) (interior.{u1} α _inst_1 (Set.iInter.{u1, succ u2} α ι (fun (i : ι) => f i))) (Set.iInter.{u1, succ u2} α ι (fun (i : ι) => interior.{u1} α _inst_1 (f i)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_2 : Finite.{succ u1} ι] (f : ι -> (Set.{u2} α)), Eq.{succ u2} (Set.{u2} α) (interior.{u2} α _inst_1 (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => f i))) (Set.interᵢ.{u2, succ u1} α ι (fun (i : ι) => interior.{u2} α _inst_1 (f i)))
-Case conversion may be inaccurate. Consider using '#align interior_Inter interior_interᵢₓ'. -/
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_2 : Finite.{succ u1} ι] (f : ι -> (Set.{u2} α)), Eq.{succ u2} (Set.{u2} α) (interior.{u2} α _inst_1 (Set.iInter.{u2, succ u1} α ι (fun (i : ι) => f i))) (Set.iInter.{u2, succ u1} α ι (fun (i : ι) => interior.{u2} α _inst_1 (f i)))
+Case conversion may be inaccurate. Consider using '#align interior_Inter interior_iInterₓ'. -/
 @[simp]
-theorem interior_interᵢ {ι : Type _} [Finite ι] (f : ι → Set α) :
+theorem interior_iInter {ι : Type _} [Finite ι] (f : ι → Set α) :
     interior (⋂ i, f i) = ⋂ i, interior (f i) :=
   by
   cases nonempty_fintype ι
   convert finset.univ.interior_Inter f <;> simp
-#align interior_Inter interior_interᵢ
+#align interior_Inter interior_iInter
 
 /- warning: interior_union_is_closed_of_interior_empty -> interior_union_isClosed_of_interior_empty is a dubious translation:
 lean 3 declaration is
@@ -593,32 +593,32 @@ theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _)(_ : t
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
 #align is_open_iff_forall_mem_open isOpen_iff_forall_mem_open
 
-#print interior_interᵢ_subset /-
-theorem interior_interᵢ_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ ⋂ i, interior (s i) :=
-  subset_interᵢ fun i => interior_mono <| interᵢ_subset _ _
-#align interior_Inter_subset interior_interᵢ_subset
+#print interior_iInter_subset /-
+theorem interior_iInter_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ ⋂ i, interior (s i) :=
+  subset_iInter fun i => interior_mono <| iInter_subset _ _
+#align interior_Inter_subset interior_iInter_subset
 -/
 
 /- warning: interior_Inter₂_subset -> interior_Inter₂_subset is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {ι : Sort.{u2}} [_inst_1 : TopologicalSpace.{u1} α] (p : ι -> Sort.{u3}) (s : forall (i : ι), (p i) -> (Set.{u1} α)), HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (interior.{u1} α _inst_1 (Set.interᵢ.{u1, u2} α ι (fun (i : ι) => Set.interᵢ.{u1, u3} α (p i) (fun (j : p i) => s i j)))) (Set.interᵢ.{u1, u2} α ι (fun (i : ι) => Set.interᵢ.{u1, u3} α (p i) (fun (j : p i) => interior.{u1} α _inst_1 (s i j))))
+  forall {α : Type.{u1}} {ι : Sort.{u2}} [_inst_1 : TopologicalSpace.{u1} α] (p : ι -> Sort.{u3}) (s : forall (i : ι), (p i) -> (Set.{u1} α)), HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) (interior.{u1} α _inst_1 (Set.iInter.{u1, u2} α ι (fun (i : ι) => Set.iInter.{u1, u3} α (p i) (fun (j : p i) => s i j)))) (Set.iInter.{u1, u2} α ι (fun (i : ι) => Set.iInter.{u1, u3} α (p i) (fun (j : p i) => interior.{u1} α _inst_1 (s i j))))
 but is expected to have type
-  forall {α : Type.{u2}} {ι : Sort.{u3}} [_inst_1 : TopologicalSpace.{u2} α] (p : ι -> Sort.{u1}) (s : forall (i : ι), (p i) -> (Set.{u2} α)), HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (interior.{u2} α _inst_1 (Set.interᵢ.{u2, u3} α ι (fun (i : ι) => Set.interᵢ.{u2, u1} α (p i) (fun (j : p i) => s i j)))) (Set.interᵢ.{u2, u3} α ι (fun (i : ι) => Set.interᵢ.{u2, u1} α (p i) (fun (j : p i) => interior.{u2} α _inst_1 (s i j))))
+  forall {α : Type.{u2}} {ι : Sort.{u3}} [_inst_1 : TopologicalSpace.{u2} α] (p : ι -> Sort.{u1}) (s : forall (i : ι), (p i) -> (Set.{u2} α)), HasSubset.Subset.{u2} (Set.{u2} α) (Set.instHasSubsetSet.{u2} α) (interior.{u2} α _inst_1 (Set.iInter.{u2, u3} α ι (fun (i : ι) => Set.iInter.{u2, u1} α (p i) (fun (j : p i) => s i j)))) (Set.iInter.{u2, u3} α ι (fun (i : ι) => Set.iInter.{u2, u1} α (p i) (fun (j : p i) => interior.{u2} α _inst_1 (s i j))))
 Case conversion may be inaccurate. Consider using '#align interior_Inter₂_subset interior_Inter₂_subsetₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 theorem interior_Inter₂_subset (p : ι → Sort _) (s : ∀ i, p i → Set α) :
     interior (⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), interior (s i j) :=
-  (interior_interᵢ_subset _).trans <| interᵢ_mono fun i => interior_interᵢ_subset _
+  (interior_iInter_subset _).trans <| iInter_mono fun i => interior_iInter_subset _
 #align interior_Inter₂_subset interior_Inter₂_subset
 
-#print interior_interₛ_subset /-
-theorem interior_interₛ_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s :=
+#print interior_sInter_subset /-
+theorem interior_sInter_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s :=
   calc
     interior (⋂₀ S) = interior (⋂ s ∈ S, s) := by rw [sInter_eq_bInter]
     _ ⊆ ⋂ s ∈ S, interior s := interior_Inter₂_subset _ _
     
-#align interior_sInter_subset interior_interₛ_subset
+#align interior_sInter_subset interior_sInter_subset
 -/
 
 /-!
@@ -636,13 +636,13 @@ def closure (s : Set α) : Set α :=
 #print isClosed_closure /-
 @[simp]
 theorem isClosed_closure {s : Set α} : IsClosed (closure s) :=
-  isClosed_interₛ fun t ⟨h₁, h₂⟩ => h₁
+  isClosed_sInter fun t ⟨h₁, h₂⟩ => h₁
 #align is_closed_closure isClosed_closure
 -/
 
 #print subset_closure /-
 theorem subset_closure {s : Set α} : s ⊆ closure s :=
-  subset_interₛ fun t ⟨h₁, h₂⟩ => h₂
+  subset_sInter fun t ⟨h₁, h₂⟩ => h₂
 #align subset_closure subset_closure
 -/
 
@@ -654,7 +654,7 @@ theorem not_mem_of_not_mem_closure {s : Set α} {P : α} (hP : P ∉ closure s)
 
 #print closure_minimal /-
 theorem closure_minimal {s t : Set α} (h₁ : s ⊆ t) (h₂ : IsClosed t) : closure s ⊆ t :=
-  interₛ_subset_of_mem ⟨h₂, h₁⟩
+  sInter_subset_of_mem ⟨h₂, h₁⟩
 #align closure_minimal closure_minimal
 -/
 
@@ -814,34 +814,34 @@ theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure
     ((monotone_closure α).le_map_sup s t)
 #align closure_union closure_union
 
-/- warning: finset.closure_bUnion -> Finset.closure_bunionᵢ is a dubious translation:
+/- warning: finset.closure_bUnion -> Finset.closure_biUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} (s : Finset.{u2} ι) (f : ι -> (Set.{u1} α)), Eq.{succ u1} (Set.{u1} α) (closure.{u1} α _inst_1 (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 s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i)))) (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 s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => closure.{u1} α _inst_1 (f i))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} (s : Finset.{u2} ι) (f : ι -> (Set.{u1} α)), Eq.{succ u1} (Set.{u1} α) (closure.{u1} α _inst_1 (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 s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => f i)))) (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 s) (fun (H : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i s) => closure.{u1} α _inst_1 (f i))))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} (s : Finset.{u1} ι) (f : ι -> (Set.{u2} α)), Eq.{succ u2} (Set.{u2} α) (closure.{u2} α _inst_1 (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 s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => f i)))) (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 s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => closure.{u2} α _inst_1 (f i))))
-Case conversion may be inaccurate. Consider using '#align finset.closure_bUnion Finset.closure_bunionᵢₓ'. -/
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} (s : Finset.{u1} ι) (f : ι -> (Set.{u2} α)), Eq.{succ u2} (Set.{u2} α) (closure.{u2} α _inst_1 (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 s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => f i)))) (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 s) (fun (H : Membership.mem.{u1, u1} ι (Finset.{u1} ι) (Finset.instMembershipFinset.{u1} ι) i s) => closure.{u2} α _inst_1 (f i))))
+Case conversion may be inaccurate. Consider using '#align finset.closure_bUnion Finset.closure_biUnionₓ'. -/
 @[simp]
-theorem Finset.closure_bunionᵢ {ι : Type _} (s : Finset ι) (f : ι → Set α) :
+theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α) :
     closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
   classical
     refine' s.induction_on (by simp) _
     intro i s h₁ h₂
     simp [h₂]
-#align finset.closure_bUnion Finset.closure_bunionᵢ
+#align finset.closure_bUnion Finset.closure_biUnion
 
-/- warning: closure_Union -> closure_unionᵢ is a dubious translation:
+/- warning: closure_Union -> closure_iUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_2 : Finite.{succ u2} ι] (f : ι -> (Set.{u1} α)), Eq.{succ u1} (Set.{u1} α) (closure.{u1} α _inst_1 (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => f i))) (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => closure.{u1} α _inst_1 (f i)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_2 : Finite.{succ u2} ι] (f : ι -> (Set.{u1} α)), Eq.{succ u1} (Set.{u1} α) (closure.{u1} α _inst_1 (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => f i))) (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => closure.{u1} α _inst_1 (f i)))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_2 : Finite.{succ u1} ι] (f : ι -> (Set.{u2} α)), Eq.{succ u2} (Set.{u2} α) (closure.{u2} α _inst_1 (Set.unionᵢ.{u2, succ u1} α ι (fun (i : ι) => f i))) (Set.unionᵢ.{u2, succ u1} α ι (fun (i : ι) => closure.{u2} α _inst_1 (f i)))
-Case conversion may be inaccurate. Consider using '#align closure_Union closure_unionᵢₓ'. -/
+  forall {α : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_2 : Finite.{succ u1} ι] (f : ι -> (Set.{u2} α)), Eq.{succ u2} (Set.{u2} α) (closure.{u2} α _inst_1 (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => f i))) (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => closure.{u2} α _inst_1 (f i)))
+Case conversion may be inaccurate. Consider using '#align closure_Union closure_iUnionₓ'. -/
 @[simp]
-theorem closure_unionᵢ {ι : Type _} [Finite ι] (f : ι → Set α) :
+theorem closure_iUnion {ι : Type _} [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) :=
   by
   cases nonempty_fintype ι
   convert finset.univ.closure_bUnion f <;> simp
-#align closure_Union closure_unionᵢ
+#align closure_Union closure_iUnion
 
 #print interior_subset_closure /-
 theorem interior_subset_closure {s : Set α} : interior s ⊆ closure s :=
@@ -1377,21 +1377,21 @@ scoped[Topology] notation "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
 
 /- warning: nhds_def -> nhds_def is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (nhds.{u1} α _inst_1 a) (infᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => infᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) => Filter.principal.{u1} α s)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (nhds.{u1} α _inst_1 a) (iInf.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) => Filter.principal.{u1} α s)))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (nhds.{u1} α _inst_1 a) (infᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => infᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) => Filter.principal.{u1} α s)))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (nhds.{u1} α _inst_1 a) (iInf.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) => Filter.principal.{u1} α s)))
 Case conversion may be inaccurate. Consider using '#align nhds_def nhds_defₓ'. -/
 theorem nhds_def (a : α) : 𝓝 a = ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 s := by rw [nhds]
 #align nhds_def nhds_def
 
 /- warning: nhds_def' -> nhds_def' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (nhds.{u1} α _inst_1 a) (infᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => infᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (IsOpen.{u1} α _inst_1 s) (fun (hs : IsOpen.{u1} α _inst_1 s) => infᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (fun (ha : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) => Filter.principal.{u1} α s))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (nhds.{u1} α _inst_1 a) (iInf.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (IsOpen.{u1} α _inst_1 s) (fun (hs : IsOpen.{u1} α _inst_1 s) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasInf.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (fun (ha : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) => Filter.principal.{u1} α s))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (nhds.{u1} α _inst_1 a) (infᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => infᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (IsOpen.{u1} α _inst_1 s) (fun (hs : IsOpen.{u1} α _inst_1 s) => infᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (fun (ha : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) => Filter.principal.{u1} α s))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (a : α), Eq.{succ u1} (Filter.{u1} α) (nhds.{u1} α _inst_1 a) (iInf.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Set.{u1} α) (fun (s : Set.{u1} α) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (IsOpen.{u1} α _inst_1 s) (fun (hs : IsOpen.{u1} α _inst_1 s) => iInf.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toInfSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (fun (ha : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) => Filter.principal.{u1} α s))))
 Case conversion may be inaccurate. Consider using '#align nhds_def' nhds_def'ₓ'. -/
 theorem nhds_def' (a : α) : 𝓝 a = ⨅ (s : Set α) (hs : IsOpen s) (ha : a ∈ s), 𝓟 s := by
-  simp only [nhds_def, mem_set_of_eq, and_comm' (a ∈ _), infᵢ_and]
+  simp only [nhds_def, mem_set_of_eq, and_comm' (a ∈ _), iInf_and]
 #align nhds_def' nhds_def'
 
 #print nhds_basis_opens /-
@@ -1439,7 +1439,7 @@ Case conversion may be inaccurate. Consider using '#align nhds_le_of_le nhds_le_
 /-- To show a filter is above the neighborhood filter at `a`, it suffices to show that it is above
 the principal filter of some open set `s` containing `a`. -/
 theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 a ≤ f := by
-  rw [nhds_def] <;> exact infᵢ_le_of_le s (infᵢ_le_of_le ⟨h, o⟩ sf)
+  rw [nhds_def] <;> exact iInf_le_of_le s (iInf_le_of_le ⟨h, o⟩ sf)
 #align nhds_le_of_le nhds_le_of_le
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
@@ -1459,13 +1459,13 @@ theorem eventually_nhds_iff {a : α} {p : α → Prop} :
 
 /- warning: map_nhds -> map_nhds is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {f : α -> β}, Eq.{succ u2} (Filter.{u2} β) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a)) (infᵢ.{u2, succ u1} (Filter.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.completeLattice.{u2} β))) (Set.{u1} α) (fun (s : Set.{u1} α) => infᵢ.{u2, 0} (Filter.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.completeLattice.{u2} β))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) => Filter.principal.{u2} β (Set.image.{u1, u2} α β f s))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {f : α -> β}, Eq.{succ u2} (Filter.{u2} β) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a)) (iInf.{u2, succ u1} (Filter.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.completeLattice.{u2} β))) (Set.{u1} α) (fun (s : Set.{u1} α) => iInf.{u2, 0} (Filter.{u2} β) (ConditionallyCompleteLattice.toHasInf.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.completeLattice.{u2} β))) (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) => Filter.principal.{u2} β (Set.image.{u1, u2} α β f s))))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {f : α -> β}, Eq.{succ u2} (Filter.{u2} β) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a)) (infᵢ.{u2, succ u1} (Filter.{u2} β) (ConditionallyCompleteLattice.toInfSet.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.instCompleteLatticeFilter.{u2} β))) (Set.{u1} α) (fun (s : Set.{u1} α) => infᵢ.{u2, 0} (Filter.{u2} β) (ConditionallyCompleteLattice.toInfSet.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.instCompleteLatticeFilter.{u2} β))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) => Filter.principal.{u2} β (Set.image.{u1, u2} α β f s))))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {a : α} {f : α -> β}, Eq.{succ u2} (Filter.{u2} β) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a)) (iInf.{u2, succ u1} (Filter.{u2} β) (ConditionallyCompleteLattice.toInfSet.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.instCompleteLatticeFilter.{u2} β))) (Set.{u1} α) (fun (s : Set.{u1} α) => iInf.{u2, 0} (Filter.{u2} β) (ConditionallyCompleteLattice.toInfSet.{u2} (Filter.{u2} β) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} β) (Filter.instCompleteLatticeFilter.{u2} β))) (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) (fun (H : Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) s (setOf.{u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) (IsOpen.{u1} α _inst_1 s)))) => Filter.principal.{u2} β (Set.image.{u1, u2} α β f s))))
 Case conversion may be inaccurate. Consider using '#align map_nhds map_nhdsₓ'. -/
 theorem map_nhds {a : α} {f : α → β} :
     map f (𝓝 a) = ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 (image f s) :=
-  ((nhds_basis_opens a).map f).eq_binfᵢ
+  ((nhds_basis_opens a).map f).eq_biInf
 #align map_nhds map_nhds
 
 #print mem_of_mem_nhds /-
@@ -1532,15 +1532,15 @@ theorem exists_open_set_nhds {s U : Set α} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
   have := fun x hx => (nhds_basis_opens x).mem_iff.1 (h x hx)
   choose! Z hZ hZU using this; choose hZmem hZo using hZ
   exact
-    ⟨⋃ x ∈ s, Z x, fun x hx => mem_bUnion hx (hZmem x hx), isOpen_bunionᵢ hZo, Union₂_subset hZU⟩
+    ⟨⋃ x ∈ s, Z x, fun x hx => mem_bUnion hx (hZmem x hx), isOpen_biUnion hZo, Union₂_subset hZU⟩
 #align exists_open_set_nhds exists_open_set_nhds
 -/
 
 /- warning: exists_open_set_nhds' -> exists_open_set_nhds' is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {U : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (supᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) α (fun (x : α) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (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) => nhds.{u1} α _inst_1 x)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (V : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s V) (And (IsOpen.{u1} α _inst_1 V) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) V U))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {U : Set.{u1} α}, (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) U (iSup.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) α (fun (x : α) => iSup.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))) (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) => nhds.{u1} α _inst_1 x)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (V : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s V) (And (IsOpen.{u1} α _inst_1 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] {s : Set.{u1} α} {U : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) U (supᵢ.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) α (fun (x : α) => supᵢ.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) => nhds.{u1} α _inst_1 x)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (V : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s V) (And (IsOpen.{u1} α _inst_1 V) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) V U))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {U : Set.{u1} α}, (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) U (iSup.{u1, succ u1} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) α (fun (x : α) => iSup.{u1, 0} (Filter.{u1} α) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} α) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) (fun (H : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) => nhds.{u1} α _inst_1 x)))) -> (Exists.{succ u1} (Set.{u1} α) (fun (V : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s V) (And (IsOpen.{u1} α _inst_1 V) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) V U))))
 Case conversion may be inaccurate. Consider using '#align exists_open_set_nhds' exists_open_set_nhds'ₓ'. -/
 /-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of s:
 it contains an open set containing `s`. -/
@@ -2111,7 +2111,7 @@ of a sequence is closed. -/
 theorem isClosed_setOf_clusterPt {f : Filter α} : IsClosed { x | ClusterPt x f } :=
   by
   simp only [ClusterPt, inf_ne_bot_iff_frequently_left, set_of_forall, imp_iff_not_or]
-  refine' isClosed_interᵢ fun p => IsClosed.union _ _ <;> apply isClosed_compl_iff.2
+  refine' isClosed_iInter fun p => IsClosed.union _ _ <;> apply isClosed_compl_iff.2
   exacts[isOpen_setOf_eventually_nhds, isOpen_const]
 #align is_closed_set_of_cluster_pt isClosed_setOf_clusterPt
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce86f4e05e9a9b8da5e316b22c76ce76440c56a1

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
 
 ! This file was ported from Lean 3 source module topology.basic
-! leanprover-community/mathlib commit 88b8a77d63a702923d9bee05e9e454ebc22aa766
+! leanprover-community/mathlib commit e8da5f215e815d9ed3455f0216ef52b53e05438a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -239,12 +239,6 @@ theorem isOpen_interᵢ [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)
 #align is_open_Inter isOpen_interᵢ
 -/
 
-#print isOpen_interᵢ_prop /-
-theorem isOpen_interᵢ_prop {p : Prop} {s : p → Set α} (h : ∀ h : p, IsOpen (s h)) :
-    IsOpen (interᵢ s) := by by_cases p <;> simp [*]
-#align is_open_Inter_prop isOpen_interᵢ_prop
--/
-
 #print isOpen_binterᵢ_finset /-
 theorem isOpen_binterᵢ_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
@@ -400,22 +394,13 @@ theorem isClosed_bunionᵢ {s : Set β} {f : β → Set α} (hs : s.Finite) :
 #align is_closed_bUnion isClosed_bunionᵢ
 -/
 
-/- warning: is_closed_Union -> isClosed_unionᵢ is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : Finite.{succ u2} β] {s : β -> (Set.{u1} α)}, (forall (i : β), IsClosed.{u1} α _inst_1 (s i)) -> (IsClosed.{u1} α _inst_1 (Set.unionᵢ.{u1, succ u2} α β (fun (i : β) => s i)))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Sort.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : Finite.{u2} β] {s : β -> (Set.{u1} α)}, (forall (i : β), IsClosed.{u1} α _inst_1 (s i)) -> (IsClosed.{u1} α _inst_1 (Set.unionᵢ.{u1, u2} α β (fun (i : β) => s i)))
-Case conversion may be inaccurate. Consider using '#align is_closed_Union isClosed_unionᵢₓ'. -/
-theorem isClosed_unionᵢ [Finite β] {s : β → Set α} (h : ∀ i, IsClosed (s i)) :
+#print isClosed_unionᵢ /-
+theorem isClosed_unionᵢ [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
     IsClosed (⋃ i, s i) :=
-  suffices IsClosed (⋃ (i : β) (hi : i ∈ @univ β), s i) by convert this <;> simp [Set.ext_iff]
-  isClosed_bunionᵢ finite_univ fun i _ => h i
+  by
+  simp only [← isOpen_compl_iff, compl_Union] at *
+  exact isOpen_interᵢ h
 #align is_closed_Union isClosed_unionᵢ
-
-#print isClosed_unionᵢ_prop /-
-theorem isClosed_unionᵢ_prop {p : Prop} {s : p → Set α} (h : ∀ h : p, IsClosed (s h)) :
-    IsClosed (unionᵢ s) := by by_cases p <;> simp [*]
-#align is_closed_Union_prop isClosed_unionᵢ_prop
 -/
 
 #print isClosed_imp /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/02ba8949f486ebecf93fe7460f1ed0564b5e442c

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
 
 ! This file was ported from Lean 3 source module topology.basic
-! leanprover-community/mathlib commit bcfa726826abd57587355b4b5b7e78ad6527b7e4
+! leanprover-community/mathlib commit 88b8a77d63a702923d9bee05e9e454ebc22aa766
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -233,16 +233,11 @@ theorem isOpen_binterᵢ {s : Set β} {f : β → Set α} (hs : s.Finite) :
 #align is_open_bInter isOpen_binterᵢ
 -/
 
-/- warning: is_open_Inter -> isOpen_interᵢ is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : Finite.{succ u2} β] {s : β -> (Set.{u1} α)}, (forall (i : β), IsOpen.{u1} α _inst_1 (s i)) -> (IsOpen.{u1} α _inst_1 (Set.interᵢ.{u1, succ u2} α β (fun (i : β) => s i)))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Sort.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : Finite.{u2} β] {s : β -> (Set.{u1} α)}, (forall (i : β), IsOpen.{u1} α _inst_1 (s i)) -> (IsOpen.{u1} α _inst_1 (Set.interᵢ.{u1, u2} α β (fun (i : β) => s i)))
-Case conversion may be inaccurate. Consider using '#align is_open_Inter isOpen_interᵢₓ'. -/
-theorem isOpen_interᵢ [Finite β] {s : β → Set α} (h : ∀ i, IsOpen (s i)) : IsOpen (⋂ i, s i) :=
-  suffices IsOpen (⋂ (i : β) (hi : i ∈ @univ β), s i) by simpa
-  isOpen_binterᵢ finite_univ fun i _ => h i
+#print isOpen_interᵢ /-
+theorem isOpen_interᵢ [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) : IsOpen (⋂ i, s i) :=
+  isOpen_interₛ (finite_range _) (forall_range_iff.2 h)
 #align is_open_Inter isOpen_interᵢ
+-/
 
 #print isOpen_interᵢ_prop /-
 theorem isOpen_interᵢ_prop {p : Prop} {s : p → Set α} (h : ∀ h : p, IsOpen (s h)) :

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

@@ -1626,12 +1626,12 @@ theorem Filter.EventuallyEq.eq_of_nhds {f g : α → β} {a : α} (h : f =ᶠ[
 #align filter.eventually_eq.eq_of_nhds Filter.EventuallyEq.eq_of_nhds
 -/
 
-#print eventually_eventuallyLe_nhds /-
+#print eventually_eventuallyLE_nhds /-
 @[simp]
-theorem eventually_eventuallyLe_nhds [LE β] {f g : α → β} {a : α} :
+theorem eventually_eventuallyLE_nhds [LE β] {f g : α → β} {a : α} :
     (∀ᶠ y in 𝓝 a, f ≤ᶠ[𝓝 y] g) ↔ f ≤ᶠ[𝓝 a] g :=
   eventually_eventually_nhds
-#align eventually_eventually_le_nhds eventually_eventuallyLe_nhds
+#align eventually_eventually_le_nhds eventually_eventuallyLE_nhds
 -/
 
 #print Filter.EventuallyEq.eventuallyEq_nhds /-
@@ -1643,13 +1643,13 @@ theorem Filter.EventuallyEq.eventuallyEq_nhds {f g : α → β} {a : α} (h : f
 #align filter.eventually_eq.eventually_eq_nhds Filter.EventuallyEq.eventuallyEq_nhds
 -/
 
-#print Filter.EventuallyLe.eventuallyLe_nhds /-
+#print Filter.EventuallyLE.eventuallyLE_nhds /-
 /-- If `f x ≤ g x` in a neighbourhood of `a`, then for `y` sufficiently close to `a` we have
 `f x ≤ g x` in a neighbourhood of `y`. -/
-theorem Filter.EventuallyLe.eventuallyLe_nhds [LE β] {f g : α → β} {a : α} (h : f ≤ᶠ[𝓝 a] g) :
+theorem Filter.EventuallyLE.eventuallyLE_nhds [LE β] {f g : α → β} {a : α} (h : f ≤ᶠ[𝓝 a] g) :
     ∀ᶠ y in 𝓝 a, f ≤ᶠ[𝓝 y] g :=
   h.eventually_nhds
-#align filter.eventually_le.eventually_le_nhds Filter.EventuallyLe.eventuallyLe_nhds
+#align filter.eventually_le.eventually_le_nhds Filter.EventuallyLE.eventuallyLE_nhds
 -/
 
 #print all_mem_nhds /-

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

@@ -87,8 +87,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} (T : Set.{u1} (Set.{u1} α)), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α)) T) -> (forall (A : Set.{u1} (Set.{u1} α)), (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) A T) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (Set.interₛ.{u1} α A) T)) -> (forall (A : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) A T) -> (forall (B : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) B T) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) A B) T))) -> (TopologicalSpace.{u1} α)
 Case conversion may be inaccurate. Consider using '#align topological_space.of_closed TopologicalSpace.ofClosedₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (A «expr ⊆ » T) -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (A B «expr ∈ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A «expr ⊆ » T) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (A B «expr ∈ » T) -/
 /-- A constructor for topologies by specifying the closed sets,
 and showing that they satisfy the appropriate conditions. -/
 def TopologicalSpace.ofClosed {α : Type u} (T : Set (Set α)) (empty_mem : ∅ ∈ T)
@@ -450,7 +450,7 @@ def interior (s : Set α) : Set α :=
 #align interior interior
 -/
 
-/- ./././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) -/
 theorem mem_interior {s : Set α} {x : α} :
     x ∈ interior s ↔ ∃ (t : _)(_ : t ⊆ s), IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc', and_left_comm]
@@ -608,7 +608,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} {s : Set.{u1} α} [_inst_1 : TopologicalSpace.{u1} α], Iff (IsOpen.{u1} α _inst_1 s) (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s) (And (IsOpen.{u1} α _inst_1 t) (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t)))))
 Case conversion may be inaccurate. Consider using '#align is_open_iff_forall_mem_open isOpen_iff_forall_mem_openₓ'. -/
-/- ./././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) -/
 theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ (t : _)(_ : t ⊆ s), IsOpen t ∧ x ∈ t :=
   by rw [← subset_interior_iff_isOpen] <;> simp only [subset_def, mem_interior]
 #align is_open_iff_forall_mem_open isOpen_iff_forall_mem_open
@@ -1462,7 +1462,7 @@ theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟
   rw [nhds_def] <;> exact infᵢ_le_of_le s (infᵢ_le_of_le ⟨h, o⟩ sf)
 #align nhds_le_of_le nhds_le_of_le
 
-/- ./././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) -/
 theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ (t : _)(_ : t ⊆ s), IsOpen t ∧ a ∈ t :=
   (nhds_basis_opens a).mem_iff.trans
     ⟨fun ⟨t, ⟨hat, ht⟩, hts⟩ => ⟨t, hts, ht, hat⟩, fun ⟨t, hts, ht, hat⟩ => ⟨t, ⟨hat, ht⟩, hts⟩⟩
@@ -2466,7 +2466,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {f : β -> α} {l : Filter.{u2} β} {s : Set.{u2} β} {a : α}, (forall (x : β), (Not (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s)) -> (Eq.{succ u1} α (f x) a)) -> (Iff (Filter.Tendsto.{u2, u1} β α f (Inf.inf.{u2} (Filter.{u2} β) (Filter.instInfFilter.{u2} β) l (Filter.principal.{u2} β s)) (nhds.{u1} α _inst_1 a)) (Filter.Tendsto.{u2, u1} β α f l (nhds.{u1} α _inst_1 a)))
 Case conversion may be inaccurate. Consider using '#align tendsto_inf_principal_nhds_iff_of_forall_eq tendsto_inf_principal_nhds_iff_of_forall_eqₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x «expr ∉ » s) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x «expr ∉ » s) -/
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
 Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/
 theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : β → α} {l : Filter β} {s : Set β} {a : α}

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

@@ -1766,9 +1766,9 @@ def ClusterPt (x : α) (F : Filter α) : Prop :=
 
 /- warning: cluster_pt.ne_bot -> ClusterPt.neBot is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {F : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x F) -> (Filter.NeBot.{u1} α (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhds.{u1} α _inst_1 x) F))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {F : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x F) -> (Filter.NeBot.{u1} α (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhds.{u1} α _inst_1 x) F))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {F : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x F) -> (Filter.NeBot.{u1} α (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhds.{u1} α _inst_1 x) F))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {F : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x F) -> (Filter.NeBot.{u1} α (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhds.{u1} α _inst_1 x) F))
 Case conversion may be inaccurate. Consider using '#align cluster_pt.ne_bot ClusterPt.neBotₓ'. -/
 theorem ClusterPt.neBot {x : α} {F : Filter α} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) :=
   h
@@ -1859,9 +1859,9 @@ theorem ClusterPt.mono {x : α} {f g : Filter α} (H : ClusterPt x f) (h : f ≤
 
 /- warning: cluster_pt.of_inf_left -> ClusterPt.of_inf_left is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) f g)) -> (ClusterPt.{u1} α _inst_1 x f)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) f g)) -> (ClusterPt.{u1} α _inst_1 x f)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) f g)) -> (ClusterPt.{u1} α _inst_1 x f)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) f g)) -> (ClusterPt.{u1} α _inst_1 x f)
 Case conversion may be inaccurate. Consider using '#align cluster_pt.of_inf_left ClusterPt.of_inf_leftₓ'. -/
 theorem ClusterPt.of_inf_left {x : α} {f g : Filter α} (H : ClusterPt x <| f ⊓ g) : ClusterPt x f :=
   H.mono inf_le_left
@@ -1869,9 +1869,9 @@ theorem ClusterPt.of_inf_left {x : α} {f g : Filter α} (H : ClusterPt x <| f 
 
 /- warning: cluster_pt.of_inf_right -> ClusterPt.of_inf_right is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) f g)) -> (ClusterPt.{u1} α _inst_1 x g)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) f g)) -> (ClusterPt.{u1} α _inst_1 x g)
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) f g)) -> (ClusterPt.{u1} α _inst_1 x g)
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {x : α} {f : Filter.{u1} α} {g : Filter.{u1} α}, (ClusterPt.{u1} α _inst_1 x (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) f g)) -> (ClusterPt.{u1} α _inst_1 x g)
 Case conversion may be inaccurate. Consider using '#align cluster_pt.of_inf_right ClusterPt.of_inf_rightₓ'. -/
 theorem ClusterPt.of_inf_right {x : α} {f g : Filter α} (H : ClusterPt x <| f ⊓ g) :
     ClusterPt x g :=
@@ -1936,9 +1936,9 @@ def AccPt (x : α) (F : Filter α) : Prop :=
 
 /- warning: acc_iff_cluster -> acc_iff_cluster is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (x : α) (F : Filter.{u1} α), Iff (AccPt.{u1} α _inst_1 x F) (ClusterPt.{u1} α _inst_1 x (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (Filter.principal.{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) x))) F))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (x : α) (F : Filter.{u1} α), Iff (AccPt.{u1} α _inst_1 x F) (ClusterPt.{u1} α _inst_1 x (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (Filter.principal.{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) x))) F))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (x : α) (F : Filter.{u1} α), Iff (AccPt.{u1} α _inst_1 x F) (ClusterPt.{u1} α _inst_1 x (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (Filter.principal.{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) x))) F))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] (x : α) (F : Filter.{u1} α), Iff (AccPt.{u1} α _inst_1 x F) (ClusterPt.{u1} α _inst_1 x (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (Filter.principal.{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) x))) F))
 Case conversion may be inaccurate. Consider using '#align acc_iff_cluster acc_iff_clusterₓ'. -/
 theorem acc_iff_cluster (x : α) (F : Filter α) : AccPt x F ↔ ClusterPt x (𝓟 ({x}ᶜ) ⊓ F) := by
   rw [AccPt, nhdsWithin, ClusterPt, inf_assoc]
@@ -2144,9 +2144,9 @@ theorem mem_closure_iff_clusterPt {s : Set α} {a : α} : a ∈ closure s ↔ Cl
 
 /- warning: mem_closure_iff_nhds_ne_bot -> mem_closure_iff_nhds_neBot is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {a : α} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α _inst_1 s)) (Ne.{succ u1} (Filter.{u1} α) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhds.{u1} α _inst_1 a) (Filter.principal.{u1} α s)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toHasBot.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))
+  forall {α : Type.{u1}} {a : α} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α _inst_1 s)) (Ne.{succ u1} (Filter.{u1} α) (Inf.inf.{u1} (Filter.{u1} α) (Filter.hasInf.{u1} α) (nhds.{u1} α _inst_1 a) (Filter.principal.{u1} α s)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toHasBot.{u1} (Filter.{u1} α) (Filter.completeLattice.{u1} α))))
 but is expected to have type
-  forall {α : Type.{u1}} {a : α} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α _inst_1 s)) (Ne.{succ u1} (Filter.{u1} α) (HasInf.inf.{u1} (Filter.{u1} α) (Filter.instHasInfFilter.{u1} α) (nhds.{u1} α _inst_1 a) (Filter.principal.{u1} α s)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toBot.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))
+  forall {α : Type.{u1}} {a : α} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α _inst_1 s)) (Ne.{succ u1} (Filter.{u1} α) (Inf.inf.{u1} (Filter.{u1} α) (Filter.instInfFilter.{u1} α) (nhds.{u1} α _inst_1 a) (Filter.principal.{u1} α s)) (Bot.bot.{u1} (Filter.{u1} α) (CompleteLattice.toBot.{u1} (Filter.{u1} α) (Filter.instCompleteLatticeFilter.{u1} α))))
 Case conversion may be inaccurate. Consider using '#align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_neBotₓ'. -/
 theorem mem_closure_iff_nhds_neBot {s : Set α} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ :=
   mem_closure_iff_clusterPt.trans neBot_iff
@@ -2462,9 +2462,9 @@ theorem mem_closure_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set
 
 /- warning: tendsto_inf_principal_nhds_iff_of_forall_eq -> tendsto_inf_principal_nhds_iff_of_forall_eq is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {f : β -> α} {l : Filter.{u2} β} {s : Set.{u2} β} {a : α}, (forall (x : β), (Not (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s)) -> (Eq.{succ u1} α (f x) a)) -> (Iff (Filter.Tendsto.{u2, u1} β α f (HasInf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) l (Filter.principal.{u2} β s)) (nhds.{u1} α _inst_1 a)) (Filter.Tendsto.{u2, u1} β α f l (nhds.{u1} α _inst_1 a)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {f : β -> α} {l : Filter.{u2} β} {s : Set.{u2} β} {a : α}, (forall (x : β), (Not (Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x s)) -> (Eq.{succ u1} α (f x) a)) -> (Iff (Filter.Tendsto.{u2, u1} β α f (Inf.inf.{u2} (Filter.{u2} β) (Filter.hasInf.{u2} β) l (Filter.principal.{u2} β s)) (nhds.{u1} α _inst_1 a)) (Filter.Tendsto.{u2, u1} β α f l (nhds.{u1} α _inst_1 a)))
 but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {f : β -> α} {l : Filter.{u2} β} {s : Set.{u2} β} {a : α}, (forall (x : β), (Not (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s)) -> (Eq.{succ u1} α (f x) a)) -> (Iff (Filter.Tendsto.{u2, u1} β α f (HasInf.inf.{u2} (Filter.{u2} β) (Filter.instHasInfFilter.{u2} β) l (Filter.principal.{u2} β s)) (nhds.{u1} α _inst_1 a)) (Filter.Tendsto.{u2, u1} β α f l (nhds.{u1} α _inst_1 a)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] {f : β -> α} {l : Filter.{u2} β} {s : Set.{u2} β} {a : α}, (forall (x : β), (Not (Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x s)) -> (Eq.{succ u1} α (f x) a)) -> (Iff (Filter.Tendsto.{u2, u1} β α f (Inf.inf.{u2} (Filter.{u2} β) (Filter.instInfFilter.{u2} β) l (Filter.principal.{u2} β s)) (nhds.{u1} α _inst_1 a)) (Filter.Tendsto.{u2, u1} β α f l (nhds.{u1} α _inst_1 a)))
 Case conversion may be inaccurate. Consider using '#align tendsto_inf_principal_nhds_iff_of_forall_eq tendsto_inf_principal_nhds_iff_of_forall_eqₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (x «expr ∉ » s) -/
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.

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

A PR accompanying #12339.

Zulip discussion

@@ -648,8 +648,8 @@ theorem dense_compl_singleton_iff_not_open :
     exact (hd.inter_open_nonempty _ ho (singleton_nonempty _)).ne_empty (inter_compl_self _)
   · refine' fun ho => dense_iff_inter_open.2 fun U hU hne => inter_compl_nonempty_iff.2 fun hUx => _
     obtain rfl : U = {x}
-    exact eq_singleton_iff_nonempty_unique_mem.2 ⟨hne, hUx⟩
-    exact ho hU
+    · exact eq_singleton_iff_nonempty_unique_mem.2 ⟨hne, hUx⟩
+    · exact ho hU
 #align dense_compl_singleton_iff_not_open dense_compl_singleton_iff_not_open
 
 /-!

@@ -12,7 +12,7 @@ import Mathlib.Tactic.FunProp
 # Basic definitions about topological spaces
 
 This file contains definitions about topology that do not require imports
-other than `Mathlib/Data/Set/Lattice`.
+other than `Mathlib.Data.Set.Lattice`.
 
 ## Main definitions
 

Follow-up to #10816.

Remaining places containing such lemmas are

• Option.bex_ne_none and Option.ball_ne_none: defined in Lean core
• Nat.decidableBallLT and Nat.decidableBallLE: defined in Lean core
• bef_def is still used in a number of places and could be renamed
• BAll.imp_{left,right}, BEx.imp_{left,right}, BEx.intro and BEx.elim

I only audited the first ~150 lemmas mentioning "ball"; too many lemmas named after Metric.ball/openBall/closedBall.

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

@@ -130,7 +130,7 @@ lemma isOpen_iff_of_cover {f : α → Set X} (ho : ∀ i, IsOpen (f i)) (hU : (
 theorem Set.Finite.isOpen_sInter {s : Set (Set X)} (hs : s.Finite) :
     (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
   Finite.induction_on hs (fun _ => by rw [sInter_empty]; exact isOpen_univ) fun _ _ ih h => by
-    simp only [sInter_insert, ball_insert_iff] at h ⊢
+    simp only [sInter_insert, forall_mem_insert] at h ⊢
     exact h.1.inter (ih h.2)
 #align is_open_sInter Set.Finite.isOpen_sInter
 

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

@@ -1117,7 +1117,7 @@ theorem acc_iff_cluster (x : X) (F : Filter X) : AccPt x F ↔ ClusterPt x (𝓟
   rw [AccPt, nhdsWithin, ClusterPt, inf_assoc]
 #align acc_iff_cluster acc_iff_cluster
 
-/-- `x` is an accumulation point of a set `C` iff it is a cluster point of `C ∖ {x}`.-/
+/-- `x` is an accumulation point of a set `C` iff it is a cluster point of `C ∖ {x}`. -/
 theorem acc_principal_iff_cluster (x : X) (C : Set X) :
     AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x})) := by
   rw [acc_iff_cluster, inf_principal, inter_comm, diff_eq]
@@ -1131,7 +1131,7 @@ theorem accPt_iff_nhds (x : X) (C : Set X) : AccPt x (𝓟 C) ↔ ∀ U ∈ 𝓝
 #align acc_pt_iff_nhds accPt_iff_nhds
 
 /-- `x` is an accumulation point of a set `C` iff
-there are points near `x` in `C` and different from `x`.-/
+there are points near `x` in `C` and different from `x`. -/
 theorem accPt_iff_frequently (x : X) (C : Set X) : AccPt x (𝓟 C) ↔ ∃ᶠ y in 𝓝 x, y ≠ x ∧ y ∈ C := by
   simp [acc_principal_iff_cluster, clusterPt_principal_iff_frequently, and_comm]
 #align acc_pt_iff_frequently accPt_iff_frequently

@@ -474,7 +474,7 @@ theorem closure_empty_iff (s : Set X) : closure s = ∅ ↔ s = ∅ :=
 
 @[simp]
 theorem closure_nonempty_iff : (closure s).Nonempty ↔ s.Nonempty := by
-  simp only [nonempty_iff_ne_empty, Ne.def, closure_empty_iff]
+  simp only [nonempty_iff_ne_empty, Ne, closure_empty_iff]
 #align closure_nonempty_iff closure_nonempty_iff
 
 alias ⟨Set.Nonempty.of_closure, Set.Nonempty.closure⟩ := closure_nonempty_iff

These are new lemmas added to the file about proper maps.

Co-authored-by: Etienne Marion <etienne.marion@ens-lyon.fr> Co-authored-by: EtienneC30 <66847262+EtienneC30@users.noreply.github.com>

@@ -1099,6 +1099,9 @@ theorem mapClusterPt_iff_ultrafilter {ι : Type*} (x : X) (F : Filter ι) (u : 
   simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
     ← le_inf_iff, exists_ultrafilter_iff, inf_comm]
 
+theorem mapClusterPt_comp {X α β : Type*} {x : X} [TopologicalSpace X] {F : Filter α} {φ : α → β}
+    {u : β → X} : MapClusterPt x F (u ∘ φ) ↔ MapClusterPt x (map φ F) u := Iff.rfl
+
 theorem mapClusterPt_of_comp {F : Filter α} {φ : β → α} {p : Filter β}
     {u : α → X} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) :
     MapClusterPt x F u := by

Standardizes the following names for distributivity laws across Finset and Set:

• inter_union_distrib_left
• inter_union_distrib_right
• union_inter_distrib_left
• union_inter_distrib_right

Makes arguments explicit in:

• Set.union_inter_distrib_right
• Set.union_inter_distrib_left
• Set.inter_union_distrib_right

Deprecates these theorem names:

• Finset.inter_distrib_left
• Finset.inter_distrib_right
• Finset.union_distrib_right
• Finset.union_distrib_left
• Set.inter_distrib_left
• Set.inter_distrib_right
• Set.union_distrib_right
• Set.union_distrib_left

Fixes use of deprecated names and implicit arguments in these files:

• Topology/Basic
• Topology/Connected/Basic
• MeasureTheory/MeasurableSpace/Basic
• MeasureTheory/Covering/Differentiation
• MeasureTheory/Constructions/BorelSpace/Basic
• Data/Set/Image
• Data/Set/Basic
• Data/PFun
• Data/Matroid/Dual
• Data/Finset/Sups
• Data/Finset/Basic
• Combinatorics/SetFamily/FourFunctions
• Combinatorics/Additive/SalemSpencer
• Counterexamples/Phillips.lean
• Archive/Imo/Imo2021Q1.lean

@@ -715,7 +715,8 @@ theorem frontier_inter_subset (s t : Set X) :
     frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t := by
   simp only [frontier_eq_closure_inter_closure, compl_inter, closure_union]
   refine' (inter_subset_inter_left _ (closure_inter_subset_inter_closure s t)).trans_eq _
-  simp only [inter_distrib_left, inter_distrib_right, inter_assoc, inter_comm (closure t)]
+  simp only [inter_union_distrib_left, union_inter_distrib_right, inter_assoc,
+    inter_comm (closure t)]
 #align frontier_inter_subset frontier_inter_subset
 
 theorem frontier_union_subset (s t : Set X) :

Classifies by adding issue number #10618 to porting notes claiming "simp can prove it".

@@ -1262,7 +1262,7 @@ theorem dense_compl_singleton (x : X) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : Se
 
 /-- If `x` is not an isolated point of a topological space, then the closure of `{x}ᶜ` is the whole
 space. -/
--- Porting note: was a `@[simp]` lemma but `simp` can prove it
+-- Porting note (#10618): was a `@[simp]` lemma but `simp` can prove it
 theorem closure_compl_singleton (x : X) [NeBot (𝓝[≠] x)] : closure {x}ᶜ = (univ : Set X) :=
   (dense_compl_singleton x).closure_eq
 #align closure_compl_singleton closure_compl_singleton

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -1758,7 +1758,6 @@ theorem tendsto_lift'_closure_nhds (hf : Continuous f) (x : X) :
 section DenseRange
 
 variable {α ι : Type*} (f : α → X) (g : X → Y)
-
 variable {f : α → X} {s : Set X}
 
 /-- A surjective map has dense range. -/

The main addition is IsTopologicalBasis.inf (see https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Inf.20of.20a.20pair.20of.20topologies/near/419989448), and I also reordered things to be in the more typical order (deducing the Pi version from the iInf version rather than the converse).

Also a few extra golfs and variations.

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Christopher Hoskin <christopher.hoskin@gmail.com> Co-authored-by: Christopher Hoskin <christopher.hoskin@overleaf.com>

@@ -1174,6 +1174,12 @@ theorem isOpen_iff_nhds : IsOpen s ↔ ∀ x ∈ s, 𝓝 x ≤ 𝓟 s :=
     _ ↔ ∀ x ∈ s, 𝓝 x ≤ 𝓟 s := by simp_rw [interior_eq_nhds, subset_def, mem_setOf]
 #align is_open_iff_nhds isOpen_iff_nhds
 
+theorem TopologicalSpace.ext_iff_nhds {t t' : TopologicalSpace X} :
+    t = t' ↔ ∀ x, @nhds _ t x = @nhds _ t' x :=
+  ⟨fun H x ↦ congrFun (congrArg _ H) _, fun H ↦ by ext; simp_rw [@isOpen_iff_nhds _ _ _, H]⟩
+
+alias ⟨_, TopologicalSpace.ext_nhds⟩ := TopologicalSpace.ext_iff_nhds
+
 theorem isOpen_iff_mem_nhds : IsOpen s ↔ ∀ x ∈ s, s ∈ 𝓝 x :=
   isOpen_iff_nhds.trans <| forall_congr' fun _ => imp_congr_right fun _ => le_principal_iff
 #align is_open_iff_mem_nhds isOpen_iff_mem_nhds

Per the style guidelines, λ is disallowed in mathlib. This is close to exhaustive; I left some tactic code alone when it seemed to me that tactic could be upstreamed soon.

Notes

• In lines I was modifying anyway, I also converted => to ↦.
• Also contains some mild in-passing indentation fixes in Mathlib/Order/SupClosed.
• Some doc comments still contained Lean 3 syntax λ x, , which I also replaced.

@@ -1857,11 +1857,11 @@ However, lemmas with this conclusion are not nice to use in practice because
   elaboration process.
   ```
   variable {M : Type*} [Add M] [TopologicalSpace M] [ContinuousAdd M]
-  example : Continuous (λ x : M, x + x) :=
-  continuous_add.comp _
+  example : Continuous (fun x : M ↦ x + x) :=
+    continuous_add.comp _
 
-  example : Continuous (λ x : M, x + x) :=
-  continuous_add.comp (continuous_id.prod_mk continuous_id)
+  example : Continuous (fun x : M ↦ x + x) :=
+    continuous_add.comp (continuous_id.prod_mk continuous_id)
   ```
   The second is a valid proof, which is accepted if you write it as
   `continuous_add.comp (continuous_id.prod_mk continuous_id : _)`
@@ -1873,7 +1873,8 @@ However, lemmas with this conclusion are not nice to use in practice because
 
 A much more convenient way to write continuity lemmas is like `Continuous.add`:
 ```
-Continuous.add {f g : X → M} (hf : Continuous f) (hg : Continuous g) : Continuous (λ x, f x + g x)
+Continuous.add {f g : X → M} (hf : Continuous f) (hg : Continuous g) :
+  Continuous (fun x ↦ f x + g x)
 ```
 The conclusion can be `Continuous (f + g)`, which is definitionally equal.
 This has the following advantages

Classifies by adding issue number #10756 to porting notes claiming anything semantically equivalent to:

• "new lemma"
• "added lemma"

@@ -169,7 +169,7 @@ theorem TopologicalSpace.ext_iff_isClosed {t₁ t₂ : TopologicalSpace X} :
 
 alias ⟨_, TopologicalSpace.ext_isClosed⟩ := TopologicalSpace.ext_iff_isClosed
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem isClosed_const {p : Prop} : IsClosed { _x : X | p } := ⟨isOpen_const (p := ¬p)⟩
 
 @[simp] theorem isClosed_empty : IsClosed (∅ : Set X) := isClosed_const
@@ -1713,7 +1713,7 @@ theorem image_closure_subset_closure_image (h : Continuous f) :
   ((mapsTo_image f s).closure h).image_subset
 #align image_closure_subset_closure_image image_closure_subset_closure_image
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem closure_image_closure (h : Continuous f) :
     closure (f '' closure s) = closure (f '' s) :=
   Subset.antisymm

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.

@@ -105,7 +105,7 @@ theorem isOpen_fold {t : TopologicalSpace X} : t.IsOpen s = IsOpen[t] s :=
 variable [TopologicalSpace X]
 
 theorem isOpen_iUnion {f : ι → Set X} (h : ∀ i, IsOpen (f i)) : IsOpen (⋃ i, f i) :=
-  isOpen_sUnion (forall_range_iff.2 h)
+  isOpen_sUnion (forall_mem_range.2 h)
 #align is_open_Union isOpen_iUnion
 
 theorem isOpen_biUnion {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
@@ -137,12 +137,12 @@ theorem Set.Finite.isOpen_sInter {s : Set (Set X)} (hs : s.Finite) :
 theorem Set.Finite.isOpen_biInter {s : Set α} {f : α → Set X} (hs : s.Finite)
     (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
-  sInter_image f s ▸ (hs.image _).isOpen_sInter (ball_image_iff.2 h)
+  sInter_image f s ▸ (hs.image _).isOpen_sInter (forall_mem_image.2 h)
 #align is_open_bInter Set.Finite.isOpen_biInter
 
 theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set X} (h : ∀ i, IsOpen (s i)) :
     IsOpen (⋂ i, s i) :=
-  (finite_range _).isOpen_sInter  (forall_range_iff.2 h)
+  (finite_range _).isOpen_sInter  (forall_mem_range.2 h)
 #align is_open_Inter isOpen_iInter_of_finite
 
 theorem isOpen_biInter_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
@@ -187,7 +187,7 @@ theorem isClosed_sInter {s : Set (Set X)} : (∀ t ∈ s, IsClosed t) → IsClos
 #align is_closed_sInter isClosed_sInter
 
 theorem isClosed_iInter {f : ι → Set X} (h : ∀ i, IsClosed (f i)) : IsClosed (⋂ i, f i) :=
-  isClosed_sInter <| forall_range_iff.2 h
+  isClosed_sInter <| forall_mem_range.2 h
 #align is_closed_Inter isClosed_iInter
 
 theorem isClosed_biInter {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i)) :

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

@@ -1721,9 +1721,8 @@ theorem closure_image_closure (h : Continuous f) :
     (closure_mono <| image_subset _ subset_closure)
 
 theorem closure_subset_preimage_closure_image (h : Continuous f) :
-    closure s ⊆ f ⁻¹' closure (f '' s) := by
-  rw [← Set.image_subset_iff]
-  exact image_closure_subset_closure_image h
+    closure s ⊆ f ⁻¹' closure (f '' s) :=
+  (mapsTo_image _ _).closure h
 #align closure_subset_preimage_closure_image closure_subset_preimage_closure_image
 
 theorem map_mem_closure {t : Set Y} (hf : Continuous f)
@@ -1737,6 +1736,15 @@ theorem Set.MapsTo.closure_left {t : Set Y} (h : MapsTo f s t)
   ht.closure_eq ▸ h.closure hc
 #align set.maps_to.closure_left Set.MapsTo.closure_left
 
+theorem Filter.Tendsto.lift'_closure (hf : Continuous f) {l l'} (h : Tendsto f l l') :
+    Tendsto f (l.lift' closure) (l'.lift' closure) :=
+  tendsto_lift'.2 fun s hs ↦ by
+    filter_upwards [mem_lift' (h hs)] using (mapsTo_preimage _ _).closure hf
+
+theorem tendsto_lift'_closure_nhds (hf : Continuous f) (x : X) :
+    Tendsto f ((𝓝 x).lift' closure) ((𝓝 (f x)).lift' closure) :=
+  (hf.tendsto x).lift'_closure hf
+
 /-!
 ### Function with dense range
 -/

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".

@@ -150,7 +150,7 @@ theorem isOpen_biInter_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈
   s.finite_toSet.isOpen_biInter h
 #align is_open_bInter_finset isOpen_biInter_finset
 
-@[simp] -- porting note: added `simp`
+@[simp] -- Porting note: added `simp`
 theorem isOpen_const {p : Prop} : IsOpen { _x : X | p } := by by_cases p <;> simp [*]
 #align is_open_const isOpen_const
 
@@ -169,7 +169,7 @@ theorem TopologicalSpace.ext_iff_isClosed {t₁ t₂ : TopologicalSpace X} :
 
 alias ⟨_, TopologicalSpace.ext_isClosed⟩ := TopologicalSpace.ext_iff_isClosed
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem isClosed_const {p : Prop} : IsClosed { _x : X | p } := ⟨isOpen_const (p := ¬p)⟩
 
 @[simp] theorem isClosed_empty : IsClosed (∅ : Set X) := isClosed_const
@@ -247,7 +247,7 @@ theorem IsClosed.not : IsClosed { a | p a } → IsOpen { a | ¬p a } :=
 ### Interior of a set
 -/
 
--- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
+-- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
 theorem mem_interior : x ∈ interior s ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_setOf_eq, and_assoc, and_left_comm]
 #align mem_interior mem_interiorₓ
@@ -820,7 +820,7 @@ theorem nhds_le_of_le {f} (h : x ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 
   rw [nhds_def]; exact iInf₂_le_of_le s ⟨h, o⟩ sf
 #align nhds_le_of_le nhds_le_of_le
 
--- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
+-- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
 theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t :=
   (nhds_basis_opens x).mem_iff.trans <| exists_congr fun _ =>
     ⟨fun h => ⟨h.2, h.1.2, h.1.1⟩, fun h => ⟨⟨h.2.2, h.2.1⟩, h.1⟩⟩
@@ -1256,7 +1256,7 @@ theorem dense_compl_singleton (x : X) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : Se
 
 /-- If `x` is not an isolated point of a topological space, then the closure of `{x}ᶜ` is the whole
 space. -/
--- porting note: was a `@[simp]` lemma but `simp` can prove it
+-- Porting note: was a `@[simp]` lemma but `simp` can prove it
 theorem closure_compl_singleton (x : X) [NeBot (𝓝[≠] x)] : closure {x}ᶜ = (univ : Set X) :=
   (dense_compl_singleton x).closure_eq
 #align closure_compl_singleton closure_compl_singleton
@@ -1713,7 +1713,7 @@ theorem image_closure_subset_closure_image (h : Continuous f) :
   ((mapsTo_image f s).closure h).image_subset
 #align image_closure_subset_closure_image image_closure_subset_closure_image
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem closure_image_closure (h : Continuous f) :
     closure (f '' closure s) = closure (f '' s) :=
   Subset.antisymm

• move definition to Topology/Defs/Basic;
• move lemmas to Topology/Baire/Lemmas;
• move instances to Topology/Baire/CompleteMetrizable and Topology/Baire/LocallyCompactRegular;
• assume [UniformSpace X] [IsCountablyGenerated (𝓤 X)] instead of [PseudoMetricSpace X] in the 1st theorem.

This way Lemmas file does not depend on analysis.

@@ -166,3 +166,11 @@ scoped[Topology] notation (name := closure_of) "closure[" t "]" => @closure _ t
 /-- Notation for `Continuous` with respect to a non-standard topologies. -/
 scoped[Topology] notation (name := Continuous_of) "Continuous[" t₁ ", " t₂ "]" =>
   @Continuous _ _ t₁ t₂
+
+/-- The property `BaireSpace α` means that the topological space `α` has the Baire property:
+any countable intersection of open dense subsets is dense.
+Formulated here when the source space is ℕ.
+Use `dense_iInter_of_isOpen` which works for any countable index type instead. -/
+class BaireSpace (X : Type*) [TopologicalSpace X] : Prop where
+  baire_property : ∀ f : ℕ → Set X, (∀ n, IsOpen (f n)) → (∀ n, Dense (f n)) → Dense (⋂ n, f n)
+#align baire_space BaireSpace

This is essentially the same as Filter.Tendsto.eventually_mem, but infers in more cases when applied.

Zulip discussion: https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/ContinuousAt.2Eeventually_mem'/near/422664457

@@ -1545,6 +1545,13 @@ theorem ContinuousAt.preimage_mem_nhds {t : Set Y} (h : ContinuousAt f x)
   h ht
 #align continuous_at.preimage_mem_nhds ContinuousAt.preimage_mem_nhds
 
+/-- If `f x ∈ s ∈ 𝓝 (f x)` for continuous `f`, then `f y ∈ s` near `x`.
+
+This is essentially `Filter.Tendsto.eventually_mem`, but infers in more cases when applied. -/
+theorem ContinuousAt.eventually_mem {f : X → Y} {x : X} (hf : ContinuousAt f x) {s : Set Y}
+    (hs : s ∈ 𝓝 (f x)) : ∀ᶠ y in 𝓝 x, f y ∈ s :=
+  hf hs
+
 /-- Deprecated, please use `not_mem_tsupport_iff_eventuallyEq` instead. -/
 @[deprecated] -- 15 January 2024
 theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {f : X → M₀} :

I changed Lean's 3 old "variables" command to Lean's 4 command "variable" in some implementation notes. I might have missed some

Co-authored-by: Omar Mohsen <36500353+OmarMohsenGit@users.noreply.github.com>

@@ -1841,7 +1841,7 @@ However, lemmas with this conclusion are not nice to use in practice because
 1. They confuse the elaborator. The following two examples fail, because of limitations in the
   elaboration process.
   ```
-  variables {M : Type*} [Add M] [TopologicalSpace M] [ContinuousAdd M]
+  variable {M : Type*} [Add M] [TopologicalSpace M] [ContinuousAdd M]
   example : Continuous (λ x : M, x + x) :=
   continuous_add.comp _
 

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>

@@ -1440,8 +1440,8 @@ Then `f` tends to `x` along `l` restricted to `s` if and only if it tends to `x`
 theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : α → X} {l : Filter α} {s : Set α}
     (h : ∀ a ∉ s, f a = x) : Tendsto f (l ⊓ 𝓟 s) (𝓝 x) ↔ Tendsto f l (𝓝 x) := by
   rw [tendsto_iff_comap, tendsto_iff_comap]
-  replace h : 𝓟 sᶜ ≤ comap f (𝓝 x)
-  · rintro U ⟨t, ht, htU⟩ x hx
+  replace h : 𝓟 sᶜ ≤ comap f (𝓝 x) := by
+    rintro U ⟨t, ht, htU⟩ x hx
     have : f x ∈ t := (h x hx).symm ▸ mem_of_mem_nhds ht
     exact htU this
   refine' ⟨fun h' => _, le_trans inf_le_left⟩

These were placed on purpose; reverting them caused breakage downstream.

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

@@ -1499,12 +1499,15 @@ variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalS
 
 open TopologicalSpace
 
-variable {f : X → Y} {s : Set X} {x : X} {y : Y}
-
-theorem continuous_def : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) :=
+-- The curly braces are intentional, so this definition works well with simp
+-- when topologies are not those provided by instances.
+theorem continuous_def {_ : TopologicalSpace X} {_ : TopologicalSpace Y} {f : X → Y} :
+    Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) :=
   ⟨fun hf => hf.1, fun h => ⟨h⟩⟩
 #align continuous_def continuous_def
 
+variable {f : X → Y} {s : Set X} {x : X} {y : Y}
+
 theorem IsOpen.preimage (hf : Continuous f) {t : Set Y} (h : IsOpen t) :
     IsOpen (f ⁻¹' t) :=
   hf.isOpen_preimage t h

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

@@ -3,8 +3,9 @@ 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, Jeremy Avigad
 -/
-import Mathlib.Tactic.Continuity
 import Mathlib.Order.SetNotation
+import Mathlib.Tactic.Continuity
+import Mathlib.Tactic.FunProp
 
 #align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
 /-!
@@ -132,6 +133,7 @@ def DenseRange {α : Type*} (f : α → X) := Dense (range f)
 
 /-- A function between topological spaces is continuous if the preimage
   of every open set is open. Registered as a structure to make sure it is not unfolded by Lean. -/
+@[fun_prop]
 structure Continuous (f : X → Y) : Prop where
   /-- The preimage of an open set under a continuous function is an open set. Use `IsOpen.preimage`
   instead. -/

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

@@ -1568,7 +1568,7 @@ theorem continuous_id : Continuous (id : X → X) :=
 #align continuous_id continuous_id
 
 -- This is needed due to reducibility issues with the `continuity` tactic.
-@[continuity]
+@[continuity, fun_prop]
 theorem continuous_id' : Continuous (fun (x : X) => x) := continuous_id
 
 theorem Continuous.comp {g : Y → Z} (hg : Continuous g) (hf : Continuous f) :
@@ -1577,7 +1577,7 @@ theorem Continuous.comp {g : Y → Z} (hg : Continuous g) (hf : Continuous f) :
 #align continuous.comp Continuous.comp
 
 -- This is needed due to reducibility issues with the `continuity` tactic.
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.comp' {g : Y → Z} (hg : Continuous g) (hf : Continuous f) :
     Continuous (fun x => g (f x)) := hg.comp hf
 
@@ -1590,6 +1590,10 @@ nonrec theorem ContinuousAt.comp {g : Y → Z} (hg : ContinuousAt g (f x))
   hg.comp hf
 #align continuous_at.comp ContinuousAt.comp
 
+@[fun_prop]
+theorem ContinuousAt.comp' {g : Y → Z} {x : X} (hg : ContinuousAt g (f x))
+    (hf : ContinuousAt f x) : ContinuousAt (fun x => g (f x)) x := ContinuousAt.comp hg hf
+
 /-- See note [comp_of_eq lemmas] -/
 theorem ContinuousAt.comp_of_eq {g : Y → Z} (hg : ContinuousAt g y)
     (hf : ContinuousAt f x) (hy : f x = y) : ContinuousAt (g ∘ f) x := by subst hy; exact hg.comp hf
@@ -1607,6 +1611,7 @@ theorem Continuous.tendsto' (hf : Continuous f) (x : X) (y : Y) (h : f x = y) :
   h ▸ hf.tendsto x
 #align continuous.tendsto' Continuous.tendsto'
 
+@[fun_prop]
 theorem Continuous.continuousAt (h : Continuous f) : ContinuousAt f x :=
   h.tendsto x
 #align continuous.continuous_at Continuous.continuousAt
@@ -1616,11 +1621,12 @@ theorem continuous_iff_continuousAt : Continuous f ↔ ∀ x, ContinuousAt f x :
     hf x <| hU.mem_nhds hx⟩
 #align continuous_iff_continuous_at continuous_iff_continuousAt
 
+@[fun_prop]
 theorem continuousAt_const : ContinuousAt (fun _ : X => y) x :=
   tendsto_const_nhds
 #align continuous_at_const continuousAt_const
 
-@[continuity]
+@[continuity, fun_prop]
 theorem continuous_const : Continuous fun _ : X => y :=
   continuous_iff_continuousAt.mpr fun _ => continuousAt_const
 #align continuous_const continuous_const
@@ -1639,6 +1645,9 @@ theorem continuousAt_id : ContinuousAt id x :=
   continuous_id.continuousAt
 #align continuous_at_id continuousAt_id
 
+@[fun_prop]
+theorem continuousAt_id' (y) : ContinuousAt (fun x : X => x) y := continuousAt_id
+
 theorem ContinuousAt.iterate {f : X → X} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ) :
     ContinuousAt f^[n] x :=
   Nat.recOn n continuousAt_id fun _n ihn ↦ ihn.comp_of_eq hf hx

@@ -1641,8 +1641,7 @@ theorem continuousAt_id : ContinuousAt id x :=
 
 theorem ContinuousAt.iterate {f : X → X} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ) :
     ContinuousAt f^[n] x :=
-  Nat.recOn n continuousAt_id fun n ihn =>
-    show ContinuousAt (f^[n] ∘ f) x from ContinuousAt.comp (hx.symm ▸ ihn) hf
+  Nat.recOn n continuousAt_id fun _n ihn ↦ ihn.comp_of_eq hf hx
 #align continuous_at.iterate ContinuousAt.iterate
 
 theorem continuous_iff_isClosed : Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s) :=

@@ -3,8 +3,8 @@ 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, Jeremy Avigad
 -/
-import Mathlib.Data.Set.Lattice
 import Mathlib.Tactic.Continuity
+import Mathlib.Order.SetNotation
 
 #align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
 /-!

• Rename (and generalize and move)

Int.units_ne_neg_self -> units_ne_neg_self
Int.neg_units_ne_self -> neg_units_ne_self

• Change the simps config for Closeds
• Add some gcongr-lemmas (currently is a bit picky about the exact statement of lemmas tagged with gcongr, so I had to add some variants that I could tag).

@@ -289,7 +289,7 @@ theorem subset_interior_iff : t ⊆ interior s ↔ ∃ U, IsOpen U ∧ t ⊆ U 
 lemma interior_subset_iff : interior s ⊆ t ↔ ∀ U, IsOpen U → U ⊆ s → U ⊆ t := by
   simp [interior]
 
-@[mono]
+@[mono, gcongr]
 theorem interior_mono (h : s ⊆ t) : interior s ⊆ interior t :=
   interior_maximal (Subset.trans interior_subset h) isOpen_interior
 #align interior_mono interior_mono
@@ -432,7 +432,7 @@ theorem IsClosed.mem_iff_closure_subset (hs : IsClosed s) :
   (hs.closure_subset_iff.trans Set.singleton_subset_iff).symm
 #align is_closed.mem_iff_closure_subset IsClosed.mem_iff_closure_subset
 
-@[mono]
+@[mono, gcongr]
 theorem closure_mono (h : s ⊆ t) : closure s ⊆ closure t :=
   closure_minimal (Subset.trans h subset_closure) isClosed_closure
 #align closure_mono closure_mono

In some cases, the order of implicit arguments changed because now they appear in a different order in variables.

Also, some definitions used greek letters for topological spaces, changed to X/Y.

In some cases, the order of implicit arguments changed because now they appear in a different order in variables.

Also, some definitions used greek letters for topological spaces, changed to X/Y.

@@ -5,8 +5,7 @@ Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
 -/
 import Mathlib.Algebra.Function.Support
 import Mathlib.Order.Filter.Lift
-import Mathlib.Order.Filter.Ultrafilter
-import Mathlib.Tactic.Continuity
+import Mathlib.Topology.Defs.Filter
 
 #align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
 
@@ -27,13 +26,11 @@ partially defined functions.
 
 ## Notation
 
+The following notation is introduced elsewhere and it heavily used in this file.
+
 * `𝓝 x`: the filter `nhds x` of neighborhoods of a point `x`;
 * `𝓟 s`: the principal filter of a set `s`;
 * `𝓝[s] x`: the filter `nhdsWithin x s` of neighborhoods of a point `x` within a set `s`;
-* `𝓝[≤] x`: the filter `nhdsWithin x (Set.Iic x)` of left-neighborhoods of `x`;
-* `𝓝[≥] x`: the filter `nhdsWithin x (Set.Ici x)` of right-neighborhoods of `x`;
-* `𝓝[<] x`: the filter `nhdsWithin x (Set.Iio x)` of punctured left-neighborhoods of `x`;
-* `𝓝[>] x`: the filter `nhdsWithin x (Set.Ioi x)` of punctured right-neighborhoods of `x`;
 * `𝓝[≠] x`: the filter `nhdsWithin x {x}ᶜ` of punctured neighborhoods of `x`.
 
 ## Implementation notes
@@ -61,22 +58,6 @@ universe u v w x
 ### Topological spaces
 -/
 
-
-/-- A topology on `X`. -/
-@[to_additive existing TopologicalSpace]
-class TopologicalSpace (X : Type u) where
-  /-- A predicate saying that a set is an open set. Use `IsOpen` in the root namespace instead. -/
-  protected IsOpen : Set X → Prop
-  /-- The set representing the whole space is an open set. Use `isOpen_univ` in the root namespace
-  instead. -/
-  protected isOpen_univ : IsOpen univ
-  /-- The intersection of two open sets is an open set. Use `IsOpen.inter` instead. -/
-  protected isOpen_inter : ∀ s t, IsOpen s → IsOpen t → IsOpen (s ∩ t)
-  /-- The union of a family of open sets is an open set. Use `isOpen_sUnion` in the root namespace
-  instead. -/
-  protected isOpen_sUnion : ∀ s, (∀ t ∈ s, IsOpen t) → IsOpen (⋃₀ s)
-#align topological_space TopologicalSpace
-
 /-- A constructor for topologies by specifying the closed sets,
 and showing that they satisfy the appropriate conditions. -/
 def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈ T)
@@ -95,14 +76,6 @@ section TopologicalSpace
 variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*}
   {x : X} {s s₁ s₂ t : Set X} {p p₁ p₂ : X → Prop}
 
-/-- `IsOpen s` means that `s` is open in the ambient topological space on `X` -/
-def IsOpen [TopologicalSpace X] : Set X → Prop := TopologicalSpace.IsOpen
-#align is_open IsOpen
-
-set_option quotPrecheck false in
-/-- Notation for `IsOpen` with respect to a non-standard topology. -/
-scoped[Topology] notation (name := IsOpen_of) "IsOpen[" t "]" => @IsOpen _ t
-
 open Topology
 
 lemma isOpen_mk {p h₁ h₂ h₃} : IsOpen[⟨p, h₁, h₂, h₃⟩] s ↔ p s := Iff.rfl
@@ -118,17 +91,6 @@ section
 
 variable [TopologicalSpace X]
 
-@[simp] theorem isOpen_univ : IsOpen (univ : Set X) := TopologicalSpace.isOpen_univ
-#align is_open_univ isOpen_univ
-
-theorem IsOpen.inter (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∩ s₂) :=
-  TopologicalSpace.isOpen_inter s₁ s₂ h₁ h₂
-#align is_open.inter IsOpen.inter
-
-theorem isOpen_sUnion {s : Set (Set X)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (⋃₀ s) :=
-  TopologicalSpace.isOpen_sUnion s h
-#align is_open_sUnion isOpen_sUnion
-
 end
 
 protected theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace X} :
@@ -196,16 +158,6 @@ theorem IsOpen.and : IsOpen { x | p₁ x } → IsOpen { x | p₂ x } → IsOpen
   IsOpen.inter
 #align is_open.and IsOpen.and
 
-/-- A set is closed if its complement is open -/
-class IsClosed (s : Set X) : Prop where
-  /-- The complement of a closed set is an open set. -/
-  isOpen_compl : IsOpen sᶜ
-#align is_closed IsClosed
-
-set_option quotPrecheck false in
-/-- Notation for `IsClosed` with respect to a non-standard topology. -/
-scoped[Topology] notation (name := IsClosed_of) "IsClosed[" t "]" => @IsClosed _ t
-
 @[simp] theorem isOpen_compl_iff : IsOpen sᶜ ↔ IsClosed s :=
   ⟨fun h => ⟨h⟩, fun h => h.isOpen_compl⟩
 #align is_open_compl_iff isOpen_compl_iff
@@ -295,11 +247,6 @@ theorem IsClosed.not : IsClosed { a | p a } → IsOpen { a | ¬p a } :=
 ### Interior of a set
 -/
 
-/-- The interior of a set `s` is the largest open subset of `s`. -/
-def interior (s : Set X) : Set X :=
-  ⋃₀ { t | IsOpen t ∧ t ⊆ s }
-#align interior interior
-
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
 theorem mem_interior : x ∈ interior s ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_setOf_eq, and_assoc, and_left_comm]
@@ -441,16 +388,6 @@ theorem Filter.HasBasis.lift'_interior_eq_self {l : Filter X} {p : ι → Prop}
 ### Closure of a set
 -/
 
-
-/-- The closure of `s` is the smallest closed set containing `s`. -/
-def closure (s : Set X) : Set X :=
-  ⋂₀ { t | IsClosed t ∧ s ⊆ t }
-#align closure closure
-
-set_option quotPrecheck false in
-/-- Notation for `closure` with respect to a non-standard topology. -/
-scoped[Topology] notation (name := closure_of) "closure[" t "]" => @closure _ t
-
 @[simp]
 theorem isClosed_closure : IsClosed (closure s) :=
   isClosed_sInter fun _ => And.left
@@ -637,11 +574,6 @@ theorem Filter.lift'_closure_eq_bot {l : Filter X} : l.lift' closure = ⊥ ↔ l
     h.symm ▸ by rw [lift'_bot (monotone_closure _), closure_empty, principal_empty]⟩
 #align filter.lift'_closure_eq_bot Filter.lift'_closure_eq_bot
 
-/-- A set is dense in a topological space if every point belongs to its closure. -/
-def Dense (s : Set X) : Prop :=
-  ∀ x, x ∈ closure s
-#align dense Dense
-
 theorem dense_iff_closure_eq : Dense s ↔ closure s = univ :=
   eq_univ_iff_forall.symm
 #align dense_iff_closure_eq dense_iff_closure_eq
@@ -724,11 +656,6 @@ theorem dense_compl_singleton_iff_not_open :
 ### Frontier of a set
 -/
 
-/-- The frontier of a set is the set of points between the closure and interior. -/
-def frontier (s : Set X) : Set X :=
-  closure s \ interior s
-#align frontier frontier
-
 @[simp]
 theorem closure_diff_interior (s : Set X) : closure s \ interior s = frontier s :=
   rfl
@@ -855,45 +782,6 @@ theorem compl_frontier_eq_union_interior :
 ### Neighborhoods
 -/
 
-/-- A set is called a neighborhood of `x` if it contains an open set around `x`. The set of all
-neighborhoods of `x` forms a filter, the neighborhood filter at `x`, is here defined as the
-infimum over the principal filters of all open sets containing `x`. -/
-irreducible_def nhds (x : X) : Filter X :=
-  ⨅ s ∈ { s : Set X | x ∈ s ∧ IsOpen s }, 𝓟 s
-#align nhds nhds
-#align nhds_def nhds_def
-
-/-- The "neighborhood within" filter. Elements of `𝓝[s] x` are sets containing the
-intersection of `s` and a neighborhood of `x`. -/
-def nhdsWithin (x : X) (s : Set X) : Filter X :=
-  nhds x ⊓ 𝓟 s
-#align nhds_within nhdsWithin
-
-section
-
-@[inherit_doc]
-scoped[Topology] notation "𝓝" => nhds
-
-@[inherit_doc]
-scoped[Topology] notation "𝓝[" s "] " x:100 => nhdsWithin x s
-
-/-- Notation for the filter of punctured neighborhoods of a point. -/
-scoped[Topology] notation3 "𝓝[≠] " x:100 => nhdsWithin x (@singleton _ (Set _) instSingletonSet x)ᶜ
-
-/-- Notation for the filter of right neighborhoods of a point. -/
-scoped[Topology] notation3 "𝓝[≥] " x:100 => nhdsWithin x (Set.Ici x)
-
-/-- Notation for the filter of left neighborhoods of a point. -/
-scoped[Topology] notation3 "𝓝[≤] " x:100 => nhdsWithin x (Set.Iic x)
-
-/-- Notation for the filter of punctured right neighborhoods of a point. -/
-scoped[Topology] notation3 "𝓝[>] " x:100 => nhdsWithin x (Set.Ioi x)
-
-/-- Notation for the filter of punctured left neighborhoods of a point. -/
-scoped[Topology] notation3 "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
-
-end
-
 theorem nhds_def' (x : X) : 𝓝 x = ⨅ (s : Set X) (_ : IsOpen s) (_ : x ∈ s), 𝓟 s := by
   simp only [nhds_def, mem_setOf_eq, @and_comm (x ∈ _), iInf_and]
 #align nhds_def' nhds_def'
@@ -1131,13 +1019,6 @@ In this section we define [cluster points](https://en.wikipedia.org/wiki/Limit_p
 -/
 
 
-/-- A point `x` is a cluster point of a filter `F` if `𝓝 x ⊓ F ≠ ⊥`. Also known as
-an accumulation point or a limit point, but beware that terminology varies. This
-is *not* the same as asking `𝓝[≠] x ⊓ F ≠ ⊥`. See `mem_closure_iff_clusterPt` in particular. -/
-def ClusterPt (x : X) (F : Filter X) : Prop :=
-  NeBot (𝓝 x ⊓ F)
-#align cluster_pt ClusterPt
-
 theorem ClusterPt.neBot {F : Filter X} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) :=
   h
 #align cluster_pt.ne_bot ClusterPt.neBot
@@ -1203,12 +1084,6 @@ theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
     ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
   simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]
 
-/-- A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point
-of `map u F`. -/
-def MapClusterPt {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) : Prop :=
-  ClusterPt x (map u F)
-#align map_cluster_pt MapClusterPt
-
 theorem mapClusterPt_def {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
     MapClusterPt x F u ↔ ClusterPt x (map u F) := Iff.rfl
 
@@ -1234,11 +1109,6 @@ theorem mapClusterPt_of_comp {F : Filter α} {φ : β → α} {p : Filter β}
   exact neBot_of_le this
 #align map_cluster_pt_of_comp mapClusterPt_of_comp
 
-/-- A point `x` is an accumulation point of a filter `F` if `𝓝[≠] x ⊓ F ≠ ⊥`.-/
-def AccPt (x : X) (F : Filter X) : Prop :=
-  NeBot (𝓝[≠] x ⊓ F)
-#align acc_pt AccPt
-
 theorem acc_iff_cluster (x : X) (F : Filter X) : AccPt x F ↔ ClusterPt x (𝓟 {x}ᶜ ⊓ F) := by
   rw [AccPt, nhdsWithin, ClusterPt, inf_assoc]
 #align acc_iff_cluster acc_iff_cluster
@@ -1596,25 +1466,6 @@ section lim
 -- "Lim"
 set_option linter.uppercaseLean3 false
 
-/-- If `f` is a filter, then `Filter.lim f` is a limit of the filter, if it exists. -/
-noncomputable def lim [Nonempty X] (f : Filter X) : X :=
-  Classical.epsilon fun x => f ≤ 𝓝 x
-#align Lim lim
-
-/--
-If `F` is an ultrafilter, then `Filter.Ultrafilter.lim F` is a limit of the filter, if it exists.
-Note that dot notation `F.lim` can be used for `F : Filter.Ultrafilter X`.
--/
-noncomputable nonrec def Ultrafilter.lim (F : Ultrafilter X) : X :=
-  @lim X _ (nonempty_of_neBot F) F
-#align ultrafilter.Lim Ultrafilter.lim
-
-/-- If `f` is a filter in `α` and `g : α → X` is a function, then `limUnder f g` is a limit of `g`
-at `f`, if it exists. -/
-noncomputable def limUnder [Nonempty X] (f : Filter α) (g : α → X) : X :=
-  lim (f.map g)
-#align lim limUnder
-
 /-- If a filter `f` is majorated by some `𝓝 x`, then it is majorated by `𝓝 (Filter.lim f)`. We
 formulate this lemma with a `[Nonempty X]` argument of `lim` derived from `h` to make it useful for
 types without a `[Nonempty X]` instance. Because of the built-in proof irrelevance, Lean will unify
@@ -1648,19 +1499,6 @@ variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalS
 
 open TopologicalSpace
 
-/-- A function between topological spaces is continuous if the preimage
-  of every open set is open. Registered as a structure to make sure it is not unfolded by Lean. -/
-structure Continuous (f : X → Y) : Prop where
-  /-- The preimage of an open set under a continuous function is an open set. Use `IsOpen.preimage`
-  instead. -/
-  isOpen_preimage : ∀ s, IsOpen s → IsOpen (f ⁻¹' s)
-#align continuous Continuous
-
-set_option quotPrecheck false in
-/-- Notation for `Continuous` with respect to a non-standard topologies. -/
-scoped[Topology] notation (name := Continuous_of) "Continuous[" t₁ ", " t₂ "]" =>
-  @Continuous _ _ t₁ t₂
-
 variable {f : X → Y} {s : Set X} {x : X} {y : Y}
 
 theorem continuous_def : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) :=
@@ -1680,12 +1518,6 @@ theorem Continuous.congr {g : X → Y} (h : Continuous f) (h' : ∀ x, f x = g x
   continuous_congr h' |>.mp h
 #align continuous.congr Continuous.congr
 
-/-- A function between topological spaces is continuous at a point `x₀`
-if `f x` tends to `f x₀` when `x` tends to `x₀`. -/
-def ContinuousAt (f : X → Y) (x : X) :=
-  Tendsto f (𝓝 x) (𝓝 (f x))
-#align continuous_at ContinuousAt
-
 theorem ContinuousAt.tendsto (h : ContinuousAt f x) :
     Tendsto f (𝓝 x) (𝓝 (f x)) :=
   h
@@ -1895,10 +1727,6 @@ section DenseRange
 
 variable {α ι : Type*} (f : α → X) (g : X → Y)
 
-/-- `f : α → X` has dense range if its range (image) is a dense subset of `X`. -/
-def DenseRange := Dense (range f)
-#align dense_range DenseRange
-
 variable {f : α → X} {s : Set X}
 
 /-- A surjective map has dense range. -/

@@ -425,6 +425,18 @@ theorem interior_sInter_subset (S : Set (Set X)) : interior (⋂₀ S) ⊆ ⋂ s
     _ ⊆ ⋂ s ∈ S, interior s := interior_iInter₂_subset _ _
 #align interior_sInter_subset interior_sInter_subset
 
+theorem Filter.HasBasis.lift'_interior {l : Filter X} {p : ι → Prop} {s : ι → Set X}
+    (h : l.HasBasis p s) : (l.lift' interior).HasBasis p fun i => interior (s i) :=
+  h.lift' fun _ _ ↦ interior_mono
+
+theorem Filter.lift'_interior_le (l : Filter X) : l.lift' interior ≤ l := fun _s hs ↦
+  mem_of_superset (mem_lift' hs) interior_subset
+
+theorem Filter.HasBasis.lift'_interior_eq_self {l : Filter X} {p : ι → Prop} {s : ι → Set X}
+    (h : l.HasBasis p s) (ho : ∀ i, p i → IsOpen (s i)) : l.lift' interior = l :=
+  le_antisymm l.lift'_interior_le <| h.lift'_interior.ge_iff.2 fun i hi ↦ by
+    simpa only [(ho i hi).interior_eq] using h.mem_of_mem hi
+
 /-!
 ### Closure of a set
 -/
@@ -902,6 +914,14 @@ theorem nhds_basis_closeds (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∉ s
     compl_surjective.exists.trans <| by simp only [isOpen_compl_iff, mem_compl_iff]⟩
 #align nhds_basis_closeds nhds_basis_closeds
 
+@[simp]
+theorem lift'_nhds_interior (x : X) : (𝓝 x).lift' interior = 𝓝 x :=
+  (nhds_basis_opens x).lift'_interior_eq_self fun _ ↦ And.right
+
+theorem Filter.HasBasis.nhds_interior {x : X} {p : ι → Prop} {s : ι → Set X}
+    (h : (𝓝 x).HasBasis p s) : (𝓝 x).HasBasis p (interior <| s ·) :=
+  lift'_nhds_interior x ▸ h.lift'_interior
+
 /-- A filter lies below the neighborhood filter at `x` iff it contains every open set around `x`. -/
 theorem le_nhds_iff {f} : f ≤ 𝓝 x ↔ ∀ s : Set X, x ∈ s → IsOpen s → s ∈ f := by simp [nhds_def]
 #align le_nhds_iff le_nhds_iff

Preliminaries for #6844

@@ -1179,18 +1179,30 @@ theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f 
   ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩
 #align ultrafilter.cluster_pt_iff Ultrafilter.clusterPt_iff
 
+theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
+    ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
+  simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]
+
 /-- A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point
 of `map u F`. -/
 def MapClusterPt {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) : Prop :=
   ClusterPt x (map u F)
 #align map_cluster_pt MapClusterPt
 
+theorem mapClusterPt_def {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
+    MapClusterPt x F u ↔ ClusterPt x (map u F) := Iff.rfl
+
 theorem mapClusterPt_iff {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
     MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s := by
   simp_rw [MapClusterPt, ClusterPt, inf_neBot_iff_frequently_left, frequently_map]
   rfl
 #align map_cluster_pt_iff mapClusterPt_iff
 
+theorem mapClusterPt_iff_ultrafilter {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
+    MapClusterPt x F u ↔ ∃ U : Ultrafilter ι, U ≤ F ∧ Tendsto u U (𝓝 x) := by
+  simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
+    ← le_inf_iff, exists_ultrafilter_iff, inf_comm]
+
 theorem mapClusterPt_of_comp {F : Filter α} {φ : β → α} {p : Filter β}
     {u : α → X} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) :
     MapClusterPt x F u := by
@@ -1432,6 +1444,10 @@ theorem isClosed_iff_clusterPt : IsClosed s ↔ ∀ a, ClusterPt a (𝓟 s) →
     _ ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s := by simp only [subset_def, mem_closure_iff_clusterPt]
 #align is_closed_iff_cluster_pt isClosed_iff_clusterPt
 
+theorem isClosed_iff_ultrafilter : IsClosed s ↔
+    ∀ x, ∀ u : Ultrafilter X, ↑u ≤ 𝓝 x → s ∈ u → x ∈ s := by
+  simp [isClosed_iff_clusterPt, ClusterPt, ← exists_ultrafilter_iff]
+
 theorem isClosed_iff_nhds :
     IsClosed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s := by
   simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_neBot_iff]

There is no NeBot in this lemma

@@ -1329,9 +1329,12 @@ theorem mem_closure_iff_clusterPt : x ∈ closure s ↔ ClusterPt x (𝓟 s) :=
   mem_closure_iff_frequently.trans clusterPt_principal_iff_frequently.symm
 #align mem_closure_iff_cluster_pt mem_closure_iff_clusterPt
 
-theorem mem_closure_iff_nhds_neBot : x ∈ closure s ↔ 𝓝 x ⊓ 𝓟 s ≠ ⊥ :=
+theorem mem_closure_iff_nhds_ne_bot : x ∈ closure s ↔ 𝓝 x ⊓ 𝓟 s ≠ ⊥ :=
   mem_closure_iff_clusterPt.trans neBot_iff
-#align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_neBot
+#align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_ne_bot
+
+@[deprecated] -- 28 January 2024
+alias mem_closure_iff_nhds_neBot := mem_closure_iff_nhds_ne_bot
 
 theorem mem_closure_iff_nhdsWithin_neBot : x ∈ closure s ↔ NeBot (𝓝[s] x) :=
   mem_closure_iff_clusterPt
@@ -1471,14 +1474,8 @@ theorem Dense.open_subset_closure_inter (hs : Dense s) (ht : IsOpen t) :
 
 theorem mem_closure_of_mem_closure_union (h : x ∈ closure (s₁ ∪ s₂))
     (h₁ : s₁ᶜ ∈ 𝓝 x) : x ∈ closure s₂ := by
-  rw [mem_closure_iff_nhds_neBot] at *
-  rwa [←
-    calc
-      𝓝 x ⊓ principal (s₁ ∪ s₂) = 𝓝 x ⊓ (principal s₁ ⊔ principal s₂) := by rw [sup_principal]
-      _ = 𝓝 x ⊓ principal s₁ ⊔ 𝓝 x ⊓ principal s₂ := inf_sup_left
-      _ = ⊥ ⊔ 𝓝 x ⊓ principal s₂ := by rw [inf_principal_eq_bot.mpr h₁]
-      _ = 𝓝 x ⊓ principal s₂ := bot_sup_eq
-      ]
+  rw [mem_closure_iff_nhds_ne_bot] at *
+  rwa [← sup_principal, inf_sup_left, inf_principal_eq_bot.mpr h₁, bot_sup_eq] at h
 #align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_union
 
 /-- The intersection of an open dense set with a dense set is a dense set. -/

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -20,8 +20,8 @@ Then `Set X` gets predicates `IsOpen`, `IsClosed` and functions `interior`, `clo
 along `F : Filter α` if `MapClusterPt x F f : ClusterPt x (map f F)`. In particular
 the notion of cluster point of a sequence `u` is `MapClusterPt x atTop u`.
 
-For topological spaces `X` and `Y`, a function `f : X → Y` and a point `a : X`,
-`ContinuousAt f a` means `f` is continuous at `a`, and global continuity is
+For topological spaces `X` and `Y`, a function `f : X → Y` and a point `x : X`,
+`ContinuousAt f x` means `f` is continuous at `x`, and global continuity is
 `Continuous f`. There is also a version of continuity `PContinuous` for
 partially defined functions.
 
@@ -93,7 +93,7 @@ def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ 
 section TopologicalSpace
 
 variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*}
-  {a : X} {s s₁ s₂ t : Set X} {p p₁ p₂ : X → Prop}
+  {x : X} {s s₁ s₂ t : Set X} {p p₁ p₂ : X → Prop}
 
 /-- `IsOpen s` means that `s` is open in the ambient topological space on `X` -/
 def IsOpen [TopologicalSpace X] : Set X → Prop := TopologicalSpace.IsOpen
@@ -105,7 +105,7 @@ scoped[Topology] notation (name := IsOpen_of) "IsOpen[" t "]" => @IsOpen _ t
 
 open Topology
 
-lemma isOpen_mk {p h₁ h₂ h₃} {s : Set X} : IsOpen[⟨p, h₁, h₂, h₃⟩] s ↔ p s := Iff.rfl
+lemma isOpen_mk {p h₁ h₂ h₃} : IsOpen[⟨p, h₁, h₂, h₃⟩] s ↔ p s := Iff.rfl
 #align is_open_mk isOpen_mk
 
 @[ext]
@@ -136,7 +136,7 @@ protected theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace X} :
   ⟨fun h s => h ▸ Iff.rfl, fun h => by ext; exact h _⟩
 #align topological_space_eq_iff TopologicalSpace.ext_iff
 
-theorem isOpen_fold {s : Set X} {t : TopologicalSpace X} : t.IsOpen s = IsOpen[t] s :=
+theorem isOpen_fold {t : TopologicalSpace X} : t.IsOpen s = IsOpen[t] s :=
   rfl
 #align is_open_fold isOpen_fold
 
@@ -189,10 +189,10 @@ theorem isOpen_biInter_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈
 #align is_open_bInter_finset isOpen_biInter_finset
 
 @[simp] -- porting note: added `simp`
-theorem isOpen_const {p : Prop} : IsOpen { _a : X | p } := by by_cases p <;> simp [*]
+theorem isOpen_const {p : Prop} : IsOpen { _x : X | p } := by by_cases p <;> simp [*]
 #align is_open_const isOpen_const
 
-theorem IsOpen.and : IsOpen { a | p₁ a } → IsOpen { a | p₂ a } → IsOpen { a | p₁ a ∧ p₂ a } :=
+theorem IsOpen.and : IsOpen { x | p₁ x } → IsOpen { x | p₂ x } → IsOpen { x | p₁ x ∧ p₂ x } :=
   IsOpen.inter
 #align is_open.and IsOpen.and
 
@@ -206,19 +206,19 @@ set_option quotPrecheck false in
 /-- Notation for `IsClosed` with respect to a non-standard topology. -/
 scoped[Topology] notation (name := IsClosed_of) "IsClosed[" t "]" => @IsClosed _ t
 
-@[simp] theorem isOpen_compl_iff {s : Set X} : IsOpen sᶜ ↔ IsClosed s :=
+@[simp] theorem isOpen_compl_iff : IsOpen sᶜ ↔ IsClosed s :=
   ⟨fun h => ⟨h⟩, fun h => h.isOpen_compl⟩
 #align is_open_compl_iff isOpen_compl_iff
 
 theorem TopologicalSpace.ext_iff_isClosed {t₁ t₂ : TopologicalSpace X} :
     t₁ = t₂ ↔ ∀ s, IsClosed[t₁] s ↔ IsClosed[t₂] s := by
   rw [TopologicalSpace.ext_iff, compl_surjective.forall]
-  simp only [@isOpen_compl_iff _ t₁, @isOpen_compl_iff _ t₂]
+  simp only [@isOpen_compl_iff _ _ t₁, @isOpen_compl_iff _ _ t₂]
 
 alias ⟨_, TopologicalSpace.ext_isClosed⟩ := TopologicalSpace.ext_iff_isClosed
 
 -- porting note: new lemma
-theorem isClosed_const {p : Prop} : IsClosed { _a : X | p } := ⟨isOpen_const (p := ¬p)⟩
+theorem isClosed_const {p : Prop} : IsClosed { _x : X | p } := ⟨isOpen_const (p := ¬p)⟩
 
 @[simp] theorem isClosed_empty : IsClosed (∅ : Set X) := isClosed_const
 #align is_closed_empty isClosed_empty
@@ -251,7 +251,7 @@ theorem isClosed_compl_iff {s : Set X} : IsClosed sᶜ ↔ IsOpen s := by
 alias ⟨_, IsOpen.isClosed_compl⟩ := isClosed_compl_iff
 #align is_open.is_closed_compl IsOpen.isClosed_compl
 
-theorem IsOpen.sdiff {s t : Set X} (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) :=
+theorem IsOpen.sdiff (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) :=
   IsOpen.inter h₁ h₂.isOpen_compl
 #align is_open.sdiff IsOpen.sdiff
 
@@ -261,7 +261,7 @@ theorem IsClosed.inter (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) : IsClosed
   exact IsOpen.union h₁ h₂
 #align is_closed.inter IsClosed.inter
 
-theorem IsClosed.sdiff {s t : Set X} (h₁ : IsClosed s) (h₂ : IsOpen t) : IsClosed (s \ t) :=
+theorem IsClosed.sdiff (h₁ : IsClosed s) (h₂ : IsOpen t) : IsClosed (s \ t) :=
   IsClosed.inter h₁ (isClosed_compl_iff.mpr h₂)
 #align is_closed.sdiff IsClosed.sdiff
 
@@ -301,40 +301,40 @@ def interior (s : Set X) : Set X :=
 #align interior interior
 
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
-theorem mem_interior {s : Set X} {x : X} : x ∈ interior s ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := by
+theorem mem_interior : x ∈ interior s ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_setOf_eq, and_assoc, and_left_comm]
 #align mem_interior mem_interiorₓ
 
 @[simp]
-theorem isOpen_interior {s : Set X} : IsOpen (interior s) :=
+theorem isOpen_interior : IsOpen (interior s) :=
   isOpen_sUnion fun _ => And.left
 #align is_open_interior isOpen_interior
 
-theorem interior_subset {s : Set X} : interior s ⊆ s :=
+theorem interior_subset : interior s ⊆ s :=
   sUnion_subset fun _ => And.right
 #align interior_subset interior_subset
 
-theorem interior_maximal {s t : Set X} (h₁ : t ⊆ s) (h₂ : IsOpen t) : t ⊆ interior s :=
+theorem interior_maximal (h₁ : t ⊆ s) (h₂ : IsOpen t) : t ⊆ interior s :=
   subset_sUnion_of_mem ⟨h₂, h₁⟩
 #align interior_maximal interior_maximal
 
-theorem IsOpen.interior_eq {s : Set X} (h : IsOpen s) : interior s = s :=
+theorem IsOpen.interior_eq (h : IsOpen s) : interior s = s :=
   interior_subset.antisymm (interior_maximal (Subset.refl s) h)
 #align is_open.interior_eq IsOpen.interior_eq
 
-theorem interior_eq_iff_isOpen {s : Set X} : interior s = s ↔ IsOpen s :=
+theorem interior_eq_iff_isOpen : interior s = s ↔ IsOpen s :=
   ⟨fun h => h ▸ isOpen_interior, IsOpen.interior_eq⟩
 #align interior_eq_iff_is_open interior_eq_iff_isOpen
 
-theorem subset_interior_iff_isOpen {s : Set X} : s ⊆ interior s ↔ IsOpen s := by
+theorem subset_interior_iff_isOpen : s ⊆ interior s ↔ IsOpen s := by
   simp only [interior_eq_iff_isOpen.symm, Subset.antisymm_iff, interior_subset, true_and]
 #align subset_interior_iff_is_open subset_interior_iff_isOpen
 
-theorem IsOpen.subset_interior_iff {s t : Set X} (h₁ : IsOpen s) : s ⊆ interior t ↔ s ⊆ t :=
+theorem IsOpen.subset_interior_iff (h₁ : IsOpen s) : s ⊆ interior t ↔ s ⊆ t :=
   ⟨fun h => Subset.trans h interior_subset, fun h₂ => interior_maximal h₂ h₁⟩
 #align is_open.subset_interior_iff IsOpen.subset_interior_iff
 
-theorem subset_interior_iff {s t : Set X} : t ⊆ interior s ↔ ∃ U, IsOpen U ∧ t ⊆ U ∧ U ⊆ s :=
+theorem subset_interior_iff : t ⊆ interior s ↔ ∃ U, IsOpen U ∧ t ⊆ U ∧ U ⊆ s :=
   ⟨fun h => ⟨interior s, isOpen_interior, h, interior_subset⟩, fun ⟨_U, hU, htU, hUs⟩ =>
     htU.trans (interior_maximal hUs hU)⟩
 #align subset_interior_iff subset_interior_iff
@@ -343,7 +343,7 @@ lemma interior_subset_iff : interior s ⊆ t ↔ ∀ U, IsOpen U → U ⊆ s →
   simp [interior]
 
 @[mono]
-theorem interior_mono {s t : Set X} (h : s ⊆ t) : interior s ⊆ interior t :=
+theorem interior_mono (h : s ⊆ t) : interior s ⊆ interior t :=
   interior_maximal (Subset.trans interior_subset h) isOpen_interior
 #align interior_mono interior_mono
 
@@ -358,17 +358,17 @@ theorem interior_univ : interior (univ : Set X) = univ :=
 #align interior_univ interior_univ
 
 @[simp]
-theorem interior_eq_univ {s : Set X} : interior s = univ ↔ s = univ :=
+theorem interior_eq_univ : interior s = univ ↔ s = univ :=
   ⟨fun h => univ_subset_iff.mp <| h.symm.trans_le interior_subset, fun h => h.symm ▸ interior_univ⟩
 #align interior_eq_univ interior_eq_univ
 
 @[simp]
-theorem interior_interior {s : Set X} : interior (interior s) = interior s :=
+theorem interior_interior : interior (interior s) = interior s :=
   isOpen_interior.interior_eq
 #align interior_interior interior_interior
 
 @[simp]
-theorem interior_inter {s t : Set X} : interior (s ∩ t) = interior s ∩ interior t :=
+theorem interior_inter : interior (s ∩ t) = interior s ∩ interior t :=
   (Monotone.map_inf_le (fun _ _ ↦ interior_mono) s t).antisymm <|
     interior_maximal (inter_subset_inter interior_subset interior_subset) <|
       isOpen_interior.inter isOpen_interior
@@ -394,7 +394,7 @@ theorem interior_iInter_of_finite [Finite ι] (f : ι → Set X) :
   rw [← sInter_range, (finite_range f).interior_sInter, biInter_range]
 #align interior_Inter interior_iInter_of_finite
 
-theorem interior_union_isClosed_of_interior_empty {s t : Set X} (h₁ : IsClosed s)
+theorem interior_union_isClosed_of_interior_empty (h₁ : IsClosed s)
     (h₂ : interior t = ∅) : interior (s ∪ t) = interior s :=
   have : interior (s ∪ t) ⊆ s := fun x ⟨u, ⟨(hu₁ : IsOpen u), (hu₂ : u ⊆ s ∪ t)⟩, (hx₁ : x ∈ u)⟩ =>
     by_contradiction fun hx₂ : x ∉ s =>
@@ -440,51 +440,51 @@ set_option quotPrecheck false in
 scoped[Topology] notation (name := closure_of) "closure[" t "]" => @closure _ t
 
 @[simp]
-theorem isClosed_closure {s : Set X} : IsClosed (closure s) :=
+theorem isClosed_closure : IsClosed (closure s) :=
   isClosed_sInter fun _ => And.left
 #align is_closed_closure isClosed_closure
 
-theorem subset_closure {s : Set X} : s ⊆ closure s :=
+theorem subset_closure : s ⊆ closure s :=
   subset_sInter fun _ => And.right
 #align subset_closure subset_closure
 
-theorem not_mem_of_not_mem_closure {s : Set X} {P : X} (hP : P ∉ closure s) : P ∉ s := fun h =>
+theorem not_mem_of_not_mem_closure {P : X} (hP : P ∉ closure s) : P ∉ s := fun h =>
   hP (subset_closure h)
 #align not_mem_of_not_mem_closure not_mem_of_not_mem_closure
 
-theorem closure_minimal {s t : Set X} (h₁ : s ⊆ t) (h₂ : IsClosed t) : closure s ⊆ t :=
+theorem closure_minimal (h₁ : s ⊆ t) (h₂ : IsClosed t) : closure s ⊆ t :=
   sInter_subset_of_mem ⟨h₂, h₁⟩
 #align closure_minimal closure_minimal
 
-theorem Disjoint.closure_left {s t : Set X} (hd : Disjoint s t) (ht : IsOpen t) :
+theorem Disjoint.closure_left (hd : Disjoint s t) (ht : IsOpen t) :
     Disjoint (closure s) t :=
   disjoint_compl_left.mono_left <| closure_minimal hd.subset_compl_right ht.isClosed_compl
 #align disjoint.closure_left Disjoint.closure_left
 
-theorem Disjoint.closure_right {s t : Set X} (hd : Disjoint s t) (hs : IsOpen s) :
+theorem Disjoint.closure_right (hd : Disjoint s t) (hs : IsOpen s) :
     Disjoint s (closure t) :=
   (hd.symm.closure_left hs).symm
 #align disjoint.closure_right Disjoint.closure_right
 
-theorem IsClosed.closure_eq {s : Set X} (h : IsClosed s) : closure s = s :=
+theorem IsClosed.closure_eq (h : IsClosed s) : closure s = s :=
   Subset.antisymm (closure_minimal (Subset.refl s) h) subset_closure
 #align is_closed.closure_eq IsClosed.closure_eq
 
-theorem IsClosed.closure_subset {s : Set X} (hs : IsClosed s) : closure s ⊆ s :=
+theorem IsClosed.closure_subset (hs : IsClosed s) : closure s ⊆ s :=
   closure_minimal (Subset.refl _) hs
 #align is_closed.closure_subset IsClosed.closure_subset
 
-theorem IsClosed.closure_subset_iff {s t : Set X} (h₁ : IsClosed t) : closure s ⊆ t ↔ s ⊆ t :=
+theorem IsClosed.closure_subset_iff (h₁ : IsClosed t) : closure s ⊆ t ↔ s ⊆ t :=
   ⟨Subset.trans subset_closure, fun h => closure_minimal h h₁⟩
 #align is_closed.closure_subset_iff IsClosed.closure_subset_iff
 
-theorem IsClosed.mem_iff_closure_subset {s : Set X} (hs : IsClosed s) {x : X} :
+theorem IsClosed.mem_iff_closure_subset (hs : IsClosed s) :
     x ∈ s ↔ closure ({x} : Set X) ⊆ s :=
   (hs.closure_subset_iff.trans Set.singleton_subset_iff).symm
 #align is_closed.mem_iff_closure_subset IsClosed.mem_iff_closure_subset
 
 @[mono]
-theorem closure_mono {s t : Set X} (h : s ⊆ t) : closure s ⊆ closure t :=
+theorem closure_mono (h : s ⊆ t) : closure s ⊆ closure t :=
   closure_minimal (Subset.trans h subset_closure) isClosed_closure
 #align closure_mono closure_mono
 
@@ -492,7 +492,7 @@ theorem monotone_closure (X : Type*) [TopologicalSpace X] : Monotone (@closure X
   closure_mono
 #align monotone_closure monotone_closure
 
-theorem diff_subset_closure_iff {s t : Set X} : s \ t ⊆ closure t ↔ s ⊆ closure t := by
+theorem diff_subset_closure_iff : s \ t ⊆ closure t ↔ s ⊆ closure t := by
   rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]
 #align diff_subset_closure_iff diff_subset_closure_iff
 
@@ -501,15 +501,15 @@ theorem closure_inter_subset_inter_closure (s t : Set X) :
   (monotone_closure X).map_inf_le s t
 #align closure_inter_subset_inter_closure closure_inter_subset_inter_closure
 
-theorem isClosed_of_closure_subset {s : Set X} (h : closure s ⊆ s) : IsClosed s := by
+theorem isClosed_of_closure_subset (h : closure s ⊆ s) : IsClosed s := by
   rw [subset_closure.antisymm h]; exact isClosed_closure
 #align is_closed_of_closure_subset isClosed_of_closure_subset
 
-theorem closure_eq_iff_isClosed {s : Set X} : closure s = s ↔ IsClosed s :=
+theorem closure_eq_iff_isClosed : closure s = s ↔ IsClosed s :=
   ⟨fun h => h ▸ isClosed_closure, IsClosed.closure_eq⟩
 #align closure_eq_iff_is_closed closure_eq_iff_isClosed
 
-theorem closure_subset_iff_isClosed {s : Set X} : closure s ⊆ s ↔ IsClosed s :=
+theorem closure_subset_iff_isClosed : closure s ⊆ s ↔ IsClosed s :=
   ⟨isClosed_of_closure_subset, IsClosed.closure_subset⟩
 #align closure_subset_iff_is_closed closure_subset_iff_isClosed
 
@@ -524,7 +524,7 @@ theorem closure_empty_iff (s : Set X) : closure s = ∅ ↔ s = ∅ :=
 #align closure_empty_iff closure_empty_iff
 
 @[simp]
-theorem closure_nonempty_iff {s : Set X} : (closure s).Nonempty ↔ s.Nonempty := by
+theorem closure_nonempty_iff : (closure s).Nonempty ↔ s.Nonempty := by
   simp only [nonempty_iff_ne_empty, Ne.def, closure_empty_iff]
 #align closure_nonempty_iff closure_nonempty_iff
 
@@ -538,17 +538,17 @@ theorem closure_univ : closure (univ : Set X) = univ :=
 #align closure_univ closure_univ
 
 @[simp]
-theorem closure_closure {s : Set X} : closure (closure s) = closure s :=
+theorem closure_closure : closure (closure s) = closure s :=
   isClosed_closure.closure_eq
 #align closure_closure closure_closure
 
-theorem closure_eq_compl_interior_compl {s : Set X} : closure s = (interior sᶜ)ᶜ := by
+theorem closure_eq_compl_interior_compl : closure s = (interior sᶜ)ᶜ := by
   rw [interior, closure, compl_sUnion, compl_image_set_of]
   simp only [compl_subset_compl, isOpen_compl_iff]
 #align closure_eq_compl_interior_compl closure_eq_compl_interior_compl
 
 @[simp]
-theorem closure_union {s t : Set X} : closure (s ∪ t) = closure s ∪ closure t := by
+theorem closure_union : closure (s ∪ t) = closure s ∪ closure t := by
   simp [closure_eq_compl_interior_compl, compl_inter]
 #align closure_union closure_union
 
@@ -572,22 +572,22 @@ theorem closure_iUnion_of_finite [Finite ι] (f : ι → Set X) :
   rw [← sUnion_range, (finite_range _).closure_sUnion, biUnion_range]
 #align closure_Union closure_iUnion_of_finite
 
-theorem interior_subset_closure {s : Set X} : interior s ⊆ closure s :=
+theorem interior_subset_closure : interior s ⊆ closure s :=
   Subset.trans interior_subset subset_closure
 #align interior_subset_closure interior_subset_closure
 
 @[simp]
-theorem interior_compl {s : Set X} : interior sᶜ = (closure s)ᶜ := by
+theorem interior_compl : interior sᶜ = (closure s)ᶜ := by
   simp [closure_eq_compl_interior_compl]
 #align interior_compl interior_compl
 
 @[simp]
-theorem closure_compl {s : Set X} : closure sᶜ = (interior s)ᶜ := by
+theorem closure_compl : closure sᶜ = (interior s)ᶜ := by
   simp [closure_eq_compl_interior_compl]
 #align closure_compl closure_compl
 
-theorem mem_closure_iff {s : Set X} {a : X} :
-    a ∈ closure s ↔ ∀ o, IsOpen o → a ∈ o → (o ∩ s).Nonempty :=
+theorem mem_closure_iff :
+    x ∈ closure s ↔ ∀ o, IsOpen o → x ∈ o → (o ∩ s).Nonempty :=
   ⟨fun h o oo ao =>
     by_contradiction fun os =>
       have : s ⊆ oᶜ := fun x xs xo => os ⟨x, xo, xs⟩
@@ -598,7 +598,7 @@ theorem mem_closure_iff {s : Set X} {a : X} :
       hc (h₂ hs)⟩
 #align mem_closure_iff mem_closure_iff
 
-theorem closure_inter_open_nonempty_iff {s t : Set X} (h : IsOpen t) :
+theorem closure_inter_open_nonempty_iff (h : IsOpen t) :
     (closure s ∩ t).Nonempty ↔ (s ∩ t).Nonempty :=
   ⟨fun ⟨_x, hxcs, hxt⟩ => inter_comm t s ▸ mem_closure_iff.1 hxcs t h hxt, fun h =>
     h.mono <| inf_le_inf_right t subset_closure⟩
@@ -630,24 +630,24 @@ def Dense (s : Set X) : Prop :=
   ∀ x, x ∈ closure s
 #align dense Dense
 
-theorem dense_iff_closure_eq {s : Set X} : Dense s ↔ closure s = univ :=
+theorem dense_iff_closure_eq : Dense s ↔ closure s = univ :=
   eq_univ_iff_forall.symm
 #align dense_iff_closure_eq dense_iff_closure_eq
 
 alias ⟨Dense.closure_eq, _⟩ := dense_iff_closure_eq
 #align dense.closure_eq Dense.closure_eq
 
-theorem interior_eq_empty_iff_dense_compl {s : Set X} : interior s = ∅ ↔ Dense sᶜ := by
+theorem interior_eq_empty_iff_dense_compl : interior s = ∅ ↔ Dense sᶜ := by
   rw [dense_iff_closure_eq, closure_compl, compl_univ_iff]
 #align interior_eq_empty_iff_dense_compl interior_eq_empty_iff_dense_compl
 
-theorem Dense.interior_compl {s : Set X} (h : Dense s) : interior sᶜ = ∅ :=
+theorem Dense.interior_compl (h : Dense s) : interior sᶜ = ∅ :=
   interior_eq_empty_iff_dense_compl.2 <| by rwa [compl_compl]
 #align dense.interior_compl Dense.interior_compl
 
 /-- The closure of a set `s` is dense if and only if `s` is dense. -/
 @[simp]
-theorem dense_closure {s : Set X} : Dense (closure s) ↔ Dense s := by
+theorem dense_closure : Dense (closure s) ↔ Dense s := by
   rw [Dense, Dense, closure_closure]
 #align dense_closure dense_closure
 
@@ -661,7 +661,7 @@ theorem dense_univ : Dense (univ : Set X) := fun _ => subset_closure trivial
 #align dense_univ dense_univ
 
 /-- A set is dense if and only if it has a nonempty intersection with each nonempty open set. -/
-theorem dense_iff_inter_open {s : Set X} :
+theorem dense_iff_inter_open :
     Dense s ↔ ∀ U, IsOpen U → U.Nonempty → (U ∩ s).Nonempty := by
   constructor <;> intro h
   · rintro U U_op ⟨x, x_in⟩
@@ -675,29 +675,29 @@ theorem dense_iff_inter_open {s : Set X} :
 alias ⟨Dense.inter_open_nonempty, _⟩ := dense_iff_inter_open
 #align dense.inter_open_nonempty Dense.inter_open_nonempty
 
-theorem Dense.exists_mem_open {s : Set X} (hs : Dense s) {U : Set X} (ho : IsOpen U)
+theorem Dense.exists_mem_open (hs : Dense s) {U : Set X} (ho : IsOpen U)
     (hne : U.Nonempty) : ∃ x ∈ s, x ∈ U :=
   let ⟨x, hx⟩ := hs.inter_open_nonempty U ho hne
   ⟨x, hx.2, hx.1⟩
 #align dense.exists_mem_open Dense.exists_mem_open
 
-theorem Dense.nonempty_iff {s : Set X} (hs : Dense s) : s.Nonempty ↔ Nonempty X :=
+theorem Dense.nonempty_iff (hs : Dense s) : s.Nonempty ↔ Nonempty X :=
   ⟨fun ⟨x, _⟩ => ⟨x⟩, fun ⟨x⟩ =>
     let ⟨y, hy⟩ := hs.inter_open_nonempty _ isOpen_univ ⟨x, trivial⟩
     ⟨y, hy.2⟩⟩
 #align dense.nonempty_iff Dense.nonempty_iff
 
-theorem Dense.nonempty [h : Nonempty X] {s : Set X} (hs : Dense s) : s.Nonempty :=
+theorem Dense.nonempty [h : Nonempty X] (hs : Dense s) : s.Nonempty :=
   hs.nonempty_iff.2 h
 #align dense.nonempty Dense.nonempty
 
 @[mono]
-theorem Dense.mono {s₁ s₂ : Set X} (h : s₁ ⊆ s₂) (hd : Dense s₁) : Dense s₂ := fun x =>
+theorem Dense.mono (h : s₁ ⊆ s₂) (hd : Dense s₁) : Dense s₂ := fun x =>
   closure_mono h (hd x)
 #align dense.mono Dense.mono
 
 /-- Complement to a singleton is dense if and only if the singleton is not an open set. -/
-theorem dense_compl_singleton_iff_not_open {x : X} :
+theorem dense_compl_singleton_iff_not_open :
     Dense ({x}ᶜ : Set X) ↔ ¬IsOpen ({x} : Set X) := by
   constructor
   · intro hd ho
@@ -738,11 +738,11 @@ theorem self_diff_frontier (s : Set X) : s \ frontier s = interior s := by
     inter_eq_self_of_subset_right interior_subset, empty_union]
 #align self_diff_frontier self_diff_frontier
 
-theorem frontier_eq_closure_inter_closure {s : Set X} : frontier s = closure s ∩ closure sᶜ := by
+theorem frontier_eq_closure_inter_closure : frontier s = closure s ∩ closure sᶜ := by
   rw [closure_compl, frontier, diff_eq]
 #align frontier_eq_closure_inter_closure frontier_eq_closure_inter_closure
 
-theorem frontier_subset_closure {s : Set X} : frontier s ⊆ closure s :=
+theorem frontier_subset_closure : frontier s ⊆ closure s :=
   diff_subset _ _
 #align frontier_subset_closure frontier_subset_closure
 
@@ -750,11 +750,11 @@ theorem IsClosed.frontier_subset (hs : IsClosed s) : frontier s ⊆ s :=
   frontier_subset_closure.trans hs.closure_eq.subset
 #align is_closed.frontier_subset IsClosed.frontier_subset
 
-theorem frontier_closure_subset {s : Set X} : frontier (closure s) ⊆ frontier s :=
+theorem frontier_closure_subset : frontier (closure s) ⊆ frontier s :=
   diff_subset_diff closure_closure.subset <| interior_mono subset_closure
 #align frontier_closure_subset frontier_closure_subset
 
-theorem frontier_interior_subset {s : Set X} : frontier (interior s) ⊆ frontier s :=
+theorem frontier_interior_subset : frontier (interior s) ⊆ frontier s :=
   diff_subset_diff (closure_mono interior_subset) interior_interior.symm.subset
 #align frontier_interior_subset frontier_interior_subset
 
@@ -784,25 +784,25 @@ theorem frontier_union_subset (s t : Set X) :
   simpa only [frontier_compl, ← compl_union] using frontier_inter_subset sᶜ tᶜ
 #align frontier_union_subset frontier_union_subset
 
-theorem IsClosed.frontier_eq {s : Set X} (hs : IsClosed s) : frontier s = s \ interior s := by
+theorem IsClosed.frontier_eq (hs : IsClosed s) : frontier s = s \ interior s := by
   rw [frontier, hs.closure_eq]
 #align is_closed.frontier_eq IsClosed.frontier_eq
 
-theorem IsOpen.frontier_eq {s : Set X} (hs : IsOpen s) : frontier s = closure s \ s := by
+theorem IsOpen.frontier_eq (hs : IsOpen s) : frontier s = closure s \ s := by
   rw [frontier, hs.interior_eq]
 #align is_open.frontier_eq IsOpen.frontier_eq
 
-theorem IsOpen.inter_frontier_eq {s : Set X} (hs : IsOpen s) : s ∩ frontier s = ∅ := by
+theorem IsOpen.inter_frontier_eq (hs : IsOpen s) : s ∩ frontier s = ∅ := by
   rw [hs.frontier_eq, inter_diff_self]
 #align is_open.inter_frontier_eq IsOpen.inter_frontier_eq
 
 /-- The frontier of a set is closed. -/
-theorem isClosed_frontier {s : Set X} : IsClosed (frontier s) := by
+theorem isClosed_frontier : IsClosed (frontier s) := by
   rw [frontier_eq_closure_inter_closure]; exact IsClosed.inter isClosed_closure isClosed_closure
 #align is_closed_frontier isClosed_frontier
 
 /-- The frontier of a closed set has no interior point. -/
-theorem interior_frontier {s : Set X} (h : IsClosed s) : interior (frontier s) = ∅ := by
+theorem interior_frontier (h : IsClosed s) : interior (frontier s) = ∅ := by
   have A : frontier s = s \ interior s := h.frontier_eq
   have B : interior (frontier s) ⊆ interior s := by rw [A]; exact interior_mono (diff_subset _ _)
   have C : interior (frontier s) ⊆ frontier s := interior_subset
@@ -828,12 +828,12 @@ theorem Disjoint.frontier_right (hs : IsOpen s) (hd : Disjoint s t) : Disjoint s
   (hd.symm.frontier_left hs).symm
 #align disjoint.frontier_right Disjoint.frontier_right
 
-theorem frontier_eq_inter_compl_interior {s : Set X} :
+theorem frontier_eq_inter_compl_interior :
     frontier s = (interior s)ᶜ ∩ (interior sᶜ)ᶜ := by
   rw [← frontier_compl, ← closure_compl, ← diff_eq, closure_diff_interior]
 #align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interior
 
-theorem compl_frontier_eq_union_interior {s : Set X} :
+theorem compl_frontier_eq_union_interior :
     (frontier s)ᶜ = interior s ∪ interior sᶜ := by
   rw [frontier_eq_inter_compl_interior]
   simp only [compl_inter, compl_compl]
@@ -843,18 +843,18 @@ theorem compl_frontier_eq_union_interior {s : Set X} :
 ### Neighborhoods
 -/
 
-/-- A set is called a neighborhood of `a` if it contains an open set around `a`. The set of all
-neighborhoods of `a` forms a filter, the neighborhood filter at `a`, is here defined as the
-infimum over the principal filters of all open sets containing `a`. -/
-irreducible_def nhds (a : X) : Filter X :=
-  ⨅ s ∈ { s : Set X | a ∈ s ∧ IsOpen s }, 𝓟 s
+/-- A set is called a neighborhood of `x` if it contains an open set around `x`. The set of all
+neighborhoods of `x` forms a filter, the neighborhood filter at `x`, is here defined as the
+infimum over the principal filters of all open sets containing `x`. -/
+irreducible_def nhds (x : X) : Filter X :=
+  ⨅ s ∈ { s : Set X | x ∈ s ∧ IsOpen s }, 𝓟 s
 #align nhds nhds
 #align nhds_def nhds_def
 
-/-- The "neighborhood within" filter. Elements of `𝓝[s] a` are sets containing the
-intersection of `s` and a neighborhood of `a`. -/
-def nhdsWithin (a : X) (s : Set X) : Filter X :=
-  nhds a ⊓ 𝓟 s
+/-- The "neighborhood within" filter. Elements of `𝓝[s] x` are sets containing the
+intersection of `s` and a neighborhood of `x`. -/
+def nhdsWithin (x : X) (s : Set X) : Filter X :=
+  nhds x ⊓ 𝓟 s
 #align nhds_within nhdsWithin
 
 section
@@ -882,164 +882,164 @@ scoped[Topology] notation3 "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
 
 end
 
-theorem nhds_def' (a : X) : 𝓝 a = ⨅ (s : Set X) (_ : IsOpen s) (_ : a ∈ s), 𝓟 s := by
-  simp only [nhds_def, mem_setOf_eq, @and_comm (a ∈ _), iInf_and]
+theorem nhds_def' (x : X) : 𝓝 x = ⨅ (s : Set X) (_ : IsOpen s) (_ : x ∈ s), 𝓟 s := by
+  simp only [nhds_def, mem_setOf_eq, @and_comm (x ∈ _), iInf_and]
 #align nhds_def' nhds_def'
 
-/-- The open sets containing `a` are a basis for the neighborhood filter. See `nhds_basis_opens'`
+/-- The open sets containing `x` are a basis for the neighborhood filter. See `nhds_basis_opens'`
 for a variant using open neighborhoods instead. -/
-theorem nhds_basis_opens (a : X) :
-    (𝓝 a).HasBasis (fun s : Set X => a ∈ s ∧ IsOpen s) fun s => s := by
+theorem nhds_basis_opens (x : X) :
+    (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsOpen s) fun s => s := by
   rw [nhds_def]
   exact hasBasis_biInf_principal
     (fun s ⟨has, hs⟩ t ⟨hat, ht⟩ =>
       ⟨s ∩ t, ⟨⟨has, hat⟩, IsOpen.inter hs ht⟩, ⟨inter_subset_left _ _, inter_subset_right _ _⟩⟩)
-    ⟨univ, ⟨mem_univ a, isOpen_univ⟩⟩
+    ⟨univ, ⟨mem_univ x, isOpen_univ⟩⟩
 #align nhds_basis_opens nhds_basis_opens
 
-theorem nhds_basis_closeds (a : X) : (𝓝 a).HasBasis (fun s : Set X => a ∉ s ∧ IsClosed s) compl :=
-  ⟨fun t => (nhds_basis_opens a).mem_iff.trans <|
+theorem nhds_basis_closeds (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∉ s ∧ IsClosed s) compl :=
+  ⟨fun t => (nhds_basis_opens x).mem_iff.trans <|
     compl_surjective.exists.trans <| by simp only [isOpen_compl_iff, mem_compl_iff]⟩
 #align nhds_basis_closeds nhds_basis_closeds
 
-/-- A filter lies below the neighborhood filter at `a` iff it contains every open set around `a`. -/
-theorem le_nhds_iff {f a} : f ≤ 𝓝 a ↔ ∀ s : Set X, a ∈ s → IsOpen s → s ∈ f := by simp [nhds_def]
+/-- A filter lies below the neighborhood filter at `x` iff it contains every open set around `x`. -/
+theorem le_nhds_iff {f} : f ≤ 𝓝 x ↔ ∀ s : Set X, x ∈ s → IsOpen s → s ∈ f := by simp [nhds_def]
 #align le_nhds_iff le_nhds_iff
 
-/-- To show a filter is above the neighborhood filter at `a`, it suffices to show that it is above
-the principal filter of some open set `s` containing `a`. -/
-theorem nhds_le_of_le {f a} {s : Set X} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 a ≤ f := by
+/-- To show a filter is above the neighborhood filter at `x`, it suffices to show that it is above
+the principal filter of some open set `s` containing `x`. -/
+theorem nhds_le_of_le {f} (h : x ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 x ≤ f := by
   rw [nhds_def]; exact iInf₂_le_of_le s ⟨h, o⟩ sf
 #align nhds_le_of_le nhds_le_of_le
 
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
-theorem mem_nhds_iff {a : X} {s : Set X} : s ∈ 𝓝 a ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ a ∈ t :=
-  (nhds_basis_opens a).mem_iff.trans <| exists_congr fun _ =>
+theorem mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t :=
+  (nhds_basis_opens x).mem_iff.trans <| exists_congr fun _ =>
     ⟨fun h => ⟨h.2, h.1.2, h.1.1⟩, fun h => ⟨⟨h.2.2, h.2.1⟩, h.1⟩⟩
 #align mem_nhds_iff mem_nhds_iffₓ
 
-/-- A predicate is true in a neighborhood of `a` iff it is true for all the points in an open set
-containing `a`. -/
-theorem eventually_nhds_iff {a : X} {p : X → Prop} :
-    (∀ᶠ x in 𝓝 a, p x) ↔ ∃ t : Set X, (∀ x ∈ t, p x) ∧ IsOpen t ∧ a ∈ t :=
+/-- A predicate is true in a neighborhood of `x` iff it is true for all the points in an open set
+containing `x`. -/
+theorem eventually_nhds_iff {p : X → Prop} :
+    (∀ᶠ x in 𝓝 x, p x) ↔ ∃ t : Set X, (∀ x ∈ t, p x) ∧ IsOpen t ∧ x ∈ t :=
   mem_nhds_iff.trans <| by simp only [subset_def, exists_prop, mem_setOf_eq]
 #align eventually_nhds_iff eventually_nhds_iff
 
-theorem mem_interior_iff_mem_nhds {s : Set X} {a : X} : a ∈ interior s ↔ s ∈ 𝓝 a :=
+theorem mem_interior_iff_mem_nhds : x ∈ interior s ↔ s ∈ 𝓝 x :=
   mem_interior.trans mem_nhds_iff.symm
 #align mem_interior_iff_mem_nhds mem_interior_iff_mem_nhds
 
-theorem map_nhds {a : X} {f : X → α} :
-    map f (𝓝 a) = ⨅ s ∈ { s : Set X | a ∈ s ∧ IsOpen s }, 𝓟 (f '' s) :=
-  ((nhds_basis_opens a).map f).eq_biInf
+theorem map_nhds {f : X → α} :
+    map f (𝓝 x) = ⨅ s ∈ { s : Set X | x ∈ s ∧ IsOpen s }, 𝓟 (f '' s) :=
+  ((nhds_basis_opens x).map f).eq_biInf
 #align map_nhds map_nhds
 
-theorem mem_of_mem_nhds {a : X} {s : Set X} : s ∈ 𝓝 a → a ∈ s := fun H =>
+theorem mem_of_mem_nhds : s ∈ 𝓝 x → x ∈ s := fun H =>
   let ⟨_t, ht, _, hs⟩ := mem_nhds_iff.1 H; ht hs
 #align mem_of_mem_nhds mem_of_mem_nhds
 
-/-- If a predicate is true in a neighborhood of `a`, then it is true for `a`. -/
-theorem Filter.Eventually.self_of_nhds {p : X → Prop} {a : X} (h : ∀ᶠ y in 𝓝 a, p y) : p a :=
+/-- If a predicate is true in a neighborhood of `x`, then it is true for `x`. -/
+theorem Filter.Eventually.self_of_nhds {p : X → Prop} (h : ∀ᶠ y in 𝓝 x, p y) : p x :=
   mem_of_mem_nhds h
 #align filter.eventually.self_of_nhds Filter.Eventually.self_of_nhds
 
-theorem IsOpen.mem_nhds {a : X} {s : Set X} (hs : IsOpen s) (ha : a ∈ s) : s ∈ 𝓝 a :=
-  mem_nhds_iff.2 ⟨s, Subset.refl _, hs, ha⟩
+theorem IsOpen.mem_nhds (hs : IsOpen s) (hx : x ∈ s) : s ∈ 𝓝 x :=
+  mem_nhds_iff.2 ⟨s, Subset.refl _, hs, hx⟩
 #align is_open.mem_nhds IsOpen.mem_nhds
 
-protected theorem IsOpen.mem_nhds_iff {a : X} {s : Set X} (hs : IsOpen s) : s ∈ 𝓝 a ↔ a ∈ s :=
-  ⟨mem_of_mem_nhds, fun ha => mem_nhds_iff.2 ⟨s, Subset.rfl, hs, ha⟩⟩
+protected theorem IsOpen.mem_nhds_iff (hs : IsOpen s) : s ∈ 𝓝 x ↔ x ∈ s :=
+  ⟨mem_of_mem_nhds, fun hx => mem_nhds_iff.2 ⟨s, Subset.rfl, hs, hx⟩⟩
 #align is_open.mem_nhds_iff IsOpen.mem_nhds_iff
 
-theorem IsClosed.compl_mem_nhds {a : X} {s : Set X} (hs : IsClosed s) (ha : a ∉ s) : sᶜ ∈ 𝓝 a :=
-  hs.isOpen_compl.mem_nhds (mem_compl ha)
+theorem IsClosed.compl_mem_nhds (hs : IsClosed s) (hx : x ∉ s) : sᶜ ∈ 𝓝 x :=
+  hs.isOpen_compl.mem_nhds (mem_compl hx)
 #align is_closed.compl_mem_nhds IsClosed.compl_mem_nhds
 
-theorem IsOpen.eventually_mem {a : X} {s : Set X} (hs : IsOpen s) (ha : a ∈ s) :
-    ∀ᶠ x in 𝓝 a, x ∈ s :=
-  IsOpen.mem_nhds hs ha
+theorem IsOpen.eventually_mem (hs : IsOpen s) (hx : x ∈ s) :
+    ∀ᶠ x in 𝓝 x, x ∈ s :=
+  IsOpen.mem_nhds hs hx
 #align is_open.eventually_mem IsOpen.eventually_mem
 
-/-- The open neighborhoods of `a` are a basis for the neighborhood filter. See `nhds_basis_opens`
-for a variant using open sets around `a` instead. -/
-theorem nhds_basis_opens' (a : X) :
-    (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsOpen s) fun x => x := by
-  convert nhds_basis_opens a using 2
+/-- The open neighborhoods of `x` are a basis for the neighborhood filter. See `nhds_basis_opens`
+for a variant using open sets around `x` instead. -/
+theorem nhds_basis_opens' (x : X) :
+    (𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsOpen s) fun x => x := by
+  convert nhds_basis_opens x using 2
   exact and_congr_left_iff.2 IsOpen.mem_nhds_iff
 #align nhds_basis_opens' nhds_basis_opens'
 
 /-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of `s`:
 it contains an open set containing `s`. -/
-theorem exists_open_set_nhds {s U : Set X} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
+theorem exists_open_set_nhds {U : Set X} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
     ∃ V : Set X, s ⊆ V ∧ IsOpen V ∧ V ⊆ U :=
   ⟨interior U, fun x hx => mem_interior_iff_mem_nhds.2 <| h x hx, isOpen_interior, interior_subset⟩
 #align exists_open_set_nhds exists_open_set_nhds
 
 /-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of s:
 it contains an open set containing `s`. -/
-theorem exists_open_set_nhds' {s U : Set X} (h : U ∈ ⨆ x ∈ s, 𝓝 x) :
+theorem exists_open_set_nhds' {U : Set X} (h : U ∈ ⨆ x ∈ s, 𝓝 x) :
     ∃ V : Set X, s ⊆ V ∧ IsOpen V ∧ V ⊆ U :=
   exists_open_set_nhds (by simpa using h)
 #align exists_open_set_nhds' exists_open_set_nhds'
 
-/-- If a predicate is true in a neighbourhood of `a`, then for `y` sufficiently close
-to `a` this predicate is true in a neighbourhood of `y`. -/
-theorem Filter.Eventually.eventually_nhds {p : X → Prop} {a : X} (h : ∀ᶠ y in 𝓝 a, p y) :
-    ∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝 y, p x :=
+/-- If a predicate is true in a neighbourhood of `x`, then for `y` sufficiently close
+to `x` this predicate is true in a neighbourhood of `y`. -/
+theorem Filter.Eventually.eventually_nhds {p : X → Prop} (h : ∀ᶠ y in 𝓝 x, p y) :
+    ∀ᶠ y in 𝓝 x, ∀ᶠ x in 𝓝 y, p x :=
   let ⟨t, htp, hto, ha⟩ := eventually_nhds_iff.1 h
   eventually_nhds_iff.2 ⟨t, fun _x hx => eventually_nhds_iff.2 ⟨t, htp, hto, hx⟩, hto, ha⟩
 #align filter.eventually.eventually_nhds Filter.Eventually.eventually_nhds
 
 @[simp]
-theorem eventually_eventually_nhds {p : X → Prop} {a : X} :
-    (∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝 y, p x) ↔ ∀ᶠ x in 𝓝 a, p x :=
+theorem eventually_eventually_nhds {p : X → Prop} :
+    (∀ᶠ y in 𝓝 x, ∀ᶠ x in 𝓝 y, p x) ↔ ∀ᶠ x in 𝓝 x, p x :=
   ⟨fun h => h.self_of_nhds, fun h => h.eventually_nhds⟩
 #align eventually_eventually_nhds eventually_eventually_nhds
 
 @[simp]
-theorem frequently_frequently_nhds {p : X → Prop} {a : X} :
-    (∃ᶠ y in 𝓝 a, ∃ᶠ x in 𝓝 y, p x) ↔ ∃ᶠ x in 𝓝 a, p x := by
+theorem frequently_frequently_nhds {p : X → Prop} :
+    (∃ᶠ x' in 𝓝 x, ∃ᶠ x'' in 𝓝 x', p x'') ↔ ∃ᶠ x in 𝓝 x, p x := by
   rw [← not_iff_not]
   simp only [not_frequently, eventually_eventually_nhds]
 #align frequently_frequently_nhds frequently_frequently_nhds
 
 @[simp]
-theorem eventually_mem_nhds {s : Set X} {a : X} : (∀ᶠ x in 𝓝 a, s ∈ 𝓝 x) ↔ s ∈ 𝓝 a :=
+theorem eventually_mem_nhds : (∀ᶠ x' in 𝓝 x, s ∈ 𝓝 x') ↔ s ∈ 𝓝 x :=
   eventually_eventually_nhds
 #align eventually_mem_nhds eventually_mem_nhds
 
 @[simp]
-theorem nhds_bind_nhds : (𝓝 a).bind 𝓝 = 𝓝 a :=
+theorem nhds_bind_nhds : (𝓝 x).bind 𝓝 = 𝓝 x :=
   Filter.ext fun _ => eventually_eventually_nhds
 #align nhds_bind_nhds nhds_bind_nhds
 
 @[simp]
-theorem eventually_eventuallyEq_nhds {f g : X → α} {a : X} :
-    (∀ᶠ y in 𝓝 a, f =ᶠ[𝓝 y] g) ↔ f =ᶠ[𝓝 a] g :=
+theorem eventually_eventuallyEq_nhds {f g : X → α} :
+    (∀ᶠ y in 𝓝 x, f =ᶠ[𝓝 y] g) ↔ f =ᶠ[𝓝 x] g :=
   eventually_eventually_nhds
 #align eventually_eventually_eq_nhds eventually_eventuallyEq_nhds
 
-theorem Filter.EventuallyEq.eq_of_nhds {f g : X → α} {a : X} (h : f =ᶠ[𝓝 a] g) : f a = g a :=
+theorem Filter.EventuallyEq.eq_of_nhds {f g : X → α} (h : f =ᶠ[𝓝 x] g) : f x = g x :=
   h.self_of_nhds
 #align filter.eventually_eq.eq_of_nhds Filter.EventuallyEq.eq_of_nhds
 
 @[simp]
-theorem eventually_eventuallyLE_nhds [LE α] {f g : X → α} {a : X} :
-    (∀ᶠ y in 𝓝 a, f ≤ᶠ[𝓝 y] g) ↔ f ≤ᶠ[𝓝 a] g :=
+theorem eventually_eventuallyLE_nhds [LE α] {f g : X → α} :
+    (∀ᶠ y in 𝓝 x, f ≤ᶠ[𝓝 y] g) ↔ f ≤ᶠ[𝓝 x] g :=
   eventually_eventually_nhds
 #align eventually_eventually_le_nhds eventually_eventuallyLE_nhds
 
-/-- If two functions are equal in a neighbourhood of `a`, then for `y` sufficiently close
-to `a` these functions are equal in a neighbourhood of `y`. -/
-theorem Filter.EventuallyEq.eventuallyEq_nhds {f g : X → α} {a : X} (h : f =ᶠ[𝓝 a] g) :
-    ∀ᶠ y in 𝓝 a, f =ᶠ[𝓝 y] g :=
+/-- If two functions are equal in a neighbourhood of `x`, then for `y` sufficiently close
+to `x` these functions are equal in a neighbourhood of `y`. -/
+theorem Filter.EventuallyEq.eventuallyEq_nhds {f g : X → α} (h : f =ᶠ[𝓝 x] g) :
+    ∀ᶠ y in 𝓝 x, f =ᶠ[𝓝 y] g :=
   h.eventually_nhds
 #align filter.eventually_eq.eventually_eq_nhds Filter.EventuallyEq.eventuallyEq_nhds
 
-/-- If `f x ≤ g x` in a neighbourhood of `a`, then for `y` sufficiently close to `a` we have
+/-- If `f x ≤ g x` in a neighbourhood of `x`, then for `y` sufficiently close to `x` we have
 `f x ≤ g x` in a neighbourhood of `y`. -/
-theorem Filter.EventuallyLE.eventuallyLE_nhds [LE α] {f g : X → α} {a : X} (h : f ≤ᶠ[𝓝 a] g) :
-    ∀ᶠ y in 𝓝 a, f ≤ᶠ[𝓝 y] g :=
+theorem Filter.EventuallyLE.eventuallyLE_nhds [LE α] {f g : X → α} (h : f ≤ᶠ[𝓝 x] g) :
+    ∀ᶠ y in 𝓝 x, f ≤ᶠ[𝓝 y] g :=
   h.eventually_nhds
 #align filter.eventually_le.eventually_le_nhds Filter.EventuallyLE.eventuallyLE_nhds
 
@@ -1053,27 +1053,27 @@ theorem all_mem_nhds_filter (x : X) (f : Set X → Set α) (hf : ∀ s t, s ⊆
   all_mem_nhds _ _ fun s t ssubt h => mem_of_superset h (hf s t ssubt)
 #align all_mem_nhds_filter all_mem_nhds_filter
 
-theorem tendsto_nhds {f : α → X} {l : Filter α} {a : X} :
-    Tendsto f l (𝓝 a) ↔ ∀ s, IsOpen s → a ∈ s → f ⁻¹' s ∈ l :=
+theorem tendsto_nhds {f : α → X} {l : Filter α} :
+    Tendsto f l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → f ⁻¹' s ∈ l :=
   all_mem_nhds_filter _ _ (fun _ _ h => preimage_mono h) _
 #align tendsto_nhds tendsto_nhds
 
-theorem tendsto_atTop_nhds [Nonempty α] [SemilatticeSup α] {f : α → X} {a : X} :
-    Tendsto f atTop (𝓝 a) ↔ ∀ U : Set X, a ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U :=
-  (atTop_basis.tendsto_iff (nhds_basis_opens a)).trans <| by
+theorem tendsto_atTop_nhds [Nonempty α] [SemilatticeSup α] {f : α → X} :
+    Tendsto f atTop (𝓝 x) ↔ ∀ U : Set X, x ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U :=
+  (atTop_basis.tendsto_iff (nhds_basis_opens x)).trans <| by
     simp only [and_imp, exists_prop, true_and_iff, mem_Ici, ge_iff_le]
 #align tendsto_at_top_nhds tendsto_atTop_nhds
 
-theorem tendsto_const_nhds {a : X} {f : Filter α} : Tendsto (fun _ : α => a) f (𝓝 a) :=
+theorem tendsto_const_nhds {f : Filter α} : Tendsto (fun _ : α => x) f (𝓝 x) :=
   tendsto_nhds.mpr fun _ _ ha => univ_mem' fun _ => ha
 #align tendsto_const_nhds tendsto_const_nhds
 
-theorem tendsto_atTop_of_eventually_const {ι : Type*} [SemilatticeSup ι] [Nonempty ι] {x : X}
+theorem tendsto_atTop_of_eventually_const {ι : Type*} [SemilatticeSup ι] [Nonempty ι]
     {u : ι → X} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : Tendsto u atTop (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atTop.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_top_of_eventually_const tendsto_atTop_of_eventually_const
 
-theorem tendsto_atBot_of_eventually_const {ι : Type*} [SemilatticeInf ι] [Nonempty ι] {x : X}
+theorem tendsto_atBot_of_eventually_const {ι : Type*} [SemilatticeInf ι] [Nonempty ι]
     {u : ι → X} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : Tendsto u atBot (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atBot.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_bot_of_eventually_const tendsto_atBot_of_eventually_const
@@ -1091,16 +1091,16 @@ theorem OrderTop.tendsto_atTop_nhds [PartialOrder α] [OrderTop α] (f : α →
 #align order_top.tendsto_at_top_nhds OrderTop.tendsto_atTop_nhds
 
 @[simp]
-instance nhds_neBot {a : X} : NeBot (𝓝 a) :=
-  neBot_of_le (pure_le_nhds a)
+instance nhds_neBot : NeBot (𝓝 x) :=
+  neBot_of_le (pure_le_nhds x)
 #align nhds_ne_bot nhds_neBot
 
-theorem tendsto_nhds_of_eventually_eq {l : Filter α} {f : α → X} {a : X} (h : ∀ᶠ x in l, f x = a) :
-    Tendsto f l (𝓝 a) :=
+theorem tendsto_nhds_of_eventually_eq {l : Filter α} {f : α → X} (h : ∀ᶠ x' in l, f x' = x) :
+    Tendsto f l (𝓝 x) :=
   tendsto_const_nhds.congr' (.symm h)
 
-theorem Filter.EventuallyEq.tendsto {l : Filter α} {f : α → X} {a : X} (hf : f =ᶠ[l] fun _ ↦ a) :
-    Tendsto f l (𝓝 a) :=
+theorem Filter.EventuallyEq.tendsto {l : Filter α} {f : α → X} (hf : f =ᶠ[l] fun _ ↦ x) :
+    Tendsto f l (𝓝 x) :=
   tendsto_nhds_of_eventually_eq hf
 
 /-!
@@ -1118,64 +1118,64 @@ def ClusterPt (x : X) (F : Filter X) : Prop :=
   NeBot (𝓝 x ⊓ F)
 #align cluster_pt ClusterPt
 
-theorem ClusterPt.neBot {x : X} {F : Filter X} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) :=
+theorem ClusterPt.neBot {F : Filter X} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) :=
   h
 #align cluster_pt.ne_bot ClusterPt.neBot
 
-theorem Filter.HasBasis.clusterPt_iff {ιa ιF} {pa : ιa → Prop} {sa : ιa → Set X} {pF : ιF → Prop}
-    {sF : ιF → Set X} {F : Filter X} (ha : (𝓝 a).HasBasis pa sa) (hF : F.HasBasis pF sF) :
-    ClusterPt a F ↔ ∀ ⦃i⦄, pa i → ∀ ⦃j⦄, pF j → (sa i ∩ sF j).Nonempty :=
-  ha.inf_basis_neBot_iff hF
+theorem Filter.HasBasis.clusterPt_iff {ιX ιF} {pX : ιX → Prop} {sX : ιX → Set X} {pF : ιF → Prop}
+    {sF : ιF → Set X} {F : Filter X} (hX : (𝓝 x).HasBasis pX sX) (hF : F.HasBasis pF sF) :
+    ClusterPt x F ↔ ∀ ⦃i⦄, pX i → ∀ ⦃j⦄, pF j → (sX i ∩ sF j).Nonempty :=
+  hX.inf_basis_neBot_iff hF
 #align filter.has_basis.cluster_pt_iff Filter.HasBasis.clusterPt_iff
 
-theorem clusterPt_iff {x : X} {F : Filter X} :
+theorem clusterPt_iff {F : Filter X} :
     ClusterPt x F ↔ ∀ ⦃U : Set X⦄, U ∈ 𝓝 x → ∀ ⦃V⦄, V ∈ F → (U ∩ V).Nonempty :=
   inf_neBot_iff
 #align cluster_pt_iff clusterPt_iff
 
-theorem clusterPt_iff_not_disjoint {x : X} {F : Filter X} :
+theorem clusterPt_iff_not_disjoint {F : Filter X} :
     ClusterPt x F ↔ ¬Disjoint (𝓝 x) F := by
   rw [disjoint_iff, ClusterPt, neBot_iff]
 
 /-- `x` is a cluster point of a set `s` if every neighbourhood of `x` meets `s` on a nonempty
 set. See also `mem_closure_iff_clusterPt`. -/
-theorem clusterPt_principal_iff {x : X} {s : Set X} :
+theorem clusterPt_principal_iff :
     ClusterPt x (𝓟 s) ↔ ∀ U ∈ 𝓝 x, (U ∩ s).Nonempty :=
   inf_principal_neBot_iff
 #align cluster_pt_principal_iff clusterPt_principal_iff
 
-theorem clusterPt_principal_iff_frequently {x : X} {s : Set X} :
+theorem clusterPt_principal_iff_frequently :
     ClusterPt x (𝓟 s) ↔ ∃ᶠ y in 𝓝 x, y ∈ s := by
   simp only [clusterPt_principal_iff, frequently_iff, Set.Nonempty, exists_prop, mem_inter_iff]
 #align cluster_pt_principal_iff_frequently clusterPt_principal_iff_frequently
 
-theorem ClusterPt.of_le_nhds {x : X} {f : Filter X} (H : f ≤ 𝓝 x) [NeBot f] : ClusterPt x f := by
+theorem ClusterPt.of_le_nhds {f : Filter X} (H : f ≤ 𝓝 x) [NeBot f] : ClusterPt x f := by
   rwa [ClusterPt, inf_eq_right.mpr H]
 #align cluster_pt.of_le_nhds ClusterPt.of_le_nhds
 
-theorem ClusterPt.of_le_nhds' {x : X} {f : Filter X} (H : f ≤ 𝓝 x) (_hf : NeBot f) :
+theorem ClusterPt.of_le_nhds' {f : Filter X} (H : f ≤ 𝓝 x) (_hf : NeBot f) :
     ClusterPt x f :=
   ClusterPt.of_le_nhds H
 #align cluster_pt.of_le_nhds' ClusterPt.of_le_nhds'
 
-theorem ClusterPt.of_nhds_le {x : X} {f : Filter X} (H : 𝓝 x ≤ f) : ClusterPt x f := by
+theorem ClusterPt.of_nhds_le {f : Filter X} (H : 𝓝 x ≤ f) : ClusterPt x f := by
   simp only [ClusterPt, inf_eq_left.mpr H, nhds_neBot]
 #align cluster_pt.of_nhds_le ClusterPt.of_nhds_le
 
-theorem ClusterPt.mono {x : X} {f g : Filter X} (H : ClusterPt x f) (h : f ≤ g) : ClusterPt x g :=
+theorem ClusterPt.mono {f g : Filter X} (H : ClusterPt x f) (h : f ≤ g) : ClusterPt x g :=
   NeBot.mono H <| inf_le_inf_left _ h
 #align cluster_pt.mono ClusterPt.mono
 
-theorem ClusterPt.of_inf_left {x : X} {f g : Filter X} (H : ClusterPt x <| f ⊓ g) : ClusterPt x f :=
+theorem ClusterPt.of_inf_left {f g : Filter X} (H : ClusterPt x <| f ⊓ g) : ClusterPt x f :=
   H.mono inf_le_left
 #align cluster_pt.of_inf_left ClusterPt.of_inf_left
 
-theorem ClusterPt.of_inf_right {x : X} {f g : Filter X} (H : ClusterPt x <| f ⊓ g) :
+theorem ClusterPt.of_inf_right {f g : Filter X} (H : ClusterPt x <| f ⊓ g) :
     ClusterPt x g :=
   H.mono inf_le_right
 #align cluster_pt.of_inf_right ClusterPt.of_inf_right
 
-theorem Ultrafilter.clusterPt_iff {x : X} {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
+theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
   ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩
 #align ultrafilter.cluster_pt_iff Ultrafilter.clusterPt_iff
 
@@ -1191,7 +1191,7 @@ theorem mapClusterPt_iff {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
   rfl
 #align map_cluster_pt_iff mapClusterPt_iff
 
-theorem mapClusterPt_of_comp {F : Filter α} {φ : β → α} {p : Filter β} {x : X}
+theorem mapClusterPt_of_comp {F : Filter α} {φ : β → α} {p : Filter β}
     {u : α → X} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) :
     MapClusterPt x F u := by
   have :=
@@ -1232,7 +1232,7 @@ theorem accPt_iff_frequently (x : X) (C : Set X) : AccPt x (𝓟 C) ↔ ∃ᶠ y
 
 /-- If `x` is an accumulation point of `F` and `F ≤ G`, then
 `x` is an accumulation point of `D`. -/
-theorem AccPt.mono {x : X} {F G : Filter X} (h : AccPt x F) (hFG : F ≤ G) : AccPt x G :=
+theorem AccPt.mono {F G : Filter X} (h : AccPt x F) (hFG : F ≤ G) : AccPt x G :=
   NeBot.mono h (inf_le_inf_left _ hFG)
 #align acc_pt.mono AccPt.mono
 
@@ -1240,16 +1240,16 @@ theorem AccPt.mono {x : X} {F G : Filter X} (h : AccPt x F) (hFG : F ≤ G) : Ac
 ### Interior, closure and frontier in terms of neighborhoods
 -/
 
-theorem interior_eq_nhds' {s : Set X} : interior s = { a | s ∈ 𝓝 a } :=
+theorem interior_eq_nhds' : interior s = { x | s ∈ 𝓝 x } :=
   Set.ext fun x => by simp only [mem_interior, mem_nhds_iff, mem_setOf_eq]
 #align interior_eq_nhds' interior_eq_nhds'
 
-theorem interior_eq_nhds {s : Set X} : interior s = { a | 𝓝 a ≤ 𝓟 s } :=
+theorem interior_eq_nhds : interior s = { x | 𝓝 x ≤ 𝓟 s } :=
   interior_eq_nhds'.trans <| by simp only [le_principal_iff]
 #align interior_eq_nhds interior_eq_nhds
 
 @[simp]
-theorem interior_mem_nhds {s : Set X} {a : X} : interior s ∈ 𝓝 a ↔ s ∈ 𝓝 a :=
+theorem interior_mem_nhds : interior s ∈ 𝓝 x ↔ s ∈ 𝓝 x :=
   ⟨fun h => mem_of_superset h interior_subset, fun h =>
     IsOpen.mem_nhds isOpen_interior (mem_interior_iff_mem_nhds.2 h)⟩
 #align interior_mem_nhds interior_mem_nhds
@@ -1262,46 +1262,46 @@ theorem isOpen_setOf_eventually_nhds {p : X → Prop} : IsOpen { x | ∀ᶠ y in
   simp only [← interior_setOf_eq, isOpen_interior]
 #align is_open_set_of_eventually_nhds isOpen_setOf_eventually_nhds
 
-theorem subset_interior_iff_nhds {s V : Set X} : s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x := by
+theorem subset_interior_iff_nhds {V : Set X} : s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x := by
   simp_rw [subset_def, mem_interior_iff_mem_nhds]
 #align subset_interior_iff_nhds subset_interior_iff_nhds
 
-theorem isOpen_iff_nhds {s : Set X} : IsOpen s ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s :=
+theorem isOpen_iff_nhds : IsOpen s ↔ ∀ x ∈ s, 𝓝 x ≤ 𝓟 s :=
   calc
     IsOpen s ↔ s ⊆ interior s := subset_interior_iff_isOpen.symm
-    _ ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s := by simp_rw [interior_eq_nhds, subset_def, mem_setOf]
+    _ ↔ ∀ x ∈ s, 𝓝 x ≤ 𝓟 s := by simp_rw [interior_eq_nhds, subset_def, mem_setOf]
 #align is_open_iff_nhds isOpen_iff_nhds
 
-theorem isOpen_iff_mem_nhds {s : Set X} : IsOpen s ↔ ∀ a ∈ s, s ∈ 𝓝 a :=
+theorem isOpen_iff_mem_nhds : IsOpen s ↔ ∀ x ∈ s, s ∈ 𝓝 x :=
   isOpen_iff_nhds.trans <| forall_congr' fun _ => imp_congr_right fun _ => le_principal_iff
 #align is_open_iff_mem_nhds isOpen_iff_mem_nhds
 
 /-- A set `s` is open iff for every point `x` in `s` and every `y` close to `x`, `y` is in `s`. -/
-theorem isOpen_iff_eventually {s : Set X} : IsOpen s ↔ ∀ x, x ∈ s → ∀ᶠ y in 𝓝 x, y ∈ s :=
+theorem isOpen_iff_eventually : IsOpen s ↔ ∀ x, x ∈ s → ∀ᶠ y in 𝓝 x, y ∈ s :=
   isOpen_iff_mem_nhds
 #align is_open_iff_eventually isOpen_iff_eventually
 
-theorem isOpen_iff_ultrafilter {s : Set X} :
+theorem isOpen_iff_ultrafilter :
     IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by
   simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]
 #align is_open_iff_ultrafilter isOpen_iff_ultrafilter
 
-theorem isOpen_singleton_iff_nhds_eq_pure (a : X) : IsOpen ({a} : Set X) ↔ 𝓝 a = pure a := by
+theorem isOpen_singleton_iff_nhds_eq_pure (x : X) : IsOpen ({x} : Set X) ↔ 𝓝 x = pure x := by
   constructor
   · intro h
-    apply le_antisymm _ (pure_le_nhds a)
+    apply le_antisymm _ (pure_le_nhds x)
     rw [le_pure_iff]
-    exact h.mem_nhds (mem_singleton a)
+    exact h.mem_nhds (mem_singleton x)
   · intro h
     simp [isOpen_iff_nhds, h]
 #align is_open_singleton_iff_nhds_eq_pure isOpen_singleton_iff_nhds_eq_pure
 
-theorem isOpen_singleton_iff_punctured_nhds (a : X) : IsOpen ({a} : Set X) ↔ 𝓝[≠] a = ⊥ := by
+theorem isOpen_singleton_iff_punctured_nhds (x : X) : IsOpen ({x} : Set X) ↔ 𝓝[≠] x = ⊥ := by
   rw [isOpen_singleton_iff_nhds_eq_pure, nhdsWithin, ← mem_iff_inf_principal_compl, ← le_pure_iff,
     nhds_neBot.le_pure_iff]
 #align is_open_singleton_iff_punctured_nhds isOpen_singleton_iff_punctured_nhds
 
-theorem mem_closure_iff_frequently {s : Set X} {a : X} : a ∈ closure s ↔ ∃ᶠ x in 𝓝 a, x ∈ s := by
+theorem mem_closure_iff_frequently : x ∈ closure s ↔ ∃ᶠ x in 𝓝 x, x ∈ s := by
   rw [Filter.Frequently, Filter.Eventually, ← mem_interior_iff_mem_nhds,
     closure_eq_compl_interior_compl, mem_compl_iff, compl_def]
 #align mem_closure_iff_frequently mem_closure_iff_frequently
@@ -1311,7 +1311,7 @@ alias ⟨_, Filter.Frequently.mem_closure⟩ := mem_closure_iff_frequently
 
 /-- A set `s` is closed iff for every point `x`, if there is a point `y` close to `x` that belongs
 to `s` then `x` is in `s`. -/
-theorem isClosed_iff_frequently {s : Set X} : IsClosed s ↔ ∀ x, (∃ᶠ y in 𝓝 x, y ∈ s) → x ∈ s := by
+theorem isClosed_iff_frequently : IsClosed s ↔ ∀ x, (∃ᶠ y in 𝓝 x, y ∈ s) → x ∈ s := by
   rw [← closure_subset_iff_isClosed]
   refine' forall_congr' fun x => _
   rw [mem_closure_iff_frequently]
@@ -1325,19 +1325,19 @@ theorem isClosed_setOf_clusterPt {f : Filter X} : IsClosed { x | ClusterPt x f }
   exacts [isOpen_setOf_eventually_nhds, isOpen_const]
 #align is_closed_set_of_cluster_pt isClosed_setOf_clusterPt
 
-theorem mem_closure_iff_clusterPt {s : Set X} {a : X} : a ∈ closure s ↔ ClusterPt a (𝓟 s) :=
+theorem mem_closure_iff_clusterPt : x ∈ closure s ↔ ClusterPt x (𝓟 s) :=
   mem_closure_iff_frequently.trans clusterPt_principal_iff_frequently.symm
 #align mem_closure_iff_cluster_pt mem_closure_iff_clusterPt
 
-theorem mem_closure_iff_nhds_neBot {s : Set X} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ :=
+theorem mem_closure_iff_nhds_neBot : x ∈ closure s ↔ 𝓝 x ⊓ 𝓟 s ≠ ⊥ :=
   mem_closure_iff_clusterPt.trans neBot_iff
 #align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_neBot
 
-theorem mem_closure_iff_nhdsWithin_neBot {s : Set X} {x : X} : x ∈ closure s ↔ NeBot (𝓝[s] x) :=
+theorem mem_closure_iff_nhdsWithin_neBot : x ∈ closure s ↔ NeBot (𝓝[s] x) :=
   mem_closure_iff_clusterPt
 #align mem_closure_iff_nhds_within_ne_bot mem_closure_iff_nhdsWithin_neBot
 
-lemma not_mem_closure_iff_nhdsWithin_eq_bot {s : Set X} {x : X} : x ∉ closure s ↔ 𝓝[s] x = ⊥ := by
+lemma not_mem_closure_iff_nhdsWithin_eq_bot : x ∉ closure s ↔ 𝓝[s] x = ⊥ := by
   rw [mem_closure_iff_nhdsWithin_neBot, not_neBot]
 
 /-- If `x` is not an isolated point of a topological space, then `{x}ᶜ` is dense in the whole
@@ -1366,81 +1366,81 @@ theorem not_isOpen_singleton (x : X) [NeBot (𝓝[≠] x)] : ¬IsOpen ({x} : Set
   dense_compl_singleton_iff_not_open.1 (dense_compl_singleton x)
 #align not_is_open_singleton not_isOpen_singleton
 
-theorem closure_eq_cluster_pts {s : Set X} : closure s = { a | ClusterPt a (𝓟 s) } :=
+theorem closure_eq_cluster_pts : closure s = { a | ClusterPt a (𝓟 s) } :=
   Set.ext fun _ => mem_closure_iff_clusterPt
 #align closure_eq_cluster_pts closure_eq_cluster_pts
 
-theorem mem_closure_iff_nhds {s : Set X} {a : X} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, (t ∩ s).Nonempty :=
+theorem mem_closure_iff_nhds : x ∈ closure s ↔ ∀ t ∈ 𝓝 x, (t ∩ s).Nonempty :=
   mem_closure_iff_clusterPt.trans clusterPt_principal_iff
 #align mem_closure_iff_nhds mem_closure_iff_nhds
 
-theorem mem_closure_iff_nhds' {s : Set X} {a : X} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, ∃ y : s, ↑y ∈ t := by
+theorem mem_closure_iff_nhds' : x ∈ closure s ↔ ∀ t ∈ 𝓝 x, ∃ y : s, ↑y ∈ t := by
   simp only [mem_closure_iff_nhds, Set.inter_nonempty_iff_exists_right, SetCoe.exists, exists_prop]
 #align mem_closure_iff_nhds' mem_closure_iff_nhds'
 
-theorem mem_closure_iff_comap_neBot {A : Set X} {x : X} :
-    x ∈ closure A ↔ NeBot (comap ((↑) : A → X) (𝓝 x)) := by
+theorem mem_closure_iff_comap_neBot :
+    x ∈ closure s ↔ NeBot (comap ((↑) : s → X) (𝓝 x)) := by
   simp_rw [mem_closure_iff_nhds, comap_neBot_iff, Set.inter_nonempty_iff_exists_right,
     SetCoe.exists, exists_prop]
 #align mem_closure_iff_comap_ne_bot mem_closure_iff_comap_neBot
 
-theorem mem_closure_iff_nhds_basis' {a : X} {p : ι → Prop} {s : ι → Set X} (h : (𝓝 a).HasBasis p s)
-    {t : Set X} : a ∈ closure t ↔ ∀ i, p i → (s i ∩ t).Nonempty :=
+theorem mem_closure_iff_nhds_basis' {p : ι → Prop} {s : ι → Set X} (h : (𝓝 x).HasBasis p s) :
+    x ∈ closure t ↔ ∀ i, p i → (s i ∩ t).Nonempty :=
   mem_closure_iff_clusterPt.trans <|
     (h.clusterPt_iff (hasBasis_principal _)).trans <| by simp only [exists_prop, forall_const]
 #align mem_closure_iff_nhds_basis' mem_closure_iff_nhds_basis'
 
-theorem mem_closure_iff_nhds_basis {a : X} {p : ι → Prop} {s : ι → Set X} (h : (𝓝 a).HasBasis p s)
-    {t : Set X} : a ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i :=
+theorem mem_closure_iff_nhds_basis {p : ι → Prop} {s : ι → Set X} (h : (𝓝 x).HasBasis p s) :
+    x ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i :=
   (mem_closure_iff_nhds_basis' h).trans <| by
     simp only [Set.Nonempty, mem_inter_iff, exists_prop, and_comm]
 #align mem_closure_iff_nhds_basis mem_closure_iff_nhds_basis
 
-theorem clusterPt_iff_forall_mem_closure {F : Filter X} {a : X} :
-    ClusterPt a F ↔ ∀ s ∈ F, a ∈ closure s := by
+theorem clusterPt_iff_forall_mem_closure {F : Filter X} :
+    ClusterPt x F ↔ ∀ s ∈ F, x ∈ closure s := by
   simp_rw [ClusterPt, inf_neBot_iff, mem_closure_iff_nhds]
   rw [forall₂_swap]
 
-theorem clusterPt_iff_lift'_closure {F : Filter X} {a : X} :
-    ClusterPt a F ↔ pure a ≤ (F.lift' closure) := by
+theorem clusterPt_iff_lift'_closure {F : Filter X} :
+    ClusterPt x F ↔ pure x ≤ (F.lift' closure) := by
   simp_rw [clusterPt_iff_forall_mem_closure,
     (hasBasis_pure _).le_basis_iff F.basis_sets.lift'_closure, id, singleton_subset_iff, true_and,
     exists_const]
 
-theorem clusterPt_iff_lift'_closure' {F : Filter X} {a : X} :
-    ClusterPt a F ↔ (F.lift' closure ⊓ pure a).NeBot := by
+theorem clusterPt_iff_lift'_closure' {F : Filter X} :
+    ClusterPt x F ↔ (F.lift' closure ⊓ pure x).NeBot := by
   rw [clusterPt_iff_lift'_closure, ← Ultrafilter.coe_pure, inf_comm, Ultrafilter.inf_neBot_iff]
 
 @[simp]
-theorem clusterPt_lift'_closure_iff {F : Filter X} {a : X} :
-    ClusterPt a (F.lift' closure) ↔ ClusterPt a F := by
+theorem clusterPt_lift'_closure_iff {F : Filter X} :
+    ClusterPt x (F.lift' closure) ↔ ClusterPt x F := by
   simp [clusterPt_iff_lift'_closure, lift'_lift'_assoc (monotone_closure X) (monotone_closure X)]
 
 /-- `x` belongs to the closure of `s` if and only if some ultrafilter
   supported on `s` converges to `x`. -/
-theorem mem_closure_iff_ultrafilter {s : Set X} {x : X} :
+theorem mem_closure_iff_ultrafilter :
     x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x := by
   simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]
 #align mem_closure_iff_ultrafilter mem_closure_iff_ultrafilter
 
-theorem isClosed_iff_clusterPt {s : Set X} : IsClosed s ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s :=
+theorem isClosed_iff_clusterPt : IsClosed s ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s :=
   calc
     IsClosed s ↔ closure s ⊆ s := closure_subset_iff_isClosed.symm
     _ ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s := by simp only [subset_def, mem_closure_iff_clusterPt]
 #align is_closed_iff_cluster_pt isClosed_iff_clusterPt
 
-theorem isClosed_iff_nhds {s : Set X} :
+theorem isClosed_iff_nhds :
     IsClosed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s := by
   simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_neBot_iff]
 #align is_closed_iff_nhds isClosed_iff_nhds
 
-lemma isClosed_iff_forall_filter {s : Set X} :
+lemma isClosed_iff_forall_filter :
     IsClosed s ↔ ∀ x, ∀ F : Filter X, F.NeBot → F ≤ 𝓟 s → F ≤ 𝓝 x → x ∈ s := by
   simp_rw [isClosed_iff_clusterPt]
   exact ⟨fun hs x F F_ne FS Fx ↦ hs _ <| NeBot.mono F_ne (le_inf Fx FS),
          fun hs x hx ↦ hs x (𝓝 x ⊓ 𝓟 s) hx inf_le_right inf_le_left⟩
 
-theorem IsClosed.interior_union_left {s t : Set X} (_ : IsClosed s) :
+theorem IsClosed.interior_union_left (_ : IsClosed s) :
     interior (s ∪ t) ⊆ s ∪ interior t := fun a ⟨u, ⟨⟨hu₁, hu₂⟩, ha⟩⟩ =>
   (Classical.em (a ∈ s)).imp_right fun h =>
     mem_interior.mpr
@@ -1448,28 +1448,28 @@ theorem IsClosed.interior_union_left {s t : Set X} (_ : IsClosed s) :
         ⟨ha, h⟩⟩
 #align is_closed.interior_union_left IsClosed.interior_union_left
 
-theorem IsClosed.interior_union_right {s t : Set X} (h : IsClosed t) :
+theorem IsClosed.interior_union_right (h : IsClosed t) :
     interior (s ∪ t) ⊆ interior s ∪ t := by
   simpa only [union_comm _ t] using h.interior_union_left
 #align is_closed.interior_union_right IsClosed.interior_union_right
 
-theorem IsOpen.inter_closure {s t : Set X} (h : IsOpen s) : s ∩ closure t ⊆ closure (s ∩ t) :=
+theorem IsOpen.inter_closure (h : IsOpen s) : s ∩ closure t ⊆ closure (s ∩ t) :=
   compl_subset_compl.mp <| by
     simpa only [← interior_compl, compl_inter] using IsClosed.interior_union_left h.isClosed_compl
 #align is_open.inter_closure IsOpen.inter_closure
 
-theorem IsOpen.closure_inter {s t : Set X} (h : IsOpen t) : closure s ∩ t ⊆ closure (s ∩ t) := by
+theorem IsOpen.closure_inter (h : IsOpen t) : closure s ∩ t ⊆ closure (s ∩ t) := by
   simpa only [inter_comm t] using h.inter_closure
 #align is_open.closure_inter IsOpen.closure_inter
 
-theorem Dense.open_subset_closure_inter {s t : Set X} (hs : Dense s) (ht : IsOpen t) :
+theorem Dense.open_subset_closure_inter (hs : Dense s) (ht : IsOpen t) :
     t ⊆ closure (t ∩ s) :=
   calc
     t = t ∩ closure s := by rw [hs.closure_eq, inter_univ]
     _ ⊆ closure (t ∩ s) := ht.inter_closure
 #align dense.open_subset_closure_inter Dense.open_subset_closure_inter
 
-theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set X} {x : X} (h : x ∈ closure (s₁ ∪ s₂))
+theorem mem_closure_of_mem_closure_union (h : x ∈ closure (s₁ ∪ s₂))
     (h₁ : s₁ᶜ ∈ 𝓝 x) : x ∈ closure s₂ := by
   rw [mem_closure_iff_nhds_neBot] at *
   rwa [←
@@ -1482,24 +1482,24 @@ theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set X} {x : X} (h : x ∈
 #align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_union
 
 /-- The intersection of an open dense set with a dense set is a dense set. -/
-theorem Dense.inter_of_isOpen_left {s t : Set X} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
+theorem Dense.inter_of_isOpen_left (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
     Dense (s ∩ t) := fun x =>
   closure_minimal hso.inter_closure isClosed_closure <| by simp [hs.closure_eq, ht.closure_eq]
 #align dense.inter_of_open_left Dense.inter_of_isOpen_left
 
 /-- The intersection of a dense set with an open dense set is a dense set. -/
-theorem Dense.inter_of_isOpen_right {s t : Set X} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
+theorem Dense.inter_of_isOpen_right (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
     Dense (s ∩ t) :=
   inter_comm t s ▸ ht.inter_of_isOpen_left hs hto
 #align dense.inter_of_open_right Dense.inter_of_isOpen_right
 
-theorem Dense.inter_nhds_nonempty {s t : Set X} (hs : Dense s) {x : X} (ht : t ∈ 𝓝 x) :
+theorem Dense.inter_nhds_nonempty (hs : Dense s) (ht : t ∈ 𝓝 x) :
     (s ∩ t).Nonempty :=
   let ⟨U, hsub, ho, hx⟩ := mem_nhds_iff.1 ht
   (hs.inter_open_nonempty U ho ⟨x, hx⟩).mono fun _y hy => ⟨hy.2, hsub hy.1⟩
 #align dense.inter_nhds_nonempty Dense.inter_nhds_nonempty
 
-theorem closure_diff {s t : Set X} : closure s \ closure t ⊆ closure (s \ t) :=
+theorem closure_diff : closure s \ closure t ⊆ closure (s \ t) :=
   calc
     closure s \ closure t = (closure t)ᶜ ∩ closure s := by simp only [diff_eq, inter_comm]
     _ ⊆ closure ((closure t)ᶜ ∩ s) := (isOpen_compl_iff.mpr <| isClosed_closure).inter_closure
@@ -1507,37 +1507,37 @@ theorem closure_diff {s t : Set X} : closure s \ closure t ⊆ closure (s \ t) :
     _ ⊆ closure (s \ t) := closure_mono <| diff_subset_diff (Subset.refl s) subset_closure
 #align closure_diff closure_diff
 
-theorem Filter.Frequently.mem_of_closed {a : X} {s : Set X} (h : ∃ᶠ x in 𝓝 a, x ∈ s)
-    (hs : IsClosed s) : a ∈ s :=
+theorem Filter.Frequently.mem_of_closed (h : ∃ᶠ x in 𝓝 x, x ∈ s)
+    (hs : IsClosed s) : x ∈ s :=
   hs.closure_subset h.mem_closure
 #align filter.frequently.mem_of_closed Filter.Frequently.mem_of_closed
 
-theorem IsClosed.mem_of_frequently_of_tendsto {f : α → X} {b : Filter α} {a : X} {s : Set X}
-    (hs : IsClosed s) (h : ∃ᶠ x in b, f x ∈ s) (hf : Tendsto f b (𝓝 a)) : a ∈ s :=
+theorem IsClosed.mem_of_frequently_of_tendsto {f : α → X} {b : Filter α}
+    (hs : IsClosed s) (h : ∃ᶠ x in b, f x ∈ s) (hf : Tendsto f b (𝓝 x)) : x ∈ s :=
   (hf.frequently <| show ∃ᶠ x in b, (fun y => y ∈ s) (f x) from h).mem_of_closed hs
 #align is_closed.mem_of_frequently_of_tendsto IsClosed.mem_of_frequently_of_tendsto
 
-theorem IsClosed.mem_of_tendsto {f : α → X} {b : Filter α} {a : X} {s : Set X} [NeBot b]
-    (hs : IsClosed s) (hf : Tendsto f b (𝓝 a)) (h : ∀ᶠ x in b, f x ∈ s) : a ∈ s :=
+theorem IsClosed.mem_of_tendsto {f : α → X} {b : Filter α} [NeBot b]
+    (hs : IsClosed s) (hf : Tendsto f b (𝓝 x)) (h : ∀ᶠ x in b, f x ∈ s) : x ∈ s :=
   hs.mem_of_frequently_of_tendsto h.frequently hf
 #align is_closed.mem_of_tendsto IsClosed.mem_of_tendsto
 
-theorem mem_closure_of_frequently_of_tendsto {f : α → X} {b : Filter α} {a : X} {s : Set X}
-    (h : ∃ᶠ x in b, f x ∈ s) (hf : Tendsto f b (𝓝 a)) : a ∈ closure s :=
+theorem mem_closure_of_frequently_of_tendsto {f : α → X} {b : Filter α}
+    (h : ∃ᶠ x in b, f x ∈ s) (hf : Tendsto f b (𝓝 x)) : x ∈ closure s :=
   (hf.frequently h).mem_closure
 #align mem_closure_of_frequently_of_tendsto mem_closure_of_frequently_of_tendsto
 
-theorem mem_closure_of_tendsto {f : α → X} {b : Filter α} {a : X} {s : Set X} [NeBot b]
-    (hf : Tendsto f b (𝓝 a)) (h : ∀ᶠ x in b, f x ∈ s) : a ∈ closure s :=
+theorem mem_closure_of_tendsto {f : α → X} {b : Filter α} [NeBot b]
+    (hf : Tendsto f b (𝓝 x)) (h : ∀ᶠ x in b, f x ∈ s) : x ∈ closure s :=
   mem_closure_of_frequently_of_tendsto h.frequently hf
 #align mem_closure_of_tendsto mem_closure_of_tendsto
 
-/-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
-Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/
-theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : α → X} {l : Filter α} {s : Set α} {a : X}
-    (h : ∀ x ∉ s, f x = a) : Tendsto f (l ⊓ 𝓟 s) (𝓝 a) ↔ Tendsto f l (𝓝 a) := by
+/-- Suppose that `f` sends the complement to `s` to a single point `x`, and `l` is some filter.
+Then `f` tends to `x` along `l` restricted to `s` if and only if it tends to `x` along `l`. -/
+theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : α → X} {l : Filter α} {s : Set α}
+    (h : ∀ a ∉ s, f a = x) : Tendsto f (l ⊓ 𝓟 s) (𝓝 x) ↔ Tendsto f l (𝓝 x) := by
   rw [tendsto_iff_comap, tendsto_iff_comap]
-  replace h : 𝓟 sᶜ ≤ comap f (𝓝 a)
+  replace h : 𝓟 sᶜ ≤ comap f (𝓝 x)
   · rintro U ⟨t, ht, htU⟩ x hx
     have : f x ∈ t := (h x hx).symm ▸ mem_of_mem_nhds ht
     exact htU this
@@ -1554,7 +1554,7 @@ theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : α → X} {l : Filter 
 In this section we define functions that return a limit of a filter (or of a function along a
 filter), if it exists, and a random point otherwise. These functions are rarely used in Mathlib,
 most of the theorems are written using `Filter.Tendsto`. One of the reasons is that
-`Filter.limUnder f g = a` is not equivalent to `Filter.Tendsto g f (𝓝 a)` unless the codomain is a
+`Filter.limUnder f g = x` is not equivalent to `Filter.Tendsto g f (𝓝 x)` unless the codomain is a
 Hausdorff space and `g` has a limit along `f`.
 -/
 
@@ -1565,7 +1565,7 @@ set_option linter.uppercaseLean3 false
 
 /-- If `f` is a filter, then `Filter.lim f` is a limit of the filter, if it exists. -/
 noncomputable def lim [Nonempty X] (f : Filter X) : X :=
-  Classical.epsilon fun a => f ≤ 𝓝 a
+  Classical.epsilon fun x => f ≤ 𝓝 x
 #align Lim lim
 
 /--
@@ -1582,19 +1582,19 @@ noncomputable def limUnder [Nonempty X] (f : Filter α) (g : α → X) : X :=
   lim (f.map g)
 #align lim limUnder
 
-/-- If a filter `f` is majorated by some `𝓝 a`, then it is majorated by `𝓝 (Filter.lim f)`. We
+/-- If a filter `f` is majorated by some `𝓝 x`, then it is majorated by `𝓝 (Filter.lim f)`. We
 formulate this lemma with a `[Nonempty X]` argument of `lim` derived from `h` to make it useful for
 types without a `[Nonempty X]` instance. Because of the built-in proof irrelevance, Lean will unify
 this instance with any other instance. -/
-theorem le_nhds_lim {f : Filter X} (h : ∃ a, f ≤ 𝓝 a) : f ≤ 𝓝 (@lim _ _ (nonempty_of_exists h) f) :=
+theorem le_nhds_lim {f : Filter X} (h : ∃ x, f ≤ 𝓝 x) : f ≤ 𝓝 (@lim _ _ (nonempty_of_exists h) f) :=
   Classical.epsilon_spec h
 #align le_nhds_Lim le_nhds_lim
 
-/-- If `g` tends to some `𝓝 a` along `f`, then it tends to `𝓝 (Filter.limUnder f g)`. We formulate
+/-- If `g` tends to some `𝓝 x` along `f`, then it tends to `𝓝 (Filter.limUnder f g)`. We formulate
 this lemma with a `[Nonempty X]` argument of `lim` derived from `h` to make it useful for types
 without a `[Nonempty X]` instance. Because of the built-in proof irrelevance, Lean will unify this
 instance with any other instance. -/
-theorem tendsto_nhds_limUnder {f : Filter α} {g : α → X} (h : ∃ a, Tendsto g f (𝓝 a)) :
+theorem tendsto_nhds_limUnder {f : Filter α} {g : α → X} (h : ∃ x, Tendsto g f (𝓝 x)) :
     Tendsto g f (𝓝 (@limUnder _ _ _ (nonempty_of_exists h) f g)) :=
   le_nhds_lim h
 #align tendsto_nhds_lim tendsto_nhds_limUnder
@@ -1628,21 +1628,22 @@ set_option quotPrecheck false in
 scoped[Topology] notation (name := Continuous_of) "Continuous[" t₁ ", " t₂ "]" =>
   @Continuous _ _ t₁ t₂
 
-theorem continuous_def {_ : TopologicalSpace X} {_ : TopologicalSpace Y} {f : X → Y} :
-    Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) :=
+variable {f : X → Y} {s : Set X} {x : X} {y : Y}
+
+theorem continuous_def : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) :=
   ⟨fun hf => hf.1, fun h => ⟨h⟩⟩
 #align continuous_def continuous_def
 
-theorem IsOpen.preimage {f : X → Y} (hf : Continuous f) {s : Set Y} (h : IsOpen s) :
-    IsOpen (f ⁻¹' s) :=
-  hf.isOpen_preimage s h
+theorem IsOpen.preimage (hf : Continuous f) {t : Set Y} (h : IsOpen t) :
+    IsOpen (f ⁻¹' t) :=
+  hf.isOpen_preimage t h
 #align is_open.preimage IsOpen.preimage
 
-theorem continuous_congr {f g : X → Y} (h : ∀ x, f x = g x) :
+theorem continuous_congr {g : X → Y} (h : ∀ x, f x = g x) :
     Continuous f ↔ Continuous g :=
   .of_eq <| congrArg _ <| funext h
 
-theorem Continuous.congr {f g : X → Y} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g :=
+theorem Continuous.congr {g : X → Y} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g :=
   continuous_congr h' |>.mp h
 #align continuous.congr Continuous.congr
 
@@ -1652,47 +1653,47 @@ def ContinuousAt (f : X → Y) (x : X) :=
   Tendsto f (𝓝 x) (𝓝 (f x))
 #align continuous_at ContinuousAt
 
-theorem ContinuousAt.tendsto {f : X → Y} {x : X} (h : ContinuousAt f x) :
+theorem ContinuousAt.tendsto (h : ContinuousAt f x) :
     Tendsto f (𝓝 x) (𝓝 (f x)) :=
   h
 #align continuous_at.tendsto ContinuousAt.tendsto
 
-theorem continuousAt_def {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x :=
+theorem continuousAt_def : ContinuousAt f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x :=
   Iff.rfl
 #align continuous_at_def continuousAt_def
 
-theorem continuousAt_congr {f g : X → Y} {x : X} (h : f =ᶠ[𝓝 x] g) :
+theorem continuousAt_congr {g : X → Y} (h : f =ᶠ[𝓝 x] g) :
     ContinuousAt f x ↔ ContinuousAt g x := by
   simp only [ContinuousAt, tendsto_congr' h, h.eq_of_nhds]
 #align continuous_at_congr continuousAt_congr
 
-theorem ContinuousAt.congr {f g : X → Y} {x : X} (hf : ContinuousAt f x) (h : f =ᶠ[𝓝 x] g) :
+theorem ContinuousAt.congr {g : X → Y} (hf : ContinuousAt f x) (h : f =ᶠ[𝓝 x] g) :
     ContinuousAt g x :=
   (continuousAt_congr h).1 hf
 #align continuous_at.congr ContinuousAt.congr
 
-theorem ContinuousAt.preimage_mem_nhds {f : X → Y} {x : X} {t : Set Y} (h : ContinuousAt f x)
+theorem ContinuousAt.preimage_mem_nhds {t : Set Y} (h : ContinuousAt f x)
     (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝 x :=
   h ht
 #align continuous_at.preimage_mem_nhds ContinuousAt.preimage_mem_nhds
 
 /-- Deprecated, please use `not_mem_tsupport_iff_eventuallyEq` instead. -/
 @[deprecated] -- 15 January 2024
-theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {a : X} {f : X → M₀} :
-    f =ᶠ[𝓝 a] 0 ↔ a ∉ closure (Function.support f) := by
+theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {f : X → M₀} :
+    f =ᶠ[𝓝 x] 0 ↔ x ∉ closure (Function.support f) := by
   rw [← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds, Function.compl_support,
     EventuallyEq, eventually_iff]
   simp only [Pi.zero_apply]
 #align eventually_eq_zero_nhds eventuallyEq_zero_nhds
 
-theorem ClusterPt.map {x : X} {la : Filter X} {lb : Filter Y} (H : ClusterPt x la) {f : X → Y}
-    (hfc : ContinuousAt f x) (hf : Tendsto f la lb) : ClusterPt (f x) lb :=
+theorem ClusterPt.map {lx : Filter X} {ly : Filter Y} (H : ClusterPt x lx)
+    (hfc : ContinuousAt f x) (hf : Tendsto f lx ly) : ClusterPt (f x) ly :=
   (NeBot.map H f).mono <| hfc.tendsto.inf hf
 #align cluster_pt.map ClusterPt.map
 
 /-- See also `interior_preimage_subset_preimage_interior`. -/
-theorem preimage_interior_subset_interior_preimage {f : X → Y} {s : Set Y} (hf : Continuous f) :
-    f ⁻¹' interior s ⊆ interior (f ⁻¹' s) :=
+theorem preimage_interior_subset_interior_preimage {t : Set Y} (hf : Continuous f) :
+    f ⁻¹' interior t ⊆ interior (f ⁻¹' t) :=
   interior_maximal (preimage_mono interior_subset) (isOpen_interior.preimage hf)
 #align preimage_interior_subset_interior_preimage preimage_interior_subset_interior_preimage
 
@@ -1705,150 +1706,150 @@ theorem continuous_id : Continuous (id : X → X) :=
 @[continuity]
 theorem continuous_id' : Continuous (fun (x : X) => x) := continuous_id
 
-theorem Continuous.comp {g : Y → Z} {f : X → Y} (hg : Continuous g) (hf : Continuous f) :
+theorem Continuous.comp {g : Y → Z} (hg : Continuous g) (hf : Continuous f) :
     Continuous (g ∘ f) :=
   continuous_def.2 fun _ h => (h.preimage hg).preimage hf
 #align continuous.comp Continuous.comp
 
 -- This is needed due to reducibility issues with the `continuity` tactic.
 @[continuity]
-theorem Continuous.comp' {g : Y → Z} {f : X → Y} (hg : Continuous g) (hf : Continuous f) :
+theorem Continuous.comp' {g : Y → Z} (hg : Continuous g) (hf : Continuous f) :
     Continuous (fun x => g (f x)) := hg.comp hf
 
 theorem Continuous.iterate {f : X → X} (h : Continuous f) (n : ℕ) : Continuous f^[n] :=
   Nat.recOn n continuous_id fun _ ihn => ihn.comp h
 #align continuous.iterate Continuous.iterate
 
-nonrec theorem ContinuousAt.comp {g : Y → Z} {f : X → Y} {x : X} (hg : ContinuousAt g (f x))
+nonrec theorem ContinuousAt.comp {g : Y → Z} (hg : ContinuousAt g (f x))
     (hf : ContinuousAt f x) : ContinuousAt (g ∘ f) x :=
   hg.comp hf
 #align continuous_at.comp ContinuousAt.comp
 
 /-- See note [comp_of_eq lemmas] -/
-theorem ContinuousAt.comp_of_eq {g : Y → Z} {f : X → Y} {x : X} {y : Y} (hg : ContinuousAt g y)
+theorem ContinuousAt.comp_of_eq {g : Y → Z} (hg : ContinuousAt g y)
     (hf : ContinuousAt f x) (hy : f x = y) : ContinuousAt (g ∘ f) x := by subst hy; exact hg.comp hf
 #align continuous_at.comp_of_eq ContinuousAt.comp_of_eq
 
-theorem Continuous.tendsto {f : X → Y} (hf : Continuous f) (x) : Tendsto f (𝓝 x) (𝓝 (f x)) :=
+theorem Continuous.tendsto (hf : Continuous f) (x) : Tendsto f (𝓝 x) (𝓝 (f x)) :=
   ((nhds_basis_opens x).tendsto_iff <| nhds_basis_opens <| f x).2 fun t ⟨hxt, ht⟩ =>
     ⟨f ⁻¹' t, ⟨hxt, ht.preimage hf⟩, Subset.rfl⟩
 #align continuous.tendsto Continuous.tendsto
 
 /-- A version of `Continuous.tendsto` that allows one to specify a simpler form of the limit.
 E.g., one can write `continuous_exp.tendsto' 0 1 exp_zero`. -/
-theorem Continuous.tendsto' {f : X → Y} (hf : Continuous f) (x : X) (y : Y) (h : f x = y) :
+theorem Continuous.tendsto' (hf : Continuous f) (x : X) (y : Y) (h : f x = y) :
     Tendsto f (𝓝 x) (𝓝 y) :=
   h ▸ hf.tendsto x
 #align continuous.tendsto' Continuous.tendsto'
 
-theorem Continuous.continuousAt {f : X → Y} {x : X} (h : Continuous f) : ContinuousAt f x :=
+theorem Continuous.continuousAt (h : Continuous f) : ContinuousAt f x :=
   h.tendsto x
 #align continuous.continuous_at Continuous.continuousAt
 
-theorem continuous_iff_continuousAt {f : X → Y} : Continuous f ↔ ∀ x, ContinuousAt f x :=
+theorem continuous_iff_continuousAt : Continuous f ↔ ∀ x, ContinuousAt f x :=
   ⟨Continuous.tendsto, fun hf => continuous_def.2 fun _U hU => isOpen_iff_mem_nhds.2 fun x hx =>
     hf x <| hU.mem_nhds hx⟩
 #align continuous_iff_continuous_at continuous_iff_continuousAt
 
-theorem continuousAt_const {x : X} {b : Y} : ContinuousAt (fun _ : X => b) x :=
+theorem continuousAt_const : ContinuousAt (fun _ : X => y) x :=
   tendsto_const_nhds
 #align continuous_at_const continuousAt_const
 
 @[continuity]
-theorem continuous_const {b : Y} : Continuous fun _ : X => b :=
+theorem continuous_const : Continuous fun _ : X => y :=
   continuous_iff_continuousAt.mpr fun _ => continuousAt_const
 #align continuous_const continuous_const
 
-theorem Filter.EventuallyEq.continuousAt {x : X} {f : X → Y} {y : Y} (h : f =ᶠ[𝓝 x] fun _ => y) :
+theorem Filter.EventuallyEq.continuousAt (h : f =ᶠ[𝓝 x] fun _ => y) :
     ContinuousAt f x :=
   (continuousAt_congr h).2 tendsto_const_nhds
 #align filter.eventually_eq.continuous_at Filter.EventuallyEq.continuousAt
 
-theorem continuous_of_const {f : X → Y} (h : ∀ x y, f x = f y) : Continuous f :=
+theorem continuous_of_const (h : ∀ x y, f x = f y) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x =>
     Filter.EventuallyEq.continuousAt <| eventually_of_forall fun y => h y x
 #align continuous_of_const continuous_of_const
 
-theorem continuousAt_id {x : X} : ContinuousAt id x :=
+theorem continuousAt_id : ContinuousAt id x :=
   continuous_id.continuousAt
 #align continuous_at_id continuousAt_id
 
-theorem ContinuousAt.iterate {f : X → X} {x : X} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ) :
+theorem ContinuousAt.iterate {f : X → X} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ) :
     ContinuousAt f^[n] x :=
   Nat.recOn n continuousAt_id fun n ihn =>
     show ContinuousAt (f^[n] ∘ f) x from ContinuousAt.comp (hx.symm ▸ ihn) hf
 #align continuous_at.iterate ContinuousAt.iterate
 
-theorem continuous_iff_isClosed {f : X → Y} : Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s) :=
+theorem continuous_iff_isClosed : Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s) :=
   continuous_def.trans <| compl_surjective.forall.trans <| by
     simp only [isOpen_compl_iff, preimage_compl]
 #align continuous_iff_is_closed continuous_iff_isClosed
 
-theorem IsClosed.preimage {f : X → Y} (hf : Continuous f) {s : Set Y} (h : IsClosed s) :
-    IsClosed (f ⁻¹' s) :=
-  continuous_iff_isClosed.mp hf s h
+theorem IsClosed.preimage (hf : Continuous f) {t : Set Y} (h : IsClosed t) :
+    IsClosed (f ⁻¹' t) :=
+  continuous_iff_isClosed.mp hf t h
 #align is_closed.preimage IsClosed.preimage
 
-theorem mem_closure_image {f : X → Y} {x : X} {s : Set X} (hf : ContinuousAt f x)
+theorem mem_closure_image (hf : ContinuousAt f x)
     (hx : x ∈ closure s) : f x ∈ closure (f '' s) :=
   mem_closure_of_frequently_of_tendsto
     ((mem_closure_iff_frequently.1 hx).mono fun _ => mem_image_of_mem _) hf
 #align mem_closure_image mem_closure_image
 
-theorem continuousAt_iff_ultrafilter {f : X → Y} {x} :
+theorem continuousAt_iff_ultrafilter :
     ContinuousAt f x ↔ ∀ g : Ultrafilter X, ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) :=
   tendsto_iff_ultrafilter f (𝓝 x) (𝓝 (f x))
 #align continuous_at_iff_ultrafilter continuousAt_iff_ultrafilter
 
-theorem continuous_iff_ultrafilter {f : X → Y} :
+theorem continuous_iff_ultrafilter :
     Continuous f ↔ ∀ (x) (g : Ultrafilter X), ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) := by
   simp only [continuous_iff_continuousAt, continuousAt_iff_ultrafilter]
 #align continuous_iff_ultrafilter continuous_iff_ultrafilter
 
-theorem Continuous.closure_preimage_subset {f : X → Y} (hf : Continuous f) (t : Set Y) :
+theorem Continuous.closure_preimage_subset (hf : Continuous f) (t : Set Y) :
     closure (f ⁻¹' t) ⊆ f ⁻¹' closure t := by
   rw [← (isClosed_closure.preimage hf).closure_eq]
   exact closure_mono (preimage_mono subset_closure)
 #align continuous.closure_preimage_subset Continuous.closure_preimage_subset
 
-theorem Continuous.frontier_preimage_subset {f : X → Y} (hf : Continuous f) (t : Set Y) :
+theorem Continuous.frontier_preimage_subset (hf : Continuous f) (t : Set Y) :
     frontier (f ⁻¹' t) ⊆ f ⁻¹' frontier t :=
   diff_subset_diff (hf.closure_preimage_subset t) (preimage_interior_subset_interior_preimage hf)
 #align continuous.frontier_preimage_subset Continuous.frontier_preimage_subset
 
 /-- If a continuous map `f` maps `s` to `t`, then it maps `closure s` to `closure t`. -/
-protected theorem Set.MapsTo.closure {s : Set X} {t : Set Y} {f : X → Y} (h : MapsTo f s t)
+protected theorem Set.MapsTo.closure {t : Set Y} (h : MapsTo f s t)
     (hc : Continuous f) : MapsTo f (closure s) (closure t) := by
   simp only [MapsTo, mem_closure_iff_clusterPt]
   exact fun x hx => hx.map hc.continuousAt (tendsto_principal_principal.2 h)
 #align set.maps_to.closure Set.MapsTo.closure
 
 /-- See also `IsClosedMap.closure_image_eq_of_continuous`. -/
-theorem image_closure_subset_closure_image {f : X → Y} {s : Set X} (h : Continuous f) :
+theorem image_closure_subset_closure_image (h : Continuous f) :
     f '' closure s ⊆ closure (f '' s) :=
   ((mapsTo_image f s).closure h).image_subset
 #align image_closure_subset_closure_image image_closure_subset_closure_image
 
 -- porting note: new lemma
-theorem closure_image_closure {f : X → Y} {s : Set X} (h : Continuous f) :
+theorem closure_image_closure (h : Continuous f) :
     closure (f '' closure s) = closure (f '' s) :=
   Subset.antisymm
     (closure_minimal (image_closure_subset_closure_image h) isClosed_closure)
     (closure_mono <| image_subset _ subset_closure)
 
-theorem closure_subset_preimage_closure_image {f : X → Y} {s : Set X} (h : Continuous f) :
+theorem closure_subset_preimage_closure_image (h : Continuous f) :
     closure s ⊆ f ⁻¹' closure (f '' s) := by
   rw [← Set.image_subset_iff]
   exact image_closure_subset_closure_image h
 #align closure_subset_preimage_closure_image closure_subset_preimage_closure_image
 
-theorem map_mem_closure {s : Set X} {t : Set Y} {f : X → Y} {a : X} (hf : Continuous f)
-    (ha : a ∈ closure s) (ht : MapsTo f s t) : f a ∈ closure t :=
-  ht.closure hf ha
+theorem map_mem_closure {t : Set Y} (hf : Continuous f)
+    (hx : x ∈ closure s) (ht : MapsTo f s t) : f x ∈ closure t :=
+  ht.closure hf hx
 #align map_mem_closure map_mem_closure
 
 /-- If a continuous map `f` maps `s` to a closed set `t`, then it maps `closure s` to `t`. -/
-theorem Set.MapsTo.closure_left {s : Set X} {t : Set Y} {f : X → Y} (h : MapsTo f s t)
+theorem Set.MapsTo.closure_left {t : Set Y} (h : MapsTo f s t)
     (hc : Continuous f) (ht : IsClosed t) : MapsTo f (closure s) t :=
   ht.closure_eq ▸ h.closure hc
 #align set.maps_to.closure_left Set.MapsTo.closure_left
@@ -1865,7 +1866,7 @@ variable {α ι : Type*} (f : α → X) (g : X → Y)
 def DenseRange := Dense (range f)
 #align dense_range DenseRange
 
-variable {f}
+variable {f : α → X} {s : Set X}
 
 /-- A surjective map has dense range. -/
 theorem Function.Surjective.denseRange (hf : Function.Surjective f) : DenseRange f := fun x => by
@@ -1884,33 +1885,33 @@ theorem DenseRange.closure_range (h : DenseRange f) : closure (range f) = univ :
   h.closure_eq
 #align dense_range.closure_range DenseRange.closure_range
 
-theorem Dense.denseRange_val {s : Set X} (h : Dense s) : DenseRange ((↑) : s → X) := by
+theorem Dense.denseRange_val (h : Dense s) : DenseRange ((↑) : s → X) := by
   simpa only [DenseRange, Subtype.range_coe_subtype]
 #align dense.dense_range_coe Dense.denseRange_val
 
-theorem Continuous.range_subset_closure_image_dense {f : X → Y} (hf : Continuous f) {s : Set X}
+theorem Continuous.range_subset_closure_image_dense {f : X → Y} (hf : Continuous f)
     (hs : Dense s) : range f ⊆ closure (f '' s) := by
   rw [← image_univ, ← hs.closure_eq]
   exact image_closure_subset_closure_image hf
 #align continuous.range_subset_closure_image_dense Continuous.range_subset_closure_image_dense
 
 /-- The image of a dense set under a continuous map with dense range is a dense set. -/
-theorem DenseRange.dense_image {f : X → Y} (hf' : DenseRange f) (hf : Continuous f) {s : Set X}
+theorem DenseRange.dense_image {f : X → Y} (hf' : DenseRange f) (hf : Continuous f)
     (hs : Dense s) : Dense (f '' s) :=
   (hf'.mono <| hf.range_subset_closure_image_dense hs).of_closure
 #align dense_range.dense_image DenseRange.dense_image
 
 /-- If `f` has dense range and `s` is an open set in the codomain of `f`, then the image of the
 preimage of `s` under `f` is dense in `s`. -/
-theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) {s : Set X}
-    (hs : IsOpen s) : s ⊆ closure (f '' (f ⁻¹' s)) := by
+theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) (hs : IsOpen s) :
+    s ⊆ closure (f '' (f ⁻¹' s)) := by
   rw [image_preimage_eq_inter_range]
   exact hf.open_subset_closure_inter hs
 #align dense_range.subset_closure_image_preimage_of_is_open DenseRange.subset_closure_image_preimage_of_isOpen
 
 /-- If a continuous map with dense range maps a dense set to a subset of `t`, then `t` is a dense
 set. -/
-theorem DenseRange.dense_of_mapsTo {f : X → Y} (hf' : DenseRange f) (hf : Continuous f) {s : Set X}
+theorem DenseRange.dense_of_mapsTo {f : X → Y} (hf' : DenseRange f) (hf : Continuous f)
     (hs : Dense s) {t : Set Y} (ht : MapsTo f s t) : Dense t :=
   (hf'.dense_image hf hs).mono ht.image_subset
 #align dense_range.dense_of_maps_to DenseRange.dense_of_mapsTo
@@ -1931,19 +1932,19 @@ theorem DenseRange.nonempty [h : Nonempty X] (hf : DenseRange f) : Nonempty α :
   hf.nonempty_iff.mpr h
 #align dense_range.nonempty DenseRange.nonempty
 
-/-- Given a function `f : X → Y` with dense range and `b : Y`, returns some `a : X`. -/
-def DenseRange.some (hf : DenseRange f) (b : X) : α :=
-  Classical.choice <| hf.nonempty_iff.mpr ⟨b⟩
+/-- Given a function `f : X → Y` with dense range and `y : Y`, returns some `x : X`. -/
+def DenseRange.some (hf : DenseRange f) (x : X) : α :=
+  Classical.choice <| hf.nonempty_iff.mpr ⟨x⟩
 #align dense_range.some DenseRange.some
 
-nonrec theorem DenseRange.exists_mem_open (hf : DenseRange f) {s : Set X} (ho : IsOpen s)
-    (hs : s.Nonempty) : ∃ a, f a ∈ s :=
+nonrec theorem DenseRange.exists_mem_open (hf : DenseRange f) (ho : IsOpen s) (hs : s.Nonempty) :
+    ∃ a, f a ∈ s :=
   exists_range_iff.1 <| hf.exists_mem_open ho hs
 #align dense_range.exists_mem_open DenseRange.exists_mem_open
 
-theorem DenseRange.mem_nhds (h : DenseRange f) {b : X} {U : Set X} (U_in : U ∈ 𝓝 b) :
-    ∃ a, f a ∈ U :=
-  let ⟨a, ha⟩ := h.exists_mem_open isOpen_interior ⟨b, mem_interior_iff_mem_nhds.2 U_in⟩
+theorem DenseRange.mem_nhds (h : DenseRange f) (hs : s ∈ 𝓝 x) :
+    ∃ a, f a ∈ s :=
+  let ⟨a, ha⟩ := h.exists_mem_open isOpen_interior ⟨x, mem_interior_iff_mem_nhds.2 hs⟩
   ⟨a, interior_subset ha⟩
 #align dense_range.mem_nhds DenseRange.mem_nhds
 

Use it to golf PrimeSpectrum.localization_comap_inducing.

@@ -210,6 +210,13 @@ scoped[Topology] notation (name := IsClosed_of) "IsClosed[" t "]" => @IsClosed _
   ⟨fun h => ⟨h⟩, fun h => h.isOpen_compl⟩
 #align is_open_compl_iff isOpen_compl_iff
 
+theorem TopologicalSpace.ext_iff_isClosed {t₁ t₂ : TopologicalSpace X} :
+    t₁ = t₂ ↔ ∀ s, IsClosed[t₁] s ↔ IsClosed[t₂] s := by
+  rw [TopologicalSpace.ext_iff, compl_surjective.forall]
+  simp only [@isOpen_compl_iff _ t₁, @isOpen_compl_iff _ t₂]
+
+alias ⟨_, TopologicalSpace.ext_isClosed⟩ := TopologicalSpace.ext_iff_isClosed
+
 -- porting note: new lemma
 theorem isClosed_const {p : Prop} : IsClosed { _a : X | p } := ⟨isOpen_const (p := ¬p)⟩
 

Use X, Y, Z for topological spaces.

@@ -13,14 +13,14 @@ import Mathlib.Tactic.Continuity
 /-!
 # Basic theory of topological spaces.
 
-The main definition is the type class `TopologicalSpace α` which endows a type `α` with a topology.
-Then `Set α` gets predicates `IsOpen`, `IsClosed` and functions `interior`, `closure` and
-`frontier`. Each point `x` of `α` gets a neighborhood filter `𝓝 x`. A filter `F` on `α` has
-`x` as a cluster point if `ClusterPt x F : 𝓝 x ⊓ F ≠ ⊥`. A map `f : ι → α` clusters at `x`
-along `F : Filter ι` if `MapClusterPt x F f : ClusterPt x (map f F)`. In particular
+The main definition is the type class `TopologicalSpace X` which endows a type `X` with a topology.
+Then `Set X` gets predicates `IsOpen`, `IsClosed` and functions `interior`, `closure` and
+`frontier`. Each point `x` of `X` gets a neighborhood filter `𝓝 x`. A filter `F` on `X` has
+`x` as a cluster point if `ClusterPt x F : 𝓝 x ⊓ F ≠ ⊥`. A map `f : α → X` clusters at `x`
+along `F : Filter α` if `MapClusterPt x F f : ClusterPt x (map f F)`. In particular
 the notion of cluster point of a sequence `u` is `MapClusterPt x atTop u`.
 
-For topological spaces `α` and `β`, a function `f : α → β` and a point `a : α`,
+For topological spaces `X` and `Y`, a function `f : X → Y` and a point `a : X`,
 `ContinuousAt f a` means `f` is continuous at `a`, and global continuity is
 `Continuous f`. There is also a version of continuity `PContinuous` for
 partially defined functions.
@@ -55,18 +55,18 @@ noncomputable section
 
 open Set Filter
 
-universe u v w
+universe u v w x
 
 /-!
 ### Topological spaces
 -/
 
 
-/-- A topology on `α`. -/
+/-- A topology on `X`. -/
 @[to_additive existing TopologicalSpace]
-class TopologicalSpace (α : Type u) where
+class TopologicalSpace (X : Type u) where
   /-- A predicate saying that a set is an open set. Use `IsOpen` in the root namespace instead. -/
-  protected IsOpen : Set α → Prop
+  protected IsOpen : Set X → Prop
   /-- The set representing the whole space is an open set. Use `isOpen_univ` in the root namespace
   instead. -/
   protected isOpen_univ : IsOpen univ
@@ -79,9 +79,9 @@ class TopologicalSpace (α : Type u) where
 
 /-- A constructor for topologies by specifying the closed sets,
 and showing that they satisfy the appropriate conditions. -/
-def TopologicalSpace.ofClosed {α : Type u} (T : Set (Set α)) (empty_mem : ∅ ∈ T)
+def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈ T)
     (sInter_mem : ∀ A, A ⊆ T → ⋂₀ A ∈ T)
-    (union_mem : ∀ A, A ∈ T → ∀ B, B ∈ T → A ∪ B ∈ T) : TopologicalSpace α where
+    (union_mem : ∀ A, A ∈ T → ∀ B, B ∈ T → A ∪ B ∈ T) : TopologicalSpace X where
   IsOpen X := Xᶜ ∈ T
   isOpen_univ := by simp [empty_mem]
   isOpen_inter s t hs ht := by simpa only [compl_inter] using union_mem sᶜ hs tᶜ ht
@@ -92,10 +92,11 @@ def TopologicalSpace.ofClosed {α : Type u} (T : Set (Set α)) (empty_mem : ∅
 
 section TopologicalSpace
 
-variable {α : Type u} {β : Type v} {ι : Sort w} {a : α} {s s₁ s₂ t : Set α} {p p₁ p₂ : α → Prop}
+variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*}
+  {a : X} {s s₁ s₂ t : Set X} {p p₁ p₂ : X → Prop}
 
-/-- `IsOpen s` means that `s` is open in the ambient topological space on `α` -/
-def IsOpen [TopologicalSpace α] : Set α → Prop := TopologicalSpace.IsOpen
+/-- `IsOpen s` means that `s` is open in the ambient topological space on `X` -/
+def IsOpen [TopologicalSpace X] : Set X → Prop := TopologicalSpace.IsOpen
 #align is_open IsOpen
 
 set_option quotPrecheck false in
@@ -104,48 +105,48 @@ scoped[Topology] notation (name := IsOpen_of) "IsOpen[" t "]" => @IsOpen _ t
 
 open Topology
 
-lemma isOpen_mk {p h₁ h₂ h₃} {s : Set α} : IsOpen[⟨p, h₁, h₂, h₃⟩] s ↔ p s := Iff.rfl
+lemma isOpen_mk {p h₁ h₂ h₃} {s : Set X} : IsOpen[⟨p, h₁, h₂, h₃⟩] s ↔ p s := Iff.rfl
 #align is_open_mk isOpen_mk
 
 @[ext]
 protected theorem TopologicalSpace.ext :
-    ∀ {f g : TopologicalSpace α}, IsOpen[f] = IsOpen[g] → f = g
+    ∀ {f g : TopologicalSpace X}, IsOpen[f] = IsOpen[g] → f = g
   | ⟨_, _, _, _⟩, ⟨_, _, _, _⟩, rfl => rfl
 #align topological_space_eq TopologicalSpace.ext
 
 section
 
-variable [TopologicalSpace α]
+variable [TopologicalSpace X]
 
-@[simp] theorem isOpen_univ : IsOpen (univ : Set α) := TopologicalSpace.isOpen_univ
+@[simp] theorem isOpen_univ : IsOpen (univ : Set X) := TopologicalSpace.isOpen_univ
 #align is_open_univ isOpen_univ
 
 theorem IsOpen.inter (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∩ s₂) :=
   TopologicalSpace.isOpen_inter s₁ s₂ h₁ h₂
 #align is_open.inter IsOpen.inter
 
-theorem isOpen_sUnion {s : Set (Set α)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (⋃₀ s) :=
+theorem isOpen_sUnion {s : Set (Set X)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (⋃₀ s) :=
   TopologicalSpace.isOpen_sUnion s h
 #align is_open_sUnion isOpen_sUnion
 
 end
 
-protected theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace α} :
+protected theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace X} :
     t = t' ↔ ∀ s, IsOpen[t] s ↔ IsOpen[t'] s :=
   ⟨fun h s => h ▸ Iff.rfl, fun h => by ext; exact h _⟩
 #align topological_space_eq_iff TopologicalSpace.ext_iff
 
-theorem isOpen_fold {s : Set α} {t : TopologicalSpace α} : t.IsOpen s = IsOpen[t] s :=
+theorem isOpen_fold {s : Set X} {t : TopologicalSpace X} : t.IsOpen s = IsOpen[t] s :=
   rfl
 #align is_open_fold isOpen_fold
 
-variable [TopologicalSpace α]
+variable [TopologicalSpace X]
 
-theorem isOpen_iUnion {f : ι → Set α} (h : ∀ i, IsOpen (f i)) : IsOpen (⋃ i, f i) :=
+theorem isOpen_iUnion {f : ι → Set X} (h : ∀ i, IsOpen (f i)) : IsOpen (⋃ i, f i) :=
   isOpen_sUnion (forall_range_iff.2 h)
 #align is_open_Union isOpen_iUnion
 
-theorem isOpen_biUnion {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
+theorem isOpen_biUnion {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋃ i ∈ s, f i) :=
   isOpen_iUnion fun i => isOpen_iUnion fun hi => h i hi
 #align is_open_bUnion isOpen_biUnion
@@ -154,41 +155,41 @@ theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ 
   rw [union_eq_iUnion]; exact isOpen_iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align is_open.union IsOpen.union
 
-lemma isOpen_iff_of_cover {ι : Type*} {f : ι → Set α} (ho : ∀ i, IsOpen (f i))
-    (hU : (⋃ i, f i) = univ) : IsOpen s ↔ ∀ i, IsOpen (f i ∩ s) := by
+lemma isOpen_iff_of_cover {f : α → Set X} (ho : ∀ i, IsOpen (f i)) (hU : (⋃ i, f i) = univ) :
+    IsOpen s ↔ ∀ i, IsOpen (f i ∩ s) := by
   refine ⟨fun h i ↦ (ho i).inter h, fun h ↦ ?_⟩
   rw [← s.inter_univ, inter_comm, ← hU, iUnion_inter]
   exact isOpen_iUnion fun i ↦ h i
 
-@[simp] theorem isOpen_empty : IsOpen (∅ : Set α) := by
+@[simp] theorem isOpen_empty : IsOpen (∅ : Set X) := by
   rw [← sUnion_empty]; exact isOpen_sUnion fun a => False.elim
 #align is_open_empty isOpen_empty
 
-theorem Set.Finite.isOpen_sInter {s : Set (Set α)} (hs : s.Finite) :
+theorem Set.Finite.isOpen_sInter {s : Set (Set X)} (hs : s.Finite) :
     (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
   Finite.induction_on hs (fun _ => by rw [sInter_empty]; exact isOpen_univ) fun _ _ ih h => by
     simp only [sInter_insert, ball_insert_iff] at h ⊢
     exact h.1.inter (ih h.2)
 #align is_open_sInter Set.Finite.isOpen_sInter
 
-theorem Set.Finite.isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finite)
+theorem Set.Finite.isOpen_biInter {s : Set α} {f : α → Set X} (hs : s.Finite)
     (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
   sInter_image f s ▸ (hs.image _).isOpen_sInter (ball_image_iff.2 h)
 #align is_open_bInter Set.Finite.isOpen_biInter
 
-theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) :
+theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set X} (h : ∀ i, IsOpen (s i)) :
     IsOpen (⋂ i, s i) :=
   (finite_range _).isOpen_sInter  (forall_range_iff.2 h)
 #align is_open_Inter isOpen_iInter_of_finite
 
-theorem isOpen_biInter_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
+theorem isOpen_biInter_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
   s.finite_toSet.isOpen_biInter h
 #align is_open_bInter_finset isOpen_biInter_finset
 
 @[simp] -- porting note: added `simp`
-theorem isOpen_const {p : Prop} : IsOpen { _a : α | p } := by by_cases p <;> simp [*]
+theorem isOpen_const {p : Prop} : IsOpen { _a : X | p } := by by_cases p <;> simp [*]
 #align is_open_const isOpen_const
 
 theorem IsOpen.and : IsOpen { a | p₁ a } → IsOpen { a | p₂ a } → IsOpen { a | p₁ a ∧ p₂ a } :=
@@ -196,7 +197,7 @@ theorem IsOpen.and : IsOpen { a | p₁ a } → IsOpen { a | p₂ a } → IsOpen
 #align is_open.and IsOpen.and
 
 /-- A set is closed if its complement is open -/
-class IsClosed (s : Set α) : Prop where
+class IsClosed (s : Set X) : Prop where
   /-- The complement of a closed set is an open set. -/
   isOpen_compl : IsOpen sᶜ
 #align is_closed IsClosed
@@ -205,45 +206,45 @@ set_option quotPrecheck false in
 /-- Notation for `IsClosed` with respect to a non-standard topology. -/
 scoped[Topology] notation (name := IsClosed_of) "IsClosed[" t "]" => @IsClosed _ t
 
-@[simp] theorem isOpen_compl_iff {s : Set α} : IsOpen sᶜ ↔ IsClosed s :=
+@[simp] theorem isOpen_compl_iff {s : Set X} : IsOpen sᶜ ↔ IsClosed s :=
   ⟨fun h => ⟨h⟩, fun h => h.isOpen_compl⟩
 #align is_open_compl_iff isOpen_compl_iff
 
 -- porting note: new lemma
-theorem isClosed_const {p : Prop} : IsClosed { _a : α | p } := ⟨isOpen_const (p := ¬p)⟩
+theorem isClosed_const {p : Prop} : IsClosed { _a : X | p } := ⟨isOpen_const (p := ¬p)⟩
 
-@[simp] theorem isClosed_empty : IsClosed (∅ : Set α) := isClosed_const
+@[simp] theorem isClosed_empty : IsClosed (∅ : Set X) := isClosed_const
 #align is_closed_empty isClosed_empty
 
-@[simp] theorem isClosed_univ : IsClosed (univ : Set α) := isClosed_const
+@[simp] theorem isClosed_univ : IsClosed (univ : Set X) := isClosed_const
 #align is_closed_univ isClosed_univ
 
 theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂) := by
   simpa only [← isOpen_compl_iff, compl_union] using IsOpen.inter
 #align is_closed.union IsClosed.union
 
-theorem isClosed_sInter {s : Set (Set α)} : (∀ t ∈ s, IsClosed t) → IsClosed (⋂₀ s) := by
+theorem isClosed_sInter {s : Set (Set X)} : (∀ t ∈ s, IsClosed t) → IsClosed (⋂₀ s) := by
   simpa only [← isOpen_compl_iff, compl_sInter, sUnion_image] using isOpen_biUnion
 #align is_closed_sInter isClosed_sInter
 
-theorem isClosed_iInter {f : ι → Set α} (h : ∀ i, IsClosed (f i)) : IsClosed (⋂ i, f i) :=
+theorem isClosed_iInter {f : ι → Set X} (h : ∀ i, IsClosed (f i)) : IsClosed (⋂ i, f i) :=
   isClosed_sInter <| forall_range_iff.2 h
 #align is_closed_Inter isClosed_iInter
 
-theorem isClosed_biInter {s : Set β} {f : β → Set α} (h : ∀ i ∈ s, IsClosed (f i)) :
+theorem isClosed_biInter {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i)) :
     IsClosed (⋂ i ∈ s, f i) :=
   isClosed_iInter fun i => isClosed_iInter <| h i
 #align is_closed_bInter isClosed_biInter
 
 @[simp]
-theorem isClosed_compl_iff {s : Set α} : IsClosed sᶜ ↔ IsOpen s := by
+theorem isClosed_compl_iff {s : Set X} : IsClosed sᶜ ↔ IsOpen s := by
   rw [← isOpen_compl_iff, compl_compl]
 #align is_closed_compl_iff isClosed_compl_iff
 
 alias ⟨_, IsOpen.isClosed_compl⟩ := isClosed_compl_iff
 #align is_open.is_closed_compl IsOpen.isClosed_compl
 
-theorem IsOpen.sdiff {s t : Set α} (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) :=
+theorem IsOpen.sdiff {s t : Set X} (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) :=
   IsOpen.inter h₁ h₂.isOpen_compl
 #align is_open.sdiff IsOpen.sdiff
 
@@ -253,28 +254,28 @@ theorem IsClosed.inter (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) : IsClosed
   exact IsOpen.union h₁ h₂
 #align is_closed.inter IsClosed.inter
 
-theorem IsClosed.sdiff {s t : Set α} (h₁ : IsClosed s) (h₂ : IsOpen t) : IsClosed (s \ t) :=
+theorem IsClosed.sdiff {s t : Set X} (h₁ : IsClosed s) (h₂ : IsOpen t) : IsClosed (s \ t) :=
   IsClosed.inter h₁ (isClosed_compl_iff.mpr h₂)
 #align is_closed.sdiff IsClosed.sdiff
 
-theorem Set.Finite.isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite)
+theorem Set.Finite.isClosed_biUnion {s : Set α} {f : α → Set X} (hs : s.Finite)
     (h : ∀ i ∈ s, IsClosed (f i)) :
     IsClosed (⋃ i ∈ s, f i) := by
   simp only [← isOpen_compl_iff, compl_iUnion] at *
   exact hs.isOpen_biInter h
 #align is_closed_bUnion Set.Finite.isClosed_biUnion
 
-lemma isClosed_biUnion_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsClosed (f i)) :
+lemma isClosed_biUnion_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i)) :
     IsClosed (⋃ i ∈ s, f i) :=
   s.finite_toSet.isClosed_biUnion h
 
-theorem isClosed_iUnion_of_finite [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
+theorem isClosed_iUnion_of_finite [Finite ι] {s : ι → Set X} (h : ∀ i, IsClosed (s i)) :
     IsClosed (⋃ i, s i) := by
   simp only [← isOpen_compl_iff, compl_iUnion] at *
   exact isOpen_iInter_of_finite h
 #align is_closed_Union isClosed_iUnion_of_finite
 
-theorem isClosed_imp {p q : α → Prop} (hp : IsOpen { x | p x }) (hq : IsClosed { x | q x }) :
+theorem isClosed_imp {p q : X → Prop} (hp : IsOpen { x | p x }) (hq : IsClosed { x | q x }) :
     IsClosed { x | p x → q x } := by
   simpa only [imp_iff_not_or] using hp.isClosed_compl.union hq
 #align is_closed_imp isClosed_imp
@@ -288,45 +289,45 @@ theorem IsClosed.not : IsClosed { a | p a } → IsOpen { a | ¬p a } :=
 -/
 
 /-- The interior of a set `s` is the largest open subset of `s`. -/
-def interior (s : Set α) : Set α :=
+def interior (s : Set X) : Set X :=
   ⋃₀ { t | IsOpen t ∧ t ⊆ s }
 #align interior interior
 
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
-theorem mem_interior {s : Set α} {x : α} : x ∈ interior s ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := by
+theorem mem_interior {s : Set X} {x : X} : x ∈ interior s ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := by
   simp only [interior, mem_sUnion, mem_setOf_eq, and_assoc, and_left_comm]
 #align mem_interior mem_interiorₓ
 
 @[simp]
-theorem isOpen_interior {s : Set α} : IsOpen (interior s) :=
+theorem isOpen_interior {s : Set X} : IsOpen (interior s) :=
   isOpen_sUnion fun _ => And.left
 #align is_open_interior isOpen_interior
 
-theorem interior_subset {s : Set α} : interior s ⊆ s :=
+theorem interior_subset {s : Set X} : interior s ⊆ s :=
   sUnion_subset fun _ => And.right
 #align interior_subset interior_subset
 
-theorem interior_maximal {s t : Set α} (h₁ : t ⊆ s) (h₂ : IsOpen t) : t ⊆ interior s :=
+theorem interior_maximal {s t : Set X} (h₁ : t ⊆ s) (h₂ : IsOpen t) : t ⊆ interior s :=
   subset_sUnion_of_mem ⟨h₂, h₁⟩
 #align interior_maximal interior_maximal
 
-theorem IsOpen.interior_eq {s : Set α} (h : IsOpen s) : interior s = s :=
+theorem IsOpen.interior_eq {s : Set X} (h : IsOpen s) : interior s = s :=
   interior_subset.antisymm (interior_maximal (Subset.refl s) h)
 #align is_open.interior_eq IsOpen.interior_eq
 
-theorem interior_eq_iff_isOpen {s : Set α} : interior s = s ↔ IsOpen s :=
+theorem interior_eq_iff_isOpen {s : Set X} : interior s = s ↔ IsOpen s :=
   ⟨fun h => h ▸ isOpen_interior, IsOpen.interior_eq⟩
 #align interior_eq_iff_is_open interior_eq_iff_isOpen
 
-theorem subset_interior_iff_isOpen {s : Set α} : s ⊆ interior s ↔ IsOpen s := by
+theorem subset_interior_iff_isOpen {s : Set X} : s ⊆ interior s ↔ IsOpen s := by
   simp only [interior_eq_iff_isOpen.symm, Subset.antisymm_iff, interior_subset, true_and]
 #align subset_interior_iff_is_open subset_interior_iff_isOpen
 
-theorem IsOpen.subset_interior_iff {s t : Set α} (h₁ : IsOpen s) : s ⊆ interior t ↔ s ⊆ t :=
+theorem IsOpen.subset_interior_iff {s t : Set X} (h₁ : IsOpen s) : s ⊆ interior t ↔ s ⊆ t :=
   ⟨fun h => Subset.trans h interior_subset, fun h₂ => interior_maximal h₂ h₁⟩
 #align is_open.subset_interior_iff IsOpen.subset_interior_iff
 
-theorem subset_interior_iff {s t : Set α} : t ⊆ interior s ↔ ∃ U, IsOpen U ∧ t ⊆ U ∧ U ⊆ s :=
+theorem subset_interior_iff {s t : Set X} : t ⊆ interior s ↔ ∃ U, IsOpen U ∧ t ⊆ U ∧ U ⊆ s :=
   ⟨fun h => ⟨interior s, isOpen_interior, h, interior_subset⟩, fun ⟨_U, hU, htU, hUs⟩ =>
     htU.trans (interior_maximal hUs hU)⟩
 #align subset_interior_iff subset_interior_iff
@@ -335,58 +336,58 @@ lemma interior_subset_iff : interior s ⊆ t ↔ ∀ U, IsOpen U → U ⊆ s →
   simp [interior]
 
 @[mono]
-theorem interior_mono {s t : Set α} (h : s ⊆ t) : interior s ⊆ interior t :=
+theorem interior_mono {s t : Set X} (h : s ⊆ t) : interior s ⊆ interior t :=
   interior_maximal (Subset.trans interior_subset h) isOpen_interior
 #align interior_mono interior_mono
 
 @[simp]
-theorem interior_empty : interior (∅ : Set α) = ∅ :=
+theorem interior_empty : interior (∅ : Set X) = ∅ :=
   isOpen_empty.interior_eq
 #align interior_empty interior_empty
 
 @[simp]
-theorem interior_univ : interior (univ : Set α) = univ :=
+theorem interior_univ : interior (univ : Set X) = univ :=
   isOpen_univ.interior_eq
 #align interior_univ interior_univ
 
 @[simp]
-theorem interior_eq_univ {s : Set α} : interior s = univ ↔ s = univ :=
+theorem interior_eq_univ {s : Set X} : interior s = univ ↔ s = univ :=
   ⟨fun h => univ_subset_iff.mp <| h.symm.trans_le interior_subset, fun h => h.symm ▸ interior_univ⟩
 #align interior_eq_univ interior_eq_univ
 
 @[simp]
-theorem interior_interior {s : Set α} : interior (interior s) = interior s :=
+theorem interior_interior {s : Set X} : interior (interior s) = interior s :=
   isOpen_interior.interior_eq
 #align interior_interior interior_interior
 
 @[simp]
-theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ interior t :=
+theorem interior_inter {s t : Set X} : interior (s ∩ t) = interior s ∩ interior t :=
   (Monotone.map_inf_le (fun _ _ ↦ interior_mono) s t).antisymm <|
     interior_maximal (inter_subset_inter interior_subset interior_subset) <|
       isOpen_interior.inter isOpen_interior
 #align interior_inter interior_inter
 
-theorem Set.Finite.interior_biInter {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set α) :
+theorem Set.Finite.interior_biInter {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set X) :
     interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) :=
   hs.induction_on (by simp) <| by intros; simp [*]
 
-theorem Set.Finite.interior_sInter {S : Set (Set α)} (hS : S.Finite) :
+theorem Set.Finite.interior_sInter {S : Set (Set X)} (hS : S.Finite) :
     interior (⋂₀ S) = ⋂ s ∈ S, interior s := by
   rw [sInter_eq_biInter, hS.interior_biInter]
 
 @[simp]
-theorem Finset.interior_iInter {ι : Type*} (s : Finset ι) (f : ι → Set α) :
+theorem Finset.interior_iInter {ι : Type*} (s : Finset ι) (f : ι → Set X) :
     interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) :=
   s.finite_toSet.interior_biInter f
 #align finset.interior_Inter Finset.interior_iInter
 
 @[simp]
-theorem interior_iInter_of_finite [Finite ι] (f : ι → Set α) :
+theorem interior_iInter_of_finite [Finite ι] (f : ι → Set X) :
     interior (⋂ i, f i) = ⋂ i, interior (f i) := by
   rw [← sInter_range, (finite_range f).interior_sInter, biInter_range]
 #align interior_Inter interior_iInter_of_finite
 
-theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClosed s)
+theorem interior_union_isClosed_of_interior_empty {s t : Set X} (h₁ : IsClosed s)
     (h₂ : interior t = ∅) : interior (s ∪ t) = interior s :=
   have : interior (s ∪ t) ⊆ s := fun x ⟨u, ⟨(hu₁ : IsOpen u), (hu₂ : u ⊆ s ∪ t)⟩, (hx₁ : x ∈ u)⟩ =>
     by_contradiction fun hx₂ : x ∉ s =>
@@ -402,16 +403,16 @@ theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ t, t ⊆ s 
   simp only [subset_def, mem_interior]
 #align is_open_iff_forall_mem_open isOpen_iff_forall_mem_open
 
-theorem interior_iInter_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ ⋂ i, interior (s i) :=
+theorem interior_iInter_subset (s : ι → Set X) : interior (⋂ i, s i) ⊆ ⋂ i, interior (s i) :=
   subset_iInter fun _ => interior_mono <| iInter_subset _ _
 #align interior_Inter_subset interior_iInter_subset
 
-theorem interior_iInter₂_subset (p : ι → Sort*) (s : ∀ i, p i → Set α) :
+theorem interior_iInter₂_subset (p : ι → Sort*) (s : ∀ i, p i → Set X) :
     interior (⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), interior (s i j) :=
   (interior_iInter_subset _).trans <| iInter_mono fun _ => interior_iInter_subset _
 #align interior_Inter₂_subset interior_iInter₂_subset
 
-theorem interior_sInter_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s :=
+theorem interior_sInter_subset (S : Set (Set X)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s :=
   calc
     interior (⋂₀ S) = interior (⋂ s ∈ S, s) := by rw [sInter_eq_biInter]
     _ ⊆ ⋂ s ∈ S, interior s := interior_iInter₂_subset _ _
@@ -423,7 +424,7 @@ theorem interior_sInter_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂
 
 
 /-- The closure of `s` is the smallest closed set containing `s`. -/
-def closure (s : Set α) : Set α :=
+def closure (s : Set X) : Set X :=
   ⋂₀ { t | IsClosed t ∧ s ⊆ t }
 #align closure closure
 
@@ -432,91 +433,91 @@ set_option quotPrecheck false in
 scoped[Topology] notation (name := closure_of) "closure[" t "]" => @closure _ t
 
 @[simp]
-theorem isClosed_closure {s : Set α} : IsClosed (closure s) :=
+theorem isClosed_closure {s : Set X} : IsClosed (closure s) :=
   isClosed_sInter fun _ => And.left
 #align is_closed_closure isClosed_closure
 
-theorem subset_closure {s : Set α} : s ⊆ closure s :=
+theorem subset_closure {s : Set X} : s ⊆ closure s :=
   subset_sInter fun _ => And.right
 #align subset_closure subset_closure
 
-theorem not_mem_of_not_mem_closure {s : Set α} {P : α} (hP : P ∉ closure s) : P ∉ s := fun h =>
+theorem not_mem_of_not_mem_closure {s : Set X} {P : X} (hP : P ∉ closure s) : P ∉ s := fun h =>
   hP (subset_closure h)
 #align not_mem_of_not_mem_closure not_mem_of_not_mem_closure
 
-theorem closure_minimal {s t : Set α} (h₁ : s ⊆ t) (h₂ : IsClosed t) : closure s ⊆ t :=
+theorem closure_minimal {s t : Set X} (h₁ : s ⊆ t) (h₂ : IsClosed t) : closure s ⊆ t :=
   sInter_subset_of_mem ⟨h₂, h₁⟩
 #align closure_minimal closure_minimal
 
-theorem Disjoint.closure_left {s t : Set α} (hd : Disjoint s t) (ht : IsOpen t) :
+theorem Disjoint.closure_left {s t : Set X} (hd : Disjoint s t) (ht : IsOpen t) :
     Disjoint (closure s) t :=
   disjoint_compl_left.mono_left <| closure_minimal hd.subset_compl_right ht.isClosed_compl
 #align disjoint.closure_left Disjoint.closure_left
 
-theorem Disjoint.closure_right {s t : Set α} (hd : Disjoint s t) (hs : IsOpen s) :
+theorem Disjoint.closure_right {s t : Set X} (hd : Disjoint s t) (hs : IsOpen s) :
     Disjoint s (closure t) :=
   (hd.symm.closure_left hs).symm
 #align disjoint.closure_right Disjoint.closure_right
 
-theorem IsClosed.closure_eq {s : Set α} (h : IsClosed s) : closure s = s :=
+theorem IsClosed.closure_eq {s : Set X} (h : IsClosed s) : closure s = s :=
   Subset.antisymm (closure_minimal (Subset.refl s) h) subset_closure
 #align is_closed.closure_eq IsClosed.closure_eq
 
-theorem IsClosed.closure_subset {s : Set α} (hs : IsClosed s) : closure s ⊆ s :=
+theorem IsClosed.closure_subset {s : Set X} (hs : IsClosed s) : closure s ⊆ s :=
   closure_minimal (Subset.refl _) hs
 #align is_closed.closure_subset IsClosed.closure_subset
 
-theorem IsClosed.closure_subset_iff {s t : Set α} (h₁ : IsClosed t) : closure s ⊆ t ↔ s ⊆ t :=
+theorem IsClosed.closure_subset_iff {s t : Set X} (h₁ : IsClosed t) : closure s ⊆ t ↔ s ⊆ t :=
   ⟨Subset.trans subset_closure, fun h => closure_minimal h h₁⟩
 #align is_closed.closure_subset_iff IsClosed.closure_subset_iff
 
-theorem IsClosed.mem_iff_closure_subset {s : Set α} (hs : IsClosed s) {x : α} :
-    x ∈ s ↔ closure ({x} : Set α) ⊆ s :=
+theorem IsClosed.mem_iff_closure_subset {s : Set X} (hs : IsClosed s) {x : X} :
+    x ∈ s ↔ closure ({x} : Set X) ⊆ s :=
   (hs.closure_subset_iff.trans Set.singleton_subset_iff).symm
 #align is_closed.mem_iff_closure_subset IsClosed.mem_iff_closure_subset
 
 @[mono]
-theorem closure_mono {s t : Set α} (h : s ⊆ t) : closure s ⊆ closure t :=
+theorem closure_mono {s t : Set X} (h : s ⊆ t) : closure s ⊆ closure t :=
   closure_minimal (Subset.trans h subset_closure) isClosed_closure
 #align closure_mono closure_mono
 
-theorem monotone_closure (α : Type*) [TopologicalSpace α] : Monotone (@closure α _) := fun _ _ =>
+theorem monotone_closure (X : Type*) [TopologicalSpace X] : Monotone (@closure X _) := fun _ _ =>
   closure_mono
 #align monotone_closure monotone_closure
 
-theorem diff_subset_closure_iff {s t : Set α} : s \ t ⊆ closure t ↔ s ⊆ closure t := by
+theorem diff_subset_closure_iff {s t : Set X} : s \ t ⊆ closure t ↔ s ⊆ closure t := by
   rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]
 #align diff_subset_closure_iff diff_subset_closure_iff
 
-theorem closure_inter_subset_inter_closure (s t : Set α) :
+theorem closure_inter_subset_inter_closure (s t : Set X) :
     closure (s ∩ t) ⊆ closure s ∩ closure t :=
-  (monotone_closure α).map_inf_le s t
+  (monotone_closure X).map_inf_le s t
 #align closure_inter_subset_inter_closure closure_inter_subset_inter_closure
 
-theorem isClosed_of_closure_subset {s : Set α} (h : closure s ⊆ s) : IsClosed s := by
+theorem isClosed_of_closure_subset {s : Set X} (h : closure s ⊆ s) : IsClosed s := by
   rw [subset_closure.antisymm h]; exact isClosed_closure
 #align is_closed_of_closure_subset isClosed_of_closure_subset
 
-theorem closure_eq_iff_isClosed {s : Set α} : closure s = s ↔ IsClosed s :=
+theorem closure_eq_iff_isClosed {s : Set X} : closure s = s ↔ IsClosed s :=
   ⟨fun h => h ▸ isClosed_closure, IsClosed.closure_eq⟩
 #align closure_eq_iff_is_closed closure_eq_iff_isClosed
 
-theorem closure_subset_iff_isClosed {s : Set α} : closure s ⊆ s ↔ IsClosed s :=
+theorem closure_subset_iff_isClosed {s : Set X} : closure s ⊆ s ↔ IsClosed s :=
   ⟨isClosed_of_closure_subset, IsClosed.closure_subset⟩
 #align closure_subset_iff_is_closed closure_subset_iff_isClosed
 
 @[simp]
-theorem closure_empty : closure (∅ : Set α) = ∅ :=
+theorem closure_empty : closure (∅ : Set X) = ∅ :=
   isClosed_empty.closure_eq
 #align closure_empty closure_empty
 
 @[simp]
-theorem closure_empty_iff (s : Set α) : closure s = ∅ ↔ s = ∅ :=
+theorem closure_empty_iff (s : Set X) : closure s = ∅ ↔ s = ∅ :=
   ⟨subset_eq_empty subset_closure, fun h => h.symm ▸ closure_empty⟩
 #align closure_empty_iff closure_empty_iff
 
 @[simp]
-theorem closure_nonempty_iff {s : Set α} : (closure s).Nonempty ↔ s.Nonempty := by
+theorem closure_nonempty_iff {s : Set X} : (closure s).Nonempty ↔ s.Nonempty := by
   simp only [nonempty_iff_ne_empty, Ne.def, closure_empty_iff]
 #align closure_nonempty_iff closure_nonempty_iff
 
@@ -525,60 +526,60 @@ alias ⟨Set.Nonempty.of_closure, Set.Nonempty.closure⟩ := closure_nonempty_if
 #align set.nonempty.closure Set.Nonempty.closure
 
 @[simp]
-theorem closure_univ : closure (univ : Set α) = univ :=
+theorem closure_univ : closure (univ : Set X) = univ :=
   isClosed_univ.closure_eq
 #align closure_univ closure_univ
 
 @[simp]
-theorem closure_closure {s : Set α} : closure (closure s) = closure s :=
+theorem closure_closure {s : Set X} : closure (closure s) = closure s :=
   isClosed_closure.closure_eq
 #align closure_closure closure_closure
 
-theorem closure_eq_compl_interior_compl {s : Set α} : closure s = (interior sᶜ)ᶜ := by
+theorem closure_eq_compl_interior_compl {s : Set X} : closure s = (interior sᶜ)ᶜ := by
   rw [interior, closure, compl_sUnion, compl_image_set_of]
   simp only [compl_subset_compl, isOpen_compl_iff]
 #align closure_eq_compl_interior_compl closure_eq_compl_interior_compl
 
 @[simp]
-theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure t := by
+theorem closure_union {s t : Set X} : closure (s ∪ t) = closure s ∪ closure t := by
   simp [closure_eq_compl_interior_compl, compl_inter]
 #align closure_union closure_union
 
-theorem Set.Finite.closure_biUnion {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set α) :
+theorem Set.Finite.closure_biUnion {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set X) :
     closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
   simp [closure_eq_compl_interior_compl, hs.interior_biInter]
 
-theorem Set.Finite.closure_sUnion {S : Set (Set α)} (hS : S.Finite) :
+theorem Set.Finite.closure_sUnion {S : Set (Set X)} (hS : S.Finite) :
     closure (⋃₀ S) = ⋃ s ∈ S, closure s := by
   rw [sUnion_eq_biUnion, hS.closure_biUnion]
 
 @[simp]
-theorem Finset.closure_biUnion {ι : Type*} (s : Finset ι) (f : ι → Set α) :
+theorem Finset.closure_biUnion {ι : Type*} (s : Finset ι) (f : ι → Set X) :
     closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) :=
   s.finite_toSet.closure_biUnion f
 #align finset.closure_bUnion Finset.closure_biUnion
 
 @[simp]
-theorem closure_iUnion_of_finite [Finite ι] (f : ι → Set α) :
+theorem closure_iUnion_of_finite [Finite ι] (f : ι → Set X) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by
   rw [← sUnion_range, (finite_range _).closure_sUnion, biUnion_range]
 #align closure_Union closure_iUnion_of_finite
 
-theorem interior_subset_closure {s : Set α} : interior s ⊆ closure s :=
+theorem interior_subset_closure {s : Set X} : interior s ⊆ closure s :=
   Subset.trans interior_subset subset_closure
 #align interior_subset_closure interior_subset_closure
 
 @[simp]
-theorem interior_compl {s : Set α} : interior sᶜ = (closure s)ᶜ := by
+theorem interior_compl {s : Set X} : interior sᶜ = (closure s)ᶜ := by
   simp [closure_eq_compl_interior_compl]
 #align interior_compl interior_compl
 
 @[simp]
-theorem closure_compl {s : Set α} : closure sᶜ = (interior s)ᶜ := by
+theorem closure_compl {s : Set X} : closure sᶜ = (interior s)ᶜ := by
   simp [closure_eq_compl_interior_compl]
 #align closure_compl closure_compl
 
-theorem mem_closure_iff {s : Set α} {a : α} :
+theorem mem_closure_iff {s : Set X} {a : X} :
     a ∈ closure s ↔ ∀ o, IsOpen o → a ∈ o → (o ∩ s).Nonempty :=
   ⟨fun h o oo ao =>
     by_contradiction fun os =>
@@ -590,56 +591,56 @@ theorem mem_closure_iff {s : Set α} {a : α} :
       hc (h₂ hs)⟩
 #align mem_closure_iff mem_closure_iff
 
-theorem closure_inter_open_nonempty_iff {s t : Set α} (h : IsOpen t) :
+theorem closure_inter_open_nonempty_iff {s t : Set X} (h : IsOpen t) :
     (closure s ∩ t).Nonempty ↔ (s ∩ t).Nonempty :=
   ⟨fun ⟨_x, hxcs, hxt⟩ => inter_comm t s ▸ mem_closure_iff.1 hxcs t h hxt, fun h =>
     h.mono <| inf_le_inf_right t subset_closure⟩
 #align closure_inter_open_nonempty_iff closure_inter_open_nonempty_iff
 
-theorem Filter.le_lift'_closure (l : Filter α) : l ≤ l.lift' closure :=
+theorem Filter.le_lift'_closure (l : Filter X) : l ≤ l.lift' closure :=
   le_lift'.2 fun _ h => mem_of_superset h subset_closure
 #align filter.le_lift'_closure Filter.le_lift'_closure
 
-theorem Filter.HasBasis.lift'_closure {l : Filter α} {p : ι → Prop} {s : ι → Set α}
+theorem Filter.HasBasis.lift'_closure {l : Filter X} {p : ι → Prop} {s : ι → Set X}
     (h : l.HasBasis p s) : (l.lift' closure).HasBasis p fun i => closure (s i) :=
-  h.lift' (monotone_closure α)
+  h.lift' (monotone_closure X)
 #align filter.has_basis.lift'_closure Filter.HasBasis.lift'_closure
 
-theorem Filter.HasBasis.lift'_closure_eq_self {l : Filter α} {p : ι → Prop} {s : ι → Set α}
+theorem Filter.HasBasis.lift'_closure_eq_self {l : Filter X} {p : ι → Prop} {s : ι → Set X}
     (h : l.HasBasis p s) (hc : ∀ i, p i → IsClosed (s i)) : l.lift' closure = l :=
   le_antisymm (h.ge_iff.2 fun i hi => (hc i hi).closure_eq ▸ mem_lift' (h.mem_of_mem hi))
     l.le_lift'_closure
 #align filter.has_basis.lift'_closure_eq_self Filter.HasBasis.lift'_closure_eq_self
 
 @[simp]
-theorem Filter.lift'_closure_eq_bot {l : Filter α} : l.lift' closure = ⊥ ↔ l = ⊥ :=
+theorem Filter.lift'_closure_eq_bot {l : Filter X} : l.lift' closure = ⊥ ↔ l = ⊥ :=
   ⟨fun h => bot_unique <| h ▸ l.le_lift'_closure, fun h =>
     h.symm ▸ by rw [lift'_bot (monotone_closure _), closure_empty, principal_empty]⟩
 #align filter.lift'_closure_eq_bot Filter.lift'_closure_eq_bot
 
 /-- A set is dense in a topological space if every point belongs to its closure. -/
-def Dense (s : Set α) : Prop :=
+def Dense (s : Set X) : Prop :=
   ∀ x, x ∈ closure s
 #align dense Dense
 
-theorem dense_iff_closure_eq {s : Set α} : Dense s ↔ closure s = univ :=
+theorem dense_iff_closure_eq {s : Set X} : Dense s ↔ closure s = univ :=
   eq_univ_iff_forall.symm
 #align dense_iff_closure_eq dense_iff_closure_eq
 
 alias ⟨Dense.closure_eq, _⟩ := dense_iff_closure_eq
 #align dense.closure_eq Dense.closure_eq
 
-theorem interior_eq_empty_iff_dense_compl {s : Set α} : interior s = ∅ ↔ Dense sᶜ := by
+theorem interior_eq_empty_iff_dense_compl {s : Set X} : interior s = ∅ ↔ Dense sᶜ := by
   rw [dense_iff_closure_eq, closure_compl, compl_univ_iff]
 #align interior_eq_empty_iff_dense_compl interior_eq_empty_iff_dense_compl
 
-theorem Dense.interior_compl {s : Set α} (h : Dense s) : interior sᶜ = ∅ :=
+theorem Dense.interior_compl {s : Set X} (h : Dense s) : interior sᶜ = ∅ :=
   interior_eq_empty_iff_dense_compl.2 <| by rwa [compl_compl]
 #align dense.interior_compl Dense.interior_compl
 
 /-- The closure of a set `s` is dense if and only if `s` is dense. -/
 @[simp]
-theorem dense_closure {s : Set α} : Dense (closure s) ↔ Dense s := by
+theorem dense_closure {s : Set X} : Dense (closure s) ↔ Dense s := by
   rw [Dense, Dense, closure_closure]
 #align dense_closure dense_closure
 
@@ -649,11 +650,11 @@ alias ⟨Dense.of_closure, _⟩ := dense_closure
 #align dense.closure Dense.closure
 
 @[simp]
-theorem dense_univ : Dense (univ : Set α) := fun _ => subset_closure trivial
+theorem dense_univ : Dense (univ : Set X) := fun _ => subset_closure trivial
 #align dense_univ dense_univ
 
 /-- A set is dense if and only if it has a nonempty intersection with each nonempty open set. -/
-theorem dense_iff_inter_open {s : Set α} :
+theorem dense_iff_inter_open {s : Set X} :
     Dense s ↔ ∀ U, IsOpen U → U.Nonempty → (U ∩ s).Nonempty := by
   constructor <;> intro h
   · rintro U U_op ⟨x, x_in⟩
@@ -667,30 +668,30 @@ theorem dense_iff_inter_open {s : Set α} :
 alias ⟨Dense.inter_open_nonempty, _⟩ := dense_iff_inter_open
 #align dense.inter_open_nonempty Dense.inter_open_nonempty
 
-theorem Dense.exists_mem_open {s : Set α} (hs : Dense s) {U : Set α} (ho : IsOpen U)
+theorem Dense.exists_mem_open {s : Set X} (hs : Dense s) {U : Set X} (ho : IsOpen U)
     (hne : U.Nonempty) : ∃ x ∈ s, x ∈ U :=
   let ⟨x, hx⟩ := hs.inter_open_nonempty U ho hne
   ⟨x, hx.2, hx.1⟩
 #align dense.exists_mem_open Dense.exists_mem_open
 
-theorem Dense.nonempty_iff {s : Set α} (hs : Dense s) : s.Nonempty ↔ Nonempty α :=
+theorem Dense.nonempty_iff {s : Set X} (hs : Dense s) : s.Nonempty ↔ Nonempty X :=
   ⟨fun ⟨x, _⟩ => ⟨x⟩, fun ⟨x⟩ =>
     let ⟨y, hy⟩ := hs.inter_open_nonempty _ isOpen_univ ⟨x, trivial⟩
     ⟨y, hy.2⟩⟩
 #align dense.nonempty_iff Dense.nonempty_iff
 
-theorem Dense.nonempty [h : Nonempty α] {s : Set α} (hs : Dense s) : s.Nonempty :=
+theorem Dense.nonempty [h : Nonempty X] {s : Set X} (hs : Dense s) : s.Nonempty :=
   hs.nonempty_iff.2 h
 #align dense.nonempty Dense.nonempty
 
 @[mono]
-theorem Dense.mono {s₁ s₂ : Set α} (h : s₁ ⊆ s₂) (hd : Dense s₁) : Dense s₂ := fun x =>
+theorem Dense.mono {s₁ s₂ : Set X} (h : s₁ ⊆ s₂) (hd : Dense s₁) : Dense s₂ := fun x =>
   closure_mono h (hd x)
 #align dense.mono Dense.mono
 
 /-- Complement to a singleton is dense if and only if the singleton is not an open set. -/
-theorem dense_compl_singleton_iff_not_open {x : α} :
-    Dense ({x}ᶜ : Set α) ↔ ¬IsOpen ({x} : Set α) := by
+theorem dense_compl_singleton_iff_not_open {x : X} :
+    Dense ({x}ᶜ : Set X) ↔ ¬IsOpen ({x} : Set X) := by
   constructor
   · intro hd ho
     exact (hd.inter_open_nonempty _ ho (singleton_nonempty _)).ne_empty (inter_compl_self _)
@@ -705,12 +706,12 @@ theorem dense_compl_singleton_iff_not_open {x : α} :
 -/
 
 /-- The frontier of a set is the set of points between the closure and interior. -/
-def frontier (s : Set α) : Set α :=
+def frontier (s : Set X) : Set X :=
   closure s \ interior s
 #align frontier frontier
 
 @[simp]
-theorem closure_diff_interior (s : Set α) : closure s \ interior s = frontier s :=
+theorem closure_diff_interior (s : Set X) : closure s \ interior s = frontier s :=
   rfl
 #align closure_diff_interior closure_diff_interior
 
@@ -720,21 +721,21 @@ lemma disjoint_interior_frontier : Disjoint (interior s) (frontier s) := by
     ← inter_assoc, inter_comm, ← inter_assoc, compl_inter_self, empty_inter]
 
 @[simp]
-theorem closure_diff_frontier (s : Set α) : closure s \ frontier s = interior s := by
+theorem closure_diff_frontier (s : Set X) : closure s \ frontier s = interior s := by
   rw [frontier, diff_diff_right_self, inter_eq_self_of_subset_right interior_subset_closure]
 #align closure_diff_frontier closure_diff_frontier
 
 @[simp]
-theorem self_diff_frontier (s : Set α) : s \ frontier s = interior s := by
+theorem self_diff_frontier (s : Set X) : s \ frontier s = interior s := by
   rw [frontier, diff_diff_right, diff_eq_empty.2 subset_closure,
     inter_eq_self_of_subset_right interior_subset, empty_union]
 #align self_diff_frontier self_diff_frontier
 
-theorem frontier_eq_closure_inter_closure {s : Set α} : frontier s = closure s ∩ closure sᶜ := by
+theorem frontier_eq_closure_inter_closure {s : Set X} : frontier s = closure s ∩ closure sᶜ := by
   rw [closure_compl, frontier, diff_eq]
 #align frontier_eq_closure_inter_closure frontier_eq_closure_inter_closure
 
-theorem frontier_subset_closure {s : Set α} : frontier s ⊆ closure s :=
+theorem frontier_subset_closure {s : Set X} : frontier s ⊆ closure s :=
   diff_subset _ _
 #align frontier_subset_closure frontier_subset_closure
 
@@ -742,59 +743,59 @@ theorem IsClosed.frontier_subset (hs : IsClosed s) : frontier s ⊆ s :=
   frontier_subset_closure.trans hs.closure_eq.subset
 #align is_closed.frontier_subset IsClosed.frontier_subset
 
-theorem frontier_closure_subset {s : Set α} : frontier (closure s) ⊆ frontier s :=
+theorem frontier_closure_subset {s : Set X} : frontier (closure s) ⊆ frontier s :=
   diff_subset_diff closure_closure.subset <| interior_mono subset_closure
 #align frontier_closure_subset frontier_closure_subset
 
-theorem frontier_interior_subset {s : Set α} : frontier (interior s) ⊆ frontier s :=
+theorem frontier_interior_subset {s : Set X} : frontier (interior s) ⊆ frontier s :=
   diff_subset_diff (closure_mono interior_subset) interior_interior.symm.subset
 #align frontier_interior_subset frontier_interior_subset
 
 /-- The complement of a set has the same frontier as the original set. -/
 @[simp]
-theorem frontier_compl (s : Set α) : frontier sᶜ = frontier s := by
+theorem frontier_compl (s : Set X) : frontier sᶜ = frontier s := by
   simp only [frontier_eq_closure_inter_closure, compl_compl, inter_comm]
 #align frontier_compl frontier_compl
 
 @[simp]
-theorem frontier_univ : frontier (univ : Set α) = ∅ := by simp [frontier]
+theorem frontier_univ : frontier (univ : Set X) = ∅ := by simp [frontier]
 #align frontier_univ frontier_univ
 
 @[simp]
-theorem frontier_empty : frontier (∅ : Set α) = ∅ := by simp [frontier]
+theorem frontier_empty : frontier (∅ : Set X) = ∅ := by simp [frontier]
 #align frontier_empty frontier_empty
 
-theorem frontier_inter_subset (s t : Set α) :
+theorem frontier_inter_subset (s t : Set X) :
     frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t := by
   simp only [frontier_eq_closure_inter_closure, compl_inter, closure_union]
   refine' (inter_subset_inter_left _ (closure_inter_subset_inter_closure s t)).trans_eq _
   simp only [inter_distrib_left, inter_distrib_right, inter_assoc, inter_comm (closure t)]
 #align frontier_inter_subset frontier_inter_subset
 
-theorem frontier_union_subset (s t : Set α) :
+theorem frontier_union_subset (s t : Set X) :
     frontier (s ∪ t) ⊆ frontier s ∩ closure tᶜ ∪ closure sᶜ ∩ frontier t := by
   simpa only [frontier_compl, ← compl_union] using frontier_inter_subset sᶜ tᶜ
 #align frontier_union_subset frontier_union_subset
 
-theorem IsClosed.frontier_eq {s : Set α} (hs : IsClosed s) : frontier s = s \ interior s := by
+theorem IsClosed.frontier_eq {s : Set X} (hs : IsClosed s) : frontier s = s \ interior s := by
   rw [frontier, hs.closure_eq]
 #align is_closed.frontier_eq IsClosed.frontier_eq
 
-theorem IsOpen.frontier_eq {s : Set α} (hs : IsOpen s) : frontier s = closure s \ s := by
+theorem IsOpen.frontier_eq {s : Set X} (hs : IsOpen s) : frontier s = closure s \ s := by
   rw [frontier, hs.interior_eq]
 #align is_open.frontier_eq IsOpen.frontier_eq
 
-theorem IsOpen.inter_frontier_eq {s : Set α} (hs : IsOpen s) : s ∩ frontier s = ∅ := by
+theorem IsOpen.inter_frontier_eq {s : Set X} (hs : IsOpen s) : s ∩ frontier s = ∅ := by
   rw [hs.frontier_eq, inter_diff_self]
 #align is_open.inter_frontier_eq IsOpen.inter_frontier_eq
 
 /-- The frontier of a set is closed. -/
-theorem isClosed_frontier {s : Set α} : IsClosed (frontier s) := by
+theorem isClosed_frontier {s : Set X} : IsClosed (frontier s) := by
   rw [frontier_eq_closure_inter_closure]; exact IsClosed.inter isClosed_closure isClosed_closure
 #align is_closed_frontier isClosed_frontier
 
 /-- The frontier of a closed set has no interior point. -/
-theorem interior_frontier {s : Set α} (h : IsClosed s) : interior (frontier s) = ∅ := by
+theorem interior_frontier {s : Set X} (h : IsClosed s) : interior (frontier s) = ∅ := by
   have A : frontier s = s \ interior s := h.frontier_eq
   have B : interior (frontier s) ⊆ interior s := by rw [A]; exact interior_mono (diff_subset _ _)
   have C : interior (frontier s) ⊆ frontier s := interior_subset
@@ -803,11 +804,11 @@ theorem interior_frontier {s : Set α} (h : IsClosed s) : interior (frontier s)
   rwa [inter_diff_self, subset_empty_iff] at this
 #align interior_frontier interior_frontier
 
-theorem closure_eq_interior_union_frontier (s : Set α) : closure s = interior s ∪ frontier s :=
+theorem closure_eq_interior_union_frontier (s : Set X) : closure s = interior s ∪ frontier s :=
   (union_diff_cancel interior_subset_closure).symm
 #align closure_eq_interior_union_frontier closure_eq_interior_union_frontier
 
-theorem closure_eq_self_union_frontier (s : Set α) : closure s = s ∪ frontier s :=
+theorem closure_eq_self_union_frontier (s : Set X) : closure s = s ∪ frontier s :=
   (union_diff_cancel' interior_subset subset_closure).symm
 #align closure_eq_self_union_frontier closure_eq_self_union_frontier
 
@@ -820,12 +821,12 @@ theorem Disjoint.frontier_right (hs : IsOpen s) (hd : Disjoint s t) : Disjoint s
   (hd.symm.frontier_left hs).symm
 #align disjoint.frontier_right Disjoint.frontier_right
 
-theorem frontier_eq_inter_compl_interior {s : Set α} :
+theorem frontier_eq_inter_compl_interior {s : Set X} :
     frontier s = (interior s)ᶜ ∩ (interior sᶜ)ᶜ := by
   rw [← frontier_compl, ← closure_compl, ← diff_eq, closure_diff_interior]
 #align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interior
 
-theorem compl_frontier_eq_union_interior {s : Set α} :
+theorem compl_frontier_eq_union_interior {s : Set X} :
     (frontier s)ᶜ = interior s ∪ interior sᶜ := by
   rw [frontier_eq_inter_compl_interior]
   simp only [compl_inter, compl_compl]
@@ -838,14 +839,14 @@ theorem compl_frontier_eq_union_interior {s : Set α} :
 /-- A set is called a neighborhood of `a` if it contains an open set around `a`. The set of all
 neighborhoods of `a` forms a filter, the neighborhood filter at `a`, is here defined as the
 infimum over the principal filters of all open sets containing `a`. -/
-irreducible_def nhds (a : α) : Filter α :=
-  ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 s
+irreducible_def nhds (a : X) : Filter X :=
+  ⨅ s ∈ { s : Set X | a ∈ s ∧ IsOpen s }, 𝓟 s
 #align nhds nhds
 #align nhds_def nhds_def
 
 /-- The "neighborhood within" filter. Elements of `𝓝[s] a` are sets containing the
 intersection of `s` and a neighborhood of `a`. -/
-def nhdsWithin (a : α) (s : Set α) : Filter α :=
+def nhdsWithin (a : X) (s : Set X) : Filter X :=
   nhds a ⊓ 𝓟 s
 #align nhds_within nhdsWithin
 
@@ -874,14 +875,14 @@ scoped[Topology] notation3 "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
 
 end
 
-theorem nhds_def' (a : α) : 𝓝 a = ⨅ (s : Set α) (_ : IsOpen s) (_ : a ∈ s), 𝓟 s := by
+theorem nhds_def' (a : X) : 𝓝 a = ⨅ (s : Set X) (_ : IsOpen s) (_ : a ∈ s), 𝓟 s := by
   simp only [nhds_def, mem_setOf_eq, @and_comm (a ∈ _), iInf_and]
 #align nhds_def' nhds_def'
 
 /-- The open sets containing `a` are a basis for the neighborhood filter. See `nhds_basis_opens'`
 for a variant using open neighborhoods instead. -/
-theorem nhds_basis_opens (a : α) :
-    (𝓝 a).HasBasis (fun s : Set α => a ∈ s ∧ IsOpen s) fun s => s := by
+theorem nhds_basis_opens (a : X) :
+    (𝓝 a).HasBasis (fun s : Set X => a ∈ s ∧ IsOpen s) fun s => s := by
   rw [nhds_def]
   exact hasBasis_biInf_principal
     (fun s ⟨has, hs⟩ t ⟨hat, ht⟩ =>
@@ -889,114 +890,114 @@ theorem nhds_basis_opens (a : α) :
     ⟨univ, ⟨mem_univ a, isOpen_univ⟩⟩
 #align nhds_basis_opens nhds_basis_opens
 
-theorem nhds_basis_closeds (a : α) : (𝓝 a).HasBasis (fun s : Set α => a ∉ s ∧ IsClosed s) compl :=
+theorem nhds_basis_closeds (a : X) : (𝓝 a).HasBasis (fun s : Set X => a ∉ s ∧ IsClosed s) compl :=
   ⟨fun t => (nhds_basis_opens a).mem_iff.trans <|
     compl_surjective.exists.trans <| by simp only [isOpen_compl_iff, mem_compl_iff]⟩
 #align nhds_basis_closeds nhds_basis_closeds
 
 /-- A filter lies below the neighborhood filter at `a` iff it contains every open set around `a`. -/
-theorem le_nhds_iff {f a} : f ≤ 𝓝 a ↔ ∀ s : Set α, a ∈ s → IsOpen s → s ∈ f := by simp [nhds_def]
+theorem le_nhds_iff {f a} : f ≤ 𝓝 a ↔ ∀ s : Set X, a ∈ s → IsOpen s → s ∈ f := by simp [nhds_def]
 #align le_nhds_iff le_nhds_iff
 
 /-- To show a filter is above the neighborhood filter at `a`, it suffices to show that it is above
 the principal filter of some open set `s` containing `a`. -/
-theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 a ≤ f := by
+theorem nhds_le_of_le {f a} {s : Set X} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 a ≤ f := by
   rw [nhds_def]; exact iInf₂_le_of_le s ⟨h, o⟩ sf
 #align nhds_le_of_le nhds_le_of_le
 
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
-theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ a ∈ t :=
+theorem mem_nhds_iff {a : X} {s : Set X} : s ∈ 𝓝 a ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ a ∈ t :=
   (nhds_basis_opens a).mem_iff.trans <| exists_congr fun _ =>
     ⟨fun h => ⟨h.2, h.1.2, h.1.1⟩, fun h => ⟨⟨h.2.2, h.2.1⟩, h.1⟩⟩
 #align mem_nhds_iff mem_nhds_iffₓ
 
 /-- A predicate is true in a neighborhood of `a` iff it is true for all the points in an open set
 containing `a`. -/
-theorem eventually_nhds_iff {a : α} {p : α → Prop} :
-    (∀ᶠ x in 𝓝 a, p x) ↔ ∃ t : Set α, (∀ x ∈ t, p x) ∧ IsOpen t ∧ a ∈ t :=
+theorem eventually_nhds_iff {a : X} {p : X → Prop} :
+    (∀ᶠ x in 𝓝 a, p x) ↔ ∃ t : Set X, (∀ x ∈ t, p x) ∧ IsOpen t ∧ a ∈ t :=
   mem_nhds_iff.trans <| by simp only [subset_def, exists_prop, mem_setOf_eq]
 #align eventually_nhds_iff eventually_nhds_iff
 
-theorem mem_interior_iff_mem_nhds {s : Set α} {a : α} : a ∈ interior s ↔ s ∈ 𝓝 a :=
+theorem mem_interior_iff_mem_nhds {s : Set X} {a : X} : a ∈ interior s ↔ s ∈ 𝓝 a :=
   mem_interior.trans mem_nhds_iff.symm
 #align mem_interior_iff_mem_nhds mem_interior_iff_mem_nhds
 
-theorem map_nhds {a : α} {f : α → β} :
-    map f (𝓝 a) = ⨅ s ∈ { s : Set α | a ∈ s ∧ IsOpen s }, 𝓟 (image f s) :=
+theorem map_nhds {a : X} {f : X → α} :
+    map f (𝓝 a) = ⨅ s ∈ { s : Set X | a ∈ s ∧ IsOpen s }, 𝓟 (f '' s) :=
   ((nhds_basis_opens a).map f).eq_biInf
 #align map_nhds map_nhds
 
-theorem mem_of_mem_nhds {a : α} {s : Set α} : s ∈ 𝓝 a → a ∈ s := fun H =>
+theorem mem_of_mem_nhds {a : X} {s : Set X} : s ∈ 𝓝 a → a ∈ s := fun H =>
   let ⟨_t, ht, _, hs⟩ := mem_nhds_iff.1 H; ht hs
 #align mem_of_mem_nhds mem_of_mem_nhds
 
 /-- If a predicate is true in a neighborhood of `a`, then it is true for `a`. -/
-theorem Filter.Eventually.self_of_nhds {p : α → Prop} {a : α} (h : ∀ᶠ y in 𝓝 a, p y) : p a :=
+theorem Filter.Eventually.self_of_nhds {p : X → Prop} {a : X} (h : ∀ᶠ y in 𝓝 a, p y) : p a :=
   mem_of_mem_nhds h
 #align filter.eventually.self_of_nhds Filter.Eventually.self_of_nhds
 
-theorem IsOpen.mem_nhds {a : α} {s : Set α} (hs : IsOpen s) (ha : a ∈ s) : s ∈ 𝓝 a :=
+theorem IsOpen.mem_nhds {a : X} {s : Set X} (hs : IsOpen s) (ha : a ∈ s) : s ∈ 𝓝 a :=
   mem_nhds_iff.2 ⟨s, Subset.refl _, hs, ha⟩
 #align is_open.mem_nhds IsOpen.mem_nhds
 
-protected theorem IsOpen.mem_nhds_iff {a : α} {s : Set α} (hs : IsOpen s) : s ∈ 𝓝 a ↔ a ∈ s :=
+protected theorem IsOpen.mem_nhds_iff {a : X} {s : Set X} (hs : IsOpen s) : s ∈ 𝓝 a ↔ a ∈ s :=
   ⟨mem_of_mem_nhds, fun ha => mem_nhds_iff.2 ⟨s, Subset.rfl, hs, ha⟩⟩
 #align is_open.mem_nhds_iff IsOpen.mem_nhds_iff
 
-theorem IsClosed.compl_mem_nhds {a : α} {s : Set α} (hs : IsClosed s) (ha : a ∉ s) : sᶜ ∈ 𝓝 a :=
+theorem IsClosed.compl_mem_nhds {a : X} {s : Set X} (hs : IsClosed s) (ha : a ∉ s) : sᶜ ∈ 𝓝 a :=
   hs.isOpen_compl.mem_nhds (mem_compl ha)
 #align is_closed.compl_mem_nhds IsClosed.compl_mem_nhds
 
-theorem IsOpen.eventually_mem {a : α} {s : Set α} (hs : IsOpen s) (ha : a ∈ s) :
+theorem IsOpen.eventually_mem {a : X} {s : Set X} (hs : IsOpen s) (ha : a ∈ s) :
     ∀ᶠ x in 𝓝 a, x ∈ s :=
   IsOpen.mem_nhds hs ha
 #align is_open.eventually_mem IsOpen.eventually_mem
 
 /-- The open neighborhoods of `a` are a basis for the neighborhood filter. See `nhds_basis_opens`
 for a variant using open sets around `a` instead. -/
-theorem nhds_basis_opens' (a : α) :
-    (𝓝 a).HasBasis (fun s : Set α => s ∈ 𝓝 a ∧ IsOpen s) fun x => x := by
+theorem nhds_basis_opens' (a : X) :
+    (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsOpen s) fun x => x := by
   convert nhds_basis_opens a using 2
   exact and_congr_left_iff.2 IsOpen.mem_nhds_iff
 #align nhds_basis_opens' nhds_basis_opens'
 
 /-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of `s`:
 it contains an open set containing `s`. -/
-theorem exists_open_set_nhds {s U : Set α} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
-    ∃ V : Set α, s ⊆ V ∧ IsOpen V ∧ V ⊆ U :=
+theorem exists_open_set_nhds {s U : Set X} (h : ∀ x ∈ s, U ∈ 𝓝 x) :
+    ∃ V : Set X, s ⊆ V ∧ IsOpen V ∧ V ⊆ U :=
   ⟨interior U, fun x hx => mem_interior_iff_mem_nhds.2 <| h x hx, isOpen_interior, interior_subset⟩
 #align exists_open_set_nhds exists_open_set_nhds
 
 /-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of s:
 it contains an open set containing `s`. -/
-theorem exists_open_set_nhds' {s U : Set α} (h : U ∈ ⨆ x ∈ s, 𝓝 x) :
-    ∃ V : Set α, s ⊆ V ∧ IsOpen V ∧ V ⊆ U :=
+theorem exists_open_set_nhds' {s U : Set X} (h : U ∈ ⨆ x ∈ s, 𝓝 x) :
+    ∃ V : Set X, s ⊆ V ∧ IsOpen V ∧ V ⊆ U :=
   exists_open_set_nhds (by simpa using h)
 #align exists_open_set_nhds' exists_open_set_nhds'
 
 /-- If a predicate is true in a neighbourhood of `a`, then for `y` sufficiently close
 to `a` this predicate is true in a neighbourhood of `y`. -/
-theorem Filter.Eventually.eventually_nhds {p : α → Prop} {a : α} (h : ∀ᶠ y in 𝓝 a, p y) :
+theorem Filter.Eventually.eventually_nhds {p : X → Prop} {a : X} (h : ∀ᶠ y in 𝓝 a, p y) :
     ∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝 y, p x :=
   let ⟨t, htp, hto, ha⟩ := eventually_nhds_iff.1 h
   eventually_nhds_iff.2 ⟨t, fun _x hx => eventually_nhds_iff.2 ⟨t, htp, hto, hx⟩, hto, ha⟩
 #align filter.eventually.eventually_nhds Filter.Eventually.eventually_nhds
 
 @[simp]
-theorem eventually_eventually_nhds {p : α → Prop} {a : α} :
+theorem eventually_eventually_nhds {p : X → Prop} {a : X} :
     (∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝 y, p x) ↔ ∀ᶠ x in 𝓝 a, p x :=
   ⟨fun h => h.self_of_nhds, fun h => h.eventually_nhds⟩
 #align eventually_eventually_nhds eventually_eventually_nhds
 
 @[simp]
-theorem frequently_frequently_nhds {p : α → Prop} {a : α} :
+theorem frequently_frequently_nhds {p : X → Prop} {a : X} :
     (∃ᶠ y in 𝓝 a, ∃ᶠ x in 𝓝 y, p x) ↔ ∃ᶠ x in 𝓝 a, p x := by
   rw [← not_iff_not]
   simp only [not_frequently, eventually_eventually_nhds]
 #align frequently_frequently_nhds frequently_frequently_nhds
 
 @[simp]
-theorem eventually_mem_nhds {s : Set α} {a : α} : (∀ᶠ x in 𝓝 a, s ∈ 𝓝 x) ↔ s ∈ 𝓝 a :=
+theorem eventually_mem_nhds {s : Set X} {a : X} : (∀ᶠ x in 𝓝 a, s ∈ 𝓝 x) ↔ s ∈ 𝓝 a :=
   eventually_eventually_nhds
 #align eventually_mem_nhds eventually_mem_nhds
 
@@ -1006,93 +1007,92 @@ theorem nhds_bind_nhds : (𝓝 a).bind 𝓝 = 𝓝 a :=
 #align nhds_bind_nhds nhds_bind_nhds
 
 @[simp]
-theorem eventually_eventuallyEq_nhds {f g : α → β} {a : α} :
+theorem eventually_eventuallyEq_nhds {f g : X → α} {a : X} :
     (∀ᶠ y in 𝓝 a, f =ᶠ[𝓝 y] g) ↔ f =ᶠ[𝓝 a] g :=
   eventually_eventually_nhds
 #align eventually_eventually_eq_nhds eventually_eventuallyEq_nhds
 
-theorem Filter.EventuallyEq.eq_of_nhds {f g : α → β} {a : α} (h : f =ᶠ[𝓝 a] g) : f a = g a :=
+theorem Filter.EventuallyEq.eq_of_nhds {f g : X → α} {a : X} (h : f =ᶠ[𝓝 a] g) : f a = g a :=
   h.self_of_nhds
 #align filter.eventually_eq.eq_of_nhds Filter.EventuallyEq.eq_of_nhds
 
 @[simp]
-theorem eventually_eventuallyLE_nhds [LE β] {f g : α → β} {a : α} :
+theorem eventually_eventuallyLE_nhds [LE α] {f g : X → α} {a : X} :
     (∀ᶠ y in 𝓝 a, f ≤ᶠ[𝓝 y] g) ↔ f ≤ᶠ[𝓝 a] g :=
   eventually_eventually_nhds
 #align eventually_eventually_le_nhds eventually_eventuallyLE_nhds
 
 /-- If two functions are equal in a neighbourhood of `a`, then for `y` sufficiently close
 to `a` these functions are equal in a neighbourhood of `y`. -/
-theorem Filter.EventuallyEq.eventuallyEq_nhds {f g : α → β} {a : α} (h : f =ᶠ[𝓝 a] g) :
+theorem Filter.EventuallyEq.eventuallyEq_nhds {f g : X → α} {a : X} (h : f =ᶠ[𝓝 a] g) :
     ∀ᶠ y in 𝓝 a, f =ᶠ[𝓝 y] g :=
   h.eventually_nhds
 #align filter.eventually_eq.eventually_eq_nhds Filter.EventuallyEq.eventuallyEq_nhds
 
 /-- If `f x ≤ g x` in a neighbourhood of `a`, then for `y` sufficiently close to `a` we have
 `f x ≤ g x` in a neighbourhood of `y`. -/
-theorem Filter.EventuallyLE.eventuallyLE_nhds [LE β] {f g : α → β} {a : α} (h : f ≤ᶠ[𝓝 a] g) :
+theorem Filter.EventuallyLE.eventuallyLE_nhds [LE α] {f g : X → α} {a : X} (h : f ≤ᶠ[𝓝 a] g) :
     ∀ᶠ y in 𝓝 a, f ≤ᶠ[𝓝 y] g :=
   h.eventually_nhds
 #align filter.eventually_le.eventually_le_nhds Filter.EventuallyLE.eventuallyLE_nhds
 
-theorem all_mem_nhds (x : α) (P : Set α → Prop) (hP : ∀ s t, s ⊆ t → P s → P t) :
+theorem all_mem_nhds (x : X) (P : Set X → Prop) (hP : ∀ s t, s ⊆ t → P s → P t) :
     (∀ s ∈ 𝓝 x, P s) ↔ ∀ s, IsOpen s → x ∈ s → P s :=
   ((nhds_basis_opens x).forall_iff hP).trans <| by simp only [@and_comm (x ∈ _), and_imp]
 #align all_mem_nhds all_mem_nhds
 
-theorem all_mem_nhds_filter (x : α) (f : Set α → Set β) (hf : ∀ s t, s ⊆ t → f s ⊆ f t)
-    (l : Filter β) : (∀ s ∈ 𝓝 x, f s ∈ l) ↔ ∀ s, IsOpen s → x ∈ s → f s ∈ l :=
+theorem all_mem_nhds_filter (x : X) (f : Set X → Set α) (hf : ∀ s t, s ⊆ t → f s ⊆ f t)
+    (l : Filter α) : (∀ s ∈ 𝓝 x, f s ∈ l) ↔ ∀ s, IsOpen s → x ∈ s → f s ∈ l :=
   all_mem_nhds _ _ fun s t ssubt h => mem_of_superset h (hf s t ssubt)
 #align all_mem_nhds_filter all_mem_nhds_filter
 
-theorem tendsto_nhds {f : β → α} {l : Filter β} {a : α} :
+theorem tendsto_nhds {f : α → X} {l : Filter α} {a : X} :
     Tendsto f l (𝓝 a) ↔ ∀ s, IsOpen s → a ∈ s → f ⁻¹' s ∈ l :=
   all_mem_nhds_filter _ _ (fun _ _ h => preimage_mono h) _
 #align tendsto_nhds tendsto_nhds
 
-theorem tendsto_atTop_nhds [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} :
-    Tendsto f atTop (𝓝 a) ↔ ∀ U : Set α, a ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U :=
+theorem tendsto_atTop_nhds [Nonempty α] [SemilatticeSup α] {f : α → X} {a : X} :
+    Tendsto f atTop (𝓝 a) ↔ ∀ U : Set X, a ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U :=
   (atTop_basis.tendsto_iff (nhds_basis_opens a)).trans <| by
     simp only [and_imp, exists_prop, true_and_iff, mem_Ici, ge_iff_le]
 #align tendsto_at_top_nhds tendsto_atTop_nhds
 
-theorem tendsto_const_nhds {a : α} {f : Filter β} : Tendsto (fun _ : β => a) f (𝓝 a) :=
+theorem tendsto_const_nhds {a : X} {f : Filter α} : Tendsto (fun _ : α => a) f (𝓝 a) :=
   tendsto_nhds.mpr fun _ _ ha => univ_mem' fun _ => ha
 #align tendsto_const_nhds tendsto_const_nhds
 
-theorem tendsto_atTop_of_eventually_const {ι : Type*} [SemilatticeSup ι] [Nonempty ι] {x : α}
-    {u : ι → α} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : Tendsto u atTop (𝓝 x) :=
+theorem tendsto_atTop_of_eventually_const {ι : Type*} [SemilatticeSup ι] [Nonempty ι] {x : X}
+    {u : ι → X} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : Tendsto u atTop (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atTop.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_top_of_eventually_const tendsto_atTop_of_eventually_const
 
-theorem tendsto_atBot_of_eventually_const {ι : Type*} [SemilatticeInf ι] [Nonempty ι] {x : α}
-    {u : ι → α} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : Tendsto u atBot (𝓝 x) :=
+theorem tendsto_atBot_of_eventually_const {ι : Type*} [SemilatticeInf ι] [Nonempty ι] {x : X}
+    {u : ι → X} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : Tendsto u atBot (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atBot.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_bot_of_eventually_const tendsto_atBot_of_eventually_const
 
-theorem pure_le_nhds : pure ≤ (𝓝 : α → Filter α) := fun _ _ hs => mem_pure.2 <| mem_of_mem_nhds hs
+theorem pure_le_nhds : pure ≤ (𝓝 : X → Filter X) := fun _ _ hs => mem_pure.2 <| mem_of_mem_nhds hs
 #align pure_le_nhds pure_le_nhds
 
-theorem tendsto_pure_nhds {α : Type*} [TopologicalSpace β] (f : α → β) (a : α) :
-    Tendsto f (pure a) (𝓝 (f a)) :=
+theorem tendsto_pure_nhds (f : α → X) (a : α) : Tendsto f (pure a) (𝓝 (f a)) :=
   (tendsto_pure_pure f a).mono_right (pure_le_nhds _)
 #align tendsto_pure_nhds tendsto_pure_nhds
 
-theorem OrderTop.tendsto_atTop_nhds {α : Type*} [PartialOrder α] [OrderTop α] [TopologicalSpace β]
-    (f : α → β) : Tendsto f atTop (𝓝 (f ⊤)) :=
+theorem OrderTop.tendsto_atTop_nhds [PartialOrder α] [OrderTop α] (f : α → X) :
+    Tendsto f atTop (𝓝 (f ⊤)) :=
   (tendsto_atTop_pure f).mono_right (pure_le_nhds _)
 #align order_top.tendsto_at_top_nhds OrderTop.tendsto_atTop_nhds
 
 @[simp]
-instance nhds_neBot {a : α} : NeBot (𝓝 a) :=
+instance nhds_neBot {a : X} : NeBot (𝓝 a) :=
   neBot_of_le (pure_le_nhds a)
 #align nhds_ne_bot nhds_neBot
 
-theorem tendsto_nhds_of_eventually_eq {l : Filter β} {f : β → α} {a : α} (h : ∀ᶠ x in l, f x = a) :
+theorem tendsto_nhds_of_eventually_eq {l : Filter α} {f : α → X} {a : X} (h : ∀ᶠ x in l, f x = a) :
     Tendsto f l (𝓝 a) :=
   tendsto_const_nhds.congr' (.symm h)
 
-theorem Filter.EventuallyEq.tendsto {l : Filter β} {f : β → α} {a : α} (hf : f =ᶠ[l] fun _ ↦ a) :
+theorem Filter.EventuallyEq.tendsto {l : Filter α} {f : α → X} {a : X} (hf : f =ᶠ[l] fun _ ↦ a) :
     Tendsto f l (𝓝 a) :=
   tendsto_nhds_of_eventually_eq hf
 
@@ -1107,85 +1107,85 @@ In this section we define [cluster points](https://en.wikipedia.org/wiki/Limit_p
 /-- A point `x` is a cluster point of a filter `F` if `𝓝 x ⊓ F ≠ ⊥`. Also known as
 an accumulation point or a limit point, but beware that terminology varies. This
 is *not* the same as asking `𝓝[≠] x ⊓ F ≠ ⊥`. See `mem_closure_iff_clusterPt` in particular. -/
-def ClusterPt (x : α) (F : Filter α) : Prop :=
+def ClusterPt (x : X) (F : Filter X) : Prop :=
   NeBot (𝓝 x ⊓ F)
 #align cluster_pt ClusterPt
 
-theorem ClusterPt.neBot {x : α} {F : Filter α} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) :=
+theorem ClusterPt.neBot {x : X} {F : Filter X} (h : ClusterPt x F) : NeBot (𝓝 x ⊓ F) :=
   h
 #align cluster_pt.ne_bot ClusterPt.neBot
 
-theorem Filter.HasBasis.clusterPt_iff {ιa ιF} {pa : ιa → Prop} {sa : ιa → Set α} {pF : ιF → Prop}
-    {sF : ιF → Set α} {F : Filter α} (ha : (𝓝 a).HasBasis pa sa) (hF : F.HasBasis pF sF) :
+theorem Filter.HasBasis.clusterPt_iff {ιa ιF} {pa : ιa → Prop} {sa : ιa → Set X} {pF : ιF → Prop}
+    {sF : ιF → Set X} {F : Filter X} (ha : (𝓝 a).HasBasis pa sa) (hF : F.HasBasis pF sF) :
     ClusterPt a F ↔ ∀ ⦃i⦄, pa i → ∀ ⦃j⦄, pF j → (sa i ∩ sF j).Nonempty :=
   ha.inf_basis_neBot_iff hF
 #align filter.has_basis.cluster_pt_iff Filter.HasBasis.clusterPt_iff
 
-theorem clusterPt_iff {x : α} {F : Filter α} :
-    ClusterPt x F ↔ ∀ ⦃U : Set α⦄, U ∈ 𝓝 x → ∀ ⦃V⦄, V ∈ F → (U ∩ V).Nonempty :=
+theorem clusterPt_iff {x : X} {F : Filter X} :
+    ClusterPt x F ↔ ∀ ⦃U : Set X⦄, U ∈ 𝓝 x → ∀ ⦃V⦄, V ∈ F → (U ∩ V).Nonempty :=
   inf_neBot_iff
 #align cluster_pt_iff clusterPt_iff
 
-theorem clusterPt_iff_not_disjoint {x : α} {F : Filter α} :
+theorem clusterPt_iff_not_disjoint {x : X} {F : Filter X} :
     ClusterPt x F ↔ ¬Disjoint (𝓝 x) F := by
   rw [disjoint_iff, ClusterPt, neBot_iff]
 
 /-- `x` is a cluster point of a set `s` if every neighbourhood of `x` meets `s` on a nonempty
 set. See also `mem_closure_iff_clusterPt`. -/
-theorem clusterPt_principal_iff {x : α} {s : Set α} :
+theorem clusterPt_principal_iff {x : X} {s : Set X} :
     ClusterPt x (𝓟 s) ↔ ∀ U ∈ 𝓝 x, (U ∩ s).Nonempty :=
   inf_principal_neBot_iff
 #align cluster_pt_principal_iff clusterPt_principal_iff
 
-theorem clusterPt_principal_iff_frequently {x : α} {s : Set α} :
+theorem clusterPt_principal_iff_frequently {x : X} {s : Set X} :
     ClusterPt x (𝓟 s) ↔ ∃ᶠ y in 𝓝 x, y ∈ s := by
   simp only [clusterPt_principal_iff, frequently_iff, Set.Nonempty, exists_prop, mem_inter_iff]
 #align cluster_pt_principal_iff_frequently clusterPt_principal_iff_frequently
 
-theorem ClusterPt.of_le_nhds {x : α} {f : Filter α} (H : f ≤ 𝓝 x) [NeBot f] : ClusterPt x f := by
+theorem ClusterPt.of_le_nhds {x : X} {f : Filter X} (H : f ≤ 𝓝 x) [NeBot f] : ClusterPt x f := by
   rwa [ClusterPt, inf_eq_right.mpr H]
 #align cluster_pt.of_le_nhds ClusterPt.of_le_nhds
 
-theorem ClusterPt.of_le_nhds' {x : α} {f : Filter α} (H : f ≤ 𝓝 x) (_hf : NeBot f) :
+theorem ClusterPt.of_le_nhds' {x : X} {f : Filter X} (H : f ≤ 𝓝 x) (_hf : NeBot f) :
     ClusterPt x f :=
   ClusterPt.of_le_nhds H
 #align cluster_pt.of_le_nhds' ClusterPt.of_le_nhds'
 
-theorem ClusterPt.of_nhds_le {x : α} {f : Filter α} (H : 𝓝 x ≤ f) : ClusterPt x f := by
+theorem ClusterPt.of_nhds_le {x : X} {f : Filter X} (H : 𝓝 x ≤ f) : ClusterPt x f := by
   simp only [ClusterPt, inf_eq_left.mpr H, nhds_neBot]
 #align cluster_pt.of_nhds_le ClusterPt.of_nhds_le
 
-theorem ClusterPt.mono {x : α} {f g : Filter α} (H : ClusterPt x f) (h : f ≤ g) : ClusterPt x g :=
+theorem ClusterPt.mono {x : X} {f g : Filter X} (H : ClusterPt x f) (h : f ≤ g) : ClusterPt x g :=
   NeBot.mono H <| inf_le_inf_left _ h
 #align cluster_pt.mono ClusterPt.mono
 
-theorem ClusterPt.of_inf_left {x : α} {f g : Filter α} (H : ClusterPt x <| f ⊓ g) : ClusterPt x f :=
+theorem ClusterPt.of_inf_left {x : X} {f g : Filter X} (H : ClusterPt x <| f ⊓ g) : ClusterPt x f :=
   H.mono inf_le_left
 #align cluster_pt.of_inf_left ClusterPt.of_inf_left
 
-theorem ClusterPt.of_inf_right {x : α} {f g : Filter α} (H : ClusterPt x <| f ⊓ g) :
+theorem ClusterPt.of_inf_right {x : X} {f g : Filter X} (H : ClusterPt x <| f ⊓ g) :
     ClusterPt x g :=
   H.mono inf_le_right
 #align cluster_pt.of_inf_right ClusterPt.of_inf_right
 
-theorem Ultrafilter.clusterPt_iff {x : α} {f : Ultrafilter α} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
+theorem Ultrafilter.clusterPt_iff {x : X} {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
   ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩
 #align ultrafilter.cluster_pt_iff Ultrafilter.clusterPt_iff
 
 /-- A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point
 of `map u F`. -/
-def MapClusterPt {ι : Type*} (x : α) (F : Filter ι) (u : ι → α) : Prop :=
+def MapClusterPt {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) : Prop :=
   ClusterPt x (map u F)
 #align map_cluster_pt MapClusterPt
 
-theorem mapClusterPt_iff {ι : Type*} (x : α) (F : Filter ι) (u : ι → α) :
+theorem mapClusterPt_iff {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
     MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s := by
   simp_rw [MapClusterPt, ClusterPt, inf_neBot_iff_frequently_left, frequently_map]
   rfl
 #align map_cluster_pt_iff mapClusterPt_iff
 
-theorem mapClusterPt_of_comp {ι δ : Type*} {F : Filter ι} {φ : δ → ι} {p : Filter δ} {x : α}
-    {u : ι → α} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) :
+theorem mapClusterPt_of_comp {F : Filter α} {φ : β → α} {p : Filter β} {x : X}
+    {u : α → X} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) :
     MapClusterPt x F u := by
   have :=
     calc
@@ -1196,36 +1196,36 @@ theorem mapClusterPt_of_comp {ι δ : Type*} {F : Filter ι} {φ : δ → ι} {p
 #align map_cluster_pt_of_comp mapClusterPt_of_comp
 
 /-- A point `x` is an accumulation point of a filter `F` if `𝓝[≠] x ⊓ F ≠ ⊥`.-/
-def AccPt (x : α) (F : Filter α) : Prop :=
+def AccPt (x : X) (F : Filter X) : Prop :=
   NeBot (𝓝[≠] x ⊓ F)
 #align acc_pt AccPt
 
-theorem acc_iff_cluster (x : α) (F : Filter α) : AccPt x F ↔ ClusterPt x (𝓟 {x}ᶜ ⊓ F) := by
+theorem acc_iff_cluster (x : X) (F : Filter X) : AccPt x F ↔ ClusterPt x (𝓟 {x}ᶜ ⊓ F) := by
   rw [AccPt, nhdsWithin, ClusterPt, inf_assoc]
 #align acc_iff_cluster acc_iff_cluster
 
 /-- `x` is an accumulation point of a set `C` iff it is a cluster point of `C ∖ {x}`.-/
-theorem acc_principal_iff_cluster (x : α) (C : Set α) :
+theorem acc_principal_iff_cluster (x : X) (C : Set X) :
     AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x})) := by
   rw [acc_iff_cluster, inf_principal, inter_comm, diff_eq]
 #align acc_principal_iff_cluster acc_principal_iff_cluster
 
 /-- `x` is an accumulation point of a set `C` iff every neighborhood
 of `x` contains a point of `C` other than `x`. -/
-theorem accPt_iff_nhds (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x := by
+theorem accPt_iff_nhds (x : X) (C : Set X) : AccPt x (𝓟 C) ↔ ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x := by
   simp [acc_principal_iff_cluster, clusterPt_principal_iff, Set.Nonempty, exists_prop, and_assoc,
     @and_comm (¬_ = x)]
 #align acc_pt_iff_nhds accPt_iff_nhds
 
 /-- `x` is an accumulation point of a set `C` iff
 there are points near `x` in `C` and different from `x`.-/
-theorem accPt_iff_frequently (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ∃ᶠ y in 𝓝 x, y ≠ x ∧ y ∈ C := by
+theorem accPt_iff_frequently (x : X) (C : Set X) : AccPt x (𝓟 C) ↔ ∃ᶠ y in 𝓝 x, y ≠ x ∧ y ∈ C := by
   simp [acc_principal_iff_cluster, clusterPt_principal_iff_frequently, and_comm]
 #align acc_pt_iff_frequently accPt_iff_frequently
 
 /-- If `x` is an accumulation point of `F` and `F ≤ G`, then
 `x` is an accumulation point of `D`. -/
-theorem AccPt.mono {x : α} {F G : Filter α} (h : AccPt x F) (hFG : F ≤ G) : AccPt x G :=
+theorem AccPt.mono {x : X} {F G : Filter X} (h : AccPt x F) (hFG : F ≤ G) : AccPt x G :=
   NeBot.mono h (inf_le_inf_left _ hFG)
 #align acc_pt.mono AccPt.mono
 
@@ -1233,53 +1233,53 @@ theorem AccPt.mono {x : α} {F G : Filter α} (h : AccPt x F) (hFG : F ≤ G) :
 ### Interior, closure and frontier in terms of neighborhoods
 -/
 
-theorem interior_eq_nhds' {s : Set α} : interior s = { a | s ∈ 𝓝 a } :=
+theorem interior_eq_nhds' {s : Set X} : interior s = { a | s ∈ 𝓝 a } :=
   Set.ext fun x => by simp only [mem_interior, mem_nhds_iff, mem_setOf_eq]
 #align interior_eq_nhds' interior_eq_nhds'
 
-theorem interior_eq_nhds {s : Set α} : interior s = { a | 𝓝 a ≤ 𝓟 s } :=
+theorem interior_eq_nhds {s : Set X} : interior s = { a | 𝓝 a ≤ 𝓟 s } :=
   interior_eq_nhds'.trans <| by simp only [le_principal_iff]
 #align interior_eq_nhds interior_eq_nhds
 
 @[simp]
-theorem interior_mem_nhds {s : Set α} {a : α} : interior s ∈ 𝓝 a ↔ s ∈ 𝓝 a :=
+theorem interior_mem_nhds {s : Set X} {a : X} : interior s ∈ 𝓝 a ↔ s ∈ 𝓝 a :=
   ⟨fun h => mem_of_superset h interior_subset, fun h =>
     IsOpen.mem_nhds isOpen_interior (mem_interior_iff_mem_nhds.2 h)⟩
 #align interior_mem_nhds interior_mem_nhds
 
-theorem interior_setOf_eq {p : α → Prop} : interior { x | p x } = { x | ∀ᶠ y in 𝓝 x, p y } :=
+theorem interior_setOf_eq {p : X → Prop} : interior { x | p x } = { x | ∀ᶠ y in 𝓝 x, p y } :=
   interior_eq_nhds'
 #align interior_set_of_eq interior_setOf_eq
 
-theorem isOpen_setOf_eventually_nhds {p : α → Prop} : IsOpen { x | ∀ᶠ y in 𝓝 x, p y } := by
+theorem isOpen_setOf_eventually_nhds {p : X → Prop} : IsOpen { x | ∀ᶠ y in 𝓝 x, p y } := by
   simp only [← interior_setOf_eq, isOpen_interior]
 #align is_open_set_of_eventually_nhds isOpen_setOf_eventually_nhds
 
-theorem subset_interior_iff_nhds {s V : Set α} : s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x := by
+theorem subset_interior_iff_nhds {s V : Set X} : s ⊆ interior V ↔ ∀ x ∈ s, V ∈ 𝓝 x := by
   simp_rw [subset_def, mem_interior_iff_mem_nhds]
 #align subset_interior_iff_nhds subset_interior_iff_nhds
 
-theorem isOpen_iff_nhds {s : Set α} : IsOpen s ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s :=
+theorem isOpen_iff_nhds {s : Set X} : IsOpen s ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s :=
   calc
     IsOpen s ↔ s ⊆ interior s := subset_interior_iff_isOpen.symm
     _ ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s := by simp_rw [interior_eq_nhds, subset_def, mem_setOf]
 #align is_open_iff_nhds isOpen_iff_nhds
 
-theorem isOpen_iff_mem_nhds {s : Set α} : IsOpen s ↔ ∀ a ∈ s, s ∈ 𝓝 a :=
+theorem isOpen_iff_mem_nhds {s : Set X} : IsOpen s ↔ ∀ a ∈ s, s ∈ 𝓝 a :=
   isOpen_iff_nhds.trans <| forall_congr' fun _ => imp_congr_right fun _ => le_principal_iff
 #align is_open_iff_mem_nhds isOpen_iff_mem_nhds
 
 /-- A set `s` is open iff for every point `x` in `s` and every `y` close to `x`, `y` is in `s`. -/
-theorem isOpen_iff_eventually {s : Set α} : IsOpen s ↔ ∀ x, x ∈ s → ∀ᶠ y in 𝓝 x, y ∈ s :=
+theorem isOpen_iff_eventually {s : Set X} : IsOpen s ↔ ∀ x, x ∈ s → ∀ᶠ y in 𝓝 x, y ∈ s :=
   isOpen_iff_mem_nhds
 #align is_open_iff_eventually isOpen_iff_eventually
 
-theorem isOpen_iff_ultrafilter {s : Set α} :
-    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter α), ↑l ≤ 𝓝 x → s ∈ l := by
+theorem isOpen_iff_ultrafilter {s : Set X} :
+    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by
   simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]
 #align is_open_iff_ultrafilter isOpen_iff_ultrafilter
 
-theorem isOpen_singleton_iff_nhds_eq_pure (a : α) : IsOpen ({a} : Set α) ↔ 𝓝 a = pure a := by
+theorem isOpen_singleton_iff_nhds_eq_pure (a : X) : IsOpen ({a} : Set X) ↔ 𝓝 a = pure a := by
   constructor
   · intro h
     apply le_antisymm _ (pure_le_nhds a)
@@ -1289,13 +1289,12 @@ theorem isOpen_singleton_iff_nhds_eq_pure (a : α) : IsOpen ({a} : Set α) ↔ 
     simp [isOpen_iff_nhds, h]
 #align is_open_singleton_iff_nhds_eq_pure isOpen_singleton_iff_nhds_eq_pure
 
-theorem isOpen_singleton_iff_punctured_nhds {α : Type*} [TopologicalSpace α] (a : α) :
-    IsOpen ({a} : Set α) ↔ 𝓝[≠] a = ⊥ := by
+theorem isOpen_singleton_iff_punctured_nhds (a : X) : IsOpen ({a} : Set X) ↔ 𝓝[≠] a = ⊥ := by
   rw [isOpen_singleton_iff_nhds_eq_pure, nhdsWithin, ← mem_iff_inf_principal_compl, ← le_pure_iff,
     nhds_neBot.le_pure_iff]
 #align is_open_singleton_iff_punctured_nhds isOpen_singleton_iff_punctured_nhds
 
-theorem mem_closure_iff_frequently {s : Set α} {a : α} : a ∈ closure s ↔ ∃ᶠ x in 𝓝 a, x ∈ s := by
+theorem mem_closure_iff_frequently {s : Set X} {a : X} : a ∈ closure s ↔ ∃ᶠ x in 𝓝 a, x ∈ s := by
   rw [Filter.Frequently, Filter.Eventually, ← mem_interior_iff_mem_nhds,
     closure_eq_compl_interior_compl, mem_compl_iff, compl_def]
 #align mem_closure_iff_frequently mem_closure_iff_frequently
@@ -1305,7 +1304,7 @@ alias ⟨_, Filter.Frequently.mem_closure⟩ := mem_closure_iff_frequently
 
 /-- A set `s` is closed iff for every point `x`, if there is a point `y` close to `x` that belongs
 to `s` then `x` is in `s`. -/
-theorem isClosed_iff_frequently {s : Set α} : IsClosed s ↔ ∀ x, (∃ᶠ y in 𝓝 x, y ∈ s) → x ∈ s := by
+theorem isClosed_iff_frequently {s : Set X} : IsClosed s ↔ ∀ x, (∃ᶠ y in 𝓝 x, y ∈ s) → x ∈ s := by
   rw [← closure_subset_iff_isClosed]
   refine' forall_congr' fun x => _
   rw [mem_closure_iff_frequently]
@@ -1313,30 +1312,30 @@ theorem isClosed_iff_frequently {s : Set α} : IsClosed s ↔ ∀ x, (∃ᶠ y i
 
 /-- The set of cluster points of a filter is closed. In particular, the set of limit points
 of a sequence is closed. -/
-theorem isClosed_setOf_clusterPt {f : Filter α} : IsClosed { x | ClusterPt x f } := by
+theorem isClosed_setOf_clusterPt {f : Filter X} : IsClosed { x | ClusterPt x f } := by
   simp only [ClusterPt, inf_neBot_iff_frequently_left, setOf_forall, imp_iff_not_or]
   refine' isClosed_iInter fun p => IsClosed.union _ _ <;> apply isClosed_compl_iff.2
   exacts [isOpen_setOf_eventually_nhds, isOpen_const]
 #align is_closed_set_of_cluster_pt isClosed_setOf_clusterPt
 
-theorem mem_closure_iff_clusterPt {s : Set α} {a : α} : a ∈ closure s ↔ ClusterPt a (𝓟 s) :=
+theorem mem_closure_iff_clusterPt {s : Set X} {a : X} : a ∈ closure s ↔ ClusterPt a (𝓟 s) :=
   mem_closure_iff_frequently.trans clusterPt_principal_iff_frequently.symm
 #align mem_closure_iff_cluster_pt mem_closure_iff_clusterPt
 
-theorem mem_closure_iff_nhds_neBot {s : Set α} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ :=
+theorem mem_closure_iff_nhds_neBot {s : Set X} : a ∈ closure s ↔ 𝓝 a ⊓ 𝓟 s ≠ ⊥ :=
   mem_closure_iff_clusterPt.trans neBot_iff
 #align mem_closure_iff_nhds_ne_bot mem_closure_iff_nhds_neBot
 
-theorem mem_closure_iff_nhdsWithin_neBot {s : Set α} {x : α} : x ∈ closure s ↔ NeBot (𝓝[s] x) :=
+theorem mem_closure_iff_nhdsWithin_neBot {s : Set X} {x : X} : x ∈ closure s ↔ NeBot (𝓝[s] x) :=
   mem_closure_iff_clusterPt
 #align mem_closure_iff_nhds_within_ne_bot mem_closure_iff_nhdsWithin_neBot
 
-lemma not_mem_closure_iff_nhdsWithin_eq_bot {s : Set α} {x : α} : x ∉ closure s ↔ 𝓝[s] x = ⊥ := by
+lemma not_mem_closure_iff_nhdsWithin_eq_bot {s : Set X} {x : X} : x ∉ closure s ↔ 𝓝[s] x = ⊥ := by
   rw [mem_closure_iff_nhdsWithin_neBot, not_neBot]
 
 /-- If `x` is not an isolated point of a topological space, then `{x}ᶜ` is dense in the whole
 space. -/
-theorem dense_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : Set α) := by
+theorem dense_compl_singleton (x : X) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : Set X) := by
   intro y
   rcases eq_or_ne y x with (rfl | hne)
   · rwa [mem_closure_iff_nhdsWithin_neBot]
@@ -1346,95 +1345,95 @@ theorem dense_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : S
 /-- If `x` is not an isolated point of a topological space, then the closure of `{x}ᶜ` is the whole
 space. -/
 -- porting note: was a `@[simp]` lemma but `simp` can prove it
-theorem closure_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : closure {x}ᶜ = (univ : Set α) :=
+theorem closure_compl_singleton (x : X) [NeBot (𝓝[≠] x)] : closure {x}ᶜ = (univ : Set X) :=
   (dense_compl_singleton x).closure_eq
 #align closure_compl_singleton closure_compl_singleton
 
 /-- If `x` is not an isolated point of a topological space, then the interior of `{x}` is empty. -/
 @[simp]
-theorem interior_singleton (x : α) [NeBot (𝓝[≠] x)] : interior {x} = (∅ : Set α) :=
+theorem interior_singleton (x : X) [NeBot (𝓝[≠] x)] : interior {x} = (∅ : Set X) :=
   interior_eq_empty_iff_dense_compl.2 (dense_compl_singleton x)
 #align interior_singleton interior_singleton
 
-theorem not_isOpen_singleton (x : α) [NeBot (𝓝[≠] x)] : ¬IsOpen ({x} : Set α) :=
+theorem not_isOpen_singleton (x : X) [NeBot (𝓝[≠] x)] : ¬IsOpen ({x} : Set X) :=
   dense_compl_singleton_iff_not_open.1 (dense_compl_singleton x)
 #align not_is_open_singleton not_isOpen_singleton
 
-theorem closure_eq_cluster_pts {s : Set α} : closure s = { a | ClusterPt a (𝓟 s) } :=
+theorem closure_eq_cluster_pts {s : Set X} : closure s = { a | ClusterPt a (𝓟 s) } :=
   Set.ext fun _ => mem_closure_iff_clusterPt
 #align closure_eq_cluster_pts closure_eq_cluster_pts
 
-theorem mem_closure_iff_nhds {s : Set α} {a : α} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, (t ∩ s).Nonempty :=
+theorem mem_closure_iff_nhds {s : Set X} {a : X} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, (t ∩ s).Nonempty :=
   mem_closure_iff_clusterPt.trans clusterPt_principal_iff
 #align mem_closure_iff_nhds mem_closure_iff_nhds
 
-theorem mem_closure_iff_nhds' {s : Set α} {a : α} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, ∃ y : s, ↑y ∈ t := by
+theorem mem_closure_iff_nhds' {s : Set X} {a : X} : a ∈ closure s ↔ ∀ t ∈ 𝓝 a, ∃ y : s, ↑y ∈ t := by
   simp only [mem_closure_iff_nhds, Set.inter_nonempty_iff_exists_right, SetCoe.exists, exists_prop]
 #align mem_closure_iff_nhds' mem_closure_iff_nhds'
 
-theorem mem_closure_iff_comap_neBot {A : Set α} {x : α} :
-    x ∈ closure A ↔ NeBot (comap ((↑) : A → α) (𝓝 x)) := by
+theorem mem_closure_iff_comap_neBot {A : Set X} {x : X} :
+    x ∈ closure A ↔ NeBot (comap ((↑) : A → X) (𝓝 x)) := by
   simp_rw [mem_closure_iff_nhds, comap_neBot_iff, Set.inter_nonempty_iff_exists_right,
     SetCoe.exists, exists_prop]
 #align mem_closure_iff_comap_ne_bot mem_closure_iff_comap_neBot
 
-theorem mem_closure_iff_nhds_basis' {a : α} {p : ι → Prop} {s : ι → Set α} (h : (𝓝 a).HasBasis p s)
-    {t : Set α} : a ∈ closure t ↔ ∀ i, p i → (s i ∩ t).Nonempty :=
+theorem mem_closure_iff_nhds_basis' {a : X} {p : ι → Prop} {s : ι → Set X} (h : (𝓝 a).HasBasis p s)
+    {t : Set X} : a ∈ closure t ↔ ∀ i, p i → (s i ∩ t).Nonempty :=
   mem_closure_iff_clusterPt.trans <|
     (h.clusterPt_iff (hasBasis_principal _)).trans <| by simp only [exists_prop, forall_const]
 #align mem_closure_iff_nhds_basis' mem_closure_iff_nhds_basis'
 
-theorem mem_closure_iff_nhds_basis {a : α} {p : ι → Prop} {s : ι → Set α} (h : (𝓝 a).HasBasis p s)
-    {t : Set α} : a ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i :=
+theorem mem_closure_iff_nhds_basis {a : X} {p : ι → Prop} {s : ι → Set X} (h : (𝓝 a).HasBasis p s)
+    {t : Set X} : a ∈ closure t ↔ ∀ i, p i → ∃ y ∈ t, y ∈ s i :=
   (mem_closure_iff_nhds_basis' h).trans <| by
     simp only [Set.Nonempty, mem_inter_iff, exists_prop, and_comm]
 #align mem_closure_iff_nhds_basis mem_closure_iff_nhds_basis
 
-theorem clusterPt_iff_forall_mem_closure {F : Filter α} {a : α} :
+theorem clusterPt_iff_forall_mem_closure {F : Filter X} {a : X} :
     ClusterPt a F ↔ ∀ s ∈ F, a ∈ closure s := by
   simp_rw [ClusterPt, inf_neBot_iff, mem_closure_iff_nhds]
   rw [forall₂_swap]
 
-theorem clusterPt_iff_lift'_closure {F : Filter α} {a : α} :
+theorem clusterPt_iff_lift'_closure {F : Filter X} {a : X} :
     ClusterPt a F ↔ pure a ≤ (F.lift' closure) := by
   simp_rw [clusterPt_iff_forall_mem_closure,
     (hasBasis_pure _).le_basis_iff F.basis_sets.lift'_closure, id, singleton_subset_iff, true_and,
     exists_const]
 
-theorem clusterPt_iff_lift'_closure' {F : Filter α} {a : α} :
+theorem clusterPt_iff_lift'_closure' {F : Filter X} {a : X} :
     ClusterPt a F ↔ (F.lift' closure ⊓ pure a).NeBot := by
   rw [clusterPt_iff_lift'_closure, ← Ultrafilter.coe_pure, inf_comm, Ultrafilter.inf_neBot_iff]
 
 @[simp]
-theorem clusterPt_lift'_closure_iff {F : Filter α} {a : α} :
+theorem clusterPt_lift'_closure_iff {F : Filter X} {a : X} :
     ClusterPt a (F.lift' closure) ↔ ClusterPt a F := by
-  simp [clusterPt_iff_lift'_closure, lift'_lift'_assoc (monotone_closure α) (monotone_closure α)]
+  simp [clusterPt_iff_lift'_closure, lift'_lift'_assoc (monotone_closure X) (monotone_closure X)]
 
 /-- `x` belongs to the closure of `s` if and only if some ultrafilter
   supported on `s` converges to `x`. -/
-theorem mem_closure_iff_ultrafilter {s : Set α} {x : α} :
-    x ∈ closure s ↔ ∃ u : Ultrafilter α, s ∈ u ∧ ↑u ≤ 𝓝 x := by
+theorem mem_closure_iff_ultrafilter {s : Set X} {x : X} :
+    x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x := by
   simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]
 #align mem_closure_iff_ultrafilter mem_closure_iff_ultrafilter
 
-theorem isClosed_iff_clusterPt {s : Set α} : IsClosed s ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s :=
+theorem isClosed_iff_clusterPt {s : Set X} : IsClosed s ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s :=
   calc
     IsClosed s ↔ closure s ⊆ s := closure_subset_iff_isClosed.symm
     _ ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s := by simp only [subset_def, mem_closure_iff_clusterPt]
 #align is_closed_iff_cluster_pt isClosed_iff_clusterPt
 
-theorem isClosed_iff_nhds {s : Set α} :
+theorem isClosed_iff_nhds {s : Set X} :
     IsClosed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s := by
   simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_neBot_iff]
 #align is_closed_iff_nhds isClosed_iff_nhds
 
-lemma isClosed_iff_forall_filter {s : Set α} :
-    IsClosed s ↔ ∀ x, ∀ F : Filter α, F.NeBot → F ≤ 𝓟 s → F ≤ 𝓝 x → x ∈ s := by
+lemma isClosed_iff_forall_filter {s : Set X} :
+    IsClosed s ↔ ∀ x, ∀ F : Filter X, F.NeBot → F ≤ 𝓟 s → F ≤ 𝓝 x → x ∈ s := by
   simp_rw [isClosed_iff_clusterPt]
   exact ⟨fun hs x F F_ne FS Fx ↦ hs _ <| NeBot.mono F_ne (le_inf Fx FS),
          fun hs x hx ↦ hs x (𝓝 x ⊓ 𝓟 s) hx inf_le_right inf_le_left⟩
 
-theorem IsClosed.interior_union_left {s t : Set α} (_ : IsClosed s) :
+theorem IsClosed.interior_union_left {s t : Set X} (_ : IsClosed s) :
     interior (s ∪ t) ⊆ s ∪ interior t := fun a ⟨u, ⟨⟨hu₁, hu₂⟩, ha⟩⟩ =>
   (Classical.em (a ∈ s)).imp_right fun h =>
     mem_interior.mpr
@@ -1442,28 +1441,28 @@ theorem IsClosed.interior_union_left {s t : Set α} (_ : IsClosed s) :
         ⟨ha, h⟩⟩
 #align is_closed.interior_union_left IsClosed.interior_union_left
 
-theorem IsClosed.interior_union_right {s t : Set α} (h : IsClosed t) :
+theorem IsClosed.interior_union_right {s t : Set X} (h : IsClosed t) :
     interior (s ∪ t) ⊆ interior s ∪ t := by
   simpa only [union_comm _ t] using h.interior_union_left
 #align is_closed.interior_union_right IsClosed.interior_union_right
 
-theorem IsOpen.inter_closure {s t : Set α} (h : IsOpen s) : s ∩ closure t ⊆ closure (s ∩ t) :=
+theorem IsOpen.inter_closure {s t : Set X} (h : IsOpen s) : s ∩ closure t ⊆ closure (s ∩ t) :=
   compl_subset_compl.mp <| by
     simpa only [← interior_compl, compl_inter] using IsClosed.interior_union_left h.isClosed_compl
 #align is_open.inter_closure IsOpen.inter_closure
 
-theorem IsOpen.closure_inter {s t : Set α} (h : IsOpen t) : closure s ∩ t ⊆ closure (s ∩ t) := by
+theorem IsOpen.closure_inter {s t : Set X} (h : IsOpen t) : closure s ∩ t ⊆ closure (s ∩ t) := by
   simpa only [inter_comm t] using h.inter_closure
 #align is_open.closure_inter IsOpen.closure_inter
 
-theorem Dense.open_subset_closure_inter {s t : Set α} (hs : Dense s) (ht : IsOpen t) :
+theorem Dense.open_subset_closure_inter {s t : Set X} (hs : Dense s) (ht : IsOpen t) :
     t ⊆ closure (t ∩ s) :=
   calc
     t = t ∩ closure s := by rw [hs.closure_eq, inter_univ]
     _ ⊆ closure (t ∩ s) := ht.inter_closure
 #align dense.open_subset_closure_inter Dense.open_subset_closure_inter
 
-theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x ∈ closure (s₁ ∪ s₂))
+theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set X} {x : X} (h : x ∈ closure (s₁ ∪ s₂))
     (h₁ : s₁ᶜ ∈ 𝓝 x) : x ∈ closure s₂ := by
   rw [mem_closure_iff_nhds_neBot] at *
   rwa [←
@@ -1476,24 +1475,24 @@ theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x 
 #align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_union
 
 /-- The intersection of an open dense set with a dense set is a dense set. -/
-theorem Dense.inter_of_isOpen_left {s t : Set α} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
+theorem Dense.inter_of_isOpen_left {s t : Set X} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
     Dense (s ∩ t) := fun x =>
   closure_minimal hso.inter_closure isClosed_closure <| by simp [hs.closure_eq, ht.closure_eq]
 #align dense.inter_of_open_left Dense.inter_of_isOpen_left
 
 /-- The intersection of a dense set with an open dense set is a dense set. -/
-theorem Dense.inter_of_isOpen_right {s t : Set α} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
+theorem Dense.inter_of_isOpen_right {s t : Set X} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
     Dense (s ∩ t) :=
   inter_comm t s ▸ ht.inter_of_isOpen_left hs hto
 #align dense.inter_of_open_right Dense.inter_of_isOpen_right
 
-theorem Dense.inter_nhds_nonempty {s t : Set α} (hs : Dense s) {x : α} (ht : t ∈ 𝓝 x) :
+theorem Dense.inter_nhds_nonempty {s t : Set X} (hs : Dense s) {x : X} (ht : t ∈ 𝓝 x) :
     (s ∩ t).Nonempty :=
   let ⟨U, hsub, ho, hx⟩ := mem_nhds_iff.1 ht
   (hs.inter_open_nonempty U ho ⟨x, hx⟩).mono fun _y hy => ⟨hy.2, hsub hy.1⟩
 #align dense.inter_nhds_nonempty Dense.inter_nhds_nonempty
 
-theorem closure_diff {s t : Set α} : closure s \ closure t ⊆ closure (s \ t) :=
+theorem closure_diff {s t : Set X} : closure s \ closure t ⊆ closure (s \ t) :=
   calc
     closure s \ closure t = (closure t)ᶜ ∩ closure s := by simp only [diff_eq, inter_comm]
     _ ⊆ closure ((closure t)ᶜ ∩ s) := (isOpen_compl_iff.mpr <| isClosed_closure).inter_closure
@@ -1501,35 +1500,35 @@ theorem closure_diff {s t : Set α} : closure s \ closure t ⊆ closure (s \ t)
     _ ⊆ closure (s \ t) := closure_mono <| diff_subset_diff (Subset.refl s) subset_closure
 #align closure_diff closure_diff
 
-theorem Filter.Frequently.mem_of_closed {a : α} {s : Set α} (h : ∃ᶠ x in 𝓝 a, x ∈ s)
+theorem Filter.Frequently.mem_of_closed {a : X} {s : Set X} (h : ∃ᶠ x in 𝓝 a, x ∈ s)
     (hs : IsClosed s) : a ∈ s :=
   hs.closure_subset h.mem_closure
 #align filter.frequently.mem_of_closed Filter.Frequently.mem_of_closed
 
-theorem IsClosed.mem_of_frequently_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set α}
+theorem IsClosed.mem_of_frequently_of_tendsto {f : α → X} {b : Filter α} {a : X} {s : Set X}
     (hs : IsClosed s) (h : ∃ᶠ x in b, f x ∈ s) (hf : Tendsto f b (𝓝 a)) : a ∈ s :=
   (hf.frequently <| show ∃ᶠ x in b, (fun y => y ∈ s) (f x) from h).mem_of_closed hs
 #align is_closed.mem_of_frequently_of_tendsto IsClosed.mem_of_frequently_of_tendsto
 
-theorem IsClosed.mem_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set α} [NeBot b]
+theorem IsClosed.mem_of_tendsto {f : α → X} {b : Filter α} {a : X} {s : Set X} [NeBot b]
     (hs : IsClosed s) (hf : Tendsto f b (𝓝 a)) (h : ∀ᶠ x in b, f x ∈ s) : a ∈ s :=
   hs.mem_of_frequently_of_tendsto h.frequently hf
 #align is_closed.mem_of_tendsto IsClosed.mem_of_tendsto
 
-theorem mem_closure_of_frequently_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set α}
+theorem mem_closure_of_frequently_of_tendsto {f : α → X} {b : Filter α} {a : X} {s : Set X}
     (h : ∃ᶠ x in b, f x ∈ s) (hf : Tendsto f b (𝓝 a)) : a ∈ closure s :=
   (hf.frequently h).mem_closure
 #align mem_closure_of_frequently_of_tendsto mem_closure_of_frequently_of_tendsto
 
-theorem mem_closure_of_tendsto {f : β → α} {b : Filter β} {a : α} {s : Set α} [NeBot b]
+theorem mem_closure_of_tendsto {f : α → X} {b : Filter α} {a : X} {s : Set X} [NeBot b]
     (hf : Tendsto f b (𝓝 a)) (h : ∀ᶠ x in b, f x ∈ s) : a ∈ closure s :=
   mem_closure_of_frequently_of_tendsto h.frequently hf
 #align mem_closure_of_tendsto mem_closure_of_tendsto
 
 /-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.
 Then `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/
-theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : β → α} {l : Filter β} {s : Set β} {a : α}
-    (h : ∀ (x) (_ : x ∉ s), f x = a) : Tendsto f (l ⊓ 𝓟 s) (𝓝 a) ↔ Tendsto f l (𝓝 a) := by
+theorem tendsto_inf_principal_nhds_iff_of_forall_eq {f : α → X} {l : Filter α} {s : Set α} {a : X}
+    (h : ∀ x ∉ s, f x = a) : Tendsto f (l ⊓ 𝓟 s) (𝓝 a) ↔ Tendsto f l (𝓝 a) := by
   rw [tendsto_iff_comap, tendsto_iff_comap]
   replace h : 𝓟 sᶜ ≤ comap f (𝓝 a)
   · rintro U ⟨t, ht, htU⟩ x hx
@@ -1558,37 +1557,37 @@ section lim
 set_option linter.uppercaseLean3 false
 
 /-- If `f` is a filter, then `Filter.lim f` is a limit of the filter, if it exists. -/
-noncomputable def lim [Nonempty α] (f : Filter α) : α :=
+noncomputable def lim [Nonempty X] (f : Filter X) : X :=
   Classical.epsilon fun a => f ≤ 𝓝 a
 #align Lim lim
 
 /--
 If `F` is an ultrafilter, then `Filter.Ultrafilter.lim F` is a limit of the filter, if it exists.
-Note that dot notation `F.lim` can be used for `F : Filter.Ultrafilter α`.
+Note that dot notation `F.lim` can be used for `F : Filter.Ultrafilter X`.
 -/
-noncomputable nonrec def Ultrafilter.lim (F : Ultrafilter α) : α :=
-  @lim α _ (nonempty_of_neBot F) F
+noncomputable nonrec def Ultrafilter.lim (F : Ultrafilter X) : X :=
+  @lim X _ (nonempty_of_neBot F) F
 #align ultrafilter.Lim Ultrafilter.lim
 
-/-- If `f` is a filter in `β` and `g : β → α` is a function, then `limUnder f g` is a limit of `g`
+/-- If `f` is a filter in `α` and `g : α → X` is a function, then `limUnder f g` is a limit of `g`
 at `f`, if it exists. -/
-noncomputable def limUnder [Nonempty α] (f : Filter β) (g : β → α) : α :=
+noncomputable def limUnder [Nonempty X] (f : Filter α) (g : α → X) : X :=
   lim (f.map g)
 #align lim limUnder
 
 /-- If a filter `f` is majorated by some `𝓝 a`, then it is majorated by `𝓝 (Filter.lim f)`. We
-formulate this lemma with a `[Nonempty α]` argument of `lim` derived from `h` to make it useful for
-types without a `[Nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify
+formulate this lemma with a `[Nonempty X]` argument of `lim` derived from `h` to make it useful for
+types without a `[Nonempty X]` instance. Because of the built-in proof irrelevance, Lean will unify
 this instance with any other instance. -/
-theorem le_nhds_lim {f : Filter α} (h : ∃ a, f ≤ 𝓝 a) : f ≤ 𝓝 (@lim _ _ (nonempty_of_exists h) f) :=
+theorem le_nhds_lim {f : Filter X} (h : ∃ a, f ≤ 𝓝 a) : f ≤ 𝓝 (@lim _ _ (nonempty_of_exists h) f) :=
   Classical.epsilon_spec h
 #align le_nhds_Lim le_nhds_lim
 
 /-- If `g` tends to some `𝓝 a` along `f`, then it tends to `𝓝 (Filter.limUnder f g)`. We formulate
-this lemma with a `[Nonempty α]` argument of `lim` derived from `h` to make it useful for types
-without a `[Nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify this
+this lemma with a `[Nonempty X]` argument of `lim` derived from `h` to make it useful for types
+without a `[Nonempty X]` instance. Because of the built-in proof irrelevance, Lean will unify this
 instance with any other instance. -/
-theorem tendsto_nhds_limUnder {f : Filter β} {g : β → α} (h : ∃ a, Tendsto g f (𝓝 a)) :
+theorem tendsto_nhds_limUnder {f : Filter α} {g : α → X} (h : ∃ a, Tendsto g f (𝓝 a)) :
     Tendsto g f (𝓝 (@limUnder _ _ _ (nonempty_of_exists h) f g)) :=
   le_nhds_lim h
 #align tendsto_nhds_lim tendsto_nhds_limUnder
@@ -1605,15 +1604,13 @@ open Topology
 
 section Continuous
 
-variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
-
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
 open TopologicalSpace
 
 /-- A function between topological spaces is continuous if the preimage
   of every open set is open. Registered as a structure to make sure it is not unfolded by Lean. -/
-structure Continuous (f : α → β) : Prop where
+structure Continuous (f : X → Y) : Prop where
   /-- The preimage of an open set under a continuous function is an open set. Use `IsOpen.preimage`
   instead. -/
   isOpen_preimage : ∀ s, IsOpen s → IsOpen (f ⁻¹' s)
@@ -1624,227 +1621,227 @@ set_option quotPrecheck false in
 scoped[Topology] notation (name := Continuous_of) "Continuous[" t₁ ", " t₂ "]" =>
   @Continuous _ _ t₁ t₂
 
-theorem continuous_def {_ : TopologicalSpace α} {_ : TopologicalSpace β} {f : α → β} :
+theorem continuous_def {_ : TopologicalSpace X} {_ : TopologicalSpace Y} {f : X → Y} :
     Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) :=
   ⟨fun hf => hf.1, fun h => ⟨h⟩⟩
 #align continuous_def continuous_def
 
-theorem IsOpen.preimage {f : α → β} (hf : Continuous f) {s : Set β} (h : IsOpen s) :
+theorem IsOpen.preimage {f : X → Y} (hf : Continuous f) {s : Set Y} (h : IsOpen s) :
     IsOpen (f ⁻¹' s) :=
   hf.isOpen_preimage s h
 #align is_open.preimage IsOpen.preimage
 
-theorem continuous_congr {f g : α → β} (h : ∀ x, f x = g x) :
+theorem continuous_congr {f g : X → Y} (h : ∀ x, f x = g x) :
     Continuous f ↔ Continuous g :=
   .of_eq <| congrArg _ <| funext h
 
-theorem Continuous.congr {f g : α → β} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g :=
+theorem Continuous.congr {f g : X → Y} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g :=
   continuous_congr h' |>.mp h
 #align continuous.congr Continuous.congr
 
 /-- A function between topological spaces is continuous at a point `x₀`
 if `f x` tends to `f x₀` when `x` tends to `x₀`. -/
-def ContinuousAt (f : α → β) (x : α) :=
+def ContinuousAt (f : X → Y) (x : X) :=
   Tendsto f (𝓝 x) (𝓝 (f x))
 #align continuous_at ContinuousAt
 
-theorem ContinuousAt.tendsto {f : α → β} {x : α} (h : ContinuousAt f x) :
+theorem ContinuousAt.tendsto {f : X → Y} {x : X} (h : ContinuousAt f x) :
     Tendsto f (𝓝 x) (𝓝 (f x)) :=
   h
 #align continuous_at.tendsto ContinuousAt.tendsto
 
-theorem continuousAt_def {f : α → β} {x : α} : ContinuousAt f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x :=
+theorem continuousAt_def {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ A ∈ 𝓝 (f x), f ⁻¹' A ∈ 𝓝 x :=
   Iff.rfl
 #align continuous_at_def continuousAt_def
 
-theorem continuousAt_congr {f g : α → β} {x : α} (h : f =ᶠ[𝓝 x] g) :
+theorem continuousAt_congr {f g : X → Y} {x : X} (h : f =ᶠ[𝓝 x] g) :
     ContinuousAt f x ↔ ContinuousAt g x := by
   simp only [ContinuousAt, tendsto_congr' h, h.eq_of_nhds]
 #align continuous_at_congr continuousAt_congr
 
-theorem ContinuousAt.congr {f g : α → β} {x : α} (hf : ContinuousAt f x) (h : f =ᶠ[𝓝 x] g) :
+theorem ContinuousAt.congr {f g : X → Y} {x : X} (hf : ContinuousAt f x) (h : f =ᶠ[𝓝 x] g) :
     ContinuousAt g x :=
   (continuousAt_congr h).1 hf
 #align continuous_at.congr ContinuousAt.congr
 
-theorem ContinuousAt.preimage_mem_nhds {f : α → β} {x : α} {t : Set β} (h : ContinuousAt f x)
+theorem ContinuousAt.preimage_mem_nhds {f : X → Y} {x : X} {t : Set Y} (h : ContinuousAt f x)
     (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝 x :=
   h ht
 #align continuous_at.preimage_mem_nhds ContinuousAt.preimage_mem_nhds
 
 /-- Deprecated, please use `not_mem_tsupport_iff_eventuallyEq` instead. -/
 @[deprecated] -- 15 January 2024
-theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {a : α} {f : α → M₀} :
+theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {a : X} {f : X → M₀} :
     f =ᶠ[𝓝 a] 0 ↔ a ∉ closure (Function.support f) := by
   rw [← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds, Function.compl_support,
     EventuallyEq, eventually_iff]
   simp only [Pi.zero_apply]
 #align eventually_eq_zero_nhds eventuallyEq_zero_nhds
 
-theorem ClusterPt.map {x : α} {la : Filter α} {lb : Filter β} (H : ClusterPt x la) {f : α → β}
+theorem ClusterPt.map {x : X} {la : Filter X} {lb : Filter Y} (H : ClusterPt x la) {f : X → Y}
     (hfc : ContinuousAt f x) (hf : Tendsto f la lb) : ClusterPt (f x) lb :=
   (NeBot.map H f).mono <| hfc.tendsto.inf hf
 #align cluster_pt.map ClusterPt.map
 
 /-- See also `interior_preimage_subset_preimage_interior`. -/
-theorem preimage_interior_subset_interior_preimage {f : α → β} {s : Set β} (hf : Continuous f) :
+theorem preimage_interior_subset_interior_preimage {f : X → Y} {s : Set Y} (hf : Continuous f) :
     f ⁻¹' interior s ⊆ interior (f ⁻¹' s) :=
   interior_maximal (preimage_mono interior_subset) (isOpen_interior.preimage hf)
 #align preimage_interior_subset_interior_preimage preimage_interior_subset_interior_preimage
 
 @[continuity]
-theorem continuous_id : Continuous (id : α → α) :=
+theorem continuous_id : Continuous (id : X → X) :=
   continuous_def.2 fun _ => id
 #align continuous_id continuous_id
 
 -- This is needed due to reducibility issues with the `continuity` tactic.
 @[continuity]
-theorem continuous_id' : Continuous (fun (x : α) => x) := continuous_id
+theorem continuous_id' : Continuous (fun (x : X) => x) := continuous_id
 
-theorem Continuous.comp {g : β → γ} {f : α → β} (hg : Continuous g) (hf : Continuous f) :
+theorem Continuous.comp {g : Y → Z} {f : X → Y} (hg : Continuous g) (hf : Continuous f) :
     Continuous (g ∘ f) :=
   continuous_def.2 fun _ h => (h.preimage hg).preimage hf
 #align continuous.comp Continuous.comp
 
 -- This is needed due to reducibility issues with the `continuity` tactic.
 @[continuity]
-theorem Continuous.comp' {g : β → γ} {f : α → β} (hg : Continuous g) (hf : Continuous f) :
+theorem Continuous.comp' {g : Y → Z} {f : X → Y} (hg : Continuous g) (hf : Continuous f) :
     Continuous (fun x => g (f x)) := hg.comp hf
 
-theorem Continuous.iterate {f : α → α} (h : Continuous f) (n : ℕ) : Continuous f^[n] :=
+theorem Continuous.iterate {f : X → X} (h : Continuous f) (n : ℕ) : Continuous f^[n] :=
   Nat.recOn n continuous_id fun _ ihn => ihn.comp h
 #align continuous.iterate Continuous.iterate
 
-nonrec theorem ContinuousAt.comp {g : β → γ} {f : α → β} {x : α} (hg : ContinuousAt g (f x))
+nonrec theorem ContinuousAt.comp {g : Y → Z} {f : X → Y} {x : X} (hg : ContinuousAt g (f x))
     (hf : ContinuousAt f x) : ContinuousAt (g ∘ f) x :=
   hg.comp hf
 #align continuous_at.comp ContinuousAt.comp
 
 /-- See note [comp_of_eq lemmas] -/
-theorem ContinuousAt.comp_of_eq {g : β → γ} {f : α → β} {x : α} {y : β} (hg : ContinuousAt g y)
+theorem ContinuousAt.comp_of_eq {g : Y → Z} {f : X → Y} {x : X} {y : Y} (hg : ContinuousAt g y)
     (hf : ContinuousAt f x) (hy : f x = y) : ContinuousAt (g ∘ f) x := by subst hy; exact hg.comp hf
 #align continuous_at.comp_of_eq ContinuousAt.comp_of_eq
 
-theorem Continuous.tendsto {f : α → β} (hf : Continuous f) (x) : Tendsto f (𝓝 x) (𝓝 (f x)) :=
+theorem Continuous.tendsto {f : X → Y} (hf : Continuous f) (x) : Tendsto f (𝓝 x) (𝓝 (f x)) :=
   ((nhds_basis_opens x).tendsto_iff <| nhds_basis_opens <| f x).2 fun t ⟨hxt, ht⟩ =>
     ⟨f ⁻¹' t, ⟨hxt, ht.preimage hf⟩, Subset.rfl⟩
 #align continuous.tendsto Continuous.tendsto
 
 /-- A version of `Continuous.tendsto` that allows one to specify a simpler form of the limit.
 E.g., one can write `continuous_exp.tendsto' 0 1 exp_zero`. -/
-theorem Continuous.tendsto' {f : α → β} (hf : Continuous f) (x : α) (y : β) (h : f x = y) :
+theorem Continuous.tendsto' {f : X → Y} (hf : Continuous f) (x : X) (y : Y) (h : f x = y) :
     Tendsto f (𝓝 x) (𝓝 y) :=
   h ▸ hf.tendsto x
 #align continuous.tendsto' Continuous.tendsto'
 
-theorem Continuous.continuousAt {f : α → β} {x : α} (h : Continuous f) : ContinuousAt f x :=
+theorem Continuous.continuousAt {f : X → Y} {x : X} (h : Continuous f) : ContinuousAt f x :=
   h.tendsto x
 #align continuous.continuous_at Continuous.continuousAt
 
-theorem continuous_iff_continuousAt {f : α → β} : Continuous f ↔ ∀ x, ContinuousAt f x :=
+theorem continuous_iff_continuousAt {f : X → Y} : Continuous f ↔ ∀ x, ContinuousAt f x :=
   ⟨Continuous.tendsto, fun hf => continuous_def.2 fun _U hU => isOpen_iff_mem_nhds.2 fun x hx =>
     hf x <| hU.mem_nhds hx⟩
 #align continuous_iff_continuous_at continuous_iff_continuousAt
 
-theorem continuousAt_const {x : α} {b : β} : ContinuousAt (fun _ : α => b) x :=
+theorem continuousAt_const {x : X} {b : Y} : ContinuousAt (fun _ : X => b) x :=
   tendsto_const_nhds
 #align continuous_at_const continuousAt_const
 
 @[continuity]
-theorem continuous_const {b : β} : Continuous fun _ : α => b :=
+theorem continuous_const {b : Y} : Continuous fun _ : X => b :=
   continuous_iff_continuousAt.mpr fun _ => continuousAt_const
 #align continuous_const continuous_const
 
-theorem Filter.EventuallyEq.continuousAt {x : α} {f : α → β} {y : β} (h : f =ᶠ[𝓝 x] fun _ => y) :
+theorem Filter.EventuallyEq.continuousAt {x : X} {f : X → Y} {y : Y} (h : f =ᶠ[𝓝 x] fun _ => y) :
     ContinuousAt f x :=
   (continuousAt_congr h).2 tendsto_const_nhds
 #align filter.eventually_eq.continuous_at Filter.EventuallyEq.continuousAt
 
-theorem continuous_of_const {f : α → β} (h : ∀ x y, f x = f y) : Continuous f :=
+theorem continuous_of_const {f : X → Y} (h : ∀ x y, f x = f y) : Continuous f :=
   continuous_iff_continuousAt.mpr fun x =>
     Filter.EventuallyEq.continuousAt <| eventually_of_forall fun y => h y x
 #align continuous_of_const continuous_of_const
 
-theorem continuousAt_id {x : α} : ContinuousAt id x :=
+theorem continuousAt_id {x : X} : ContinuousAt id x :=
   continuous_id.continuousAt
 #align continuous_at_id continuousAt_id
 
-theorem ContinuousAt.iterate {f : α → α} {x : α} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ) :
+theorem ContinuousAt.iterate {f : X → X} {x : X} (hf : ContinuousAt f x) (hx : f x = x) (n : ℕ) :
     ContinuousAt f^[n] x :=
   Nat.recOn n continuousAt_id fun n ihn =>
     show ContinuousAt (f^[n] ∘ f) x from ContinuousAt.comp (hx.symm ▸ ihn) hf
 #align continuous_at.iterate ContinuousAt.iterate
 
-theorem continuous_iff_isClosed {f : α → β} : Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s) :=
+theorem continuous_iff_isClosed {f : X → Y} : Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s) :=
   continuous_def.trans <| compl_surjective.forall.trans <| by
     simp only [isOpen_compl_iff, preimage_compl]
 #align continuous_iff_is_closed continuous_iff_isClosed
 
-theorem IsClosed.preimage {f : α → β} (hf : Continuous f) {s : Set β} (h : IsClosed s) :
+theorem IsClosed.preimage {f : X → Y} (hf : Continuous f) {s : Set Y} (h : IsClosed s) :
     IsClosed (f ⁻¹' s) :=
   continuous_iff_isClosed.mp hf s h
 #align is_closed.preimage IsClosed.preimage
 
-theorem mem_closure_image {f : α → β} {x : α} {s : Set α} (hf : ContinuousAt f x)
+theorem mem_closure_image {f : X → Y} {x : X} {s : Set X} (hf : ContinuousAt f x)
     (hx : x ∈ closure s) : f x ∈ closure (f '' s) :=
   mem_closure_of_frequently_of_tendsto
     ((mem_closure_iff_frequently.1 hx).mono fun _ => mem_image_of_mem _) hf
 #align mem_closure_image mem_closure_image
 
-theorem continuousAt_iff_ultrafilter {f : α → β} {x} :
-    ContinuousAt f x ↔ ∀ g : Ultrafilter α, ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) :=
+theorem continuousAt_iff_ultrafilter {f : X → Y} {x} :
+    ContinuousAt f x ↔ ∀ g : Ultrafilter X, ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) :=
   tendsto_iff_ultrafilter f (𝓝 x) (𝓝 (f x))
 #align continuous_at_iff_ultrafilter continuousAt_iff_ultrafilter
 
-theorem continuous_iff_ultrafilter {f : α → β} :
-    Continuous f ↔ ∀ (x) (g : Ultrafilter α), ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) := by
+theorem continuous_iff_ultrafilter {f : X → Y} :
+    Continuous f ↔ ∀ (x) (g : Ultrafilter X), ↑g ≤ 𝓝 x → Tendsto f g (𝓝 (f x)) := by
   simp only [continuous_iff_continuousAt, continuousAt_iff_ultrafilter]
 #align continuous_iff_ultrafilter continuous_iff_ultrafilter
 
-theorem Continuous.closure_preimage_subset {f : α → β} (hf : Continuous f) (t : Set β) :
+theorem Continuous.closure_preimage_subset {f : X → Y} (hf : Continuous f) (t : Set Y) :
     closure (f ⁻¹' t) ⊆ f ⁻¹' closure t := by
   rw [← (isClosed_closure.preimage hf).closure_eq]
   exact closure_mono (preimage_mono subset_closure)
 #align continuous.closure_preimage_subset Continuous.closure_preimage_subset
 
-theorem Continuous.frontier_preimage_subset {f : α → β} (hf : Continuous f) (t : Set β) :
+theorem Continuous.frontier_preimage_subset {f : X → Y} (hf : Continuous f) (t : Set Y) :
     frontier (f ⁻¹' t) ⊆ f ⁻¹' frontier t :=
   diff_subset_diff (hf.closure_preimage_subset t) (preimage_interior_subset_interior_preimage hf)
 #align continuous.frontier_preimage_subset Continuous.frontier_preimage_subset
 
 /-- If a continuous map `f` maps `s` to `t`, then it maps `closure s` to `closure t`. -/
-protected theorem Set.MapsTo.closure {s : Set α} {t : Set β} {f : α → β} (h : MapsTo f s t)
+protected theorem Set.MapsTo.closure {s : Set X} {t : Set Y} {f : X → Y} (h : MapsTo f s t)
     (hc : Continuous f) : MapsTo f (closure s) (closure t) := by
   simp only [MapsTo, mem_closure_iff_clusterPt]
   exact fun x hx => hx.map hc.continuousAt (tendsto_principal_principal.2 h)
 #align set.maps_to.closure Set.MapsTo.closure
 
 /-- See also `IsClosedMap.closure_image_eq_of_continuous`. -/
-theorem image_closure_subset_closure_image {f : α → β} {s : Set α} (h : Continuous f) :
+theorem image_closure_subset_closure_image {f : X → Y} {s : Set X} (h : Continuous f) :
     f '' closure s ⊆ closure (f '' s) :=
   ((mapsTo_image f s).closure h).image_subset
 #align image_closure_subset_closure_image image_closure_subset_closure_image
 
 -- porting note: new lemma
-theorem closure_image_closure {f : α → β} {s : Set α} (h : Continuous f) :
+theorem closure_image_closure {f : X → Y} {s : Set X} (h : Continuous f) :
     closure (f '' closure s) = closure (f '' s) :=
   Subset.antisymm
     (closure_minimal (image_closure_subset_closure_image h) isClosed_closure)
     (closure_mono <| image_subset _ subset_closure)
 
-theorem closure_subset_preimage_closure_image {f : α → β} {s : Set α} (h : Continuous f) :
+theorem closure_subset_preimage_closure_image {f : X → Y} {s : Set X} (h : Continuous f) :
     closure s ⊆ f ⁻¹' closure (f '' s) := by
   rw [← Set.image_subset_iff]
   exact image_closure_subset_closure_image h
 #align closure_subset_preimage_closure_image closure_subset_preimage_closure_image
 
-theorem map_mem_closure {s : Set α} {t : Set β} {f : α → β} {a : α} (hf : Continuous f)
+theorem map_mem_closure {s : Set X} {t : Set Y} {f : X → Y} {a : X} (hf : Continuous f)
     (ha : a ∈ closure s) (ht : MapsTo f s t) : f a ∈ closure t :=
   ht.closure hf ha
 #align map_mem_closure map_mem_closure
 
 /-- If a continuous map `f` maps `s` to a closed set `t`, then it maps `closure s` to `t`. -/
-theorem Set.MapsTo.closure_left {s : Set α} {t : Set β} {f : α → β} (h : MapsTo f s t)
+theorem Set.MapsTo.closure_left {s : Set X} {t : Set Y} {f : X → Y} (h : MapsTo f s t)
     (hc : Continuous f) (ht : IsClosed t) : MapsTo f (closure s) t :=
   ht.closure_eq ▸ h.closure hc
 #align set.maps_to.closure_left Set.MapsTo.closure_left
@@ -1855,9 +1852,9 @@ theorem Set.MapsTo.closure_left {s : Set α} {t : Set β} {f : α → β} (h : M
 
 section DenseRange
 
-variable {κ ι : Type*} (f : κ → β) (g : β → γ)
+variable {α ι : Type*} (f : α → X) (g : X → Y)
 
-/-- `f : ι → β` has dense range if its range (image) is a dense subset of β. -/
+/-- `f : α → X` has dense range if its range (image) is a dense subset of `X`. -/
 def DenseRange := Dense (range f)
 #align dense_range DenseRange
 
@@ -1868,7 +1865,7 @@ theorem Function.Surjective.denseRange (hf : Function.Surjective f) : DenseRange
   simp [hf.range_eq]
 #align function.surjective.dense_range Function.Surjective.denseRange
 
-theorem denseRange_id : DenseRange (id : α → α) :=
+theorem denseRange_id : DenseRange (id : X → X) :=
   Function.Surjective.denseRange Function.surjective_id
 #align dense_range_id denseRange_id
 
@@ -1880,25 +1877,25 @@ theorem DenseRange.closure_range (h : DenseRange f) : closure (range f) = univ :
   h.closure_eq
 #align dense_range.closure_range DenseRange.closure_range
 
-theorem Dense.denseRange_val {s : Set α} (h : Dense s) : DenseRange ((↑) : s → α) := by
+theorem Dense.denseRange_val {s : Set X} (h : Dense s) : DenseRange ((↑) : s → X) := by
   simpa only [DenseRange, Subtype.range_coe_subtype]
 #align dense.dense_range_coe Dense.denseRange_val
 
-theorem Continuous.range_subset_closure_image_dense {f : α → β} (hf : Continuous f) {s : Set α}
+theorem Continuous.range_subset_closure_image_dense {f : X → Y} (hf : Continuous f) {s : Set X}
     (hs : Dense s) : range f ⊆ closure (f '' s) := by
   rw [← image_univ, ← hs.closure_eq]
   exact image_closure_subset_closure_image hf
 #align continuous.range_subset_closure_image_dense Continuous.range_subset_closure_image_dense
 
 /-- The image of a dense set under a continuous map with dense range is a dense set. -/
-theorem DenseRange.dense_image {f : α → β} (hf' : DenseRange f) (hf : Continuous f) {s : Set α}
+theorem DenseRange.dense_image {f : X → Y} (hf' : DenseRange f) (hf : Continuous f) {s : Set X}
     (hs : Dense s) : Dense (f '' s) :=
   (hf'.mono <| hf.range_subset_closure_image_dense hs).of_closure
 #align dense_range.dense_image DenseRange.dense_image
 
 /-- If `f` has dense range and `s` is an open set in the codomain of `f`, then the image of the
 preimage of `s` under `f` is dense in `s`. -/
-theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) {s : Set β}
+theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) {s : Set X}
     (hs : IsOpen s) : s ⊆ closure (f '' (f ⁻¹' s)) := by
   rw [image_preimage_eq_inter_range]
   exact hf.open_subset_closure_inter hs
@@ -1906,38 +1903,38 @@ theorem DenseRange.subset_closure_image_preimage_of_isOpen (hf : DenseRange f) {
 
 /-- If a continuous map with dense range maps a dense set to a subset of `t`, then `t` is a dense
 set. -/
-theorem DenseRange.dense_of_mapsTo {f : α → β} (hf' : DenseRange f) (hf : Continuous f) {s : Set α}
-    (hs : Dense s) {t : Set β} (ht : MapsTo f s t) : Dense t :=
+theorem DenseRange.dense_of_mapsTo {f : X → Y} (hf' : DenseRange f) (hf : Continuous f) {s : Set X}
+    (hs : Dense s) {t : Set Y} (ht : MapsTo f s t) : Dense t :=
   (hf'.dense_image hf hs).mono ht.image_subset
 #align dense_range.dense_of_maps_to DenseRange.dense_of_mapsTo
 
 /-- Composition of a continuous map with dense range and a function with dense range has dense
 range. -/
-theorem DenseRange.comp {g : β → γ} {f : κ → β} (hg : DenseRange g) (hf : DenseRange f)
+theorem DenseRange.comp {g : Y → Z} {f : α → Y} (hg : DenseRange g) (hf : DenseRange f)
     (cg : Continuous g) : DenseRange (g ∘ f) := by
   rw [DenseRange, range_comp]
   exact hg.dense_image cg hf
 #align dense_range.comp DenseRange.comp
 
-nonrec theorem DenseRange.nonempty_iff (hf : DenseRange f) : Nonempty κ ↔ Nonempty β :=
+nonrec theorem DenseRange.nonempty_iff (hf : DenseRange f) : Nonempty α ↔ Nonempty X :=
   range_nonempty_iff_nonempty.symm.trans hf.nonempty_iff
 #align dense_range.nonempty_iff DenseRange.nonempty_iff
 
-theorem DenseRange.nonempty [h : Nonempty β] (hf : DenseRange f) : Nonempty κ :=
+theorem DenseRange.nonempty [h : Nonempty X] (hf : DenseRange f) : Nonempty α :=
   hf.nonempty_iff.mpr h
 #align dense_range.nonempty DenseRange.nonempty
 
-/-- Given a function `f : α → β` with dense range and `b : β`, returns some `a : α`. -/
-def DenseRange.some (hf : DenseRange f) (b : β) : κ :=
+/-- Given a function `f : X → Y` with dense range and `b : Y`, returns some `a : X`. -/
+def DenseRange.some (hf : DenseRange f) (b : X) : α :=
   Classical.choice <| hf.nonempty_iff.mpr ⟨b⟩
 #align dense_range.some DenseRange.some
 
-nonrec theorem DenseRange.exists_mem_open (hf : DenseRange f) {s : Set β} (ho : IsOpen s)
+nonrec theorem DenseRange.exists_mem_open (hf : DenseRange f) {s : Set X} (ho : IsOpen s)
     (hs : s.Nonempty) : ∃ a, f a ∈ s :=
   exists_range_iff.1 <| hf.exists_mem_open ho hs
 #align dense_range.exists_mem_open DenseRange.exists_mem_open
 
-theorem DenseRange.mem_nhds {f : κ → β} (h : DenseRange f) {b : β} {U : Set β} (U_in : U ∈ 𝓝 b) :
+theorem DenseRange.mem_nhds (h : DenseRange f) {b : X} {U : Set X} (U_in : U ∈ 𝓝 b) :
     ∃ a, f a ∈ U :=
   let ⟨a, ha⟩ := h.exists_mem_open isOpen_interior ⟨b, mem_interior_iff_mem_nhds.2 U_in⟩
   ⟨a, interior_subset ha⟩
@@ -1991,7 +1988,7 @@ This has the following advantages
 
 As an example for a unary operation, we have `Continuous.neg`.
 ```
-Continuous.neg {f : α → G} (hf : Continuous f) : Continuous (fun x ↦ -f x)
+Continuous.neg {f : X → G} (hf : Continuous f) : Continuous (fun x ↦ -f x)
 ```
 For unary functions, the elaborator is not confused when applying the traditional lemma
 (like `continuous_neg`), but it's still convenient to have the short version available (compare
@@ -2063,7 +2060,7 @@ Only after elaborating the two `ContinuousAt` arguments, Lean will try to unify
 which is often easy after having chosen the correct functions for `f` and `g`.
 Here is an example that shows the difference:
 ```
-example [TopologicalSpace α] [TopologicalSpace β] {x₀ : α} (f : α → α → β)
+example [TopologicalSpace X] [TopologicalSpace Y] {x₀ : X} (f : X → X → Y)
     (hf : ContinuousAt (Function.uncurry f) (x₀, x₀)) :
     ContinuousAt (fun x ↦ f x x) x₀ :=
   -- hf.comp (continuousAt_id.prod continuousAt_id) -- type mismatch

• multiplicativise contMDiff_of_support, contMDiffWithinAt_of_not_mem and contMDiffAt_of_not_mem and use to_additive
• golf extend_one with the multiplicative version
• generalise these lemmas to manifolds with zero/one
• slight clean-up: remove unused variables and opens
• slight drive-by golfing of one proof
• deprecate eventuallyEq_zero_nhds in favor of not_mem_tsupport_iff_eventuallyEq

Addresses the post-merge review comments in #9669.

Co-authored-by: ADedecker <anatolededecker@gmail.com> Co-authored-by: grunweg <grunweg@posteo.de>

@@ -1672,6 +1672,8 @@ theorem ContinuousAt.preimage_mem_nhds {f : α → β} {x : α} {t : Set β} (h
   h ht
 #align continuous_at.preimage_mem_nhds ContinuousAt.preimage_mem_nhds
 
+/-- Deprecated, please use `not_mem_tsupport_iff_eventuallyEq` instead. -/
+@[deprecated] -- 15 January 2024
 theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {a : α} {f : α → M₀} :
     f =ᶠ[𝓝 a] 0 ↔ a ∉ closure (Function.support f) := by
   rw [← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds, Function.compl_support,

From sphere-eversion; I just cleaned it up slightly and submitted it.

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

@@ -154,6 +154,12 @@ theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ 
   rw [union_eq_iUnion]; exact isOpen_iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)
 #align is_open.union IsOpen.union
 
+lemma isOpen_iff_of_cover {ι : Type*} {f : ι → Set α} (ho : ∀ i, IsOpen (f i))
+    (hU : (⋃ i, f i) = univ) : IsOpen s ↔ ∀ i, IsOpen (f i ∩ s) := by
+  refine ⟨fun h i ↦ (ho i).inter h, fun h ↦ ?_⟩
+  rw [← s.inter_univ, inter_comm, ← hU, iUnion_inter]
+  exact isOpen_iUnion fun i ↦ h i
+
 @[simp] theorem isOpen_empty : IsOpen (∅ : Set α) := by
   rw [← sUnion_empty]; exact isOpen_sUnion fun a => False.elim
 #align is_open_empty isOpen_empty

• Generalize interior_iInter_of_finite and closure_iUnion_of_finite to functions from Sort*.
• Add Set.Finite.interior_biInter, Set.Finite.interior_sInter, Set.Finite.closure_biUnion, and Set.Finite.closure_sUnion.
• Move closure_eq_compl_interior_compl up to golf.

@@ -355,28 +355,29 @@ theorem interior_interior {s : Set α} : interior (interior s) = interior s :=
 
 @[simp]
 theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ interior t :=
-  Subset.antisymm
-    (subset_inter (interior_mono <| inter_subset_left s t)
-      (interior_mono <| inter_subset_right s t))
-    (interior_maximal (inter_subset_inter interior_subset interior_subset) <|
-      IsOpen.inter isOpen_interior isOpen_interior)
+  (Monotone.map_inf_le (fun _ _ ↦ interior_mono) s t).antisymm <|
+    interior_maximal (inter_subset_inter interior_subset interior_subset) <|
+      isOpen_interior.inter isOpen_interior
 #align interior_inter interior_inter
 
+theorem Set.Finite.interior_biInter {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set α) :
+    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) :=
+  hs.induction_on (by simp) <| by intros; simp [*]
+
+theorem Set.Finite.interior_sInter {S : Set (Set α)} (hS : S.Finite) :
+    interior (⋂₀ S) = ⋂ s ∈ S, interior s := by
+  rw [sInter_eq_biInter, hS.interior_biInter]
+
 @[simp]
 theorem Finset.interior_iInter {ι : Type*} (s : Finset ι) (f : ι → Set α) :
-    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by
-  classical
-    refine' s.induction_on (by simp) _
-    intro i s _ h₂
-    simp [h₂]
+    interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) :=
+  s.finite_toSet.interior_biInter f
 #align finset.interior_Inter Finset.interior_iInter
 
--- todo: generalize to `ι : Sort*`
 @[simp]
-theorem interior_iInter_of_finite {ι : Type*} [Finite ι] (f : ι → Set α) :
+theorem interior_iInter_of_finite [Finite ι] (f : ι → Set α) :
     interior (⋂ i, f i) = ⋂ i, interior (f i) := by
-  cases nonempty_fintype ι
-  convert Finset.univ.interior_iInter f <;> simp
+  rw [← sInter_range, (finite_range f).interior_sInter, biInter_range]
 #align interior_Inter interior_iInter_of_finite
 
 theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClosed s)
@@ -527,39 +528,40 @@ theorem closure_closure {s : Set α} : closure (closure s) = closure s :=
   isClosed_closure.closure_eq
 #align closure_closure closure_closure
 
+theorem closure_eq_compl_interior_compl {s : Set α} : closure s = (interior sᶜ)ᶜ := by
+  rw [interior, closure, compl_sUnion, compl_image_set_of]
+  simp only [compl_subset_compl, isOpen_compl_iff]
+#align closure_eq_compl_interior_compl closure_eq_compl_interior_compl
+
 @[simp]
-theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure t :=
-  Subset.antisymm
-    (closure_minimal (union_subset_union subset_closure subset_closure) <|
-      IsClosed.union isClosed_closure isClosed_closure)
-    ((monotone_closure α).le_map_sup s t)
+theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure t := by
+  simp [closure_eq_compl_interior_compl, compl_inter]
 #align closure_union closure_union
 
+theorem Set.Finite.closure_biUnion {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set α) :
+    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
+  simp [closure_eq_compl_interior_compl, hs.interior_biInter]
+
+theorem Set.Finite.closure_sUnion {S : Set (Set α)} (hS : S.Finite) :
+    closure (⋃₀ S) = ⋃ s ∈ S, closure s := by
+  rw [sUnion_eq_biUnion, hS.closure_biUnion]
+
 @[simp]
 theorem Finset.closure_biUnion {ι : Type*} (s : Finset ι) (f : ι → Set α) :
-    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
-  classical
-    refine' s.induction_on (by simp) _
-    intro i s _ h₂
-    simp [h₂]
+    closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) :=
+  s.finite_toSet.closure_biUnion f
 #align finset.closure_bUnion Finset.closure_biUnion
 
 @[simp]
-theorem closure_iUnion_of_finite {ι : Type*} [Finite ι] (f : ι → Set α) :
+theorem closure_iUnion_of_finite [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by
-  cases nonempty_fintype ι
-  convert Finset.univ.closure_biUnion f <;> simp
+  rw [← sUnion_range, (finite_range _).closure_sUnion, biUnion_range]
 #align closure_Union closure_iUnion_of_finite
 
 theorem interior_subset_closure {s : Set α} : interior s ⊆ closure s :=
   Subset.trans interior_subset subset_closure
 #align interior_subset_closure interior_subset_closure
 
-theorem closure_eq_compl_interior_compl {s : Set α} : closure s = (interior sᶜ)ᶜ := by
-  rw [interior, closure, compl_sUnion, compl_image_set_of]
-  simp only [compl_subset_compl, isOpen_compl_iff]
-#align closure_eq_compl_interior_compl closure_eq_compl_interior_compl
-
 @[simp]
 theorem interior_compl {s : Set α} : interior sᶜ = (closure s)ᶜ := by
   simp [closure_eq_compl_interior_compl]

It's just used for two declarations; it's clearer without.

@@ -51,9 +51,6 @@ Topology in mathlib heavily uses filters (even more than in Bourbaki). See expla
 topological space, interior, closure, frontier, neighborhood, continuity, continuous function
 -/
 
-set_option autoImplicit true
-
-
 noncomputable section
 
 open Set Filter
@@ -1083,11 +1080,11 @@ instance nhds_neBot {a : α} : NeBot (𝓝 a) :=
   neBot_of_le (pure_le_nhds a)
 #align nhds_ne_bot nhds_neBot
 
-theorem tendsto_nhds_of_eventually_eq {f : β → α} {a : α} (h : ∀ᶠ x in l, f x = a) :
+theorem tendsto_nhds_of_eventually_eq {l : Filter β} {f : β → α} {a : α} (h : ∀ᶠ x in l, f x = a) :
     Tendsto f l (𝓝 a) :=
   tendsto_const_nhds.congr' (.symm h)
 
-theorem Filter.EventuallyEq.tendsto {f : β → α} {a : α} (hf : f =ᶠ[l] fun _ ↦ a) :
+theorem Filter.EventuallyEq.tendsto {l : Filter β} {f : β → α} {a : α} (hf : f =ᶠ[l] fun _ ↦ a) :
     Tendsto f l (𝓝 a) :=
   tendsto_nhds_of_eventually_eq hf
 

Subset of #9319

@@ -901,7 +901,7 @@ theorem nhds_le_of_le {f a} {s : Set α} (h : a ∈ s) (o : IsOpen s) (sf : 𝓟
 
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t ⊆ s, _`
 theorem mem_nhds_iff {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ a ∈ t :=
-  (nhds_basis_opens a).mem_iff.trans <| exists_congr <| fun _ =>
+  (nhds_basis_opens a).mem_iff.trans <| exists_congr fun _ =>
     ⟨fun h => ⟨h.2, h.1.2, h.1.1⟩, fun h => ⟨⟨h.2.2, h.2.1⟩, h.1⟩⟩
 #align mem_nhds_iff mem_nhds_iffₓ
 

This way we can easily change the definition so that it works for topological vector spaces without generalizing any of the theorems right away.

@@ -1326,6 +1326,9 @@ theorem mem_closure_iff_nhdsWithin_neBot {s : Set α} {x : α} : x ∈ closure s
   mem_closure_iff_clusterPt
 #align mem_closure_iff_nhds_within_ne_bot mem_closure_iff_nhdsWithin_neBot
 
+lemma not_mem_closure_iff_nhdsWithin_eq_bot {s : Set α} {x : α} : x ∉ closure s ↔ 𝓝[s] x = ⊥ := by
+  rw [mem_closure_iff_nhdsWithin_neBot, not_neBot]
+
 /-- If `x` is not an isolated point of a topological space, then `{x}ᶜ` is dense in the whole
 space. -/
 theorem dense_compl_singleton (x : α) [NeBot (𝓝[≠] x)] : Dense ({x}ᶜ : Set α) := by

We use the standard definition, with respect to the preferred charts at each point.

Open-ness of the interior is non-trivial, hence left to a future PR:

• for instance, in finite dimensions this requires e.g. knowing the homology of spheres, which mathlib doesn't yet have.

Co-authored-by: Winston Yin <winstonyin@gmail.com>

@@ -709,6 +709,11 @@ theorem closure_diff_interior (s : Set α) : closure s \ interior s = frontier s
   rfl
 #align closure_diff_interior closure_diff_interior
 
+/-- Interior and frontier are disjoint. -/
+lemma disjoint_interior_frontier : Disjoint (interior s) (frontier s) := by
+  rw [disjoint_iff_inter_eq_empty, ← closure_diff_interior, diff_eq,
+    ← inter_assoc, inter_comm, ← inter_assoc, compl_inter_self, empty_inter]
+
 @[simp]
 theorem closure_diff_frontier (s : Set α) : closure s \ frontier s = interior s := by
   rw [frontier, diff_diff_right_self, inter_eq_self_of_subset_right interior_subset_closure]

Function.support is a very basic definition. Nevertheless, it is a pretty heavy import because it imports most objects a support lemma can be written about.

This PR reverses the dependencies between those objects and Function.support, so that the latter can become a much more lightweight import.

Only two import could not easily be reversed, namely the ones to Data.Set.Finite and Order.ConditionallyCompleteLattice.Basic, so I created two new files instead.

I credit:

• Yury for https://github.com/leanprover-community/mathlib/pull/6791
• Oliver for https://github.com/leanprover-community/mathlib/pull/7439

@@ -3,9 +3,9 @@ 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, Jeremy Avigad
 -/
-import Mathlib.Order.Filter.Ultrafilter
-import Mathlib.Algebra.Support
+import Mathlib.Algebra.Function.Support
 import Mathlib.Order.Filter.Lift
+import Mathlib.Order.Filter.Ultrafilter
 import Mathlib.Tactic.Continuity
 
 #align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"

• Written by @PatrickMassot in #7851
• Needed for #7851

@@ -1415,6 +1415,12 @@ theorem isClosed_iff_nhds {s : Set α} :
   simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_neBot_iff]
 #align is_closed_iff_nhds isClosed_iff_nhds
 
+lemma isClosed_iff_forall_filter {s : Set α} :
+    IsClosed s ↔ ∀ x, ∀ F : Filter α, F.NeBot → F ≤ 𝓟 s → F ≤ 𝓝 x → x ∈ s := by
+  simp_rw [isClosed_iff_clusterPt]
+  exact ⟨fun hs x F F_ne FS Fx ↦ hs _ <| NeBot.mono F_ne (le_inf Fx FS),
+         fun hs x hx ↦ hs x (𝓝 x ⊓ 𝓟 s) hx inf_le_right inf_le_left⟩
+
 theorem IsClosed.interior_union_left {s t : Set α} (_ : IsClosed s) :
     interior (s ∪ t) ⊆ s ∪ interior t := fun a ⟨u, ⟨⟨hu₁, hu₂⟩, ha⟩⟩ =>
   (Classical.em (a ∈ s)).imp_right fun h =>

@@ -812,7 +812,7 @@ theorem Disjoint.frontier_right (hs : IsOpen s) (hd : Disjoint s t) : Disjoint s
 
 theorem frontier_eq_inter_compl_interior {s : Set α} :
     frontier s = (interior s)ᶜ ∩ (interior sᶜ)ᶜ := by
-  rw [← frontier_compl, ← closure_compl]; rfl
+  rw [← frontier_compl, ← closure_compl, ← diff_eq, closure_diff_interior]
 #align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interior
 
 theorem compl_frontier_eq_union_interior {s : Set α} :
@@ -1197,7 +1197,7 @@ theorem acc_iff_cluster (x : α) (F : Filter α) : AccPt x F ↔ ClusterPt x (
 /-- `x` is an accumulation point of a set `C` iff it is a cluster point of `C ∖ {x}`.-/
 theorem acc_principal_iff_cluster (x : α) (C : Set α) :
     AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x})) := by
-  rw [acc_iff_cluster, inf_principal, inter_comm]; rfl
+  rw [acc_iff_cluster, inf_principal, inter_comm, diff_eq]
 #align acc_principal_iff_cluster acc_principal_iff_cluster
 
 /-- `x` is an accumulation point of a set `C` iff every neighborhood
@@ -1252,7 +1252,7 @@ theorem subset_interior_iff_nhds {s V : Set α} : s ⊆ interior V ↔ ∀ x ∈
 theorem isOpen_iff_nhds {s : Set α} : IsOpen s ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s :=
   calc
     IsOpen s ↔ s ⊆ interior s := subset_interior_iff_isOpen.symm
-    _ ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s := by rw [interior_eq_nhds]; rfl
+    _ ↔ ∀ a ∈ s, 𝓝 a ≤ 𝓟 s := by simp_rw [interior_eq_nhds, subset_def, mem_setOf]
 #align is_open_iff_nhds isOpen_iff_nhds
 
 theorem isOpen_iff_mem_nhds {s : Set α} : IsOpen s ↔ ∀ a ∈ s, s ∈ 𝓝 a :=
@@ -1287,7 +1287,7 @@ theorem isOpen_singleton_iff_punctured_nhds {α : Type*} [TopologicalSpace α] (
 
 theorem mem_closure_iff_frequently {s : Set α} {a : α} : a ∈ closure s ↔ ∃ᶠ x in 𝓝 a, x ∈ s := by
   rw [Filter.Frequently, Filter.Eventually, ← mem_interior_iff_mem_nhds,
-      closure_eq_compl_interior_compl]; rfl
+    closure_eq_compl_interior_compl, mem_compl_iff, compl_def]
 #align mem_closure_iff_frequently mem_closure_iff_frequently
 
 alias ⟨_, Filter.Frequently.mem_closure⟩ := mem_closure_iff_frequently
@@ -1655,7 +1655,9 @@ theorem ContinuousAt.preimage_mem_nhds {f : α → β} {x : α} {t : Set β} (h
 
 theorem eventuallyEq_zero_nhds {M₀} [Zero M₀] {a : α} {f : α → M₀} :
     f =ᶠ[𝓝 a] 0 ↔ a ∉ closure (Function.support f) := by
-  rw [← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds, Function.compl_support]; rfl
+  rw [← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds, Function.compl_support,
+    EventuallyEq, eventually_iff]
+  simp only [Pi.zero_apply]
 #align eventually_eq_zero_nhds eventuallyEq_zero_nhds
 
 theorem ClusterPt.map {x : α} {la : Filter α} {lb : Filter β} (H : ClusterPt x la) {f : α → β}

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

@@ -1457,16 +1457,16 @@ theorem mem_closure_of_mem_closure_union {s₁ s₂ : Set α} {x : α} (h : x 
 #align mem_closure_of_mem_closure_union mem_closure_of_mem_closure_union
 
 /-- The intersection of an open dense set with a dense set is a dense set. -/
-theorem Dense.inter_of_open_left {s t : Set α} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
+theorem Dense.inter_of_isOpen_left {s t : Set α} (hs : Dense s) (ht : Dense t) (hso : IsOpen s) :
     Dense (s ∩ t) := fun x =>
   closure_minimal hso.inter_closure isClosed_closure <| by simp [hs.closure_eq, ht.closure_eq]
-#align dense.inter_of_open_left Dense.inter_of_open_left
+#align dense.inter_of_open_left Dense.inter_of_isOpen_left
 
 /-- The intersection of a dense set with an open dense set is a dense set. -/
-theorem Dense.inter_of_open_right {s t : Set α} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
+theorem Dense.inter_of_isOpen_right {s t : Set α} (hs : Dense s) (ht : Dense t) (hto : IsOpen t) :
     Dense (s ∩ t) :=
-  inter_comm t s ▸ ht.inter_of_open_left hs hto
-#align dense.inter_of_open_right Dense.inter_of_open_right
+  inter_comm t s ▸ ht.inter_of_isOpen_left hs hto
+#align dense.inter_of_open_right Dense.inter_of_isOpen_right
 
 theorem Dense.inter_nhds_nonempty {s t : Set α} (hs : Dense s) {x : α} (ht : t ∈ 𝓝 x) :
     (s ∩ t).Nonempty :=

The point is to get delaborators for free. This was previously blocked on notation3 not supporting scoped.

@@ -848,19 +848,19 @@ scoped[Topology] notation "𝓝" => nhds
 scoped[Topology] notation "𝓝[" s "] " x:100 => nhdsWithin x s
 
 /-- Notation for the filter of punctured neighborhoods of a point. -/
-scoped[Topology] notation "𝓝[≠] " x:100 => nhdsWithin x (@singleton _ (Set _) instSingletonSet x)ᶜ
+scoped[Topology] notation3 "𝓝[≠] " x:100 => nhdsWithin x (@singleton _ (Set _) instSingletonSet x)ᶜ
 
 /-- Notation for the filter of right neighborhoods of a point. -/
-scoped[Topology] notation "𝓝[≥] " x:100 => nhdsWithin x (Set.Ici x)
+scoped[Topology] notation3 "𝓝[≥] " x:100 => nhdsWithin x (Set.Ici x)
 
 /-- Notation for the filter of left neighborhoods of a point. -/
-scoped[Topology] notation "𝓝[≤] " x:100 => nhdsWithin x (Set.Iic x)
+scoped[Topology] notation3 "𝓝[≤] " x:100 => nhdsWithin x (Set.Iic x)
 
 /-- Notation for the filter of punctured right neighborhoods of a point. -/
-scoped[Topology] notation "𝓝[>] " x:100 => nhdsWithin x (Set.Ioi x)
+scoped[Topology] notation3 "𝓝[>] " x:100 => nhdsWithin x (Set.Ioi x)
 
 /-- Notation for the filter of punctured left neighborhoods of a point. -/
-scoped[Topology] notation "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
+scoped[Topology] notation3 "𝓝[<] " x:100 => nhdsWithin x (Set.Iio x)
 
 end
 

PR contents

This is the supremum of

• #8284
• #8056
• #8023
• #8332
• #8226 (already approved)
• #7834 (already approved)

along with some minor fixes from failures on nightly-testing as Mathlib master is merged into it.

Note that some PRs for changes that are already compatible with the current toolchain and will be necessary have already been split out: #8380.

I am hopeful that in future we will be able to progressively merge adaptation PRs into a bump/v4.X.0 branch, so we never end up with a "big merge" like this. However one of these adaptation PRs (#8056) predates my new scheme for combined CI, and it wasn't possible to keep that PR viable in the meantime.

Lean PRs involved in this bump

In particular this includes adjustments for the Lean PRs

• leanprover/lean4#2778
• leanprover/lean4#2790
• leanprover/lean4#2783
• leanprover/lean4#2825
• leanprover/lean4#2722

leanprover/lean4#2778

We can get rid of all the

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue [lean4#2220](https://github.com/leanprover/lean4/pull/2220)

macros across Mathlib (and in any projects that want to write natural number powers of reals).

leanprover/lean4#2722

Changes the default behaviour of simp to (config := {decide := false}). This makes simp (and consequentially norm_num) less powerful, but also more consistent, and less likely to blow up in long failures. This requires a variety of changes: changing some previously by simp or norm_num to decide or rfl, or adding (config := {decide := true}).

leanprover/lean4#2783

This changed the behaviour of simp so that simp [f] will only unfold "fully applied" occurrences of f. The old behaviour can be recovered with simp (config := { unfoldPartialApp := true }). We may in future add a syntax for this, e.g. simp [!f]; please provide feedback! In the meantime, we have made the following changes:

• switching to using explicit lemmas that have the intended level of application
• (config := { unfoldPartialApp := true }) in some places, to recover the old behaviour
• Using @[eqns] to manually adjust the equation lemmas for a particular definition, recovering the old behaviour just for that definition. See #8371, where we do this for Function.comp and Function.flip.

This change in Lean may require further changes down the line (e.g. adding the !f syntax, and/or upstreaming the special treatment for Function.comp and Function.flip, and/or removing this special treatment). Please keep an open and skeptical mind about these changes!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Mauricio Collares <mauricio@collares.org>

@@ -848,7 +848,7 @@ scoped[Topology] notation "𝓝" => nhds
 scoped[Topology] notation "𝓝[" s "] " x:100 => nhdsWithin x s
 
 /-- Notation for the filter of punctured neighborhoods of a point. -/
-scoped[Topology] notation "𝓝[≠] " x:100 => nhdsWithin x {x}ᶜ
+scoped[Topology] notation "𝓝[≠] " x:100 => nhdsWithin x (@singleton _ (Set _) instSingletonSet x)ᶜ
 
 /-- Notation for the filter of right neighborhoods of a point. -/
 scoped[Topology] notation "𝓝[≥] " x:100 => nhdsWithin x (Set.Ici x)

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

@@ -1195,8 +1195,9 @@ theorem acc_iff_cluster (x : α) (F : Filter α) : AccPt x F ↔ ClusterPt x (
 #align acc_iff_cluster acc_iff_cluster
 
 /-- `x` is an accumulation point of a set `C` iff it is a cluster point of `C ∖ {x}`.-/
-theorem acc_principal_iff_cluster (x : α) (C : Set α) : AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x})) :=
-  by rw [acc_iff_cluster, inf_principal, inter_comm]; rfl
+theorem acc_principal_iff_cluster (x : α) (C : Set α) :
+    AccPt x (𝓟 C) ↔ ClusterPt x (𝓟 (C \ {x})) := by
+  rw [acc_iff_cluster, inf_principal, inter_comm]; rfl
 #align acc_principal_iff_cluster acc_principal_iff_cluster
 
 /-- `x` is an accumulation point of a set `C` iff every neighborhood
@@ -1409,8 +1410,9 @@ theorem isClosed_iff_clusterPt {s : Set α} : IsClosed s ↔ ∀ a, ClusterPt a
     _ ↔ ∀ a, ClusterPt a (𝓟 s) → a ∈ s := by simp only [subset_def, mem_closure_iff_clusterPt]
 #align is_closed_iff_cluster_pt isClosed_iff_clusterPt
 
-theorem isClosed_iff_nhds {s : Set α} : IsClosed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s :=
-  by simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_neBot_iff]
+theorem isClosed_iff_nhds {s : Set α} :
+    IsClosed s ↔ ∀ x, (∀ U ∈ 𝓝 x, (U ∩ s).Nonempty) → x ∈ s := by
+  simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_neBot_iff]
 #align is_closed_iff_nhds isClosed_iff_nhds
 
 theorem IsClosed.interior_union_left {s t : Set α} (_ : IsClosed s) :

• The extra import doesn't import any additional files transitively.
• From the Sobolev project

Co-authored-by: Heather Macbeth 25316162+hrmacbeth@users.noreply.github.com

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

@@ -1082,6 +1082,10 @@ theorem tendsto_nhds_of_eventually_eq {f : β → α} {a : α} (h : ∀ᶠ x in
     Tendsto f l (𝓝 a) :=
   tendsto_const_nhds.congr' (.symm h)
 
+theorem Filter.EventuallyEq.tendsto {f : β → α} {a : α} (hf : f =ᶠ[l] fun _ ↦ a) :
+    Tendsto f l (𝓝 a) :=
+  tendsto_nhds_of_eventually_eq hf
+
 /-!
 ### Cluster points
 

Generalizing one of the proofs introduced in #7878

@@ -1609,10 +1609,12 @@ theorem IsOpen.preimage {f : α → β} (hf : Continuous f) {s : Set β} (h : Is
   hf.isOpen_preimage s h
 #align is_open.preimage IsOpen.preimage
 
-theorem Continuous.congr {f g : α → β} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g := by
-  convert h
-  ext
-  rw [h']
+theorem continuous_congr {f g : α → β} (h : ∀ x, f x = g x) :
+    Continuous f ↔ Continuous g :=
+  .of_eq <| congrArg _ <| funext h
+
+theorem Continuous.congr {f g : α → β} (h : Continuous f) (h' : ∀ x, f x = g x) : Continuous g :=
+  continuous_congr h' |>.mp h
 #align continuous.congr Continuous.congr
 
 /-- A function between topological spaces is continuous at a point `x₀`

In https://github.com/leanprover-community/mathlib/pull/19146, we defined StructureGroupoid.LocalInvariantProp.sheaf, the sheaf-of-types of functions f : M → M' (for charted spaces M, M') satisfying some local property in the sense of StructureGroupoid.LocalInvariantProp (for example continuity, differentiability, smoothness).

In this PR, in the case of smoothness, we upgrade this to a sheaf of groups if M' is a Lie group, a sheaf of abelian groups if M' is an abelian Lie group, and a sheaf of rings if M' is a "smooth ring".

Co-authored-by: Adam Topaz <github@adamtopaz.com>

Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com>

@@ -66,6 +66,7 @@ universe u v w
 
 
 /-- A topology on `α`. -/
+@[to_additive existing TopologicalSpace]
 class TopologicalSpace (α : Type u) where
   /-- A predicate saying that a set is an open set. Use `IsOpen` in the root namespace instead. -/
   protected IsOpen : Set α → Prop

@@ -401,15 +401,15 @@ theorem interior_iInter_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ 
   subset_iInter fun _ => interior_mono <| iInter_subset _ _
 #align interior_Inter_subset interior_iInter_subset
 
-theorem interior_Inter₂_subset (p : ι → Sort*) (s : ∀ i, p i → Set α) :
+theorem interior_iInter₂_subset (p : ι → Sort*) (s : ∀ i, p i → Set α) :
     interior (⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), interior (s i j) :=
   (interior_iInter_subset _).trans <| iInter_mono fun _ => interior_iInter_subset _
-#align interior_Inter₂_subset interior_Inter₂_subset
+#align interior_Inter₂_subset interior_iInter₂_subset
 
 theorem interior_sInter_subset (S : Set (Set α)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s :=
   calc
     interior (⋂₀ S) = interior (⋂ s ∈ S, s) := by rw [sInter_eq_biInter]
-    _ ⊆ ⋂ s ∈ S, interior s := interior_Inter₂_subset _ _
+    _ ⊆ ⋂ s ∈ S, interior s := interior_iInter₂_subset _ _
 #align interior_sInter_subset interior_sInter_subset
 
 /-!
@@ -547,11 +547,11 @@ theorem Finset.closure_biUnion {ι : Type*} (s : Finset ι) (f : ι → Set α)
 #align finset.closure_bUnion Finset.closure_biUnion
 
 @[simp]
-theorem closure_Union_of_finite {ι : Type*} [Finite ι] (f : ι → Set α) :
+theorem closure_iUnion_of_finite {ι : Type*} [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by
   cases nonempty_fintype ι
   convert Finset.univ.closure_biUnion f <;> simp
-#align closure_Union closure_Union_of_finite
+#align closure_Union closure_iUnion_of_finite
 
 theorem interior_subset_closure {s : Set α} : interior s ⊆ closure s :=
   Subset.trans interior_subset subset_closure

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

@@ -168,7 +168,7 @@ theorem Set.Finite.isOpen_sInter {s : Set (Set α)} (hs : s.Finite) :
 #align is_open_sInter Set.Finite.isOpen_sInter
 
 theorem Set.Finite.isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finite)
-  (h : ∀ i ∈ s, IsOpen (f i)) :
+    (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
   sInter_image f s ▸ (hs.image _).isOpen_sInter (ball_image_iff.2 h)
 #align is_open_bInter Set.Finite.isOpen_biInter
@@ -254,7 +254,7 @@ theorem IsClosed.sdiff {s t : Set α} (h₁ : IsClosed s) (h₂ : IsOpen t) : Is
 #align is_closed.sdiff IsClosed.sdiff
 
 theorem Set.Finite.isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite)
-  (h : ∀ i ∈ s, IsClosed (f i)) :
+    (h : ∀ i ∈ s, IsClosed (f i)) :
     IsClosed (⋃ i ∈ s, f i) := by
   simp only [← isOpen_compl_iff, compl_iUnion] at *
   exact hs.isOpen_biInter h
@@ -2004,8 +2004,8 @@ get something like this: (assumption `hf` could be weakened, but the important t
 of the conclusion)
 ```
 lemma ContinuousOn.comp_fract {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-  {f : X → ℝ → Y} {g : X → ℝ} (hf : Continuous ↿f) (hg : Continuous g) (h : ∀ s, f s 0 = f s 1) :
-  Continuous (fun x ↦ f x (fract (g x)))
+    {f : X → ℝ → Y} {g : X → ℝ} (hf : Continuous ↿f) (hg : Continuous g) (h : ∀ s, f s 0 = f s 1) :
+    Continuous (fun x ↦ f x (fract (g x)))
 ```
 With `ContinuousAt` you can be even more precise about what to prove in case of discontinuities,
 see e.g. `ContinuousAt.comp_div_cases`.

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

@@ -161,7 +161,7 @@ theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ 
 #align is_open_empty isOpen_empty
 
 theorem Set.Finite.isOpen_sInter {s : Set (Set α)} (hs : s.Finite) :
-  (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
+    (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
   Finite.induction_on hs (fun _ => by rw [sInter_empty]; exact isOpen_univ) fun _ _ ih h => by
     simp only [sInter_insert, ball_insert_iff] at h ⊢
     exact h.1.inter (ih h.2)
@@ -174,7 +174,7 @@ theorem Set.Finite.isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finit
 #align is_open_bInter Set.Finite.isOpen_biInter
 
 theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) :
-  IsOpen (⋂ i, s i) :=
+    IsOpen (⋂ i, s i) :=
   (finite_range _).isOpen_sInter  (forall_range_iff.2 h)
 #align is_open_Inter isOpen_iInter_of_finite
 

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.

@@ -327,6 +327,9 @@ theorem subset_interior_iff {s t : Set α} : t ⊆ interior s ↔ ∃ U, IsOpen
     htU.trans (interior_maximal hUs hU)⟩
 #align subset_interior_iff subset_interior_iff
 
+lemma interior_subset_iff : interior s ⊆ t ↔ ∀ U, IsOpen U → U ⊆ s → U ⊆ t := by
+  simp [interior]
+
 @[mono]
 theorem interior_mono {s t : Set α} (h : s ⊆ t) : interior s ⊆ interior t :=
   interior_maximal (Subset.trans interior_subset h) isOpen_interior

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.

@@ -160,24 +160,27 @@ theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ 
   rw [← sUnion_empty]; exact isOpen_sUnion fun a => False.elim
 #align is_open_empty isOpen_empty
 
-theorem isOpen_sInter {s : Set (Set α)} (hs : s.Finite) : (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
+theorem Set.Finite.isOpen_sInter {s : Set (Set α)} (hs : s.Finite) :
+  (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
   Finite.induction_on hs (fun _ => by rw [sInter_empty]; exact isOpen_univ) fun _ _ ih h => by
     simp only [sInter_insert, ball_insert_iff] at h ⊢
     exact h.1.inter (ih h.2)
-#align is_open_sInter isOpen_sInter
+#align is_open_sInter Set.Finite.isOpen_sInter
 
-theorem isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finite) (h : ∀ i ∈ s, IsOpen (f i)) :
+theorem Set.Finite.isOpen_biInter {s : Set β} {f : β → Set α} (hs : s.Finite)
+  (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
-  sInter_image f s ▸ isOpen_sInter (hs.image _) (ball_image_iff.2 h)
-#align is_open_bInter isOpen_biInter
+  sInter_image f s ▸ (hs.image _).isOpen_sInter (ball_image_iff.2 h)
+#align is_open_bInter Set.Finite.isOpen_biInter
 
-theorem isOpen_iInter [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) : IsOpen (⋂ i, s i) :=
-  isOpen_sInter (finite_range _) (forall_range_iff.2 h)
-#align is_open_Inter isOpen_iInter
+theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set α} (h : ∀ i, IsOpen (s i)) :
+  IsOpen (⋂ i, s i) :=
+  (finite_range _).isOpen_sInter  (forall_range_iff.2 h)
+#align is_open_Inter isOpen_iInter_of_finite
 
 theorem isOpen_biInter_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
-  isOpen_biInter s.finite_toSet h
+  s.finite_toSet.isOpen_biInter h
 #align is_open_bInter_finset isOpen_biInter_finset
 
 @[simp] -- porting note: added `simp`
@@ -250,17 +253,22 @@ theorem IsClosed.sdiff {s t : Set α} (h₁ : IsClosed s) (h₂ : IsOpen t) : Is
   IsClosed.inter h₁ (isClosed_compl_iff.mpr h₂)
 #align is_closed.sdiff IsClosed.sdiff
 
-theorem isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite) (h : ∀ i ∈ s, IsClosed (f i)) :
+theorem Set.Finite.isClosed_biUnion {s : Set β} {f : β → Set α} (hs : s.Finite)
+  (h : ∀ i ∈ s, IsClosed (f i)) :
     IsClosed (⋃ i ∈ s, f i) := by
   simp only [← isOpen_compl_iff, compl_iUnion] at *
-  exact isOpen_biInter hs h
-#align is_closed_bUnion isClosed_biUnion
+  exact hs.isOpen_biInter h
+#align is_closed_bUnion Set.Finite.isClosed_biUnion
 
-theorem isClosed_iUnion [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
+lemma isClosed_biUnion_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsClosed (f i)) :
+    IsClosed (⋃ i ∈ s, f i) :=
+  s.finite_toSet.isClosed_biUnion h
+
+theorem isClosed_iUnion_of_finite [Finite ι] {s : ι → Set α} (h : ∀ i, IsClosed (s i)) :
     IsClosed (⋃ i, s i) := by
   simp only [← isOpen_compl_iff, compl_iUnion] at *
-  exact isOpen_iInter h
-#align is_closed_Union isClosed_iUnion
+  exact isOpen_iInter_of_finite h
+#align is_closed_Union isClosed_iUnion_of_finite
 
 theorem isClosed_imp {p q : α → Prop} (hp : IsOpen { x | p x }) (hq : IsClosed { x | q x }) :
     IsClosed { x | p x → q x } := by
@@ -364,11 +372,11 @@ theorem Finset.interior_iInter {ι : Type*} (s : Finset ι) (f : ι → Set α)
 
 -- todo: generalize to `ι : Sort*`
 @[simp]
-theorem interior_iInter {ι : Type*} [Finite ι] (f : ι → Set α) :
+theorem interior_iInter_of_finite {ι : Type*} [Finite ι] (f : ι → Set α) :
     interior (⋂ i, f i) = ⋂ i, interior (f i) := by
   cases nonempty_fintype ι
   convert Finset.univ.interior_iInter f <;> simp
-#align interior_Inter interior_iInter
+#align interior_Inter interior_iInter_of_finite
 
 theorem interior_union_isClosed_of_interior_empty {s t : Set α} (h₁ : IsClosed s)
     (h₂ : interior t = ∅) : interior (s ∪ t) = interior s :=
@@ -536,11 +544,11 @@ theorem Finset.closure_biUnion {ι : Type*} (s : Finset ι) (f : ι → Set α)
 #align finset.closure_bUnion Finset.closure_biUnion
 
 @[simp]
-theorem closure_iUnion {ι : Type*} [Finite ι] (f : ι → Set α) :
+theorem closure_Union_of_finite {ι : Type*} [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by
   cases nonempty_fintype ι
   convert Finset.univ.closure_biUnion f <;> simp
-#align closure_Union closure_iUnion
+#align closure_Union closure_Union_of_finite
 
 theorem interior_subset_closure {s : Set α} : interior s ⊆ closure s :=
   Subset.trans interior_subset subset_closure

• After the new alias command we can now do protected alias
• alias at some point broke dot notation by unfolding (see #1022) this was fixed in #1058 but the library was not fixed up there

@@ -626,10 +626,7 @@ theorem dense_closure {s : Set α} : Dense (closure s) ↔ Dense s := by
   rw [Dense, Dense, closure_closure]
 #align dense_closure dense_closure
 
--- porting note: todo: use `alias` + `@[protected]`
-protected lemma Dense.closure {s : Set α} (h : Dense s) : Dense (closure s) :=
-  dense_closure.2 h
-
+protected alias ⟨_, Dense.closure⟩ := dense_closure
 alias ⟨Dense.of_closure, _⟩ := dense_closure
 #align dense.of_closure Dense.of_closure
 #align dense.closure Dense.closure

@@ -233,7 +233,7 @@ theorem isClosed_compl_iff {s : Set α} : IsClosed sᶜ ↔ IsOpen s := by
   rw [← isOpen_compl_iff, compl_compl]
 #align is_closed_compl_iff isClosed_compl_iff
 
-alias isClosed_compl_iff ↔ _ IsOpen.isClosed_compl
+alias ⟨_, IsOpen.isClosed_compl⟩ := isClosed_compl_iff
 #align is_open.is_closed_compl IsOpen.isClosed_compl
 
 theorem IsOpen.sdiff {s t : Set α} (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) :=
@@ -504,7 +504,7 @@ theorem closure_nonempty_iff {s : Set α} : (closure s).Nonempty ↔ s.Nonempty
   simp only [nonempty_iff_ne_empty, Ne.def, closure_empty_iff]
 #align closure_nonempty_iff closure_nonempty_iff
 
-alias closure_nonempty_iff ↔ Set.Nonempty.of_closure Set.Nonempty.closure
+alias ⟨Set.Nonempty.of_closure, Set.Nonempty.closure⟩ := closure_nonempty_iff
 #align set.nonempty.of_closure Set.Nonempty.of_closure
 #align set.nonempty.closure Set.Nonempty.closure
 
@@ -609,7 +609,7 @@ theorem dense_iff_closure_eq {s : Set α} : Dense s ↔ closure s = univ :=
   eq_univ_iff_forall.symm
 #align dense_iff_closure_eq dense_iff_closure_eq
 
-alias dense_iff_closure_eq ↔ Dense.closure_eq _
+alias ⟨Dense.closure_eq, _⟩ := dense_iff_closure_eq
 #align dense.closure_eq Dense.closure_eq
 
 theorem interior_eq_empty_iff_dense_compl {s : Set α} : interior s = ∅ ↔ Dense sᶜ := by
@@ -630,7 +630,7 @@ theorem dense_closure {s : Set α} : Dense (closure s) ↔ Dense s := by
 protected lemma Dense.closure {s : Set α} (h : Dense s) : Dense (closure s) :=
   dense_closure.2 h
 
-alias dense_closure ↔ Dense.of_closure _
+alias ⟨Dense.of_closure, _⟩ := dense_closure
 #align dense.of_closure Dense.of_closure
 #align dense.closure Dense.closure
 
@@ -650,7 +650,7 @@ theorem dense_iff_inter_open {s : Set α} :
     exact h U U_op ⟨_, x_in⟩
 #align dense_iff_inter_open dense_iff_inter_open
 
-alias dense_iff_inter_open ↔ Dense.inter_open_nonempty _
+alias ⟨Dense.inter_open_nonempty, _⟩ := dense_iff_inter_open
 #align dense.inter_open_nonempty Dense.inter_open_nonempty
 
 theorem Dense.exists_mem_open {s : Set α} (hs : Dense s) {U : Set α} (ho : IsOpen U)
@@ -1276,7 +1276,7 @@ theorem mem_closure_iff_frequently {s : Set α} {a : α} : a ∈ closure s ↔ 
       closure_eq_compl_interior_compl]; rfl
 #align mem_closure_iff_frequently mem_closure_iff_frequently
 
-alias mem_closure_iff_frequently ↔ _ Filter.Frequently.mem_closure
+alias ⟨_, Filter.Frequently.mem_closure⟩ := mem_closure_iff_frequently
 #align filter.frequently.mem_closure Filter.Frequently.mem_closure
 
 /-- A set `s` is closed iff for every point `x`, if there is a point `y` close to `x` that belongs

@@ -110,9 +110,10 @@ lemma isOpen_mk {p h₁ h₂ h₃} {s : Set α} : IsOpen[⟨p, h₁, h₂, h₃
 #align is_open_mk isOpen_mk
 
 @[ext]
-theorem topologicalSpace_eq : ∀ {f g : TopologicalSpace α}, IsOpen[f] = IsOpen[g] → f = g
+protected theorem TopologicalSpace.ext :
+    ∀ {f g : TopologicalSpace α}, IsOpen[f] = IsOpen[g] → f = g
   | ⟨_, _, _, _⟩, ⟨_, _, _, _⟩, rfl => rfl
-#align topological_space_eq topologicalSpace_eq
+#align topological_space_eq TopologicalSpace.ext
 
 section
 
@@ -131,10 +132,10 @@ theorem isOpen_sUnion {s : Set (Set α)} (h : ∀ t ∈ s, IsOpen t) : IsOpen (
 
 end
 
-theorem topologicalSpace_eq_iff {t t' : TopologicalSpace α} :
+protected theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace α} :
     t = t' ↔ ∀ s, IsOpen[t] s ↔ IsOpen[t'] s :=
   ⟨fun h s => h ▸ Iff.rfl, fun h => by ext; exact h _⟩
-#align topological_space_eq_iff topologicalSpace_eq_iff
+#align topological_space_eq_iff TopologicalSpace.ext_iff
 
 theorem isOpen_fold {s : Set α} {t : TopologicalSpace α} : t.IsOpen s = IsOpen[t] s :=
   rfl

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

• Assuming variables are in scope, but pasting the lemma in the wrong section
• Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
• Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

@@ -51,6 +51,8 @@ Topology in mathlib heavily uses filters (even more than in Bourbaki). See expla
 topological space, interior, closure, frontier, neighborhood, continuity, continuous function
 -/
 
+set_option autoImplicit true
+
 
 noncomputable section
 

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

This has nice performance benefits.

@@ -351,7 +351,7 @@ theorem interior_inter {s t : Set α} : interior (s ∩ t) = interior s ∩ inte
 #align interior_inter interior_inter
 
 @[simp]
-theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α) :
+theorem Finset.interior_iInter {ι : Type*} (s : Finset ι) (f : ι → Set α) :
     interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) := by
   classical
     refine' s.induction_on (by simp) _
@@ -359,9 +359,9 @@ theorem Finset.interior_iInter {ι : Type _} (s : Finset ι) (f : ι → Set α)
     simp [h₂]
 #align finset.interior_Inter Finset.interior_iInter
 
--- todo: generalize to `ι : Sort _`
+-- todo: generalize to `ι : Sort*`
 @[simp]
-theorem interior_iInter {ι : Type _} [Finite ι] (f : ι → Set α) :
+theorem interior_iInter {ι : Type*} [Finite ι] (f : ι → Set α) :
     interior (⋂ i, f i) = ⋂ i, interior (f i) := by
   cases nonempty_fintype ι
   convert Finset.univ.interior_iInter f <;> simp
@@ -387,7 +387,7 @@ theorem interior_iInter_subset (s : ι → Set α) : interior (⋂ i, s i) ⊆ 
   subset_iInter fun _ => interior_mono <| iInter_subset _ _
 #align interior_Inter_subset interior_iInter_subset
 
-theorem interior_Inter₂_subset (p : ι → Sort _) (s : ∀ i, p i → Set α) :
+theorem interior_Inter₂_subset (p : ι → Sort*) (s : ∀ i, p i → Set α) :
     interior (⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), interior (s i j) :=
   (interior_iInter_subset _).trans <| iInter_mono fun _ => interior_iInter_subset _
 #align interior_Inter₂_subset interior_Inter₂_subset
@@ -461,7 +461,7 @@ theorem closure_mono {s t : Set α} (h : s ⊆ t) : closure s ⊆ closure t :=
   closure_minimal (Subset.trans h subset_closure) isClosed_closure
 #align closure_mono closure_mono
 
-theorem monotone_closure (α : Type _) [TopologicalSpace α] : Monotone (@closure α _) := fun _ _ =>
+theorem monotone_closure (α : Type*) [TopologicalSpace α] : Monotone (@closure α _) := fun _ _ =>
   closure_mono
 #align monotone_closure monotone_closure
 
@@ -524,7 +524,7 @@ theorem closure_union {s t : Set α} : closure (s ∪ t) = closure s ∪ closure
 #align closure_union closure_union
 
 @[simp]
-theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α) :
+theorem Finset.closure_biUnion {ι : Type*} (s : Finset ι) (f : ι → Set α) :
     closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
   classical
     refine' s.induction_on (by simp) _
@@ -533,7 +533,7 @@ theorem Finset.closure_biUnion {ι : Type _} (s : Finset ι) (f : ι → Set α)
 #align finset.closure_bUnion Finset.closure_biUnion
 
 @[simp]
-theorem closure_iUnion {ι : Type _} [Finite ι] (f : ι → Set α) :
+theorem closure_iUnion {ι : Type*} [Finite ι] (f : ι → Set α) :
     closure (⋃ i, f i) = ⋃ i, closure (f i) := by
   cases nonempty_fintype ι
   convert Finset.univ.closure_biUnion f <;> simp
@@ -1038,12 +1038,12 @@ theorem tendsto_const_nhds {a : α} {f : Filter β} : Tendsto (fun _ : β => a)
   tendsto_nhds.mpr fun _ _ ha => univ_mem' fun _ => ha
 #align tendsto_const_nhds tendsto_const_nhds
 
-theorem tendsto_atTop_of_eventually_const {ι : Type _} [SemilatticeSup ι] [Nonempty ι] {x : α}
+theorem tendsto_atTop_of_eventually_const {ι : Type*} [SemilatticeSup ι] [Nonempty ι] {x : α}
     {u : ι → α} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : Tendsto u atTop (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atTop.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_top_of_eventually_const tendsto_atTop_of_eventually_const
 
-theorem tendsto_atBot_of_eventually_const {ι : Type _} [SemilatticeInf ι] [Nonempty ι] {x : α}
+theorem tendsto_atBot_of_eventually_const {ι : Type*} [SemilatticeInf ι] [Nonempty ι] {x : α}
     {u : ι → α} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : Tendsto u atBot (𝓝 x) :=
   Tendsto.congr' (EventuallyEq.symm (eventually_atBot.mpr ⟨i₀, h⟩)) tendsto_const_nhds
 #align tendsto_at_bot_of_eventually_const tendsto_atBot_of_eventually_const
@@ -1051,12 +1051,12 @@ theorem tendsto_atBot_of_eventually_const {ι : Type _} [SemilatticeInf ι] [Non
 theorem pure_le_nhds : pure ≤ (𝓝 : α → Filter α) := fun _ _ hs => mem_pure.2 <| mem_of_mem_nhds hs
 #align pure_le_nhds pure_le_nhds
 
-theorem tendsto_pure_nhds {α : Type _} [TopologicalSpace β] (f : α → β) (a : α) :
+theorem tendsto_pure_nhds {α : Type*} [TopologicalSpace β] (f : α → β) (a : α) :
     Tendsto f (pure a) (𝓝 (f a)) :=
   (tendsto_pure_pure f a).mono_right (pure_le_nhds _)
 #align tendsto_pure_nhds tendsto_pure_nhds
 
-theorem OrderTop.tendsto_atTop_nhds {α : Type _} [PartialOrder α] [OrderTop α] [TopologicalSpace β]
+theorem OrderTop.tendsto_atTop_nhds {α : Type*} [PartialOrder α] [OrderTop α] [TopologicalSpace β]
     (f : α → β) : Tendsto f atTop (𝓝 (f ⊤)) :=
   (tendsto_atTop_pure f).mono_right (pure_le_nhds _)
 #align order_top.tendsto_at_top_nhds OrderTop.tendsto_atTop_nhds
@@ -1148,17 +1148,17 @@ theorem Ultrafilter.clusterPt_iff {x : α} {f : Ultrafilter α} : ClusterPt x f
 
 /-- A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point
 of `map u F`. -/
-def MapClusterPt {ι : Type _} (x : α) (F : Filter ι) (u : ι → α) : Prop :=
+def MapClusterPt {ι : Type*} (x : α) (F : Filter ι) (u : ι → α) : Prop :=
   ClusterPt x (map u F)
 #align map_cluster_pt MapClusterPt
 
-theorem mapClusterPt_iff {ι : Type _} (x : α) (F : Filter ι) (u : ι → α) :
+theorem mapClusterPt_iff {ι : Type*} (x : α) (F : Filter ι) (u : ι → α) :
     MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s := by
   simp_rw [MapClusterPt, ClusterPt, inf_neBot_iff_frequently_left, frequently_map]
   rfl
 #align map_cluster_pt_iff mapClusterPt_iff
 
-theorem mapClusterPt_of_comp {ι δ : Type _} {F : Filter ι} {φ : δ → ι} {p : Filter δ} {x : α}
+theorem mapClusterPt_of_comp {ι δ : Type*} {F : Filter ι} {φ : δ → ι} {p : Filter δ} {x : α}
     {u : ι → α} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) :
     MapClusterPt x F u := by
   have :=
@@ -1262,7 +1262,7 @@ theorem isOpen_singleton_iff_nhds_eq_pure (a : α) : IsOpen ({a} : Set α) ↔ 
     simp [isOpen_iff_nhds, h]
 #align is_open_singleton_iff_nhds_eq_pure isOpen_singleton_iff_nhds_eq_pure
 
-theorem isOpen_singleton_iff_punctured_nhds {α : Type _} [TopologicalSpace α] (a : α) :
+theorem isOpen_singleton_iff_punctured_nhds {α : Type*} [TopologicalSpace α] (a : α) :
     IsOpen ({a} : Set α) ↔ 𝓝[≠] a = ⊥ := by
   rw [isOpen_singleton_iff_nhds_eq_pure, nhdsWithin, ← mem_iff_inf_principal_compl, ← le_pure_iff,
     nhds_neBot.le_pure_iff]
@@ -1568,7 +1568,7 @@ open Topology
 
 section Continuous
 
-variable {α : Type _} {β : Type _} {γ : Type _} {δ : Type _}
+variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
@@ -1812,7 +1812,7 @@ theorem Set.MapsTo.closure_left {s : Set α} {t : Set β} {f : α → β} (h : M
 
 section DenseRange
 
-variable {κ ι : Type _} (f : κ → β) (g : β → γ)
+variable {κ ι : Type*} (f : κ → β) (g : β → γ)
 
 /-- `f : ι → β` has dense range if its range (image) is a dense subset of β. -/
 def DenseRange := Dense (range f)
@@ -1921,7 +1921,7 @@ However, lemmas with this conclusion are not nice to use in practice because
 1. They confuse the elaborator. The following two examples fail, because of limitations in the
   elaboration process.
   ```
-  variables {M : Type _} [Add M] [TopologicalSpace M] [ContinuousAdd M]
+  variables {M : Type*} [Add M] [TopologicalSpace M] [ContinuousAdd M]
   example : Continuous (λ x : M, x + x) :=
   continuous_add.comp _
 
@@ -1992,7 +1992,7 @@ In this case, you want to add conditions to when a function involving `fract` is
 get something like this: (assumption `hf` could be weakened, but the important thing is the shape
 of the conclusion)
 ```
-lemma ContinuousOn.comp_fract {X Y : Type _} [TopologicalSpace X] [TopologicalSpace Y]
+lemma ContinuousOn.comp_fract {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
   {f : X → ℝ → Y} {g : X → ℝ} (hf : Continuous ↿f) (hg : Continuous g) (h : ∀ s, f s 0 = f s 1) :
   Continuous (fun x ↦ f x (fract (g x)))
 ```

This PR adds a way to characterize IsBigO along the atTop filter, for cases where we want the constant to depend on a "starting point" n₀.

It also adds the lemma tendsto_nhds_of_eventually_eq.

@@ -1066,6 +1066,10 @@ instance nhds_neBot {a : α} : NeBot (𝓝 a) :=
   neBot_of_le (pure_le_nhds a)
 #align nhds_ne_bot nhds_neBot
 
+theorem tendsto_nhds_of_eventually_eq {f : β → α} {a : α} (h : ∀ᶠ x in l, f x = a) :
+    Tendsto f l (𝓝 a) :=
+  tendsto_const_nhds.congr' (.symm h)
+
 /-!
 ### Cluster points
 

@@ -1356,6 +1356,26 @@ theorem mem_closure_iff_nhds_basis {a : α} {p : ι → Prop} {s : ι → Set α
     simp only [Set.Nonempty, mem_inter_iff, exists_prop, and_comm]
 #align mem_closure_iff_nhds_basis mem_closure_iff_nhds_basis
 
+theorem clusterPt_iff_forall_mem_closure {F : Filter α} {a : α} :
+    ClusterPt a F ↔ ∀ s ∈ F, a ∈ closure s := by
+  simp_rw [ClusterPt, inf_neBot_iff, mem_closure_iff_nhds]
+  rw [forall₂_swap]
+
+theorem clusterPt_iff_lift'_closure {F : Filter α} {a : α} :
+    ClusterPt a F ↔ pure a ≤ (F.lift' closure) := by
+  simp_rw [clusterPt_iff_forall_mem_closure,
+    (hasBasis_pure _).le_basis_iff F.basis_sets.lift'_closure, id, singleton_subset_iff, true_and,
+    exists_const]
+
+theorem clusterPt_iff_lift'_closure' {F : Filter α} {a : α} :
+    ClusterPt a F ↔ (F.lift' closure ⊓ pure a).NeBot := by
+  rw [clusterPt_iff_lift'_closure, ← Ultrafilter.coe_pure, inf_comm, Ultrafilter.inf_neBot_iff]
+
+@[simp]
+theorem clusterPt_lift'_closure_iff {F : Filter α} {a : α} :
+    ClusterPt a (F.lift' closure) ↔ ClusterPt a F := by
+  simp [clusterPt_iff_lift'_closure, lift'_lift'_assoc (monotone_closure α) (monotone_closure α)]
+
 /-- `x` belongs to the closure of `s` if and only if some ultrafilter
   supported on `s` converges to `x`. -/
 theorem mem_closure_iff_ultrafilter {s : Set α} {x : α} :
@@ -1752,6 +1772,7 @@ protected theorem Set.MapsTo.closure {s : Set α} {t : Set β} {f : α → β} (
   exact fun x hx => hx.map hc.continuousAt (tendsto_principal_principal.2 h)
 #align set.maps_to.closure Set.MapsTo.closure
 
+/-- See also `IsClosedMap.closure_image_eq_of_continuous`. -/
 theorem image_closure_subset_closure_image {f : α → β} {s : Set α} (h : Continuous f) :
     f '' closure s ⊆ closure (f '' s) :=
   ((mapsTo_image f s).closure h).image_subset