topology.bases ⟷ Mathlib.Topology.Bases

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

@@ -149,7 +149,7 @@ theorem IsTopologicalBasis.mem_nhds_iff {a : α} {s : Set α} {b : Set (Set α)}
   by
   change s ∈ (𝓝 a).sets ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s
   rw [hb.eq_generate_from, nhds_generate_from, binfi_sets_eq]
-  · simp [and_assoc', and_left_comm]
+  · simp [and_assoc, and_left_comm]
   ·
     exact fun s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩ =>
       have : a ∈ s ∩ t := ⟨hs₁, ht₁⟩
@@ -170,7 +170,7 @@ theorem IsTopologicalBasis.isOpen_iff {s : Set α} {b : Set (Set α)} (hb : IsTo
 #print TopologicalSpace.IsTopologicalBasis.nhds_hasBasis /-
 theorem IsTopologicalBasis.nhds_hasBasis {b : Set (Set α)} (hb : IsTopologicalBasis b) {a : α} :
     (𝓝 a).HasBasis (fun t : Set α => t ∈ b ∧ a ∈ t) fun t => t :=
-  ⟨fun s => hb.mem_nhds_iffₓ.trans <| by simp only [exists_prop, and_assoc']⟩
+  ⟨fun s => hb.mem_nhds_iffₓ.trans <| by simp only [exists_prop, and_assoc]⟩
 #align topological_space.is_topological_basis.nhds_has_basis TopologicalSpace.IsTopologicalBasis.nhds_hasBasis
 -/
 

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

@@ -207,7 +207,7 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
 #align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 #print TopologicalSpace.IsTopologicalBasis.open_eq_sUnion /-
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
@@ -215,7 +215,7 @@ theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopological
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 #print TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion /-
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
@@ -613,7 +613,7 @@ protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace 
 #align dense_range.separable_space DenseRange.separableSpace
 -/
 
-/- ./././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 Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}
     [SeparableSpace s] (hs : Dense s) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
@@ -621,7 +621,7 @@ theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α]
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
-/- ./././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 Dense.exists_countable_dense_subset_bot_top /-
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
@@ -929,7 +929,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds
 -/
 
-/- ./././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 TopologicalSpace.countable_cover_nhdsWithin /-
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
     (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=

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

@@ -562,8 +562,8 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
 #align is_topological_basis_pi isTopologicalBasis_pi
 -/
 
-#print isTopologicalBasis_iInf /-
-theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
+#print IsTopologicalBasis.iInf_induced /-
+theorem IsTopologicalBasis.iInf_induced {β : Type _} {ι : Type _} {X : ι → Type _}
     [t : ∀ i, TopologicalSpace (X i)] {T : ∀ i, Set (Set (X i))}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
     @IsTopologicalBasis β (⨅ i, induced (f i) (t i))
@@ -592,7 +592,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     ext1
     rw [Set.preimage_iInter]
     rfl
-#align is_topological_basis_infi isTopologicalBasis_iInf
+#align is_topological_basis_infi IsTopologicalBasis.iInf_induced
 -/
 
 #print isTopologicalBasis_singletons /-

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

@@ -264,7 +264,7 @@ theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTop
     {u : Set α} (hu : u.Nonempty) (ou : IsOpen u) : ∃ v ∈ B, Set.Nonempty v ∧ v ⊆ u :=
   by
   cases' hu with x hx
-  rw [hb.open_eq_sUnion' ou, mem_sUnion] at hx 
+  rw [hb.open_eq_sUnion' ou, mem_sUnion] at hx
   rcases hx with ⟨v, hv, hxv⟩
   exact ⟨v, hv.1, ⟨x, hxv⟩, hv.2⟩
 #align topological_space.is_topological_basis.exists_nonempty_subset TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset
@@ -302,7 +302,7 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
     rwa [hf.is_open_iff]
     refine' ⟨V, h.is_open hV, rfl⟩
   · intro a U ha hU
-    rw [hf.is_open_iff] at hU 
+    rw [hf.is_open_iff] at hU
     obtain ⟨V, hV, rfl⟩ := hU
     obtain ⟨S, hS, rfl⟩ := h.open_eq_sUnion hV
     obtain ⟨W, hW, ha⟩ := ha
@@ -316,7 +316,7 @@ theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen
     IsTopologicalBasis (⋃ i : ι, image (coe : U i → α) '' b i) :=
   by
   refine' is_topological_basis_of_open_of_nhds (fun u hu => _) _
-  · simp only [mem_Union, mem_image] at hu 
+  · simp only [mem_Union, mem_image] at hu
     rcases hu with ⟨i, s, sb, rfl⟩
     exact (Uo i).isOpenMap_subtype_val _ ((hb i).IsOpen sb)
   · intro a u ha uo

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

@@ -474,14 +474,14 @@ theorem IsSeparable.closure {s : Set α} (hs : IsSeparable s) : IsSeparable (clo
 #align topological_space.is_separable.closure TopologicalSpace.IsSeparable.closure
 -/
 
-#print TopologicalSpace.isSeparable_iUnion /-
-theorem isSeparable_iUnion {ι : Type _} [Countable ι] {s : ι → Set α}
+#print TopologicalSpace.IsSeparable.iUnion /-
+theorem TopologicalSpace.IsSeparable.iUnion {ι : Type _} [Countable ι] {s : ι → Set α}
     (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) :=
   by
   choose c hc h'c using hs
   refine' ⟨⋃ i, c i, countable_Union hc, Union_subset_iff.2 fun i => _⟩
   exact (h'c i).trans (closure_mono (subset_Union _ i))
-#align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnion
+#align topological_space.is_separable_Union TopologicalSpace.IsSeparable.iUnion
 -/
 
 #print Set.Countable.isSeparable /-
@@ -508,10 +508,11 @@ theorem isSeparable_univ_iff : IsSeparable (univ : Set α) ↔ SeparableSpace α
 #align topological_space.is_separable_univ_iff TopologicalSpace.isSeparable_univ_iff
 -/
 
-#print TopologicalSpace.isSeparable_of_separableSpace /-
-theorem isSeparable_of_separableSpace [h : SeparableSpace α] (s : Set α) : IsSeparable s :=
+#print TopologicalSpace.IsSeparable.of_separableSpace /-
+theorem TopologicalSpace.IsSeparable.of_separableSpace [h : SeparableSpace α] (s : Set α) :
+    IsSeparable s :=
   IsSeparable.mono (isSeparable_univ_iff.2 h) (subset_univ _)
-#align topological_space.is_separable_of_separable_space TopologicalSpace.isSeparable_of_separableSpace
+#align topological_space.is_separable_of_separable_space TopologicalSpace.IsSeparable.of_separableSpace
 -/
 
 #print TopologicalSpace.IsSeparable.image /-
@@ -525,13 +526,13 @@ theorem IsSeparable.image {β : Type _} [TopologicalSpace β] {s : Set α} (hs :
 #align topological_space.is_separable.image TopologicalSpace.IsSeparable.image
 -/
 
-#print TopologicalSpace.isSeparable_of_separableSpace_subtype /-
-theorem isSeparable_of_separableSpace_subtype (s : Set α) [SeparableSpace s] : IsSeparable s :=
+#print TopologicalSpace.IsSeparable.of_subtype /-
+theorem TopologicalSpace.IsSeparable.of_subtype (s : Set α) [SeparableSpace s] : IsSeparable s :=
   by
   have : IsSeparable ((coe : s → α) '' (univ : Set s)) :=
     (is_separable_of_separable_space _).image continuous_subtype_val
   simpa only [image_univ, Subtype.range_coe_subtype]
-#align topological_space.is_separable_of_separable_space_subtype TopologicalSpace.isSeparable_of_separableSpace_subtype
+#align topological_space.is_separable_of_separable_space_subtype TopologicalSpace.IsSeparable.of_subtype
 -/
 
 end TopologicalSpace

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

@@ -715,7 +715,7 @@ instance isCountablyGenerated_nhdsWithin (x : α) [IsCountablyGenerated (𝓝 x)
 variable (α)
 
 #print SecondCountableTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
 /-- A second-countable space is one with a countable basis. -/
 class SecondCountableTopology : Prop where
   is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b

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

@@ -715,7 +715,7 @@ instance isCountablyGenerated_nhdsWithin (x : α) [IsCountablyGenerated (𝓝 x)
 variable (α)
 
 #print SecondCountableTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
 /-- A second-countable space is one with a countable basis. -/
 class SecondCountableTopology : Prop where
   is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b

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

@@ -121,10 +121,10 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
 #align topological_space.is_topological_basis_of_subbasis TopologicalSpace.isTopologicalBasis_of_subbasis
 -/
 
-#print TopologicalSpace.isTopologicalBasis_of_open_of_nhds /-
+#print TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds /-
 /-- If a family of open sets `s` is such that every open neighbourhood contains some
 member of `s`, then `s` is a topological basis. -/
-theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
+theorem isTopologicalBasis_of_isOpen_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
     (h_nhds : ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u) :
     IsTopologicalBasis s :=
   by
@@ -138,7 +138,7 @@ theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u 
     rcases h_nhds a u ha hu with ⟨v, hvs, hav, hvu⟩
     rw [nhds_generate_from]
     exact iInf₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
-#align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+#align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
 -/
 
 #print TopologicalSpace.IsTopologicalBasis.mem_nhds_iff /-
@@ -272,7 +272,7 @@ theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTop
 
 #print TopologicalSpace.isTopologicalBasis_opens /-
 theorem isTopologicalBasis_opens : IsTopologicalBasis {U : Set α | IsOpen U} :=
-  isTopologicalBasis_of_open_of_nhds (by tauto) (by tauto)
+  isTopologicalBasis_of_isOpen_of_nhds (by tauto) (by tauto)
 #align topological_space.is_topological_basis_opens TopologicalSpace.isTopologicalBasis_opens
 -/
 
@@ -597,7 +597,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
 #print isTopologicalBasis_singletons /-
 theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [DiscreteTopology α] :
     IsTopologicalBasis {s | ∃ x : α, (s : Set α) = {x}} :=
-  isTopologicalBasis_of_open_of_nhds (fun u hu => isOpen_discrete _) fun x u hx u_open =>
+  isTopologicalBasis_of_isOpen_of_nhds (fun u hu => isOpen_discrete _) fun x u hx u_open =>
     ⟨{x}, ⟨x, rfl⟩, mem_singleton x, singleton_subset_iff.2 hx⟩
 #align is_topological_basis_singletons isTopologicalBasis_singletons
 -/

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

@@ -666,12 +666,12 @@ universe u
 
 variable (α : Type u) [t : TopologicalSpace α]
 
-#print TopologicalSpace.FirstCountableTopology /-
+#print FirstCountableTopology /-
 /-- A first-countable space is one in which every point has a
   countable neighborhood basis. -/
 class FirstCountableTopology : Prop where
   nhds_generated_countable : ∀ a : α, (𝓝 a).IsCountablyGenerated
-#align topological_space.first_countable_topology TopologicalSpace.FirstCountableTopology
+#align topological_space.first_countable_topology FirstCountableTopology
 -/
 
 attribute [instance] first_countable_topology.nhds_generated_countable
@@ -714,12 +714,12 @@ instance isCountablyGenerated_nhdsWithin (x : α) [IsCountablyGenerated (𝓝 x)
 
 variable (α)
 
-#print TopologicalSpace.SecondCountableTopology /-
+#print SecondCountableTopology /-
 /- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
 /-- A second-countable space is one with a countable basis. -/
 class SecondCountableTopology : Prop where
   is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b
-#align topological_space.second_countable_topology TopologicalSpace.SecondCountableTopology
+#align topological_space.second_countable_topology SecondCountableTopology
 -/
 
 variable {α}

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

@@ -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
 -/
-import Mathbin.Topology.Constructions
-import Mathbin.Topology.ContinuousOn
+import Topology.Constructions
+import Topology.ContinuousOn
 
 #align_import topology.bases from "leanprover-community/mathlib"@"f16e7a22e11fc09c71f25446ac1db23a24e8a0bd"
 
@@ -207,7 +207,7 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
 #align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 #print TopologicalSpace.IsTopologicalBasis.open_eq_sUnion /-
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
@@ -215,7 +215,7 @@ theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopological
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 #print TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion /-
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
@@ -612,7 +612,7 @@ protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace 
 #align dense_range.separable_space DenseRange.separableSpace
 -/
 
-/- ./././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 Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}
     [SeparableSpace s] (hs : Dense s) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
@@ -620,7 +620,7 @@ theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α]
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
-/- ./././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 Dense.exists_countable_dense_subset_bot_top /-
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
@@ -715,7 +715,7 @@ instance isCountablyGenerated_nhdsWithin (x : α) [IsCountablyGenerated (𝓝 x)
 variable (α)
 
 #print TopologicalSpace.SecondCountableTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
 /-- A second-countable space is one with a countable basis. -/
 class SecondCountableTopology : Prop where
   is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b
@@ -928,7 +928,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds
 -/
 
-/- ./././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 TopologicalSpace.countable_cover_nhdsWithin /-
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
     (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=

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

@@ -114,7 +114,7 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
     exact ⟨_, ⟨_, ⟨hft₁.union hft₂, union_subset ht₁b ht₂b⟩, sInter_union t₁ t₂⟩, h, subset.rfl⟩
   · rw [sUnion_image, Union₂_eq_univ_iff]
     exact fun x => ⟨∅, ⟨finite_empty, empty_subset _⟩, sInter_empty.substr <| mem_univ x⟩
-  · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩; apply isOpen_sInter
+  · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩; apply Set.Finite.isOpen_sInter
     exacts [hft, fun s hs => generate_open.basic _ <| htb hs]
   · rw [← sInter_singleton t]
     exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩

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

@@ -2,15 +2,12 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
-
-! This file was ported from Lean 3 source module topology.bases
-! leanprover-community/mathlib commit f16e7a22e11fc09c71f25446ac1db23a24e8a0bd
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Constructions
 import Mathbin.Topology.ContinuousOn
 
+#align_import topology.bases from "leanprover-community/mathlib"@"f16e7a22e11fc09c71f25446ac1db23a24e8a0bd"
+
 /-!
 # Bases of topologies. Countability axioms.
 
@@ -210,7 +207,7 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
 #align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 #print TopologicalSpace.IsTopologicalBasis.open_eq_sUnion /-
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
@@ -218,7 +215,7 @@ theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopological
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 #print TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion /-
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
@@ -615,7 +612,7 @@ protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace 
 #align dense_range.separable_space DenseRange.separableSpace
 -/
 
-/- ./././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 Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}
     [SeparableSpace s] (hs : Dense s) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
@@ -623,7 +620,7 @@ theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α]
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
-/- ./././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 Dense.exists_countable_dense_subset_bot_top /-
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
@@ -931,7 +928,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds
 -/
 
-/- ./././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 TopologicalSpace.countable_cover_nhdsWithin /-
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
     (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/1a51edf13debfcbe223fa06b1cb353b9ed9751cc

@@ -403,11 +403,11 @@ instance (priority := 100) Countable.to_separableSpace [Countable α] : Separabl
 #align topological_space.countable.to_separable_space TopologicalSpace.Countable.to_separableSpace
 -/
 
-#print TopologicalSpace.separableSpace_of_denseRange /-
-theorem separableSpace_of_denseRange {ι : Type _} [Countable ι] (u : ι → α) (hu : DenseRange u) :
-    SeparableSpace α :=
+#print TopologicalSpace.SeparableSpace.of_denseRange /-
+theorem TopologicalSpace.SeparableSpace.of_denseRange {ι : Type _} [Countable ι] (u : ι → α)
+    (hu : DenseRange u) : SeparableSpace α :=
   ⟨⟨range u, countable_range u, hu⟩⟩
-#align topological_space.separable_space_of_dense_range TopologicalSpace.separableSpace_of_denseRange
+#align topological_space.separable_space_of_dense_range TopologicalSpace.SeparableSpace.of_denseRange
 -/
 
 #print Set.PairwiseDisjoint.countable_of_isOpen /-

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

@@ -66,8 +66,6 @@ universe u
 
 variable {α : Type u} [t : TopologicalSpace α]
 
-include t
-
 #print TopologicalSpace.IsTopologicalBasis /-
 /-- A topological basis is one that satisfies the necessary conditions so that
   it suffices to take unions of the basis sets to get a topology (without taking
@@ -93,6 +91,7 @@ theorem IsTopologicalBasis.insert_empty {s : Set (Set α)} (h : IsTopologicalBas
 #align topological_space.is_topological_basis.insert_empty TopologicalSpace.IsTopologicalBasis.insert_empty
 -/
 
+#print TopologicalSpace.IsTopologicalBasis.diff_empty /-
 theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
     IsTopologicalBasis (s \ {∅}) :=
   by
@@ -105,6 +104,7 @@ theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis
     obtain rfl | he := eq_or_ne t ∅; · convert isOpen_empty
     exact generate_open.basic t ⟨ht, he⟩
 #align topological_space.is_topological_basis.diff_empty TopologicalSpace.IsTopologicalBasis.diff_empty
+-/
 
 #print TopologicalSpace.isTopologicalBasis_of_subbasis /-
 /-- If a family of sets `s` generates the topology, then intersections of finite
@@ -124,6 +124,7 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
 #align topological_space.is_topological_basis_of_subbasis TopologicalSpace.isTopologicalBasis_of_subbasis
 -/
 
+#print TopologicalSpace.isTopologicalBasis_of_open_of_nhds /-
 /-- If a family of open sets `s` is such that every open neighbourhood contains some
 member of `s`, then `s` is a topological basis. -/
 theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
@@ -141,7 +142,9 @@ theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u 
     rw [nhds_generate_from]
     exact iInf₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
 #align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+-/
 
+#print TopologicalSpace.IsTopologicalBasis.mem_nhds_iff /-
 /-- A set `s` is in the neighbourhood of `a` iff there is some basis set `t`, which
 contains `a` and is itself contained in `s`. -/
 theorem IsTopologicalBasis.mem_nhds_iff {a : α} {s : Set α} {b : Set (Set α)}
@@ -159,10 +162,13 @@ theorem IsTopologicalBasis.mem_nhds_iff {a : α} {s : Set α} {b : Set (Set α)}
   · rcases eq_univ_iff_forall.1 hb.sUnion_eq a with ⟨i, h1, h2⟩
     exact ⟨i, h2, h1⟩
 #align topological_space.is_topological_basis.mem_nhds_iff TopologicalSpace.IsTopologicalBasis.mem_nhds_iff
+-/
 
+#print TopologicalSpace.IsTopologicalBasis.isOpen_iff /-
 theorem IsTopologicalBasis.isOpen_iff {s : Set α} {b : Set (Set α)} (hb : IsTopologicalBasis b) :
     IsOpen s ↔ ∀ a ∈ s, ∃ t ∈ b, a ∈ t ∧ t ⊆ s := by simp [isOpen_iff_mem_nhds, hb.mem_nhds_iff]
 #align topological_space.is_topological_basis.is_open_iff TopologicalSpace.IsTopologicalBasis.isOpen_iff
+-/
 
 #print TopologicalSpace.IsTopologicalBasis.nhds_hasBasis /-
 theorem IsTopologicalBasis.nhds_hasBasis {b : Set (Set α)} (hb : IsTopologicalBasis b) {a : α} :
@@ -185,10 +191,12 @@ protected theorem IsTopologicalBasis.mem_nhds {a : α} {s : Set α} {b : Set (Se
 #align topological_space.is_topological_basis.mem_nhds TopologicalSpace.IsTopologicalBasis.mem_nhds
 -/
 
+#print TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_open /-
 theorem IsTopologicalBasis.exists_subset_of_mem_open {b : Set (Set α)} (hb : IsTopologicalBasis b)
     {a : α} {u : Set α} (au : a ∈ u) (ou : IsOpen u) : ∃ v ∈ b, a ∈ v ∧ v ⊆ u :=
   hb.mem_nhds_iffₓ.1 <| IsOpen.mem_nhds ou au
 #align topological_space.is_topological_basis.exists_subset_of_mem_open TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_open
+-/
 
 #print TopologicalSpace.IsTopologicalBasis.open_eq_sUnion' /-
 /-- Any open set is the union of the basis sets contained in it. -/
@@ -203,16 +211,20 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+#print TopologicalSpace.IsTopologicalBasis.open_eq_sUnion /-
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
   ⟨{s ∈ B | s ⊆ u}, fun s h => h.1, hB.open_eq_sUnion' ou⟩
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+#print TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion /-
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
   ⟨hB.open_eq_sUnion, fun ⟨S, hSB, hu⟩ => hu.symm ▸ isOpen_sUnion fun s hs => hB.IsOpen (hSB hs)⟩
 #align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion
+-/
 
 #print TopologicalSpace.IsTopologicalBasis.open_eq_iUnion /-
 theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
@@ -222,12 +234,15 @@ theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopological
 #align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
 -/
 
+#print TopologicalSpace.IsTopologicalBasis.mem_closure_iff /-
 /-- A point `a` is in the closure of `s` iff all basis sets containing `a` intersect `s`. -/
 theorem IsTopologicalBasis.mem_closure_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α}
     {a : α} : a ∈ closure s ↔ ∀ o ∈ b, a ∈ o → (o ∩ s).Nonempty :=
   (mem_closure_iff_nhds_basis' hb.nhds_hasBasis).trans <| by simp only [and_imp]
 #align topological_space.is_topological_basis.mem_closure_iff TopologicalSpace.IsTopologicalBasis.mem_closure_iff
+-/
 
+#print TopologicalSpace.IsTopologicalBasis.dense_iff /-
 /-- A set is dense iff it has non-trivial intersection with all basis sets. -/
 theorem IsTopologicalBasis.dense_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α} :
     Dense s ↔ ∀ o ∈ b, Set.Nonempty o → (o ∩ s).Nonempty :=
@@ -235,7 +250,9 @@ theorem IsTopologicalBasis.dense_iff {b : Set (Set α)} (hb : IsTopologicalBasis
   simp only [Dense, hb.mem_closure_iff]
   exact ⟨fun h o hb ⟨a, ha⟩ => h a o hb ha, fun h a o hb ha => h o hb ⟨a, ha⟩⟩
 #align topological_space.is_topological_basis.dense_iff TopologicalSpace.IsTopologicalBasis.dense_iff
+-/
 
+#print TopologicalSpace.IsTopologicalBasis.isOpenMap_iff /-
 theorem IsTopologicalBasis.isOpenMap_iff {β} [TopologicalSpace β] {B : Set (Set α)}
     (hB : IsTopologicalBasis B) {f : α → β} : IsOpenMap f ↔ ∀ s ∈ B, IsOpen (f '' s) :=
   by
@@ -243,7 +260,9 @@ theorem IsTopologicalBasis.isOpenMap_iff {β} [TopologicalSpace β] {B : Set (Se
   rw [hB.open_eq_sUnion' ho, sUnion_eq_Union, image_Union]
   exact isOpen_iUnion fun s => hf s s.2.1
 #align topological_space.is_topological_basis.is_open_map_iff TopologicalSpace.IsTopologicalBasis.isOpenMap_iff
+-/
 
+#print TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset /-
 theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTopologicalBasis B)
     {u : Set α} (hu : u.Nonempty) (ou : IsOpen u) : ∃ v ∈ B, Set.Nonempty v ∧ v ⊆ u :=
   by
@@ -252,6 +271,7 @@ theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTop
   rcases hx with ⟨v, hv, hxv⟩
   exact ⟨v, hv.1, ⟨x, hxv⟩, hv.2⟩
 #align topological_space.is_topological_basis.exists_nonempty_subset TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset
+-/
 
 #print TopologicalSpace.isTopologicalBasis_opens /-
 theorem isTopologicalBasis_opens : IsTopologicalBasis {U : Set α | IsOpen U} :=
@@ -261,6 +281,7 @@ theorem isTopologicalBasis_opens : IsTopologicalBasis {U : Set α | IsOpen U} :=
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:228:8: unsupported: ambiguous notation -/
+#print TopologicalSpace.IsTopologicalBasis.prod /-
 protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set (Set α)}
     {B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) :
     IsTopologicalBasis (image2 (· ×ˢ ·) B₁ B₂) :=
@@ -273,7 +294,9 @@ protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set
       ⟨⟨s, t⟩, ⟨⟨hs, ha⟩, ht, hb⟩, hu⟩
     exact ⟨s ×ˢ t, mem_image2_of_mem hs ht, ⟨ha, hb⟩, hu⟩
 #align topological_space.is_topological_basis.prod TopologicalSpace.IsTopologicalBasis.prod
+-/
 
+#print TopologicalSpace.IsTopologicalBasis.inducing /-
 protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α → β} {T : Set (Set β)}
     (hf : Inducing f) (h : IsTopologicalBasis T) : IsTopologicalBasis (image (preimage f) T) :=
   by
@@ -288,7 +311,9 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
     obtain ⟨W, hW, ha⟩ := ha
     refine' ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_sUnion_of_mem hW⟩
 #align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducing
+-/
 
+#print TopologicalSpace.isTopologicalBasis_of_cover /-
 theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
     (Uc : (⋃ i, U i) = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
     IsTopologicalBasis (⋃ i : ι, image (coe : U i → α) '' b i) :=
@@ -306,11 +331,14 @@ theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen
       ⟨coe '' v, mem_Union.2 ⟨i, mem_image_of_mem _ hvb⟩, mem_image_of_mem _ hav,
         image_subset_iff.2 hvu⟩
 #align topological_space.is_topological_basis_of_cover TopologicalSpace.isTopologicalBasis_of_cover
+-/
 
+#print TopologicalSpace.IsTopologicalBasis.continuous /-
 protected theorem IsTopologicalBasis.continuous {β : Type _} [TopologicalSpace β] {B : Set (Set β)}
     (hB : IsTopologicalBasis B) (f : α → β) (hf : ∀ s ∈ B, IsOpen (f ⁻¹' s)) : Continuous f := by
   rw [hB.eq_generate_from]; exact continuous_generateFrom hf
 #align topological_space.is_topological_basis.continuous TopologicalSpace.IsTopologicalBasis.continuous
+-/
 
 variable (α)
 
@@ -375,11 +403,14 @@ instance (priority := 100) Countable.to_separableSpace [Countable α] : Separabl
 #align topological_space.countable.to_separable_space TopologicalSpace.Countable.to_separableSpace
 -/
 
+#print TopologicalSpace.separableSpace_of_denseRange /-
 theorem separableSpace_of_denseRange {ι : Type _} [Countable ι] (u : ι → α) (hu : DenseRange u) :
     SeparableSpace α :=
   ⟨⟨range u, countable_range u, hu⟩⟩
 #align topological_space.separable_space_of_dense_range TopologicalSpace.separableSpace_of_denseRange
+-/
 
+#print Set.PairwiseDisjoint.countable_of_isOpen /-
 /-- In a separable space, a family of nonempty disjoint open sets is countable. -/
 theorem Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type _} {s : ι → Set α}
     {a : Set ι} (h : a.PairwiseDisjoint s) (ha : ∀ i ∈ a, IsOpen (s i))
@@ -397,13 +428,16 @@ theorem Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type
     simp only [congr_arg coe hfij, hfs j]
   exact ⟨@Encodable.ofInj _ _ u_encodable f f_inj⟩
 #align set.pairwise_disjoint.countable_of_is_open Set.PairwiseDisjoint.countable_of_isOpen
+-/
 
+#print Set.PairwiseDisjoint.countable_of_nonempty_interior /-
 /-- In a separable space, a family of disjoint sets with nonempty interiors is countable. -/
 theorem Set.PairwiseDisjoint.countable_of_nonempty_interior [SeparableSpace α] {ι : Type _}
     {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s)
     (ha : ∀ i ∈ a, (interior (s i)).Nonempty) : a.Countable :=
   (h.mono fun i => interior_subset).countable_of_isOpen (fun i hi => isOpen_interior) ha
 #align set.pairwise_disjoint.countable_of_nonempty_interior Set.PairwiseDisjoint.countable_of_nonempty_interior
+-/
 
 #print TopologicalSpace.IsSeparable /-
 /-- A set `s` in a topological space is separable if it is contained in the closure of a
@@ -423,6 +457,7 @@ theorem IsSeparable.mono {s u : Set α} (hs : IsSeparable s) (hu : u ⊆ s) : Is
 #align topological_space.is_separable.mono TopologicalSpace.IsSeparable.mono
 -/
 
+#print TopologicalSpace.IsSeparable.union /-
 theorem IsSeparable.union {s u : Set α} (hs : IsSeparable s) (hu : IsSeparable u) :
     IsSeparable (s ∪ u) := by
   rcases hs with ⟨cs, cs_count, hcs⟩
@@ -432,6 +467,7 @@ theorem IsSeparable.union {s u : Set α} (hs : IsSeparable s) (hu : IsSeparable
     union_subset (hcs.trans (closure_mono (subset_union_left _ _)))
       (hcu.trans (closure_mono (subset_union_right _ _)))
 #align topological_space.is_separable.union TopologicalSpace.IsSeparable.union
+-/
 
 #print TopologicalSpace.IsSeparable.closure /-
 theorem IsSeparable.closure {s : Set α} (hs : IsSeparable s) : IsSeparable (closure s) :=
@@ -441,6 +477,7 @@ theorem IsSeparable.closure {s : Set α} (hs : IsSeparable s) : IsSeparable (clo
 #align topological_space.is_separable.closure TopologicalSpace.IsSeparable.closure
 -/
 
+#print TopologicalSpace.isSeparable_iUnion /-
 theorem isSeparable_iUnion {ι : Type _} [Countable ι] {s : ι → Set α}
     (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) :=
   by
@@ -448,6 +485,7 @@ theorem isSeparable_iUnion {ι : Type _} [Countable ι] {s : ι → Set α}
   refine' ⟨⋃ i, c i, countable_Union hc, Union_subset_iff.2 fun i => _⟩
   exact (h'c i).trans (closure_mono (subset_Union _ i))
 #align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnion
+-/
 
 #print Set.Countable.isSeparable /-
 theorem Set.Countable.isSeparable {s : Set α} (hs : s.Countable) : IsSeparable s :=
@@ -479,6 +517,7 @@ theorem isSeparable_of_separableSpace [h : SeparableSpace α] (s : Set α) : IsS
 #align topological_space.is_separable_of_separable_space TopologicalSpace.isSeparable_of_separableSpace
 -/
 
+#print TopologicalSpace.IsSeparable.image /-
 theorem IsSeparable.image {β : Type _} [TopologicalSpace β] {s : Set α} (hs : IsSeparable s)
     {f : α → β} (hf : Continuous f) : IsSeparable (f '' s) :=
   by
@@ -487,6 +526,7 @@ theorem IsSeparable.image {β : Type _} [TopologicalSpace β] {s : Set α} (hs :
   rw [image_subset_iff]
   exact hc.trans (closure_subset_preimage_closure_image hf)
 #align topological_space.is_separable.image TopologicalSpace.IsSeparable.image
+-/
 
 #print TopologicalSpace.isSeparable_of_separableSpace_subtype /-
 theorem isSeparable_of_separableSpace_subtype (s : Set α) [SeparableSpace s] : IsSeparable s :=
@@ -501,6 +541,7 @@ end TopologicalSpace
 
 open TopologicalSpace
 
+#print isTopologicalBasis_pi /-
 theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, TopologicalSpace (X i)]
     {T : ∀ i, Set (Set (X i))} (cond : ∀ i, IsTopologicalBasis (T i)) :
     IsTopologicalBasis
@@ -521,7 +562,9 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
     exact
       ⟨_, ⟨V, I, fun i hi => hVT i, rfl⟩, fun i hi => haV i, (pi_mono fun i hi => hVt i).trans htU⟩
 #align is_topological_basis_pi isTopologicalBasis_pi
+-/
 
+#print isTopologicalBasis_iInf /-
 theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     [t : ∀ i, TopologicalSpace (X i)] {T : ∀ i, Set (Set (X i))}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
@@ -552,6 +595,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     rw [Set.preimage_iInter]
     rfl
 #align is_topological_basis_infi isTopologicalBasis_iInf
+-/
 
 #print isTopologicalBasis_singletons /-
 theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [DiscreteTopology α] :
@@ -561,6 +605,7 @@ theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [Discr
 #align is_topological_basis_singletons isTopologicalBasis_singletons
 -/
 
+#print DenseRange.separableSpace /-
 /-- If `α` is a separable space and `f : α → β` is a continuous map with dense range, then `β` is
 a separable space as well. E.g., the completion of a separable uniform space is separable. -/
 protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace α] [SeparableSpace α]
@@ -568,6 +613,7 @@ protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace 
   let ⟨s, s_cnt, s_dense⟩ := exists_countable_dense α
   ⟨⟨f '' s, Countable.image s_cnt f, h.dense_image h' s_dense⟩⟩
 #align dense_range.separable_space DenseRange.separableSpace
+-/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}
@@ -578,6 +624,7 @@ theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α]
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+#print Dense.exists_countable_dense_subset_bot_top /-
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
 dense subset `t ⊆ s` such that `t` contains bottom/top element of `α` when they exist and belong
@@ -595,6 +642,7 @@ theorem Dense.exists_countable_dense_subset_bot_top {α : Type _} [TopologicalSp
     htd.mono (subset_inter (subset_union_left _ _) hts), fun x hx hxs => ⟨Or.inr <| Or.inl hx, hxs⟩,
     fun x hx hxs => ⟨Or.inr <| Or.inr hx, hxs⟩]
 #align dense.exists_countable_dense_subset_bot_top Dense.exists_countable_dense_subset_bot_top
+-/
 
 #print separableSpace_univ /-
 instance separableSpace_univ {α : Type _} [TopologicalSpace α] [SeparableSpace α] :
@@ -621,8 +669,6 @@ universe u
 
 variable (α : Type u) [t : TopologicalSpace α]
 
-include t
-
 #print TopologicalSpace.FirstCountableTopology /-
 /-- A first-countable space is one in which every point has a
   countable neighborhood basis. -/
@@ -656,8 +702,6 @@ instance {β} [TopologicalSpace β] [FirstCountableTopology α] [FirstCountableT
 
 section Pi
 
-omit t
-
 instance {ι : Type _} {π : ι → Type _} [Countable ι] [∀ i, TopologicalSpace (π i)]
     [∀ i, FirstCountableTopology (π i)] : FirstCountableTopology (∀ i, π i) :=
   ⟨fun f => by rw [nhds_pi]; infer_instance⟩
@@ -674,7 +718,7 @@ instance isCountablyGenerated_nhdsWithin (x : α) [IsCountablyGenerated (𝓝 x)
 variable (α)
 
 #print TopologicalSpace.SecondCountableTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
 /-- A second-countable space is one with a countable basis. -/
 class SecondCountableTopology : Prop where
   is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b
@@ -770,6 +814,7 @@ instance (priority := 100) SecondCountableTopology.to_firstCountableTopology
 #align topological_space.second_countable_topology.to_first_countable_topology TopologicalSpace.SecondCountableTopology.to_firstCountableTopology
 -/
 
+#print TopologicalSpace.secondCountableTopology_induced /-
 /-- If `β` is a second-countable space, then its induced topology
 via `f` on `α` is also second-countable. -/
 theorem secondCountableTopology_induced (β) [t : TopologicalSpace β] [SecondCountableTopology β]
@@ -779,6 +824,7 @@ theorem secondCountableTopology_induced (β) [t : TopologicalSpace β] [SecondCo
   refine' { is_open_generated_countable := ⟨preimage f '' b, hb.image _, _⟩ }
   rw [Eq, induced_generateFrom_eq]
 #align topological_space.second_countable_topology_induced TopologicalSpace.secondCountableTopology_induced
+-/
 
 #print TopologicalSpace.Subtype.secondCountableTopology /-
 instance Subtype.secondCountableTopology (s : Set α) [SecondCountableTopology α] :
@@ -832,6 +878,7 @@ instance (priority := 100) SecondCountableTopology.to_separableSpace [SecondCoun
 
 variable {α}
 
+#print TopologicalSpace.secondCountableTopology_of_countable_cover /-
 /-- A countable open cover induces a second-countable topology if all open covers
 are themselves second countable. -/
 theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι → Set α}
@@ -841,7 +888,9 @@ theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι 
     is_topological_basis_of_cover Uo hc fun i => is_basis_countable_basis (U i)
   this.second_countable_topology (countable_Union fun i => (countable_countable_basis _).image _)
 #align topological_space.second_countable_topology_of_countable_cover TopologicalSpace.secondCountableTopology_of_countable_cover
+-/
 
+#print TopologicalSpace.isOpen_iUnion_countable /-
 /-- In a second-countable space, an open set, given as a union of open sets,
 is equal to the union of countably many of those sets. -/
 theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
@@ -855,6 +904,7 @@ theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Se
   rcases(is_basis_countable_basis α).exists_subset_of_mem_open xs (H _) with ⟨b, hb, xb, bs⟩
   exact ⟨_, ⟨_, rfl⟩, _, ⟨⟨⟨_, hb, _, bs⟩, rfl⟩, rfl⟩, hf _ xb⟩
 #align topological_space.is_open_Union_countable TopologicalSpace.isOpen_iUnion_countable
+-/
 
 #print TopologicalSpace.isOpen_sUnion_countable /-
 theorem isOpen_sUnion_countable [SecondCountableTopology α] (S : Set (Set α))
@@ -882,6 +932,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 -/
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (t «expr ⊆ » s) -/
+#print TopologicalSpace.countable_cover_nhdsWithin /-
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
     (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=
   by
@@ -891,13 +942,13 @@ theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set
   simp only [bUnion_image, eq_univ_iff_forall, ← preimage_Union, mem_preimage] at htU ⊢
   exact htU ⟨x, hx⟩
 #align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithin
+-/
 
 section Sigma
 
 variable {ι : Type _} {E : ι → Type _} [∀ i, TopologicalSpace (E i)]
 
-omit t
-
+#print TopologicalSpace.IsTopologicalBasis.sigma /-
 /-- In a disjoint union space `Σ i, E i`, one can form a topological basis by taking the union of
 topological bases on each of the parts of the space. -/
 theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
@@ -917,6 +968,7 @@ theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
       ⟨Sigma.mk i '' v, mem_Union.2 ⟨i, mem_image_of_mem _ vs⟩, mem_image_of_mem _ xv,
         image_subset_iff.2 hv⟩
 #align topological_space.is_topological_basis.sigma TopologicalSpace.IsTopologicalBasis.sigma
+-/
 
 /-- A countable disjoint union of second countable spaces is second countable. -/
 instance [Countable ι] [∀ i, SecondCountableTopology (E i)] : SecondCountableTopology (Σ i, E i) :=
@@ -930,10 +982,9 @@ end Sigma
 
 section Sum
 
-omit t
-
 variable {β : Type _} [TopologicalSpace α] [TopologicalSpace β]
 
+#print TopologicalSpace.IsTopologicalBasis.sum /-
 /-- In a sum space `α ⊕ β`, one can form a topological basis by taking the union of
 topological bases on each of the two components. -/
 theorem IsTopologicalBasis.sum {s : Set (Set α)} (hs : IsTopologicalBasis s) {t : Set (Set β)}
@@ -962,6 +1013,7 @@ theorem IsTopologicalBasis.sum {s : Set (Set α)} (hs : IsTopologicalBasis s) {t
         ⟨Sum.inr '' v, mem_union_right _ (mem_image_of_mem _ vs), mem_image_of_mem _ xv,
           image_subset_iff.2 vu⟩
 #align topological_space.is_topological_basis.sum TopologicalSpace.IsTopologicalBasis.sum
+-/
 
 /-- A sum type of two second countable spaces is second countable. -/
 instance [SecondCountableTopology α] [SecondCountableTopology β] :
@@ -981,8 +1033,7 @@ section Quotient
 
 variable {X : Type _} [TopologicalSpace X] {Y : Type _} [TopologicalSpace Y] {π : X → Y}
 
-omit t
-
+#print TopologicalSpace.IsTopologicalBasis.quotientMap /-
 /-- The image of a topological basis under an open quotient map is a topological basis. -/
 theorem IsTopologicalBasis.quotientMap {V : Set (Set X)} (hV : IsTopologicalBasis V)
     (h' : QuotientMap π) (h : IsOpenMap π) : IsTopologicalBasis (Set.image π '' V) :=
@@ -999,7 +1050,9 @@ theorem IsTopologicalBasis.quotientMap {V : Set (Set X)} (hV : IsTopologicalBasi
     have πZ_in_U : π '' Z ⊆ U := (Set.image_subset _ Z_in_W).trans (image_preimage_subset π U)
     exact ⟨π '' Z, ⟨Z, Z_in_V, rfl⟩, ⟨x, x_in_Z, rfl⟩, πZ_in_U⟩
 #align topological_space.is_topological_basis.quotient_map TopologicalSpace.IsTopologicalBasis.quotientMap
+-/
 
+#print QuotientMap.secondCountableTopology /-
 /-- A second countable space is mapped by an open quotient map to a second countable space. -/
 theorem QuotientMap.secondCountableTopology [SecondCountableTopology X] (h' : QuotientMap π)
     (h : IsOpenMap π) : SecondCountableTopology Y :=
@@ -1011,6 +1064,7 @@ theorem QuotientMap.secondCountableTopology [SecondCountableTopology X] (h' : Qu
         ⟨Set.image π '' V, V_countable.image (Set.image π),
           (V_generates.quotient_map h' h).eq_generateFrom⟩ }
 #align topological_space.quotient_map.second_countable_topology QuotientMap.secondCountableTopology
+-/
 
 variable {S : Setoid X}
 

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

@@ -202,13 +202,13 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
 #align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 -/
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
   ⟨{s ∈ B | s ⊆ u}, fun s h => h.1, hB.open_eq_sUnion' ou⟩
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
   ⟨hB.open_eq_sUnion, fun ⟨S, hSB, hu⟩ => hu.symm ▸ isOpen_sUnion fun s hs => hB.IsOpen (hSB hs)⟩
@@ -569,7 +569,7 @@ protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace 
   ⟨⟨f '' s, Countable.image s_cnt f, h.dense_image h' s_dense⟩⟩
 #align dense_range.separable_space DenseRange.separableSpace
 
-/- ./././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 Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}
     [SeparableSpace s] (hs : Dense s) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
@@ -577,7 +577,7 @@ theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α]
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
-/- ./././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) -/
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
 dense subset `t ⊆ s` such that `t` contains bottom/top element of `α` when they exist and belong
@@ -881,7 +881,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds
 -/
 
-/- ./././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 countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
     (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=
   by

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

@@ -110,7 +110,7 @@ theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis
 /-- If a family of sets `s` generates the topology, then intersections of finite
 subcollections of `s` form a topological basis. -/
 theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom s) :
-    IsTopologicalBasis ((fun f => ⋂₀ f) '' { f : Set (Set α) | f.Finite ∧ f ⊆ s }) :=
+    IsTopologicalBasis ((fun f => ⋂₀ f) '' {f : Set (Set α) | f.Finite ∧ f ⊆ s}) :=
   by
   refine' ⟨_, _, hs.trans (le_antisymm (le_generateFrom _) <| generate_from_anti fun t ht => _)⟩
   · rintro _ ⟨t₁, ⟨hft₁, ht₁b⟩, rfl⟩ _ ⟨t₂, ⟨hft₂, ht₂b⟩, rfl⟩ x h
@@ -193,7 +193,7 @@ theorem IsTopologicalBasis.exists_subset_of_mem_open {b : Set (Set α)} (hb : Is
 #print TopologicalSpace.IsTopologicalBasis.open_eq_sUnion' /-
 /-- Any open set is the union of the basis sets contained in it. -/
 theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
-    (ou : IsOpen u) : u = ⋃₀ { s ∈ B | s ⊆ u } :=
+    (ou : IsOpen u) : u = ⋃₀ {s ∈ B | s ⊆ u} :=
   ext fun a =>
     ⟨fun ha =>
       let ⟨b, hb, ab, bu⟩ := hB.exists_subset_of_mem_open ha ou
@@ -205,7 +205,7 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
-  ⟨{ s ∈ B | s ⊆ u }, fun s h => h.1, hB.open_eq_sUnion' ou⟩
+  ⟨{s ∈ B | s ⊆ u}, fun s h => h.1, hB.open_eq_sUnion' ou⟩
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
@@ -217,7 +217,7 @@ theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopolog
 #print TopologicalSpace.IsTopologicalBasis.open_eq_iUnion /-
 theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (β : Type u) (f : β → Set α), (u = ⋃ i, f i) ∧ ∀ i, f i ∈ B :=
-  ⟨↥({ s ∈ B | s ⊆ u }), coe, by rw [← sUnion_eq_Union]; apply hB.open_eq_sUnion' ou, fun s =>
+  ⟨↥({s ∈ B | s ⊆ u}), coe, by rw [← sUnion_eq_Union]; apply hB.open_eq_sUnion' ou, fun s =>
     And.left s.2⟩
 #align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
 -/
@@ -254,7 +254,7 @@ theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTop
 #align topological_space.is_topological_basis.exists_nonempty_subset TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset
 
 #print TopologicalSpace.isTopologicalBasis_opens /-
-theorem isTopologicalBasis_opens : IsTopologicalBasis { U : Set α | IsOpen U } :=
+theorem isTopologicalBasis_opens : IsTopologicalBasis {U : Set α | IsOpen U} :=
   isTopologicalBasis_of_open_of_nhds (by tauto) (by tauto)
 #align topological_space.is_topological_basis_opens TopologicalSpace.isTopologicalBasis_opens
 -/
@@ -504,8 +504,8 @@ open TopologicalSpace
 theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, TopologicalSpace (X i)]
     {T : ∀ i, Set (Set (X i))} (cond : ∀ i, IsTopologicalBasis (T i)) :
     IsTopologicalBasis
-      { S : Set (∀ i, X i) |
-        ∃ (U : ∀ i, Set (X i)) (F : Finset ι), (∀ i, i ∈ F → U i ∈ T i) ∧ S = (F : Set ι).pi U } :=
+      {S : Set (∀ i, X i) |
+        ∃ (U : ∀ i, Set (X i)) (F : Finset ι), (∀ i, i ∈ F → U i ∈ T i) ∧ S = (F : Set ι).pi U} :=
   by
   refine' is_topological_basis_of_open_of_nhds _ _
   · rintro _ ⟨U, F, h1, rfl⟩
@@ -526,11 +526,11 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     [t : ∀ i, TopologicalSpace (X i)] {T : ∀ i, Set (Set (X i))}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
     @IsTopologicalBasis β (⨅ i, induced (f i) (t i))
-      { S |
+      {S |
         ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
-          (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (hi : i ∈ F), f i ⁻¹' U i } :=
+          (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (hi : i ∈ F), f i ⁻¹' U i} :=
   by
-  convert(isTopologicalBasis_pi cond).Inducing (inducing_iInf_to_pi _)
+  convert (isTopologicalBasis_pi cond).Inducing (inducing_iInf_to_pi _)
   ext V
   constructor
   · rintro ⟨U, F, h1, h2⟩
@@ -555,7 +555,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
 
 #print isTopologicalBasis_singletons /-
 theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [DiscreteTopology α] :
-    IsTopologicalBasis { s | ∃ x : α, (s : Set α) = {x} } :=
+    IsTopologicalBasis {s | ∃ x : α, (s : Set α) = {x}} :=
   isTopologicalBasis_of_open_of_nhds (fun u hu => isOpen_discrete _) fun x u hx u_open =>
     ⟨{x}, ⟨x, rfl⟩, mem_singleton x, singleton_subset_iff.2 hx⟩
 #align is_topological_basis_singletons isTopologicalBasis_singletons
@@ -589,7 +589,7 @@ theorem Dense.exists_countable_dense_subset_bot_top {α : Type _} [TopologicalSp
       t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∈ s → x ∈ t) ∧ ∀ x, IsTop x → x ∈ s → x ∈ t :=
   by
   rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩
-  refine' ⟨(t ∪ ({ x | IsBot x } ∪ { x | IsTop x })) ∩ s, _, _, _, _, _⟩
+  refine' ⟨(t ∪ ({x | IsBot x} ∪ {x | IsTop x})) ∩ s, _, _, _, _, _⟩
   exacts [inter_subset_right _ _,
     (htc.union ((countable_is_bot α).union (countable_is_top α))).mono (inter_subset_left _ _),
     htd.mono (subset_inter (subset_union_left _ _) hts), fun x hx hxs => ⟨Or.inr <| Or.inl hx, hxs⟩,
@@ -674,7 +674,7 @@ instance isCountablyGenerated_nhdsWithin (x : α) [IsCountablyGenerated (𝓝 x)
 variable (α)
 
 #print TopologicalSpace.SecondCountableTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
 /-- A second-countable space is one with a countable basis. -/
 class SecondCountableTopology : Prop where
   is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b
@@ -802,13 +802,13 @@ instance {ι : Type _} {π : ι → Type _} [Countable ι] [t : ∀ a, Topologic
   constructor; refine' ⟨_, _, rfl⟩
   have :
     Set.Countable
-      { T : Set (∀ i, π i) |
+      {T : Set (∀ i, π i) |
         ∃ (I : Finset ι) (s : ∀ i : I, Set (π i)),
-          (∀ i, s i ∈ countable_basis (π i)) ∧ T = { f | ∀ i : I, f i ∈ s i } } :=
+          (∀ i, s i ∈ countable_basis (π i)) ∧ T = {f | ∀ i : I, f i ∈ s i}} :=
     by
     simp only [set_of_exists, ← exists_prop]
     refine' countable_Union fun I => countable.bUnion _ fun _ _ => countable_singleton _
-    change Set.Countable { s : ∀ i : I, Set (π i) | ∀ i, s i ∈ countable_basis (π i) }
+    change Set.Countable {s : ∀ i : I, Set (π i) | ∀ i, s i ∈ countable_basis (π i)}
     exact countable_pi fun i => countable_countable_basis _
   convert this using 1; ext1 T; constructor
   · rintro ⟨s, I, hs, rfl⟩
@@ -847,7 +847,7 @@ is equal to the union of countably many of those sets. -/
 theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
     (H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ (⋃ i ∈ T, s i) = ⋃ i, s i :=
   by
-  let B := { b ∈ countable_basis α | ∃ i, b ⊆ s i }
+  let B := {b ∈ countable_basis α | ∃ i, b ⊆ s i}
   choose f hf using fun b : B => b.2.2
   haveI : Encodable B := ((countable_countable_basis α).mono (sep_subset _ _)).toEncodable
   refine' ⟨_, countable_range f, (Union₂_subset_Union _ _).antisymm (sUnion_subset _)⟩

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

@@ -118,7 +118,7 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
   · rw [sUnion_image, Union₂_eq_univ_iff]
     exact fun x => ⟨∅, ⟨finite_empty, empty_subset _⟩, sInter_empty.substr <| mem_univ x⟩
   · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩; apply isOpen_sInter
-    exacts[hft, fun s hs => generate_open.basic _ <| htb hs]
+    exacts [hft, fun s hs => generate_open.basic _ <| htb hs]
   · rw [← sInter_singleton t]
     exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩
 #align topological_space.is_topological_basis_of_subbasis TopologicalSpace.isTopologicalBasis_of_subbasis
@@ -204,19 +204,19 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
-    (ou : IsOpen u) : ∃ (S : _)(_ : S ⊆ B), u = ⋃₀ S :=
+    (ou : IsOpen u) : ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
   ⟨{ s ∈ B | s ⊆ u }, fun s h => h.1, hB.open_eq_sUnion' ou⟩
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
-    {u : Set α} : IsOpen u ↔ ∃ (S : _)(_ : S ⊆ B), u = ⋃₀ S :=
+    {u : Set α} : IsOpen u ↔ ∃ (S : _) (_ : S ⊆ B), u = ⋃₀ S :=
   ⟨hB.open_eq_sUnion, fun ⟨S, hSB, hu⟩ => hu.symm ▸ isOpen_sUnion fun s hs => hB.IsOpen (hSB hs)⟩
 #align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion
 
 #print TopologicalSpace.IsTopologicalBasis.open_eq_iUnion /-
 theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
-    (ou : IsOpen u) : ∃ (β : Type u)(f : β → Set α), (u = ⋃ i, f i) ∧ ∀ i, f i ∈ B :=
+    (ou : IsOpen u) : ∃ (β : Type u) (f : β → Set α), (u = ⋃ i, f i) ∧ ∀ i, f i ∈ B :=
   ⟨↥({ s ∈ B | s ⊆ u }), coe, by rw [← sUnion_eq_Union]; apply hB.open_eq_sUnion' ou, fun s =>
     And.left s.2⟩
 #align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
@@ -248,7 +248,7 @@ theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTop
     {u : Set α} (hu : u.Nonempty) (ou : IsOpen u) : ∃ v ∈ B, Set.Nonempty v ∧ v ⊆ u :=
   by
   cases' hu with x hx
-  rw [hb.open_eq_sUnion' ou, mem_sUnion] at hx
+  rw [hb.open_eq_sUnion' ou, mem_sUnion] at hx 
   rcases hx with ⟨v, hv, hxv⟩
   exact ⟨v, hv.1, ⟨x, hxv⟩, hv.2⟩
 #align topological_space.is_topological_basis.exists_nonempty_subset TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset
@@ -282,7 +282,7 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
     rwa [hf.is_open_iff]
     refine' ⟨V, h.is_open hV, rfl⟩
   · intro a U ha hU
-    rw [hf.is_open_iff] at hU
+    rw [hf.is_open_iff] at hU 
     obtain ⟨V, hV, rfl⟩ := hU
     obtain ⟨S, hS, rfl⟩ := h.open_eq_sUnion hV
     obtain ⟨W, hW, ha⟩ := ha
@@ -294,7 +294,7 @@ theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen
     IsTopologicalBasis (⋃ i : ι, image (coe : U i → α) '' b i) :=
   by
   refine' is_topological_basis_of_open_of_nhds (fun u hu => _) _
-  · simp only [mem_Union, mem_image] at hu
+  · simp only [mem_Union, mem_image] at hu 
     rcases hu with ⟨i, s, sb, rfl⟩
     exact (Uo i).isOpenMap_subtype_val _ ((hb i).IsOpen sb)
   · intro a u ha uo
@@ -505,7 +505,7 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
     {T : ∀ i, Set (Set (X i))} (cond : ∀ i, IsTopologicalBasis (T i)) :
     IsTopologicalBasis
       { S : Set (∀ i, X i) |
-        ∃ (U : ∀ i, Set (X i))(F : Finset ι), (∀ i, i ∈ F → U i ∈ T i) ∧ S = (F : Set ι).pi U } :=
+        ∃ (U : ∀ i, Set (X i)) (F : Finset ι), (∀ i, i ∈ F → U i ∈ T i) ∧ S = (F : Set ι).pi U } :=
   by
   refine' is_topological_basis_of_open_of_nhds _ _
   · rintro _ ⟨U, F, h1, rfl⟩
@@ -514,7 +514,7 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
     exact (cond i).IsOpen (h1 i hi)
   · intro a U ha hU
     obtain ⟨I, t, hta, htU⟩ :
-      ∃ (I : Finset ι)(t : ∀ i : ι, Set (X i)), (∀ i, t i ∈ 𝓝 (a i)) ∧ Set.pi (↑I) t ⊆ U := by
+      ∃ (I : Finset ι) (t : ∀ i : ι, Set (X i)), (∀ i, t i ∈ 𝓝 (a i)) ∧ Set.pi (↑I) t ⊆ U := by
       rw [← Filter.mem_pi', ← nhds_pi]; exact hU.mem_nhds ha
     have : ∀ i, ∃ V ∈ T i, a i ∈ V ∧ V ⊆ t i := fun i => (cond i).mem_nhds_iffₓ.1 (hta i)
     choose V hVT haV hVt
@@ -527,7 +527,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
     @IsTopologicalBasis β (⨅ i, induced (f i) (t i))
       { S |
-        ∃ (U : ∀ i, Set (X i))(F : Finset ι),
+        ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
           (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (hi : i ∈ F), f i ⁻¹' U i } :=
   by
   convert(isTopologicalBasis_pi cond).Inducing (inducing_iInf_to_pi _)
@@ -571,7 +571,7 @@ protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace 
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}
-    [SeparableSpace s] (hs : Dense s) : ∃ (t : _)(_ : t ⊆ s), t.Countable ∧ Dense t :=
+    [SeparableSpace s] (hs : Dense s) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
   ⟨coe '' t, image_subset_iff.2 fun x _ => mem_preimage.2 <| Subtype.coe_prop _, htc.image coe,
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
@@ -585,12 +585,12 @@ to `s`. For a dense subset containing neither bot nor top elements, see
 `dense.exists_countable_dense_subset_no_bot_top`. -/
 theorem Dense.exists_countable_dense_subset_bot_top {α : Type _} [TopologicalSpace α]
     [PartialOrder α] {s : Set α} [SeparableSpace s] (hs : Dense s) :
-    ∃ (t : _)(_ : t ⊆ s),
+    ∃ (t : _) (_ : t ⊆ s),
       t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∈ s → x ∈ t) ∧ ∀ x, IsTop x → x ∈ s → x ∈ t :=
   by
   rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩
   refine' ⟨(t ∪ ({ x | IsBot x } ∪ { x | IsTop x })) ∩ s, _, _, _, _, _⟩
-  exacts[inter_subset_right _ _,
+  exacts [inter_subset_right _ _,
     (htc.union ((countable_is_bot α).union (countable_is_top α))).mono (inter_subset_left _ _),
     htd.mono (subset_inter (subset_union_left _ _) hts), fun x hx hxs => ⟨Or.inr <| Or.inl hx, hxs⟩,
     fun x hx hxs => ⟨Or.inr <| Or.inr hx, hxs⟩]
@@ -698,7 +698,7 @@ theorem exists_countable_basis [SecondCountableTopology α] :
   by
   obtain ⟨b, hb₁, hb₂⟩ := second_countable_topology.is_open_generated_countable α
   refine' ⟨_, _, not_mem_diff_of_mem _, (is_topological_basis_of_subbasis hb₂).diff_empty⟩
-  exacts[((countable_set_of_finite_subset hb₁).image _).mono (diff_subset _ _), rfl]
+  exacts [((countable_set_of_finite_subset hb₁).image _).mono (diff_subset _ _), rfl]
 #align topological_space.exists_countable_basis TopologicalSpace.exists_countable_basis
 -/
 
@@ -803,7 +803,7 @@ instance {ι : Type _} {π : ι → Type _} [Countable ι] [t : ∀ a, Topologic
   have :
     Set.Countable
       { T : Set (∀ i, π i) |
-        ∃ (I : Finset ι)(s : ∀ i : I, Set (π i)),
+        ∃ (I : Finset ι) (s : ∀ i : I, Set (π i)),
           (∀ i, s i ∈ countable_basis (π i)) ∧ T = { f | ∀ i : I, f i ∈ s i } } :=
     by
     simp only [set_of_exists, ← exists_prop]
@@ -883,12 +883,12 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
-    (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _)(_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=
+    (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _) (_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=
   by
   have : ∀ x : s, coe ⁻¹' f x ∈ 𝓝 x := fun x => preimage_coe_mem_nhds_subtype.2 (hf x x.2)
   rcases countable_cover_nhds this with ⟨t, htc, htU⟩
   refine' ⟨coe '' t, Subtype.coe_image_subset _ _, htc.image _, fun x hx => _⟩
-  simp only [bUnion_image, eq_univ_iff_forall, ← preimage_Union, mem_preimage] at htU⊢
+  simp only [bUnion_image, eq_univ_iff_forall, ← preimage_Union, mem_preimage] at htU ⊢
   exact htU ⟨x, hx⟩
 #align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithin
 
@@ -902,16 +902,16 @@ omit t
 topological bases on each of the parts of the space. -/
 theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
     (hs : ∀ i, IsTopologicalBasis (s i)) :
-    IsTopologicalBasis (⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' s i) :=
+    IsTopologicalBasis (⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σ i, E i))) '' s i) :=
   by
   apply is_topological_basis_of_open_of_nhds
   · intro u hu
-    obtain ⟨i, t, ts, rfl⟩ : ∃ (i : ι)(t : Set (E i)), t ∈ s i ∧ Sigma.mk i '' t = u := by
+    obtain ⟨i, t, ts, rfl⟩ : ∃ (i : ι) (t : Set (E i)), t ∈ s i ∧ Sigma.mk i '' t = u := by
       simpa only [mem_Union, mem_image] using hu
     exact isOpenMap_sigmaMk _ ((hs i).IsOpen ts)
   · rintro ⟨i, x⟩ u hxu u_open
     have hx : x ∈ Sigma.mk i ⁻¹' u := hxu
-    obtain ⟨v, vs, xv, hv⟩ : ∃ (v : Set (E i))(H : v ∈ s i), x ∈ v ∧ v ⊆ Sigma.mk i ⁻¹' u :=
+    obtain ⟨v, vs, xv, hv⟩ : ∃ (v : Set (E i)) (H : v ∈ s i), x ∈ v ∧ v ⊆ Sigma.mk i ⁻¹' u :=
       (hs i).exists_subset_of_mem_open hx (isOpen_sigma_iff.1 u_open i)
     exact
       ⟨Sigma.mk i '' v, mem_Union.2 ⟨i, mem_image_of_mem _ vs⟩, mem_image_of_mem _ xv,
@@ -919,9 +919,9 @@ theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
 #align topological_space.is_topological_basis.sigma TopologicalSpace.IsTopologicalBasis.sigma
 
 /-- A countable disjoint union of second countable spaces is second countable. -/
-instance [Countable ι] [∀ i, SecondCountableTopology (E i)] : SecondCountableTopology (Σi, E i) :=
+instance [Countable ι] [∀ i, SecondCountableTopology (E i)] : SecondCountableTopology (Σ i, E i) :=
   by
-  let b := ⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' countable_basis (E i)
+  let b := ⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σ i, E i))) '' countable_basis (E i)
   have A : is_topological_basis b := is_topological_basis.sigma fun i => is_basis_countable_basis _
   have B : b.countable := countable_Union fun i => countable.image (countable_countable_basis _) _
   exact A.second_countable_topology B
@@ -950,13 +950,13 @@ theorem IsTopologicalBasis.sum {s : Set (Set α)} (hs : IsTopologicalBasis s) {t
   · rintro x u hxu u_open
     cases x
     · have h'x : x ∈ Sum.inl ⁻¹' u := hxu
-      obtain ⟨v, vs, xv, vu⟩ : ∃ (v : Set α)(H : v ∈ s), x ∈ v ∧ v ⊆ Sum.inl ⁻¹' u :=
+      obtain ⟨v, vs, xv, vu⟩ : ∃ (v : Set α) (H : v ∈ s), x ∈ v ∧ v ⊆ Sum.inl ⁻¹' u :=
         hs.exists_subset_of_mem_open h'x (isOpen_sum_iff.1 u_open).1
       exact
         ⟨Sum.inl '' v, mem_union_left _ (mem_image_of_mem _ vs), mem_image_of_mem _ xv,
           image_subset_iff.2 vu⟩
     · have h'x : x ∈ Sum.inr ⁻¹' u := hxu
-      obtain ⟨v, vs, xv, vu⟩ : ∃ (v : Set β)(H : v ∈ t), x ∈ v ∧ v ⊆ Sum.inr ⁻¹' u :=
+      obtain ⟨v, vs, xv, vu⟩ : ∃ (v : Set β) (H : v ∈ t), x ∈ v ∧ v ⊆ Sum.inr ⁻¹' u :=
         ht.exists_subset_of_mem_open h'x (isOpen_sum_iff.1 u_open).2
       exact
         ⟨Sum.inr '' v, mem_union_right _ (mem_image_of_mem _ vs), mem_image_of_mem _ xv,

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

@@ -56,7 +56,7 @@ More fine grained instances for `first_countable_topology`, `separable_space`, `
 
 open Set Filter Function
 
-open Topology Filter
+open scoped Topology Filter
 
 noncomputable section
 
@@ -603,6 +603,7 @@ instance separableSpace_univ {α : Type _} [TopologicalSpace α] [SeparableSpace
 #align separable_space_univ separableSpace_univ
 -/
 
+#print exists_countable_dense_bot_top /-
 /-- If `α` is a separable topological space with a partial order, then there exists a countable
 dense set `s : set α` that contains those of both bottom and top elements of `α` that actually
 exist. For a dense set containing neither bot nor top elements, see
@@ -612,6 +613,7 @@ theorem exists_countable_dense_bot_top (α : Type _) [TopologicalSpace α] [Sepa
     ∃ s : Set α, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∈ s) ∧ ∀ x, IsTop x → x ∈ s := by
   simpa using dense_univ.exists_countable_dense_subset_bot_top
 #align exists_countable_dense_bot_top exists_countable_dense_bot_top
+-/
 
 namespace TopologicalSpace
 

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

@@ -93,12 +93,6 @@ theorem IsTopologicalBasis.insert_empty {s : Set (Set α)} (h : IsTopologicalBas
 #align topological_space.is_topological_basis.insert_empty TopologicalSpace.IsTopologicalBasis.insert_empty
 -/
 
-/- warning: topological_space.is_topological_basis.diff_empty -> TopologicalSpace.IsTopologicalBasis.diff_empty is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {s : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t s) -> (TopologicalSpace.IsTopologicalBasis.{u1} α t (SDiff.sdiff.{u1} (Set.{u1} (Set.{u1} α)) (BooleanAlgebra.toHasSdiff.{u1} (Set.{u1} (Set.{u1} α)) (Set.booleanAlgebra.{u1} (Set.{u1} α))) s (Singleton.singleton.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasSingleton.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.hasEmptyc.{u1} α)))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {s : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t s) -> (TopologicalSpace.IsTopologicalBasis.{u1} α t (SDiff.sdiff.{u1} (Set.{u1} (Set.{u1} α)) (Set.instSDiffSet.{u1} (Set.{u1} α)) s (Singleton.singleton.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instSingletonSet.{u1} (Set.{u1} α)) (EmptyCollection.emptyCollection.{u1} (Set.{u1} α) (Set.instEmptyCollectionSet.{u1} α)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.diff_empty TopologicalSpace.IsTopologicalBasis.diff_emptyₓ'. -/
 theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
     IsTopologicalBasis (s \ {∅}) :=
   by
@@ -130,12 +124,6 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
 #align topological_space.is_topological_basis_of_subbasis TopologicalSpace.isTopologicalBasis_of_subbasis
 -/
 
-/- warning: topological_space.is_topological_basis_of_open_of_nhds -> TopologicalSpace.isTopologicalBasis_of_open_of_nhds is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {s : Set.{u1} (Set.{u1} α)}, (forall (u : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) u s) -> (IsOpen.{u1} α t u)) -> (forall (a : α) (u : Set.{u1} α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a u) -> (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) v s) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) v s) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a v) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) v u))))) -> (TopologicalSpace.IsTopologicalBasis.{u1} α t s)
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {s : Set.{u1} (Set.{u1} α)}, (forall (u : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) u s) -> (IsOpen.{u1} α t u)) -> (forall (a : α) (u : Set.{u1} α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a u) -> (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) v s) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a v) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) v u))))) -> (TopologicalSpace.IsTopologicalBasis.{u1} α t s)
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_open_of_nhdsₓ'. -/
 /-- If a family of open sets `s` is such that every open neighbourhood contains some
 member of `s`, then `s` is a topological basis. -/
 theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
@@ -154,12 +142,6 @@ theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u 
     exact iInf₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
 #align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_open_of_nhds
 
-/- warning: topological_space.is_topological_basis.mem_nhds_iff -> TopologicalSpace.IsTopologicalBasis.mem_nhds_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α} {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (Iff (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) s (nhds.{u1} α t a)) (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t b) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t b) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s)))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {a : α} {s : Set.{u1} α} {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (Iff (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) s (nhds.{u1} α t a)) (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t b) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.mem_nhds_iff TopologicalSpace.IsTopologicalBasis.mem_nhds_iffₓ'. -/
 /-- A set `s` is in the neighbourhood of `a` iff there is some basis set `t`, which
 contains `a` and is itself contained in `s`. -/
 theorem IsTopologicalBasis.mem_nhds_iff {a : α} {s : Set α} {b : Set (Set α)}
@@ -178,12 +160,6 @@ theorem IsTopologicalBasis.mem_nhds_iff {a : α} {s : Set α} {b : Set (Set α)}
     exact ⟨i, h2, h1⟩
 #align topological_space.is_topological_basis.mem_nhds_iff TopologicalSpace.IsTopologicalBasis.mem_nhds_iff
 
-/- warning: topological_space.is_topological_basis.is_open_iff -> TopologicalSpace.IsTopologicalBasis.isOpen_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {s : Set.{u1} α} {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (Iff (IsOpen.{u1} α t s) (forall (a : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t b) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) t b) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s))))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {s : Set.{u1} α} {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (Iff (IsOpen.{u1} α t s) (forall (a : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) t b) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) t s))))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.is_open_iff TopologicalSpace.IsTopologicalBasis.isOpen_iffₓ'. -/
 theorem IsTopologicalBasis.isOpen_iff {s : Set α} {b : Set (Set α)} (hb : IsTopologicalBasis b) :
     IsOpen s ↔ ∀ a ∈ s, ∃ t ∈ b, a ∈ t ∧ t ⊆ s := by simp [isOpen_iff_mem_nhds, hb.mem_nhds_iff]
 #align topological_space.is_topological_basis.is_open_iff TopologicalSpace.IsTopologicalBasis.isOpen_iff
@@ -209,12 +185,6 @@ protected theorem IsTopologicalBasis.mem_nhds {a : α} {s : Set α} {b : Set (Se
 #align topological_space.is_topological_basis.mem_nhds TopologicalSpace.IsTopologicalBasis.mem_nhds
 -/
 
-/- warning: topological_space.is_topological_basis.exists_subset_of_mem_open -> TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_open is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (forall {a : α} {u : Set.{u1} α}, (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a u) -> (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) v b) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) v b) => And (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a v) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) v u)))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (forall {a : α} {u : Set.{u1} α}, (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a u) -> (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) v b) (And (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a v) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) v u)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.exists_subset_of_mem_open TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_openₓ'. -/
 theorem IsTopologicalBasis.exists_subset_of_mem_open {b : Set (Set α)} (hb : IsTopologicalBasis b)
     {a : α} {u : Set α} (au : a ∈ u) (ou : IsOpen u) : ∃ v ∈ b, a ∈ v ∧ v ⊆ u :=
   hb.mem_nhds_iffₓ.1 <| IsOpen.mem_nhds ou au
@@ -232,24 +202,12 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
 #align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 -/
 
-/- warning: topological_space.is_topological_basis.open_eq_sUnion -> TopologicalSpace.IsTopologicalBasis.open_eq_sUnion is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) (fun (H : HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) => Eq.{succ u1} (Set.{u1} α) u (Set.sUnion.{u1} α S)))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => And (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) S B) (Eq.{succ u1} (Set.{u1} α) u (Set.sUnion.{u1} α S)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnionₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _)(_ : S ⊆ B), u = ⋃₀ S :=
   ⟨{ s ∈ B | s ⊆ u }, fun s h => h.1, hB.open_eq_sUnion' ou⟩
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 
-/- warning: topological_space.is_topological_basis.open_iff_eq_sUnion -> TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, Iff (IsOpen.{u1} α t u) (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) (fun (H : HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) => Eq.{succ u1} (Set.{u1} α) u (Set.sUnion.{u1} α S)))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, Iff (IsOpen.{u1} α t u) (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => And (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) S B) (Eq.{succ u1} (Set.{u1} α) u (Set.sUnion.{u1} α S)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnionₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ (S : _)(_ : S ⊆ B), u = ⋃₀ S :=
@@ -264,24 +222,12 @@ theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopological
 #align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
 -/
 
-/- warning: topological_space.is_topological_basis.mem_closure_iff -> TopologicalSpace.IsTopologicalBasis.mem_closure_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (forall {s : Set.{u1} α} {a : α}, Iff (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a (closure.{u1} α t s)) (forall (o : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) o b) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) a o) -> (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) o s))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (forall {s : Set.{u1} α} {a : α}, Iff (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a (closure.{u1} α t s)) (forall (o : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) o b) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) a o) -> (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) o s))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.mem_closure_iff TopologicalSpace.IsTopologicalBasis.mem_closure_iffₓ'. -/
 /-- A point `a` is in the closure of `s` iff all basis sets containing `a` intersect `s`. -/
 theorem IsTopologicalBasis.mem_closure_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α}
     {a : α} : a ∈ closure s ↔ ∀ o ∈ b, a ∈ o → (o ∩ s).Nonempty :=
   (mem_closure_iff_nhds_basis' hb.nhds_hasBasis).trans <| by simp only [and_imp]
 #align topological_space.is_topological_basis.mem_closure_iff TopologicalSpace.IsTopologicalBasis.mem_closure_iff
 
-/- warning: topological_space.is_topological_basis.dense_iff -> TopologicalSpace.IsTopologicalBasis.dense_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (forall {s : Set.{u1} α}, Iff (Dense.{u1} α t s) (forall (o : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) o b) -> (Set.Nonempty.{u1} α o) -> (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) o s))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {b : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t b) -> (forall {s : Set.{u1} α}, Iff (Dense.{u1} α t s) (forall (o : Set.{u1} α), (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) o b) -> (Set.Nonempty.{u1} α o) -> (Set.Nonempty.{u1} α (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) o s))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.dense_iff TopologicalSpace.IsTopologicalBasis.dense_iffₓ'. -/
 /-- A set is dense iff it has non-trivial intersection with all basis sets. -/
 theorem IsTopologicalBasis.dense_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α} :
     Dense s ↔ ∀ o ∈ b, Set.Nonempty o → (o ∩ s).Nonempty :=
@@ -290,12 +236,6 @@ theorem IsTopologicalBasis.dense_iff {b : Set (Set α)} (hb : IsTopologicalBasis
   exact ⟨fun h o hb ⟨a, ha⟩ => h a o hb ha, fun h a o hb ha => h o hb ⟨a, ha⟩⟩
 #align topological_space.is_topological_basis.dense_iff TopologicalSpace.IsTopologicalBasis.dense_iff
 
-/- warning: topological_space.is_topological_basis.is_open_map_iff -> TopologicalSpace.IsTopologicalBasis.isOpenMap_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} β] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {f : α -> β}, Iff (IsOpenMap.{u1, u2} α β t _inst_1 f) (forall (s : Set.{u1} α), (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) s B) -> (IsOpen.{u2} β _inst_1 (Set.image.{u1, u2} α β f s))))
-but is expected to have type
-  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} β] {B : Set.{u2} (Set.{u2} α)}, (TopologicalSpace.IsTopologicalBasis.{u2} α t B) -> (forall {f : α -> β}, Iff (IsOpenMap.{u2, u1} α β t _inst_1 f) (forall (s : Set.{u2} α), (Membership.mem.{u2, u2} (Set.{u2} α) (Set.{u2} (Set.{u2} α)) (Set.instMembershipSet.{u2} (Set.{u2} α)) s B) -> (IsOpen.{u1} β _inst_1 (Set.image.{u2, u1} α β f s))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.is_open_map_iff TopologicalSpace.IsTopologicalBasis.isOpenMap_iffₓ'. -/
 theorem IsTopologicalBasis.isOpenMap_iff {β} [TopologicalSpace β] {B : Set (Set α)}
     (hB : IsTopologicalBasis B) {f : α → β} : IsOpenMap f ↔ ∀ s ∈ B, IsOpen (f '' s) :=
   by
@@ -304,12 +244,6 @@ theorem IsTopologicalBasis.isOpenMap_iff {β} [TopologicalSpace β] {B : Set (Se
   exact isOpen_iUnion fun s => hf s s.2.1
 #align topological_space.is_topological_basis.is_open_map_iff TopologicalSpace.IsTopologicalBasis.isOpenMap_iff
 
-/- warning: topological_space.is_topological_basis.exists_nonempty_subset -> TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (Set.Nonempty.{u1} α u) -> (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) v B) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.hasMem.{u1} (Set.{u1} α)) v B) => And (Set.Nonempty.{u1} α v) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) v u)))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (Set.Nonempty.{u1} α u) -> (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} α) (fun (v : Set.{u1} α) => And (Membership.mem.{u1, u1} (Set.{u1} α) (Set.{u1} (Set.{u1} α)) (Set.instMembershipSet.{u1} (Set.{u1} α)) v B) (And (Set.Nonempty.{u1} α v) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) v u)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.exists_nonempty_subset TopologicalSpace.IsTopologicalBasis.exists_nonempty_subsetₓ'. -/
 theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTopologicalBasis B)
     {u : Set α} (hu : u.Nonempty) (ou : IsOpen u) : ∃ v ∈ B, Set.Nonempty v ∧ v ⊆ u :=
   by
@@ -325,12 +259,6 @@ theorem isTopologicalBasis_opens : IsTopologicalBasis { U : Set α | IsOpen U }
 #align topological_space.is_topological_basis_opens TopologicalSpace.isTopologicalBasis_opens
 -/
 
-/- warning: topological_space.is_topological_basis.prod -> TopologicalSpace.IsTopologicalBasis.prod is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} β] {B₁ : Set.{u2} (Set.{u2} α)} {B₂ : Set.{u1} (Set.{u1} β)}, (TopologicalSpace.IsTopologicalBasis.{u2} α t B₁) -> (TopologicalSpace.IsTopologicalBasis.{u1} β _inst_1 B₂) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Prod.{u2, u1} α β) (instTopologicalSpaceProd.{u2, u1} α β t _inst_1) (Set.image2.{u2, u1, max u1 u2} (Set.{u2} α) (Set.{u1} β) (Set.{max u1 u2} (Prod.{u2, u1} α β)) (Set.prod.{u2, u1} α β) B₁ B₂))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.prod TopologicalSpace.IsTopologicalBasis.prodₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:228:8: unsupported: ambiguous notation -/
 protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set (Set α)}
@@ -346,12 +274,6 @@ protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set
     exact ⟨s ×ˢ t, mem_image2_of_mem hs ht, ⟨ha, hb⟩, hu⟩
 #align topological_space.is_topological_basis.prod TopologicalSpace.IsTopologicalBasis.prod
 
-/- warning: topological_space.is_topological_basis.inducing -> TopologicalSpace.IsTopologicalBasis.inducing is a dubious translation:
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-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} β] {f : α -> β} {T : Set.{u2} (Set.{u2} β)}, (Inducing.{u1, u2} α β t _inst_1 f) -> (TopologicalSpace.IsTopologicalBasis.{u2} β _inst_1 T) -> (TopologicalSpace.IsTopologicalBasis.{u1} α t (Set.image.{u2, u1} (Set.{u2} β) (Set.{u1} α) (Set.preimage.{u1, u2} α β f) T))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducingₓ'. -/
 protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α → β} {T : Set (Set β)}
     (hf : Inducing f) (h : IsTopologicalBasis T) : IsTopologicalBasis (image (preimage f) T) :=
   by
@@ -367,12 +289,6 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
     refine' ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_sUnion_of_mem hW⟩
 #align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducing
 
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-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis_of_cover TopologicalSpace.isTopologicalBasis_of_coverₓ'. -/
 theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
     (Uc : (⋃ i, U i) = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
     IsTopologicalBasis (⋃ i : ι, image (coe : U i → α) '' b i) :=
@@ -391,12 +307,6 @@ theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen
         image_subset_iff.2 hvu⟩
 #align topological_space.is_topological_basis_of_cover TopologicalSpace.isTopologicalBasis_of_cover
 
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-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.continuous TopologicalSpace.IsTopologicalBasis.continuousₓ'. -/
 protected theorem IsTopologicalBasis.continuous {β : Type _} [TopologicalSpace β] {B : Set (Set β)}
     (hB : IsTopologicalBasis B) (f : α → β) (hf : ∀ s ∈ B, IsOpen (f ⁻¹' s)) : Continuous f := by
   rw [hB.eq_generate_from]; exact continuous_generateFrom hf
@@ -465,23 +375,11 @@ instance (priority := 100) Countable.to_separableSpace [Countable α] : Separabl
 #align topological_space.countable.to_separable_space TopologicalSpace.Countable.to_separableSpace
 -/
 
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-Case conversion may be inaccurate. Consider using '#align topological_space.separable_space_of_dense_range TopologicalSpace.separableSpace_of_denseRangeₓ'. -/
 theorem separableSpace_of_denseRange {ι : Type _} [Countable ι] (u : ι → α) (hu : DenseRange u) :
     SeparableSpace α :=
   ⟨⟨range u, countable_range u, hu⟩⟩
 #align topological_space.separable_space_of_dense_range TopologicalSpace.separableSpace_of_denseRange
 
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-Case conversion may be inaccurate. Consider using '#align set.pairwise_disjoint.countable_of_is_open Set.PairwiseDisjoint.countable_of_isOpenₓ'. -/
 /-- In a separable space, a family of nonempty disjoint open sets is countable. -/
 theorem Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type _} {s : ι → Set α}
     {a : Set ι} (h : a.PairwiseDisjoint s) (ha : ∀ i ∈ a, IsOpen (s i))
@@ -500,12 +398,6 @@ theorem Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type
   exact ⟨@Encodable.ofInj _ _ u_encodable f f_inj⟩
 #align set.pairwise_disjoint.countable_of_is_open Set.PairwiseDisjoint.countable_of_isOpen
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align set.pairwise_disjoint.countable_of_nonempty_interior Set.PairwiseDisjoint.countable_of_nonempty_interiorₓ'. -/
 /-- In a separable space, a family of disjoint sets with nonempty interiors is countable. -/
 theorem Set.PairwiseDisjoint.countable_of_nonempty_interior [SeparableSpace α] {ι : Type _}
     {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s)
@@ -531,12 +423,6 @@ theorem IsSeparable.mono {s u : Set α} (hs : IsSeparable s) (hu : u ⊆ s) : Is
 #align topological_space.is_separable.mono TopologicalSpace.IsSeparable.mono
 -/
 
-/- warning: topological_space.is_separable.union -> TopologicalSpace.IsSeparable.union is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {s : Set.{u1} α} {u : Set.{u1} α}, (TopologicalSpace.IsSeparable.{u1} α t s) -> (TopologicalSpace.IsSeparable.{u1} α t u) -> (TopologicalSpace.IsSeparable.{u1} α t (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s u))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_separable.union TopologicalSpace.IsSeparable.unionₓ'. -/
 theorem IsSeparable.union {s u : Set α} (hs : IsSeparable s) (hu : IsSeparable u) :
     IsSeparable (s ∪ u) := by
   rcases hs with ⟨cs, cs_count, hcs⟩
@@ -555,12 +441,6 @@ theorem IsSeparable.closure {s : Set α} (hs : IsSeparable s) : IsSeparable (clo
 #align topological_space.is_separable.closure TopologicalSpace.IsSeparable.closure
 -/
 
-/- warning: topological_space.is_separable_Union -> TopologicalSpace.isSeparable_iUnion is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnionₓ'. -/
 theorem isSeparable_iUnion {ι : Type _} [Countable ι] {s : ι → Set α}
     (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) :=
   by
@@ -599,12 +479,6 @@ theorem isSeparable_of_separableSpace [h : SeparableSpace α] (s : Set α) : IsS
 #align topological_space.is_separable_of_separable_space TopologicalSpace.isSeparable_of_separableSpace
 -/
 
-/- warning: topological_space.is_separable.image -> TopologicalSpace.IsSeparable.image is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align topological_space.is_separable.image TopologicalSpace.IsSeparable.imageₓ'. -/
 theorem IsSeparable.image {β : Type _} [TopologicalSpace β] {s : Set α} (hs : IsSeparable s)
     {f : α → β} (hf : Continuous f) : IsSeparable (f '' s) :=
   by
@@ -627,12 +501,6 @@ end TopologicalSpace
 
 open TopologicalSpace
 
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 theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, TopologicalSpace (X i)]
     {T : ∀ i, Set (Set (X i))} (cond : ∀ i, IsTopologicalBasis (T i)) :
     IsTopologicalBasis
@@ -654,12 +522,6 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
       ⟨_, ⟨V, I, fun i hi => hVT i, rfl⟩, fun i hi => haV i, (pi_mono fun i hi => hVt i).trans htU⟩
 #align is_topological_basis_pi isTopologicalBasis_pi
 
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 theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     [t : ∀ i, TopologicalSpace (X i)] {T : ∀ i, Set (Set (X i))}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
@@ -699,12 +561,6 @@ theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [Discr
 #align is_topological_basis_singletons isTopologicalBasis_singletons
 -/
 
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 /-- If `α` is a separable space and `f : α → β` is a continuous map with dense range, then `β` is
 a separable space as well. E.g., the completion of a separable uniform space is separable. -/
 protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace α] [SeparableSpace α]
@@ -721,12 +577,6 @@ theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α]
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
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 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
@@ -753,12 +603,6 @@ instance separableSpace_univ {α : Type _} [TopologicalSpace α] [SeparableSpace
 #align separable_space_univ separableSpace_univ
 -/
 
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 /-- If `α` is a separable topological space with a partial order, then there exists a countable
 dense set `s : set α` that contains those of both bottom and top elements of `α` that actually
 exist. For a dense set containing neither bot nor top elements, see
@@ -924,12 +768,6 @@ instance (priority := 100) SecondCountableTopology.to_firstCountableTopology
 #align topological_space.second_countable_topology.to_first_countable_topology TopologicalSpace.SecondCountableTopology.to_firstCountableTopology
 -/
 
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 /-- If `β` is a second-countable space, then its induced topology
 via `f` on `α` is also second-countable. -/
 theorem secondCountableTopology_induced (β) [t : TopologicalSpace β] [SecondCountableTopology β]
@@ -992,12 +830,6 @@ instance (priority := 100) SecondCountableTopology.to_separableSpace [SecondCoun
 
 variable {α}
 
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 /-- A countable open cover induces a second-countable topology if all open covers
 are themselves second countable. -/
 theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι → Set α}
@@ -1008,12 +840,6 @@ theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι 
   this.second_countable_topology (countable_Union fun i => (countable_countable_basis _).image _)
 #align topological_space.second_countable_topology_of_countable_cover TopologicalSpace.secondCountableTopology_of_countable_cover
 
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-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {ι : Type.{u2}} (s : ι -> (Set.{u1} α)), (forall (i : ι), IsOpen.{u1} α t (s i)) -> (Exists.{succ u2} (Set.{u2} ι) (fun (T : Set.{u2} ι) => And (Set.Countable.{u2} ι T) (Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i T) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i T) => s i))) (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => s i)))))
-but is expected to have type
-  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u2} α t] {ι : Type.{u1}} (s : ι -> (Set.{u2} α)), (forall (i : ι), IsOpen.{u2} α t (s i)) -> (Exists.{succ u1} (Set.{u1} ι) (fun (T : Set.{u1} ι) => And (Set.Countable.{u1} ι T) (Eq.{succ u2} (Set.{u2} α) (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => Set.iUnion.{u2, 0} α (Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) i T) (fun (H : Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) i T) => s i))) (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => s i)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_open_Union_countable TopologicalSpace.isOpen_iUnion_countableₓ'. -/
 /-- In a second-countable space, an open set, given as a union of open sets,
 is equal to the union of countably many of those sets. -/
 theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
@@ -1053,12 +879,6 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds
 -/
 
-/- warning: topological_space.countable_cover_nhds_within -> TopologicalSpace.countable_cover_nhdsWithin is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (f x) (nhdsWithin.{u1} α t x s))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => f x)))))))
-but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7220 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
-Case conversion may be inaccurate. Consider using '#align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithinₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
     (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _)(_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=
@@ -1076,12 +896,6 @@ variable {ι : Type _} {E : ι → Type _} [∀ i, TopologicalSpace (E i)]
 
 omit t
 
-/- warning: topological_space.is_topological_basis.sigma -> TopologicalSpace.IsTopologicalBasis.sigma is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {E : ι -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (E i)] {s : forall (i : ι), Set.{u2} (Set.{u2} (E i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u2} (E i) (_inst_1 i) (s i)) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.topologicalSpace.{u1, u2} ι (fun (i : ι) => E i) (fun (a : ι) => _inst_1 a)) (Set.iUnion.{max u1 u2, succ u1} (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) ι (fun (i : ι) => Set.image.{u2, max u1 u2} (Set.{u2} (E i)) (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) (fun (u : Set.{u2} (E i)) => Set.image.{u2, max u1 u2} (E i) (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.mk.{u1, u2} ι (fun (i : ι) => E i) i) u) (s i))))
-but is expected to have type
-  forall {ι : Type.{u1}} {E : ι -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (E i)] {s : forall (i : ι), Set.{u2} (Set.{u2} (E i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u2} (E i) (_inst_1 i) (s i)) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (instTopologicalSpaceSigma.{u1, u2} ι (fun (i : ι) => E i) (fun (a : ι) => _inst_1 a)) (Set.iUnion.{max u1 u2, succ u1} (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) ι (fun (i : ι) => Set.image.{u2, max u1 u2} (Set.{u2} (E i)) (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) (fun (u : Set.{u2} (E i)) => Set.image.{u2, max u1 u2} (E i) (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.mk.{u1, u2} ι (fun (i : ι) => E i) i) u) (s i))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.sigma TopologicalSpace.IsTopologicalBasis.sigmaₓ'. -/
 /-- In a disjoint union space `Σ i, E i`, one can form a topological basis by taking the union of
 topological bases on each of the parts of the space. -/
 theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
@@ -1118,12 +932,6 @@ omit t
 
 variable {β : Type _} [TopologicalSpace α] [TopologicalSpace β]
 
-/- warning: topological_space.is_topological_basis.sum -> TopologicalSpace.IsTopologicalBasis.sum 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} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α _inst_1 s) -> (forall {t : Set.{u2} (Set.{u2} β)}, (TopologicalSpace.IsTopologicalBasis.{u2} β _inst_2 t) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Sum.{u1, u2} α β) (Sum.topologicalSpace.{u1, u2} α β _inst_1 _inst_2) (Union.union.{max u1 u2} (Set.{max u1 u2} (Set.{max u1 u2} (Sum.{u1, u2} α β))) (Set.hasUnion.{max u1 u2} (Set.{max u1 u2} (Sum.{u1, u2} α β))) (Set.image.{u1, max u1 u2} (Set.{u1} α) (Set.{max u1 u2} (Sum.{u1, u2} α β)) (fun (u : Set.{u1} α) => Set.image.{u1, max u1 u2} α (Sum.{u1, u2} α β) (Sum.inl.{u1, u2} α β) u) s) (Set.image.{u2, max u1 u2} (Set.{u2} β) (Set.{max u1 u2} (Sum.{u1, u2} α β)) (fun (u : Set.{u2} β) => Set.image.{u2, max u1 u2} β (Sum.{u1, u2} α β) (Sum.inr.{u1, u2} α β) u) 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} (Set.{u2} α)}, (TopologicalSpace.IsTopologicalBasis.{u2} α _inst_1 s) -> (forall {t : Set.{u1} (Set.{u1} β)}, (TopologicalSpace.IsTopologicalBasis.{u1} β _inst_2 t) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Sum.{u2, u1} α β) (instTopologicalSpaceSum.{u2, u1} α β _inst_1 _inst_2) (Union.union.{max u1 u2} (Set.{max u1 u2} (Set.{max u1 u2} (Sum.{u2, u1} α β))) (Set.instUnionSet.{max u2 u1} (Set.{max u1 u2} (Sum.{u2, u1} α β))) (Set.image.{u2, max u1 u2} (Set.{u2} α) (Set.{max u1 u2} (Sum.{u2, u1} α β)) (fun (u : Set.{u2} α) => Set.image.{u2, max u1 u2} α (Sum.{u2, u1} α β) (Sum.inl.{u2, u1} α β) u) s) (Set.image.{u1, max u2 u1} (Set.{u1} β) (Set.{max u1 u2} (Sum.{u2, u1} α β)) (fun (u : Set.{u1} β) => Set.image.{u1, max u2 u1} β (Sum.{u2, u1} α β) (Sum.inr.{u2, u1} α β) u) t))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.sum TopologicalSpace.IsTopologicalBasis.sumₓ'. -/
 /-- In a sum space `α ⊕ β`, one can form a topological basis by taking the union of
 topological bases on each of the two components. -/
 theorem IsTopologicalBasis.sum {s : Set (Set α)} (hs : IsTopologicalBasis s) {t : Set (Set β)}
@@ -1173,12 +981,6 @@ variable {X : Type _} [TopologicalSpace X] {Y : Type _} [TopologicalSpace Y] {π
 
 omit t
 
-/- warning: topological_space.is_topological_basis.quotient_map -> TopologicalSpace.IsTopologicalBasis.quotientMap is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {Y : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} Y] {π : X -> Y} {V : Set.{u1} (Set.{u1} X)}, (TopologicalSpace.IsTopologicalBasis.{u1} X _inst_1 V) -> (QuotientMap.{u1, u2} X Y _inst_1 _inst_2 π) -> (IsOpenMap.{u1, u2} X Y _inst_1 _inst_2 π) -> (TopologicalSpace.IsTopologicalBasis.{u2} Y _inst_2 (Set.image.{u1, u2} (Set.{u1} X) (Set.{u2} Y) (Set.image.{u1, u2} X Y π) V))
-but is expected to have type
-  forall {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {Y : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} Y] {π : X -> Y} {V : Set.{u2} (Set.{u2} X)}, (TopologicalSpace.IsTopologicalBasis.{u2} X _inst_1 V) -> (QuotientMap.{u2, u1} X Y _inst_1 _inst_2 π) -> (IsOpenMap.{u2, u1} X Y _inst_1 _inst_2 π) -> (TopologicalSpace.IsTopologicalBasis.{u1} Y _inst_2 (Set.image.{u2, u1} (Set.{u2} X) (Set.{u1} Y) (Set.image.{u2, u1} X Y π) V))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.quotient_map TopologicalSpace.IsTopologicalBasis.quotientMapₓ'. -/
 /-- The image of a topological basis under an open quotient map is a topological basis. -/
 theorem IsTopologicalBasis.quotientMap {V : Set (Set X)} (hV : IsTopologicalBasis V)
     (h' : QuotientMap π) (h : IsOpenMap π) : IsTopologicalBasis (Set.image π '' V) :=
@@ -1196,12 +998,6 @@ theorem IsTopologicalBasis.quotientMap {V : Set (Set X)} (hV : IsTopologicalBasi
     exact ⟨π '' Z, ⟨Z, Z_in_V, rfl⟩, ⟨x, x_in_Z, rfl⟩, πZ_in_U⟩
 #align topological_space.is_topological_basis.quotient_map TopologicalSpace.IsTopologicalBasis.quotientMap
 
-/- warning: topological_space.quotient_map.second_countable_topology -> QuotientMap.secondCountableTopology is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {Y : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} Y] {π : X -> Y} [_inst_3 : TopologicalSpace.SecondCountableTopology.{u1} X _inst_1], (QuotientMap.{u1, u2} X Y _inst_1 _inst_2 π) -> (IsOpenMap.{u1, u2} X Y _inst_1 _inst_2 π) -> (TopologicalSpace.SecondCountableTopology.{u2} Y _inst_2)
-but is expected to have type
-  forall {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {Y : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} Y] {π : X -> Y} [_inst_3 : TopologicalSpace.SecondCountableTopology.{u2} X _inst_1], (QuotientMap.{u2, u1} X Y _inst_1 _inst_2 π) -> (IsOpenMap.{u2, u1} X Y _inst_1 _inst_2 π) -> (TopologicalSpace.SecondCountableTopology.{u1} Y _inst_2)
-Case conversion may be inaccurate. Consider using '#align topological_space.quotient_map.second_countable_topology QuotientMap.secondCountableTopologyₓ'. -/
 /-- A second countable space is mapped by an open quotient map to a second countable space. -/
 theorem QuotientMap.secondCountableTopology [SecondCountableTopology X] (h' : QuotientMap π)
     (h : IsOpenMap π) : SecondCountableTopology Y :=

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

@@ -84,17 +84,12 @@ theorem IsTopologicalBasis.insert_empty {s : Set (Set α)} (h : IsTopologicalBas
     IsTopologicalBasis (insert ∅ s) :=
   by
   refine' ⟨_, by rw [sUnion_insert, empty_union, h.sUnion_eq], _⟩
-  · rintro t₁ (rfl | h₁) t₂ (rfl | h₂) x ⟨hx₁, hx₂⟩
-    · cases hx₁
-    · cases hx₁
-    · cases hx₂
+  · rintro t₁ (rfl | h₁) t₂ (rfl | h₂) x ⟨hx₁, hx₂⟩; · cases hx₁; · cases hx₁; · cases hx₂
     obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x ⟨hx₁, hx₂⟩
     exact ⟨t₃, Or.inr h₃, hs⟩
   · rw [h.eq_generate_from]
     refine' le_antisymm (le_generateFrom fun t => _) (generate_from_anti <| subset_insert ∅ s)
-    rintro (rfl | ht)
-    · convert isOpen_empty
-    · exact generate_open.basic t ht
+    rintro (rfl | ht); · convert isOpen_empty; · exact generate_open.basic t ht
 #align topological_space.is_topological_basis.insert_empty TopologicalSpace.IsTopologicalBasis.insert_empty
 -/
 
@@ -113,8 +108,7 @@ theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis
     exact ⟨t₃, ⟨h₃, nonempty.ne_empty ⟨x, hs.1⟩⟩, hs⟩
   · rw [h.eq_generate_from]
     refine' le_antisymm (generate_from_anti <| diff_subset s _) (le_generateFrom fun t ht => _)
-    obtain rfl | he := eq_or_ne t ∅
-    · convert isOpen_empty
+    obtain rfl | he := eq_or_ne t ∅; · convert isOpen_empty
     exact generate_open.basic t ⟨ht, he⟩
 #align topological_space.is_topological_basis.diff_empty TopologicalSpace.IsTopologicalBasis.diff_empty
 
@@ -129,8 +123,7 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
     exact ⟨_, ⟨_, ⟨hft₁.union hft₂, union_subset ht₁b ht₂b⟩, sInter_union t₁ t₂⟩, h, subset.rfl⟩
   · rw [sUnion_image, Union₂_eq_univ_iff]
     exact fun x => ⟨∅, ⟨finite_empty, empty_subset _⟩, sInter_empty.substr <| mem_univ x⟩
-  · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩
-    apply isOpen_sInter
+  · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩; apply isOpen_sInter
     exacts[hft, fun s hs => generate_open.basic _ <| htb hs]
   · rw [← sInter_singleton t]
     exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩
@@ -204,9 +197,7 @@ theorem IsTopologicalBasis.nhds_hasBasis {b : Set (Set α)} (hb : IsTopologicalB
 
 #print TopologicalSpace.IsTopologicalBasis.isOpen /-
 protected theorem IsTopologicalBasis.isOpen {s : Set α} {b : Set (Set α)}
-    (hb : IsTopologicalBasis b) (hs : s ∈ b) : IsOpen s :=
-  by
-  rw [hb.eq_generate_from]
+    (hb : IsTopologicalBasis b) (hs : s ∈ b) : IsOpen s := by rw [hb.eq_generate_from];
   exact generate_open.basic s hs
 #align topological_space.is_topological_basis.is_open TopologicalSpace.IsTopologicalBasis.isOpen
 -/
@@ -268,9 +259,8 @@ theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopolog
 #print TopologicalSpace.IsTopologicalBasis.open_eq_iUnion /-
 theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (β : Type u)(f : β → Set α), (u = ⋃ i, f i) ∧ ∀ i, f i ∈ B :=
-  ⟨↥({ s ∈ B | s ⊆ u }), coe, by
-    rw [← sUnion_eq_Union]
-    apply hB.open_eq_sUnion' ou, fun s => And.left s.2⟩
+  ⟨↥({ s ∈ B | s ⊆ u }), coe, by rw [← sUnion_eq_Union]; apply hB.open_eq_sUnion' ou, fun s =>
+    And.left s.2⟩
 #align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
 -/
 
@@ -656,10 +646,8 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
     exact (cond i).IsOpen (h1 i hi)
   · intro a U ha hU
     obtain ⟨I, t, hta, htU⟩ :
-      ∃ (I : Finset ι)(t : ∀ i : ι, Set (X i)), (∀ i, t i ∈ 𝓝 (a i)) ∧ Set.pi (↑I) t ⊆ U :=
-      by
-      rw [← Filter.mem_pi', ← nhds_pi]
-      exact hU.mem_nhds ha
+      ∃ (I : Finset ι)(t : ∀ i : ι, Set (X i)), (∀ i, t i ∈ 𝓝 (a i)) ∧ Set.pi (↑I) t ⊆ U := by
+      rw [← Filter.mem_pi', ← nhds_pi]; exact hU.mem_nhds ha
     have : ∀ i, ∃ V ∈ T i, a i ∈ V ∧ V ⊆ t i := fun i => (cond i).mem_nhds_iffₓ.1 (hta i)
     choose V hVT haV hVt
     exact
@@ -684,9 +672,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
   ext V
   constructor
   · rintro ⟨U, F, h1, h2⟩
-    have : (F : Set ι).pi U = ⋂ (i : ι) (hi : i ∈ F), (fun z : ∀ j, X j => z i) ⁻¹' U i :=
-      by
-      ext
+    have : (F : Set ι).pi U = ⋂ (i : ι) (hi : i ∈ F), (fun z : ∀ j, X j => z i) ⁻¹' U i := by ext;
       simp
     refine' ⟨(F : Set ι).pi U, ⟨U, F, h1, rfl⟩, _⟩
     rw [this, h2, Set.preimage_iInter]
@@ -696,9 +682,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     rfl
   · rintro ⟨U, ⟨U, F, h1, rfl⟩, h⟩
     refine' ⟨U, F, h1, _⟩
-    have : (F : Set ι).pi U = ⋂ (i : ι) (hi : i ∈ F), (fun z : ∀ j, X j => z i) ⁻¹' U i :=
-      by
-      ext
+    have : (F : Set ι).pi U = ⋂ (i : ι) (hi : i ∈ F), (fun z : ∀ j, X j => z i) ⁻¹' U i := by ext;
       simp
     rw [← h, this, Set.preimage_iInter]
     congr 1
@@ -822,9 +806,7 @@ variable {α}
 
 instance {β} [TopologicalSpace β] [FirstCountableTopology α] [FirstCountableTopology β] :
     FirstCountableTopology (α × β) :=
-  ⟨fun ⟨x, y⟩ => by
-    rw [nhds_prod_eq]
-    infer_instance⟩
+  ⟨fun ⟨x, y⟩ => by rw [nhds_prod_eq]; infer_instance⟩
 
 section Pi
 
@@ -832,9 +814,7 @@ omit t
 
 instance {ι : Type _} {π : ι → Type _} [Countable ι] [∀ i, TopologicalSpace (π i)]
     [∀ i, FirstCountableTopology (π i)] : FirstCountableTopology (∀ i, π i) :=
-  ⟨fun f => by
-    rw [nhds_pi]
-    infer_instance⟩
+  ⟨fun f => by rw [nhds_pi]; infer_instance⟩
 
 end Pi
 
@@ -979,8 +959,7 @@ instance {ι : Type _} {π : ι → Type _} [Countable ι] [t : ∀ a, Topologic
   have : t = fun a => generate_from (countable_basis (π a)) :=
     funext fun a => (is_basis_countable_basis (π a)).eq_generateFrom
   rw [this, pi_generateFrom_eq]
-  constructor
-  refine' ⟨_, _, rfl⟩
+  constructor; refine' ⟨_, _, rfl⟩
   have :
     Set.Countable
       { T : Set (∀ i, π i) |
@@ -991,13 +970,10 @@ instance {ι : Type _} {π : ι → Type _} [Countable ι] [t : ∀ a, Topologic
     refine' countable_Union fun I => countable.bUnion _ fun _ _ => countable_singleton _
     change Set.Countable { s : ∀ i : I, Set (π i) | ∀ i, s i ∈ countable_basis (π i) }
     exact countable_pi fun i => countable_countable_basis _
-  convert this using 1
-  ext1 T
-  constructor
+  convert this using 1; ext1 T; constructor
   · rintro ⟨s, I, hs, rfl⟩
     refine' ⟨I, fun i => s i, fun i => hs i i.2, _⟩
-    simp only [Set.pi, SetCoe.forall']
-    rfl
+    simp only [Set.pi, SetCoe.forall']; rfl
   · rintro ⟨I, s, hs, rfl⟩
     rcases@Subtype.surjective_restrict ι (fun i => Set (π i)) _ (fun i => i ∈ I) s with ⟨s, rfl⟩
     exact ⟨s, I, fun i hi => hs ⟨i, hi⟩, Set.ext fun f => Subtype.forall⟩
@@ -1267,9 +1243,7 @@ variable {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α 
 
 #print Inducing.secondCountableTopology /-
 protected theorem Inducing.secondCountableTopology [SecondCountableTopology β] (hf : Inducing f) :
-    SecondCountableTopology α := by
-  rw [hf.1]
-  exact second_countable_topology_induced α β f
+    SecondCountableTopology α := by rw [hf.1]; exact second_countable_topology_induced α β f
 #align inducing.second_countable_topology Inducing.secondCountableTopology
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -627,7 +627,7 @@ theorem IsSeparable.image {β : Type _} [TopologicalSpace β] {s : Set α} (hs :
 #print TopologicalSpace.isSeparable_of_separableSpace_subtype /-
 theorem isSeparable_of_separableSpace_subtype (s : Set α) [SeparableSpace s] : IsSeparable s :=
   by
-  have : is_separable ((coe : s → α) '' (univ : Set s)) :=
+  have : IsSeparable ((coe : s → α) '' (univ : Set s)) :=
     (is_separable_of_separable_space _).image continuous_subtype_val
   simpa only [image_univ, Subtype.range_coe_subtype]
 #align topological_space.is_separable_of_separable_space_subtype TopologicalSpace.isSeparable_of_separableSpace_subtype

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

@@ -739,7 +739,7 @@ theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α]
 
 /- warning: dense.exists_countable_dense_subset_bot_top -> Dense.exists_countable_dense_subset_bot_top is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PartialOrder.{u1} α] {s : Set.{u1} α} [_inst_3 : TopologicalSpace.SeparableSpace.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) _inst_1)], (Dense.{u1} α _inst_1 s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Countable.{u1} α t) (And (Dense.{u1} α _inst_1 t) (And (forall (x : α), (IsBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) x) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t)) (forall (x : α), (IsTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) x) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t)))))))
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PartialOrder.{u1} α] {s : Set.{u1} α} [_inst_3 : TopologicalSpace.SeparableSpace.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) s) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) _inst_1)], (Dense.{u1} α _inst_1 s) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Countable.{u1} α t) (And (Dense.{u1} α _inst_1 t) (And (forall (x : α), (IsBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) x) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t)) (forall (x : α), (IsTop.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) x) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t)))))))
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PartialOrder.{u1} α] {s : Set.{u1} α} [_inst_3 : TopologicalSpace.SeparableSpace.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) _inst_1)], (Dense.{u1} α _inst_1 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 (Set.Countable.{u1} α t) (And (Dense.{u1} α _inst_1 t) (And (forall (x : α), (IsBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) x) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t)) (forall (x : α), (IsTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) x) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t)))))))
 Case conversion may be inaccurate. Consider using '#align dense.exists_countable_dense_subset_bot_top Dense.exists_countable_dense_subset_bot_topₓ'. -/
@@ -769,7 +769,12 @@ instance separableSpace_univ {α : Type _} [TopologicalSpace α] [SeparableSpace
 #align separable_space_univ separableSpace_univ
 -/
 
-#print exists_countable_dense_bot_top /-
+/- warning: exists_countable_dense_bot_top -> exists_countable_dense_bot_top is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.SeparableSpace.{u1} α _inst_1] [_inst_3 : PartialOrder.{u1} α], Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Set.Countable.{u1} α s) (And (Dense.{u1} α _inst_1 s) (And (forall (x : α), (IsBot.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α _inst_3)) x) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s)) (forall (x : α), (IsTop.{u1} α (Preorder.toHasLe.{u1} α (PartialOrder.toPreorder.{u1} α _inst_3)) x) -> (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s)))))
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.SeparableSpace.{u1} α _inst_1] [_inst_3 : PartialOrder.{u1} α], Exists.{succ u1} (Set.{u1} α) (fun (s : Set.{u1} α) => And (Set.Countable.{u1} α s) (And (Dense.{u1} α _inst_1 s) (And (forall (x : α), (IsBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_3)) x) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s)) (forall (x : α), (IsTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_3)) x) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s)))))
+Case conversion may be inaccurate. Consider using '#align exists_countable_dense_bot_top exists_countable_dense_bot_topₓ'. -/
 /-- If `α` is a separable topological space with a partial order, then there exists a countable
 dense set `s : set α` that contains those of both bottom and top elements of `α` that actually
 exist. For a dense set containing neither bot nor top elements, see
@@ -779,7 +784,6 @@ theorem exists_countable_dense_bot_top (α : Type _) [TopologicalSpace α] [Sepa
     ∃ s : Set α, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∈ s) ∧ ∀ x, IsTop x → x ∈ s := by
   simpa using dense_univ.exists_countable_dense_subset_bot_top
 #align exists_countable_dense_bot_top exists_countable_dense_bot_top
--/
 
 namespace TopologicalSpace
 

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

@@ -74,7 +74,7 @@ include t
   finite intersections as well). -/
 structure IsTopologicalBasis (s : Set (Set α)) : Prop where
   exists_subset_inter : ∀ t₁ ∈ s, ∀ t₂ ∈ s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
-  unionₛ_eq : ⋃₀ s = univ
+  sUnion_eq : ⋃₀ s = univ
   eq_generateFrom : t = generateFrom s
 #align topological_space.is_topological_basis TopologicalSpace.IsTopologicalBasis
 -/
@@ -130,7 +130,7 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
   · rw [sUnion_image, Union₂_eq_univ_iff]
     exact fun x => ⟨∅, ⟨finite_empty, empty_subset _⟩, sInter_empty.substr <| mem_univ x⟩
   · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩
-    apply isOpen_interₛ
+    apply isOpen_sInter
     exacts[hft, fun s hs => generate_open.basic _ <| htb hs]
   · rw [← sInter_singleton t]
     exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩
@@ -158,7 +158,7 @@ theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u 
     refine' (@isOpen_iff_nhds α (generate_from s) u).mpr fun a ha => _
     rcases h_nhds a u ha hu with ⟨v, hvs, hav, hvu⟩
     rw [nhds_generate_from]
-    exact infᵢ₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
+    exact iInf₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
 #align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_open_of_nhds
 
 /- warning: topological_space.is_topological_basis.mem_nhds_iff -> TopologicalSpace.IsTopologicalBasis.mem_nhds_iff is a dubious translation:
@@ -229,49 +229,49 @@ theorem IsTopologicalBasis.exists_subset_of_mem_open {b : Set (Set α)} (hb : Is
   hb.mem_nhds_iffₓ.1 <| IsOpen.mem_nhds ou au
 #align topological_space.is_topological_basis.exists_subset_of_mem_open TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_open
 
-#print TopologicalSpace.IsTopologicalBasis.open_eq_unionₛ' /-
+#print TopologicalSpace.IsTopologicalBasis.open_eq_sUnion' /-
 /-- Any open set is the union of the basis sets contained in it. -/
-theorem IsTopologicalBasis.open_eq_unionₛ' {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
+theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : u = ⋃₀ { s ∈ B | s ⊆ u } :=
   ext fun a =>
     ⟨fun ha =>
       let ⟨b, hb, ab, bu⟩ := hB.exists_subset_of_mem_open ha ou
       ⟨b, ⟨hb, bu⟩, ab⟩,
       fun ⟨b, ⟨hb, bu⟩, ab⟩ => bu ab⟩
-#align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_unionₛ'
+#align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 -/
 
-/- warning: topological_space.is_topological_basis.open_eq_sUnion -> TopologicalSpace.IsTopologicalBasis.open_eq_unionₛ is a dubious translation:
+/- warning: topological_space.is_topological_basis.open_eq_sUnion -> TopologicalSpace.IsTopologicalBasis.open_eq_sUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) (fun (H : HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) => Eq.{succ u1} (Set.{u1} α) u (Set.unionₛ.{u1} α S)))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) (fun (H : HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) => Eq.{succ u1} (Set.{u1} α) u (Set.sUnion.{u1} α S)))))
 but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => And (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) S B) (Eq.{succ u1} (Set.{u1} α) u (Set.unionₛ.{u1} α S)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_unionₛₓ'. -/
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => And (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) S B) (Eq.{succ u1} (Set.{u1} α) u (Set.sUnion.{u1} α S)))))
+Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnionₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
-theorem IsTopologicalBasis.open_eq_unionₛ {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
+theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _)(_ : S ⊆ B), u = ⋃₀ S :=
-  ⟨{ s ∈ B | s ⊆ u }, fun s h => h.1, hB.open_eq_unionₛ' ou⟩
-#align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_unionₛ
+  ⟨{ s ∈ B | s ⊆ u }, fun s h => h.1, hB.open_eq_sUnion' ou⟩
+#align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 
-/- warning: topological_space.is_topological_basis.open_iff_eq_sUnion -> TopologicalSpace.IsTopologicalBasis.open_iff_eq_unionₛ is a dubious translation:
+/- warning: topological_space.is_topological_basis.open_iff_eq_sUnion -> TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, Iff (IsOpen.{u1} α t u) (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) (fun (H : HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) => Eq.{succ u1} (Set.{u1} α) u (Set.unionₛ.{u1} α S)))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, Iff (IsOpen.{u1} α t u) (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) (fun (H : HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.hasSubset.{u1} (Set.{u1} α)) S B) => Eq.{succ u1} (Set.{u1} α) u (Set.sUnion.{u1} α S)))))
 but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, Iff (IsOpen.{u1} α t u) (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => And (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) S B) (Eq.{succ u1} (Set.{u1} α) u (Set.unionₛ.{u1} α S)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_unionₛₓ'. -/
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, Iff (IsOpen.{u1} α t u) (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => And (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) S B) (Eq.{succ u1} (Set.{u1} α) u (Set.sUnion.{u1} α S)))))
+Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnionₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
-theorem IsTopologicalBasis.open_iff_eq_unionₛ {B : Set (Set α)} (hB : IsTopologicalBasis B)
+theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ (S : _)(_ : S ⊆ B), u = ⋃₀ S :=
-  ⟨hB.open_eq_unionₛ, fun ⟨S, hSB, hu⟩ => hu.symm ▸ isOpen_unionₛ fun s hs => hB.IsOpen (hSB hs)⟩
-#align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_unionₛ
+  ⟨hB.open_eq_sUnion, fun ⟨S, hSB, hu⟩ => hu.symm ▸ isOpen_sUnion fun s hs => hB.IsOpen (hSB hs)⟩
+#align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion
 
-#print TopologicalSpace.IsTopologicalBasis.open_eq_unionᵢ /-
-theorem IsTopologicalBasis.open_eq_unionᵢ {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
+#print TopologicalSpace.IsTopologicalBasis.open_eq_iUnion /-
+theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (β : Type u)(f : β → Set α), (u = ⋃ i, f i) ∧ ∀ i, f i ∈ B :=
   ⟨↥({ s ∈ B | s ⊆ u }), coe, by
     rw [← sUnion_eq_Union]
     apply hB.open_eq_sUnion' ou, fun s => And.left s.2⟩
-#align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_unionᵢ
+#align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
 -/
 
 /- warning: topological_space.is_topological_basis.mem_closure_iff -> TopologicalSpace.IsTopologicalBasis.mem_closure_iff is a dubious translation:
@@ -311,7 +311,7 @@ theorem IsTopologicalBasis.isOpenMap_iff {β} [TopologicalSpace β] {B : Set (Se
   by
   refine' ⟨fun H o ho => H _ (hB.is_open ho), fun hf o ho => _⟩
   rw [hB.open_eq_sUnion' ho, sUnion_eq_Union, image_Union]
-  exact isOpen_unionᵢ fun s => hf s s.2.1
+  exact isOpen_iUnion fun s => hf s s.2.1
 #align topological_space.is_topological_basis.is_open_map_iff TopologicalSpace.IsTopologicalBasis.isOpenMap_iff
 
 /- warning: topological_space.is_topological_basis.exists_nonempty_subset -> TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset is a dubious translation:
@@ -374,14 +374,14 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
     obtain ⟨V, hV, rfl⟩ := hU
     obtain ⟨S, hS, rfl⟩ := h.open_eq_sUnion hV
     obtain ⟨W, hW, ha⟩ := ha
-    refine' ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_unionₛ_of_mem hW⟩
+    refine' ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_sUnion_of_mem hW⟩
 #align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducing
 
 /- warning: topological_space.is_topological_basis_of_cover -> TopologicalSpace.isTopologicalBasis_of_cover is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {ι : Sort.{u2}} {U : ι -> (Set.{u1} α)}, (forall (i : ι), IsOpen.{u1} α t (U i)) -> (Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, u2} α ι (fun (i : ι) => U i)) (Set.univ.{u1} α)) -> (forall {b : forall (i : ι), Set.{u1} (Set.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (U i)) t) (b i)) -> (TopologicalSpace.IsTopologicalBasis.{u1} α t (Set.unionᵢ.{u1, u2} (Set.{u1} α) ι (fun (i : ι) => Set.image.{u1, u1} (Set.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i))) (Set.{u1} α) (Set.image.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (U i)))))))) (b i)))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {ι : Sort.{u2}} {U : ι -> (Set.{u1} α)}, (forall (i : ι), IsOpen.{u1} α t (U i)) -> (Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, u2} α ι (fun (i : ι) => U i)) (Set.univ.{u1} α)) -> (forall {b : forall (i : ι), Set.{u1} (Set.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (U i)) t) (b i)) -> (TopologicalSpace.IsTopologicalBasis.{u1} α t (Set.iUnion.{u1, u2} (Set.{u1} α) ι (fun (i : ι) => Set.image.{u1, u1} (Set.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i))) (Set.{u1} α) (Set.image.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α (HasLiftT.mk.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α (CoeTCₓ.coe.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α (coeBase.{succ u1, succ u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) α (coeSubtype.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (U i)))))))) (b i)))))
 but is expected to have type
-  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] {ι : Sort.{u1}} {U : ι -> (Set.{u2} α)}, (forall (i : ι), IsOpen.{u2} α t (U i)) -> (Eq.{succ u2} (Set.{u2} α) (Set.unionᵢ.{u2, u1} α ι (fun (i : ι) => U i)) (Set.univ.{u2} α)) -> (forall {b : forall (i : ι), Set.{u2} (Set.{u2} (Set.Elem.{u2} α (U i)))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u2} (Set.Elem.{u2} α (U i)) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i)) t) (b i)) -> (TopologicalSpace.IsTopologicalBasis.{u2} α t (Set.unionᵢ.{u2, u1} (Set.{u2} α) ι (fun (i : ι) => Set.image.{u2, u2} (Set.{u2} (Subtype.{succ u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i)))) (Set.{u2} α) (Set.image.{u2, u2} (Subtype.{succ u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i))) α (Subtype.val.{succ u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i)))) (b i)))))
+  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] {ι : Sort.{u1}} {U : ι -> (Set.{u2} α)}, (forall (i : ι), IsOpen.{u2} α t (U i)) -> (Eq.{succ u2} (Set.{u2} α) (Set.iUnion.{u2, u1} α ι (fun (i : ι) => U i)) (Set.univ.{u2} α)) -> (forall {b : forall (i : ι), Set.{u2} (Set.{u2} (Set.Elem.{u2} α (U i)))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u2} (Set.Elem.{u2} α (U i)) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i)) t) (b i)) -> (TopologicalSpace.IsTopologicalBasis.{u2} α t (Set.iUnion.{u2, u1} (Set.{u2} α) ι (fun (i : ι) => Set.image.{u2, u2} (Set.{u2} (Subtype.{succ u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i)))) (Set.{u2} α) (Set.image.{u2, u2} (Subtype.{succ u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i))) α (Subtype.val.{succ u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i)))) (b i)))))
 Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis_of_cover TopologicalSpace.isTopologicalBasis_of_coverₓ'. -/
 theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
     (Uc : (⋃ i, U i) = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
@@ -565,19 +565,19 @@ theorem IsSeparable.closure {s : Set α} (hs : IsSeparable s) : IsSeparable (clo
 #align topological_space.is_separable.closure TopologicalSpace.IsSeparable.closure
 -/
 
-/- warning: topological_space.is_separable_Union -> TopologicalSpace.isSeparable_unionᵢ is a dubious translation:
+/- warning: topological_space.is_separable_Union -> TopologicalSpace.isSeparable_iUnion is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] {s : ι -> (Set.{u1} α)}, (forall (i : ι), TopologicalSpace.IsSeparable.{u1} α t (s i)) -> (TopologicalSpace.IsSeparable.{u1} α t (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => s i)))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] {s : ι -> (Set.{u1} α)}, (forall (i : ι), TopologicalSpace.IsSeparable.{u1} α t (s i)) -> (TopologicalSpace.IsSeparable.{u1} α t (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => s i)))
 but is expected to have type
-  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {s : ι -> (Set.{u2} α)}, (forall (i : ι), TopologicalSpace.IsSeparable.{u2} α t (s i)) -> (TopologicalSpace.IsSeparable.{u2} α t (Set.unionᵢ.{u2, succ u1} α ι (fun (i : ι) => s i)))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_separable_Union TopologicalSpace.isSeparable_unionᵢₓ'. -/
-theorem isSeparable_unionᵢ {ι : Type _} [Countable ι] {s : ι → Set α}
+  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {s : ι -> (Set.{u2} α)}, (forall (i : ι), TopologicalSpace.IsSeparable.{u2} α t (s i)) -> (TopologicalSpace.IsSeparable.{u2} α t (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => s i)))
+Case conversion may be inaccurate. Consider using '#align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnionₓ'. -/
+theorem isSeparable_iUnion {ι : Type _} [Countable ι] {s : ι → Set α}
     (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) :=
   by
   choose c hc h'c using hs
   refine' ⟨⋃ i, c i, countable_Union hc, Union_subset_iff.2 fun i => _⟩
   exact (h'c i).trans (closure_mono (subset_Union _ i))
-#align topological_space.is_separable_Union TopologicalSpace.isSeparable_unionᵢ
+#align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnion
 
 #print Set.Countable.isSeparable /-
 theorem Set.Countable.isSeparable {s : Set α} (hs : s.Countable) : IsSeparable s :=
@@ -666,13 +666,13 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
       ⟨_, ⟨V, I, fun i hi => hVT i, rfl⟩, fun i hi => haV i, (pi_mono fun i hi => hVt i).trans htU⟩
 #align is_topological_basis_pi isTopologicalBasis_pi
 
-/- warning: is_topological_basis_infi -> isTopologicalBasis_infᵢ is a dubious translation:
+/- warning: is_topological_basis_infi -> isTopologicalBasis_iInf is a dubious translation:
 lean 3 declaration is
-  forall {β : Type.{u1}} {ι : Type.{u2}} {X : ι -> Type.{u3}} [t : forall (i : ι), TopologicalSpace.{u3} (X i)] {T : forall (i : ι), Set.{u3} (Set.{u3} (X i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u3} (X i) (t i) (T i)) -> (forall (f : forall (i : ι), β -> (X i)), TopologicalSpace.IsTopologicalBasis.{u1} β (infᵢ.{u1, succ u2} (TopologicalSpace.{u1} β) (ConditionallyCompleteLattice.toHasInf.{u1} (TopologicalSpace.{u1} β) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} β) (TopologicalSpace.completeLattice.{u1} β))) ι (fun (i : ι) => TopologicalSpace.induced.{u1, u3} β (X i) (f i) (t i))) (setOf.{u1} (Set.{u1} β) (fun (S : Set.{u1} β) => Exists.{max (succ u2) (succ u3)} (forall (i : ι), Set.{u3} (X i)) (fun (U : forall (i : ι), Set.{u3} (X i)) => Exists.{succ u2} (Finset.{u2} ι) (fun (F : Finset.{u2} ι) => And (forall (i : ι), (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i F) -> (Membership.Mem.{u3, u3} (Set.{u3} (X i)) (Set.{u3} (Set.{u3} (X i))) (Set.hasMem.{u3} (Set.{u3} (X i))) (U i) (T i))) (Eq.{succ u1} (Set.{u1} β) S (Set.interᵢ.{u1, succ u2} β ι (fun (i : ι) => Set.interᵢ.{u1, 0} β (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i F) (fun (hi : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i F) => Set.preimage.{u1, u3} β (X i) (f i) (U i))))))))))
+  forall {β : Type.{u1}} {ι : Type.{u2}} {X : ι -> Type.{u3}} [t : forall (i : ι), TopologicalSpace.{u3} (X i)] {T : forall (i : ι), Set.{u3} (Set.{u3} (X i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u3} (X i) (t i) (T i)) -> (forall (f : forall (i : ι), β -> (X i)), TopologicalSpace.IsTopologicalBasis.{u1} β (iInf.{u1, succ u2} (TopologicalSpace.{u1} β) (ConditionallyCompleteLattice.toHasInf.{u1} (TopologicalSpace.{u1} β) (CompleteLattice.toConditionallyCompleteLattice.{u1} (TopologicalSpace.{u1} β) (TopologicalSpace.completeLattice.{u1} β))) ι (fun (i : ι) => TopologicalSpace.induced.{u1, u3} β (X i) (f i) (t i))) (setOf.{u1} (Set.{u1} β) (fun (S : Set.{u1} β) => Exists.{max (succ u2) (succ u3)} (forall (i : ι), Set.{u3} (X i)) (fun (U : forall (i : ι), Set.{u3} (X i)) => Exists.{succ u2} (Finset.{u2} ι) (fun (F : Finset.{u2} ι) => And (forall (i : ι), (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i F) -> (Membership.Mem.{u3, u3} (Set.{u3} (X i)) (Set.{u3} (Set.{u3} (X i))) (Set.hasMem.{u3} (Set.{u3} (X i))) (U i) (T i))) (Eq.{succ u1} (Set.{u1} β) S (Set.iInter.{u1, succ u2} β ι (fun (i : ι) => Set.iInter.{u1, 0} β (Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i F) (fun (hi : Membership.Mem.{u2, u2} ι (Finset.{u2} ι) (Finset.hasMem.{u2} ι) i F) => Set.preimage.{u1, u3} β (X i) (f i) (U i))))))))))
 but is expected to have type
-  forall {β : Type.{u3}} {ι : Type.{u2}} {X : ι -> Type.{u1}} [t : forall (i : ι), TopologicalSpace.{u1} (X i)] {T : forall (i : ι), Set.{u1} (Set.{u1} (X i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u1} (X i) (t i) (T i)) -> (forall (f : forall (i : ι), β -> (X i)), TopologicalSpace.IsTopologicalBasis.{u3} β (infᵢ.{u3, succ u2} (TopologicalSpace.{u3} β) (ConditionallyCompleteLattice.toInfSet.{u3} (TopologicalSpace.{u3} β) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} β) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} β))) ι (fun (i : ι) => TopologicalSpace.induced.{u3, u1} β (X i) (f i) (t i))) (setOf.{u3} (Set.{u3} β) (fun (S : Set.{u3} β) => Exists.{max (succ u2) (succ u1)} (forall (i : ι), Set.{u1} (X i)) (fun (U : forall (i : ι), Set.{u1} (X i)) => Exists.{succ u2} (Finset.{u2} ι) (fun (F : Finset.{u2} ι) => And (forall (i : ι), (Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i F) -> (Membership.mem.{u1, u1} (Set.{u1} (X i)) (Set.{u1} (Set.{u1} (X i))) (Set.instMembershipSet.{u1} (Set.{u1} (X i))) (U i) (T i))) (Eq.{succ u3} (Set.{u3} β) S (Set.interᵢ.{u3, succ u2} β ι (fun (i : ι) => Set.interᵢ.{u3, 0} β (Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i F) (fun (hi : Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i F) => Set.preimage.{u3, u1} β (X i) (f i) (U i))))))))))
-Case conversion may be inaccurate. Consider using '#align is_topological_basis_infi isTopologicalBasis_infᵢₓ'. -/
-theorem isTopologicalBasis_infᵢ {β : Type _} {ι : Type _} {X : ι → Type _}
+  forall {β : Type.{u3}} {ι : Type.{u2}} {X : ι -> Type.{u1}} [t : forall (i : ι), TopologicalSpace.{u1} (X i)] {T : forall (i : ι), Set.{u1} (Set.{u1} (X i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u1} (X i) (t i) (T i)) -> (forall (f : forall (i : ι), β -> (X i)), TopologicalSpace.IsTopologicalBasis.{u3} β (iInf.{u3, succ u2} (TopologicalSpace.{u3} β) (ConditionallyCompleteLattice.toInfSet.{u3} (TopologicalSpace.{u3} β) (CompleteLattice.toConditionallyCompleteLattice.{u3} (TopologicalSpace.{u3} β) (TopologicalSpace.instCompleteLatticeTopologicalSpace.{u3} β))) ι (fun (i : ι) => TopologicalSpace.induced.{u3, u1} β (X i) (f i) (t i))) (setOf.{u3} (Set.{u3} β) (fun (S : Set.{u3} β) => Exists.{max (succ u2) (succ u1)} (forall (i : ι), Set.{u1} (X i)) (fun (U : forall (i : ι), Set.{u1} (X i)) => Exists.{succ u2} (Finset.{u2} ι) (fun (F : Finset.{u2} ι) => And (forall (i : ι), (Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i F) -> (Membership.mem.{u1, u1} (Set.{u1} (X i)) (Set.{u1} (Set.{u1} (X i))) (Set.instMembershipSet.{u1} (Set.{u1} (X i))) (U i) (T i))) (Eq.{succ u3} (Set.{u3} β) S (Set.iInter.{u3, succ u2} β ι (fun (i : ι) => Set.iInter.{u3, 0} β (Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i F) (fun (hi : Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i F) => Set.preimage.{u3, u1} β (X i) (f i) (U i))))))))))
+Case conversion may be inaccurate. Consider using '#align is_topological_basis_infi isTopologicalBasis_iInfₓ'. -/
+theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     [t : ∀ i, TopologicalSpace (X i)] {T : ∀ i, Set (Set (X i))}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
     @IsTopologicalBasis β (⨅ i, induced (f i) (t i))
@@ -680,7 +680,7 @@ theorem isTopologicalBasis_infᵢ {β : Type _} {ι : Type _} {X : ι → Type _
         ∃ (U : ∀ i, Set (X i))(F : Finset ι),
           (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (hi : i ∈ F), f i ⁻¹' U i } :=
   by
-  convert(isTopologicalBasis_pi cond).Inducing (inducing_infᵢ_to_pi _)
+  convert(isTopologicalBasis_pi cond).Inducing (inducing_iInf_to_pi _)
   ext V
   constructor
   · rintro ⟨U, F, h1, h2⟩
@@ -689,10 +689,10 @@ theorem isTopologicalBasis_infᵢ {β : Type _} {ι : Type _} {X : ι → Type _
       ext
       simp
     refine' ⟨(F : Set ι).pi U, ⟨U, F, h1, rfl⟩, _⟩
-    rw [this, h2, Set.preimage_interᵢ]
+    rw [this, h2, Set.preimage_iInter]
     congr 1
     ext1
-    rw [Set.preimage_interᵢ]
+    rw [Set.preimage_iInter]
     rfl
   · rintro ⟨U, ⟨U, F, h1, rfl⟩, h⟩
     refine' ⟨U, F, h1, _⟩
@@ -700,12 +700,12 @@ theorem isTopologicalBasis_infᵢ {β : Type _} {ι : Type _} {X : ι → Type _
       by
       ext
       simp
-    rw [← h, this, Set.preimage_interᵢ]
+    rw [← h, this, Set.preimage_iInter]
     congr 1
     ext1
-    rw [Set.preimage_interᵢ]
+    rw [Set.preimage_iInter]
     rfl
-#align is_topological_basis_infi isTopologicalBasis_infᵢ
+#align is_topological_basis_infi isTopologicalBasis_iInf
 
 #print isTopologicalBasis_singletons /-
 theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [DiscreteTopology α] :
@@ -1014,9 +1014,9 @@ variable {α}
 
 /- warning: topological_space.second_countable_topology_of_countable_cover -> TopologicalSpace.secondCountableTopology_of_countable_cover is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_1 : Encodable.{u2} ι] {U : ι -> (Set.{u1} α)} [_inst_2 : forall (i : ι), TopologicalSpace.SecondCountableTopology.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (U i)) t)], (forall (i : ι), IsOpen.{u1} α t (U i)) -> (Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => U i)) (Set.univ.{u1} α)) -> (TopologicalSpace.SecondCountableTopology.{u1} α t)
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {ι : Type.{u2}} [_inst_1 : Encodable.{u2} ι] {U : ι -> (Set.{u1} α)} [_inst_2 : forall (i : ι), TopologicalSpace.SecondCountableTopology.{u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (U i)) (Subtype.topologicalSpace.{u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (U i)) t)], (forall (i : ι), IsOpen.{u1} α t (U i)) -> (Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => U i)) (Set.univ.{u1} α)) -> (TopologicalSpace.SecondCountableTopology.{u1} α t)
 but is expected to have type
-  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_1 : Encodable.{u1} ι] {U : ι -> (Set.{u2} α)} [_inst_2 : forall (i : ι), TopologicalSpace.SecondCountableTopology.{u2} (Set.Elem.{u2} α (U i)) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i)) t)], (forall (i : ι), IsOpen.{u2} α t (U i)) -> (Eq.{succ u2} (Set.{u2} α) (Set.unionᵢ.{u2, succ u1} α ι (fun (i : ι) => U i)) (Set.univ.{u2} α)) -> (TopologicalSpace.SecondCountableTopology.{u2} α t)
+  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] {ι : Type.{u1}} [_inst_1 : Encodable.{u1} ι] {U : ι -> (Set.{u2} α)} [_inst_2 : forall (i : ι), TopologicalSpace.SecondCountableTopology.{u2} (Set.Elem.{u2} α (U i)) (instTopologicalSpaceSubtype.{u2} α (fun (x : α) => Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) x (U i)) t)], (forall (i : ι), IsOpen.{u2} α t (U i)) -> (Eq.{succ u2} (Set.{u2} α) (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => U i)) (Set.univ.{u2} α)) -> (TopologicalSpace.SecondCountableTopology.{u2} α t)
 Case conversion may be inaccurate. Consider using '#align topological_space.second_countable_topology_of_countable_cover TopologicalSpace.secondCountableTopology_of_countable_coverₓ'. -/
 /-- A countable open cover induces a second-countable topology if all open covers
 are themselves second countable. -/
@@ -1028,15 +1028,15 @@ theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι 
   this.second_countable_topology (countable_Union fun i => (countable_countable_basis _).image _)
 #align topological_space.second_countable_topology_of_countable_cover TopologicalSpace.secondCountableTopology_of_countable_cover
 
-/- warning: topological_space.is_open_Union_countable -> TopologicalSpace.isOpen_unionᵢ_countable is a dubious translation:
+/- warning: topological_space.is_open_Union_countable -> TopologicalSpace.isOpen_iUnion_countable is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {ι : Type.{u2}} (s : ι -> (Set.{u1} α)), (forall (i : ι), IsOpen.{u1} α t (s i)) -> (Exists.{succ u2} (Set.{u2} ι) (fun (T : Set.{u2} ι) => And (Set.Countable.{u2} ι T) (Eq.{succ u1} (Set.{u1} α) (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i T) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i T) => s i))) (Set.unionᵢ.{u1, succ u2} α ι (fun (i : ι) => s i)))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {ι : Type.{u2}} (s : ι -> (Set.{u1} α)), (forall (i : ι), IsOpen.{u1} α t (s i)) -> (Exists.{succ u2} (Set.{u2} ι) (fun (T : Set.{u2} ι) => And (Set.Countable.{u2} ι T) (Eq.{succ u1} (Set.{u1} α) (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => Set.iUnion.{u1, 0} α (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i T) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i T) => s i))) (Set.iUnion.{u1, succ u2} α ι (fun (i : ι) => s i)))))
 but is expected to have type
-  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u2} α t] {ι : Type.{u1}} (s : ι -> (Set.{u2} α)), (forall (i : ι), IsOpen.{u2} α t (s i)) -> (Exists.{succ u1} (Set.{u1} ι) (fun (T : Set.{u1} ι) => And (Set.Countable.{u1} ι T) (Eq.{succ u2} (Set.{u2} α) (Set.unionᵢ.{u2, succ u1} α ι (fun (i : ι) => Set.unionᵢ.{u2, 0} α (Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) i T) (fun (H : Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) i T) => s i))) (Set.unionᵢ.{u2, succ u1} α ι (fun (i : ι) => s i)))))
-Case conversion may be inaccurate. Consider using '#align topological_space.is_open_Union_countable TopologicalSpace.isOpen_unionᵢ_countableₓ'. -/
+  forall {α : Type.{u2}} [t : TopologicalSpace.{u2} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u2} α t] {ι : Type.{u1}} (s : ι -> (Set.{u2} α)), (forall (i : ι), IsOpen.{u2} α t (s i)) -> (Exists.{succ u1} (Set.{u1} ι) (fun (T : Set.{u1} ι) => And (Set.Countable.{u1} ι T) (Eq.{succ u2} (Set.{u2} α) (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => Set.iUnion.{u2, 0} α (Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) i T) (fun (H : Membership.mem.{u1, u1} ι (Set.{u1} ι) (Set.instMembershipSet.{u1} ι) i T) => s i))) (Set.iUnion.{u2, succ u1} α ι (fun (i : ι) => s i)))))
+Case conversion may be inaccurate. Consider using '#align topological_space.is_open_Union_countable TopologicalSpace.isOpen_iUnion_countableₓ'. -/
 /-- In a second-countable space, an open set, given as a union of open sets,
 is equal to the union of countably many of those sets. -/
-theorem isOpen_unionᵢ_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
+theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
     (H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ (⋃ i ∈ T, s i) = ⋃ i, s i :=
   by
   let B := { b ∈ countable_basis α | ∃ i, b ⊆ s i }
@@ -1046,15 +1046,15 @@ theorem isOpen_unionᵢ_countable [SecondCountableTopology α] {ι} (s : ι →
   rintro _ ⟨i, rfl⟩ x xs
   rcases(is_basis_countable_basis α).exists_subset_of_mem_open xs (H _) with ⟨b, hb, xb, bs⟩
   exact ⟨_, ⟨_, rfl⟩, _, ⟨⟨⟨_, hb, _, bs⟩, rfl⟩, rfl⟩, hf _ xb⟩
-#align topological_space.is_open_Union_countable TopologicalSpace.isOpen_unionᵢ_countable
+#align topological_space.is_open_Union_countable TopologicalSpace.isOpen_iUnion_countable
 
-#print TopologicalSpace.isOpen_unionₛ_countable /-
-theorem isOpen_unionₛ_countable [SecondCountableTopology α] (S : Set (Set α))
+#print TopologicalSpace.isOpen_sUnion_countable /-
+theorem isOpen_sUnion_countable [SecondCountableTopology α] (S : Set (Set α))
     (H : ∀ s ∈ S, IsOpen s) : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
-  let ⟨T, cT, hT⟩ := isOpen_unionᵢ_countable (fun s : S => s.1) fun s => H s.1 s.2
+  let ⟨T, cT, hT⟩ := isOpen_iUnion_countable (fun s : S => s.1) fun s => H s.1 s.2
   ⟨Subtype.val '' T, cT.image _, image_subset_iff.2 fun ⟨x, xs⟩ xt => xs, by
     rwa [sUnion_image, sUnion_eq_Union]⟩
-#align topological_space.is_open_sUnion_countable TopologicalSpace.isOpen_unionₛ_countable
+#align topological_space.is_open_sUnion_countable TopologicalSpace.isOpen_sUnion_countable
 -/
 
 #print TopologicalSpace.countable_cover_nhds /-
@@ -1075,9 +1075,9 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 
 /- warning: topological_space.countable_cover_nhds_within -> TopologicalSpace.countable_cover_nhdsWithin is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (f x) (nhdsWithin.{u1} α t x s))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => f x)))))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (f x) (nhdsWithin.{u1} α t x s))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => f x)))))))
 but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7220 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.iUnion.{u1, succ u1} α α (fun (x : α) => Set.iUnion.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7220 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
 Case conversion may be inaccurate. Consider using '#align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithinₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
@@ -1098,9 +1098,9 @@ omit t
 
 /- warning: topological_space.is_topological_basis.sigma -> TopologicalSpace.IsTopologicalBasis.sigma is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {E : ι -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (E i)] {s : forall (i : ι), Set.{u2} (Set.{u2} (E i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u2} (E i) (_inst_1 i) (s i)) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.topologicalSpace.{u1, u2} ι (fun (i : ι) => E i) (fun (a : ι) => _inst_1 a)) (Set.unionᵢ.{max u1 u2, succ u1} (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) ι (fun (i : ι) => Set.image.{u2, max u1 u2} (Set.{u2} (E i)) (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) (fun (u : Set.{u2} (E i)) => Set.image.{u2, max u1 u2} (E i) (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.mk.{u1, u2} ι (fun (i : ι) => E i) i) u) (s i))))
+  forall {ι : Type.{u1}} {E : ι -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (E i)] {s : forall (i : ι), Set.{u2} (Set.{u2} (E i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u2} (E i) (_inst_1 i) (s i)) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.topologicalSpace.{u1, u2} ι (fun (i : ι) => E i) (fun (a : ι) => _inst_1 a)) (Set.iUnion.{max u1 u2, succ u1} (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) ι (fun (i : ι) => Set.image.{u2, max u1 u2} (Set.{u2} (E i)) (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) (fun (u : Set.{u2} (E i)) => Set.image.{u2, max u1 u2} (E i) (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.mk.{u1, u2} ι (fun (i : ι) => E i) i) u) (s i))))
 but is expected to have type
-  forall {ι : Type.{u1}} {E : ι -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (E i)] {s : forall (i : ι), Set.{u2} (Set.{u2} (E i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u2} (E i) (_inst_1 i) (s i)) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (instTopologicalSpaceSigma.{u1, u2} ι (fun (i : ι) => E i) (fun (a : ι) => _inst_1 a)) (Set.unionᵢ.{max u1 u2, succ u1} (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) ι (fun (i : ι) => Set.image.{u2, max u1 u2} (Set.{u2} (E i)) (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) (fun (u : Set.{u2} (E i)) => Set.image.{u2, max u1 u2} (E i) (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.mk.{u1, u2} ι (fun (i : ι) => E i) i) u) (s i))))
+  forall {ι : Type.{u1}} {E : ι -> Type.{u2}} [_inst_1 : forall (i : ι), TopologicalSpace.{u2} (E i)] {s : forall (i : ι), Set.{u2} (Set.{u2} (E i))}, (forall (i : ι), TopologicalSpace.IsTopologicalBasis.{u2} (E i) (_inst_1 i) (s i)) -> (TopologicalSpace.IsTopologicalBasis.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (instTopologicalSpaceSigma.{u1, u2} ι (fun (i : ι) => E i) (fun (a : ι) => _inst_1 a)) (Set.iUnion.{max u1 u2, succ u1} (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) ι (fun (i : ι) => Set.image.{u2, max u1 u2} (Set.{u2} (E i)) (Set.{max u1 u2} (Sigma.{u1, u2} ι (fun (i : ι) => E i))) (fun (u : Set.{u2} (E i)) => Set.image.{u2, max u1 u2} (E i) (Sigma.{u1, u2} ι (fun (i : ι) => E i)) (Sigma.mk.{u1, u2} ι (fun (i : ι) => E i) i) u) (s i))))
 Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.sigma TopologicalSpace.IsTopologicalBasis.sigmaₓ'. -/
 /-- In a disjoint union space `Σ i, E i`, one can form a topological basis by taking the union of
 topological bases on each of the parts of the space. -/

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

@@ -1077,7 +1077,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 lean 3 declaration is
   forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (f x) (nhdsWithin.{u1} α t x s))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => f x)))))))
 but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7224 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7220 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
 Case conversion may be inaccurate. Consider using '#align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithinₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}

mathlib commit https://github.com/leanprover-community/mathlib/commit/09079525fd01b3dda35e96adaa08d2f943e1648c

@@ -844,7 +844,7 @@ instance isCountablyGenerated_nhdsWithin (x : α) [IsCountablyGenerated (𝓝 x)
 variable (α)
 
 #print TopologicalSpace.SecondCountableTopology /-
-/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`is_open_generated_countable] [] -/
 /-- A second-countable space is one with a countable basis. -/
 class SecondCountableTopology : Prop where
   is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b

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

@@ -680,7 +680,7 @@ theorem isTopologicalBasis_infᵢ {β : Type _} {ι : Type _} {X : ι → Type _
         ∃ (U : ∀ i, Set (X i))(F : Finset ι),
           (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (hi : i ∈ F), f i ⁻¹' U i } :=
   by
-  convert (isTopologicalBasis_pi cond).Inducing (inducing_infᵢ_to_pi _)
+  convert(isTopologicalBasis_pi cond).Inducing (inducing_infᵢ_to_pi _)
   ext V
   constructor
   · rintro ⟨U, F, h1, h2⟩
@@ -1077,7 +1077,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 lean 3 declaration is
   forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (f x) (nhdsWithin.{u1} α t x s))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => f x)))))))
 but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7191 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7224 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
 Case conversion may be inaccurate. Consider using '#align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithinₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}

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

@@ -1077,7 +1077,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
 lean 3 declaration is
   forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x s) -> (Membership.Mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (Filter.hasMem.{u1} α) (f x) (nhdsWithin.{u1} α t x s))) -> (Exists.{succ u1} (Set.{u1} α) (fun (t : Set.{u1} α) => Exists.{0} (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) (fun (H : HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) t s) => And (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.hasSubset.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) (fun (H : Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x t) => f x)))))))
 but is expected to have type
-  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7190 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
+  forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7191 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
 Case conversion may be inaccurate. Consider using '#align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithinₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (t «expr ⊆ » s) -/
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}

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

@@ -247,7 +247,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, (IsOpen.{u1} α t u) -> (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => And (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) S B) (Eq.{succ u1} (Set.{u1} α) u (Set.unionₛ.{u1} α S)))))
 Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_unionₛₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_eq_unionₛ {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (S : _)(_ : S ⊆ B), u = ⋃₀ S :=
   ⟨{ s ∈ B | s ⊆ u }, fun s h => h.1, hB.open_eq_unionₛ' ou⟩
@@ -259,7 +259,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] {B : Set.{u1} (Set.{u1} α)}, (TopologicalSpace.IsTopologicalBasis.{u1} α t B) -> (forall {u : Set.{u1} α}, Iff (IsOpen.{u1} α t u) (Exists.{succ u1} (Set.{u1} (Set.{u1} α)) (fun (S : Set.{u1} (Set.{u1} α)) => And (HasSubset.Subset.{u1} (Set.{u1} (Set.{u1} α)) (Set.instHasSubsetSet.{u1} (Set.{u1} α)) S B) (Eq.{succ u1} (Set.{u1} α) u (Set.unionₛ.{u1} α S)))))
 Case conversion may be inaccurate. Consider using '#align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_unionₛₓ'. -/
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (S «expr ⊆ » B) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (S «expr ⊆ » B) -/
 theorem IsTopologicalBasis.open_iff_eq_unionₛ {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ (S : _)(_ : S ⊆ B), u = ⋃₀ S :=
   ⟨hB.open_eq_unionₛ, fun ⟨S, hSB, hu⟩ => hu.symm ▸ isOpen_unionₛ fun s hs => hB.IsOpen (hSB hs)⟩
@@ -729,7 +729,7 @@ protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace 
   ⟨⟨f '' s, Countable.image s_cnt f, h.dense_image h' s_dense⟩⟩
 #align dense_range.separable_space DenseRange.separableSpace
 
-/- ./././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 Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}
     [SeparableSpace s] (hs : Dense s) : ∃ (t : _)(_ : t ⊆ s), t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
@@ -743,7 +743,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PartialOrder.{u1} α] {s : Set.{u1} α} [_inst_3 : TopologicalSpace.SeparableSpace.{u1} (Set.Elem.{u1} α s) (instTopologicalSpaceSubtype.{u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) _inst_1)], (Dense.{u1} α _inst_1 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 (Set.Countable.{u1} α t) (And (Dense.{u1} α _inst_1 t) (And (forall (x : α), (IsBot.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) x) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t)) (forall (x : α), (IsTop.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α _inst_2)) x) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t)))))))
 Case conversion may be inaccurate. Consider using '#align dense.exists_countable_dense_subset_bot_top Dense.exists_countable_dense_subset_bot_topₓ'. -/
-/- ./././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) -/
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
 dense subset `t ⊆ s` such that `t` contains bottom/top element of `α` when they exist and belong
@@ -1079,7 +1079,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [t : TopologicalSpace.{u1} α] [_inst_1 : TopologicalSpace.SecondCountableTopology.{u1} α t] {f : α -> (Set.{u1} α)} {s : Set.{u1} α}, (forall (x : α), (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x s) -> (Membership.mem.{u1, u1} (Set.{u1} α) (Filter.{u1} α) (instMembershipSetFilter.{u1} α) (f x) (nhdsWithin.{u1} α t 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 (Set.Countable.{u1} α t) (HasSubset.Subset.{u1} (Set.{u1} α) (Set.instHasSubsetSet.{u1} α) s (Set.unionᵢ.{u1, succ u1} α α (fun (x : α) => Set.unionᵢ.{u1, 0} α (Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) (fun (h._@.Mathlib.Topology.Bases._hyg.7190 : Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x t) => f x)))))))
 Case conversion may be inaccurate. Consider using '#align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithinₓ'. -/
-/- ./././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 countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
     (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ (t : _)(_ : t ⊆ s), t.Countable ∧ s ⊆ ⋃ x ∈ t, f x :=
   by

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

• consistently use the YYYY-MM-DD format
• when easily possible, put the date on the same line as the deprecated attribute
• when easily possible, format the entire declaration on the same line

Why these changes?

• consistency makes it easier for tools to parse this information
• compactness: I don't see a good reason for these declarations taking up more space than needed; as I understand it, deprecated lemmas are not supposed to be used in mathlib anyway
• putting the date on the same line as the attribute makes it easier to discover un-dated deprecations; they also ease writing a tool to replace these by a machine-readable version using leanprover/lean4#3968

@@ -553,14 +553,14 @@ theorem IsSeparable.of_subtype (s : Set α) [SeparableSpace s] : IsSeparable s :
   simpa using isSeparable_range (continuous_subtype_val (p := (· ∈ s)))
 #align topological_space.is_separable_of_separable_space_subtype TopologicalSpace.IsSeparable.of_subtype
 
-@[deprecated] -- Since 2024/02/05
+@[deprecated] -- Since 2024-02-05
 alias isSeparable_of_separableSpace_subtype := IsSeparable.of_subtype
 
 theorem IsSeparable.of_separableSpace [h : SeparableSpace α] (s : Set α) : IsSeparable s :=
   IsSeparable.mono (isSeparable_univ_iff.2 h) (subset_univ _)
 #align topological_space.is_separable_of_separable_space TopologicalSpace.IsSeparable.of_separableSpace
 
-@[deprecated] -- Since 2024/02/05
+@[deprecated] -- Since 2024-02-05
 alias isSeparable_of_separableSpace := IsSeparable.of_separableSpace
 
 end TopologicalSpace

@@ -188,15 +188,13 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
       fun ⟨_b, ⟨_, bu⟩, ab⟩ => bu ab⟩
 #align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 
--- Porting note: use `∃ S, S ⊆ B ∧ _` instead of `∃ S (_ : S ⊆ B), _`
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
-    (ou : IsOpen u) : ∃ S, S ⊆ B ∧ u = ⋃₀ S :=
+    (ou : IsOpen u) : ∃ S ⊆ B, u = ⋃₀ S :=
   ⟨{ s ∈ B | s ⊆ u }, fun _ h => h.1, hB.open_eq_sUnion' ou⟩
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 
--- Porting note: use `∃ S, S ⊆ B ∧ _` instead of `∃ S (_ : S ⊆ B), _`
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
-    {u : Set α} : IsOpen u ↔ ∃ S, S ⊆ B ∧ u = ⋃₀ S :=
+    {u : Set α} : IsOpen u ↔ ∃ S ⊆ B, u = ⋃₀ S :=
   ⟨hB.open_eq_sUnion, fun ⟨_S, hSB, hu⟩ => hu.symm ▸ isOpen_sUnion fun _s hs => hB.isOpen (hSB hs)⟩
 #align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion
 
@@ -621,7 +619,6 @@ theorem isTopologicalBasis_subtype
 
 -- Porting note: moved `DenseRange.separableSpace` up
 
--- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
 theorem Dense.exists_countable_dense_subset {α : Type*} [TopologicalSpace α] {s : Set α}
     [SeparableSpace s] (hs : Dense s) : ∃ t ⊆ s, t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
@@ -629,7 +626,6 @@ theorem Dense.exists_countable_dense_subset {α : Type*} [TopologicalSpace α] {
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
--- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
 dense subset `t ⊆ s` such that `t` contains bottom/top element of `α` when they exist and belong
@@ -900,9 +896,8 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
   exact fun x => ⟨x, mem_interior_iff_mem_nhds.2 (hf x)⟩
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds
 
--- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
-    (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ ⋃ x ∈ t, f x := by
+    (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ t ⊆ s, t.Countable ∧ s ⊆ ⋃ x ∈ t, f x := by
   have : ∀ x : s, (↑) ⁻¹' f x ∈ 𝓝 x := fun x => preimage_coe_mem_nhds_subtype.2 (hf x x.2)
   rcases countable_cover_nhds this with ⟨t, htc, htU⟩
   refine' ⟨(↑) '' t, Subtype.coe_image_subset _ _, htc.image _, fun x hx => _⟩

@@ -833,7 +833,7 @@ lemma secondCountableTopology_iInf {ι} [Countable ι] {t : ι → TopologicalSp
   exact SecondCountableTopology.mk' <|
     countable_iUnion fun i => @countable_countableBasis _ (t i) (ht i)
 
--- TODO: more fine grained instances for first_countable_topology, separable_space, t2_space, ...
+-- TODO: more fine grained instances for `FirstCountableTopology`, `SeparableSpace`, `T2Space`, ...
 instance {β : Type*} [TopologicalSpace β] [SecondCountableTopology α] [SecondCountableTopology β] :
     SecondCountableTopology (α × β) :=
   ((isBasis_countableBasis α).prod (isBasis_countableBasis β)).secondCountableTopology <|

Add firstCountableTopology_induced, Subtype.firstCountableTopology, Inducing.firstCountableTopology, and Embedding.firstCountableTopology. These are pretty literal counterparts of the existing secondCountableTopology_induced, Subtype.secondCountableTopology, Inducing.secondCountableTopology, and Embedding.secondCountableTopology.

Co-authored-by: kkytola <39528102+kkytola@users.noreply.github.com>

@@ -677,10 +677,32 @@ class _root_.FirstCountableTopology : Prop where
 
 attribute [instance] FirstCountableTopology.nhds_generated_countable
 
-namespace FirstCountableTopology
+/-- If `β` is a first-countable space, then its induced topology via `f` on `α` is also
+first-countable. -/
+theorem firstCountableTopology_induced (α β : Type*) [t : TopologicalSpace β]
+    [FirstCountableTopology β] (f : α → β) : @FirstCountableTopology α (t.induced f) :=
+  let _ := t.induced f;
+  ⟨fun x ↦ nhds_induced f x ▸ inferInstance⟩
 
 variable {α}
 
+instance Subtype.firstCountableTopology (s : Set α) [FirstCountableTopology α] :
+    FirstCountableTopology s :=
+  firstCountableTopology_induced s α (↑)
+
+protected theorem _root_.Inducing.firstCountableTopology {β : Type*}
+    [TopologicalSpace β] [FirstCountableTopology β] {f : α → β} (hf : Inducing f) :
+    FirstCountableTopology α := by
+  rw [hf.1]
+  exact firstCountableTopology_induced α β f
+
+protected theorem _root_.Embedding.firstCountableTopology {β : Type*}
+    [TopologicalSpace β] [FirstCountableTopology β] {f : α → β} (hf : Embedding f) :
+    FirstCountableTopology α :=
+  hf.1.firstCountableTopology
+
+namespace FirstCountableTopology
+
 /-- In a first-countable space, a cluster point `x` of a sequence
 is the limit of some subsequence. -/
 theorem tendsto_subseq [FirstCountableTopology α] {u : ℕ → α} {x : α}
@@ -690,8 +712,6 @@ theorem tendsto_subseq [FirstCountableTopology α] {u : ℕ → α} {x : α}
 
 end FirstCountableTopology
 
-variable {α}
-
 instance {β} [TopologicalSpace β] [FirstCountableTopology α] [FirstCountableTopology β] :
     FirstCountableTopology (α × β) :=
   ⟨fun ⟨x, y⟩ => by rw [nhds_prod_eq]; infer_instance⟩

Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -613,6 +613,12 @@ theorem isTopologicalBasis_singletons (α : Type*) [TopologicalSpace α] [Discre
     ⟨{x}, ⟨x, rfl⟩, mem_singleton x, singleton_subset_iff.2 hx⟩
 #align is_topological_basis_singletons isTopologicalBasis_singletons
 
+theorem isTopologicalBasis_subtype
+    {α : Type*} [TopologicalSpace α] {B : Set (Set α)}
+    (h : TopologicalSpace.IsTopologicalBasis B) (p : α → Prop) :
+    IsTopologicalBasis (Set.preimage (Subtype.val (p := p)) '' B) :=
+  h.inducing ⟨rfl⟩
+
 -- Porting note: moved `DenseRange.separableSpace` up
 
 -- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`

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>

@@ -119,21 +119,23 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
     exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩
 #align topological_space.is_topological_basis_of_subbasis TopologicalSpace.isTopologicalBasis_of_subbasis
 
+theorem IsTopologicalBasis.of_hasBasis_nhds {s : Set (Set α)}
+    (h_nhds : ∀ a, (𝓝 a).HasBasis (fun t ↦ t ∈ s ∧ a ∈ t) id) : IsTopologicalBasis s where
+  exists_subset_inter t₁ ht₁ t₂ ht₂ x hx := by
+    simpa only [and_assoc, (h_nhds x).mem_iff]
+      using (inter_mem ((h_nhds _).mem_of_mem ⟨ht₁, hx.1⟩) ((h_nhds _).mem_of_mem ⟨ht₂, hx.2⟩))
+  sUnion_eq := sUnion_eq_univ_iff.2 fun x ↦ (h_nhds x).ex_mem
+  eq_generateFrom := ext_nhds fun x ↦ by
+    simpa only [nhds_generateFrom, and_comm] using (h_nhds x).eq_biInf
+
 /-- If a family of open sets `s` is such that every open neighbourhood contains some
 member of `s`, then `s` is a topological basis. -/
 theorem isTopologicalBasis_of_isOpen_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
     (h_nhds : ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u) :
-    IsTopologicalBasis s := by
-  refine'
-    ⟨fun t₁ ht₁ t₂ ht₂ x hx => h_nhds _ _ hx (IsOpen.inter (h_open _ ht₁) (h_open _ ht₂)), _, _⟩
-  · refine' sUnion_eq_univ_iff.2 fun a => _
-    rcases h_nhds a univ trivial isOpen_univ with ⟨u, h₁, h₂, -⟩
-    exact ⟨u, h₁, h₂⟩
-  · refine' (le_generateFrom h_open).antisymm fun u hu => _
-    refine (@isOpen_iff_nhds α u (generateFrom s)).mpr fun a ha ↦ ?_
-    rcases h_nhds a u ha hu with ⟨v, hvs, hav, hvu⟩
-    rw [nhds_generateFrom]
-    exact iInf₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
+    IsTopologicalBasis s :=
+  .of_hasBasis_nhds <| fun a ↦
+    (nhds_basis_opens a).to_hasBasis' (by simpa [and_assoc] using h_nhds a)
+      fun t ⟨hts, hat⟩ ↦ (h_open _ hts).mem_nhds hat
 #align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
 
 /-- A set `s` is in the neighbourhood of `a` iff there is some basis set `t`, which
@@ -245,32 +247,38 @@ theorem isTopologicalBasis_opens : IsTopologicalBasis { U : Set α | IsOpen U }
   isTopologicalBasis_of_isOpen_of_nhds (by tauto) (by tauto)
 #align topological_space.is_topological_basis_opens TopologicalSpace.isTopologicalBasis_opens
 
+protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α → β} {T : Set (Set β)}
+    (hf : Inducing f) (h : IsTopologicalBasis T) : IsTopologicalBasis ((preimage f) '' T) :=
+  .of_hasBasis_nhds fun a ↦ by
+    convert (hf.basis_nhds (h.nhds_hasBasis (a := f a))).to_image_id with s
+    aesop
+#align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducing
+
+protected theorem IsTopologicalBasis.induced [s : TopologicalSpace β] (f : α → β)
+    {T : Set (Set β)} (h : IsTopologicalBasis T) :
+    IsTopologicalBasis (t := induced f s) ((preimage f) '' T) :=
+  h.inducing (t := induced f s) (inducing_induced f)
+
+protected theorem IsTopologicalBasis.inf {t₁ t₂ : TopologicalSpace β} {B₁ B₂ : Set (Set β)}
+    (h₁ : IsTopologicalBasis (t := t₁) B₁) (h₂ : IsTopologicalBasis (t := t₂) B₂) :
+    IsTopologicalBasis (t := t₁ ⊓ t₂) (image2 (· ∩ ·) B₁ B₂) := by
+  refine .of_hasBasis_nhds (t := ?_) fun a ↦ ?_
+  rw [nhds_inf (t₁ := t₁)]
+  convert ((h₁.nhds_hasBasis (t := t₁)).inf (h₂.nhds_hasBasis (t := t₂))).to_image_id
+  aesop
+
+theorem IsTopologicalBasis.inf_induced {γ} [s : TopologicalSpace β] {B₁ : Set (Set α)}
+    {B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) (f₁ : γ → α)
+    (f₂ : γ → β) :
+    IsTopologicalBasis (t := induced f₁ t ⊓ induced f₂ s) (image2 (f₁ ⁻¹' · ∩ f₂ ⁻¹' ·) B₁ B₂) := by
+  simpa only [image2_image_left, image2_image_right] using (h₁.induced f₁).inf (h₂.induced f₂)
+
 protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set (Set α)}
     {B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) :
-    IsTopologicalBasis (image2 (· ×ˢ ·) B₁ B₂) := by
-  refine' isTopologicalBasis_of_isOpen_of_nhds _ _
-  · rintro _ ⟨u₁, hu₁, u₂, hu₂, rfl⟩
-    exact (h₁.isOpen hu₁).prod (h₂.isOpen hu₂)
-  · rintro ⟨a, b⟩ u hu uo
-    rcases (h₁.nhds_hasBasis.prod_nhds h₂.nhds_hasBasis).mem_iff.1 (IsOpen.mem_nhds uo hu) with
-      ⟨⟨s, t⟩, ⟨⟨hs, ha⟩, ht, hb⟩, hu⟩
-    exact ⟨s ×ˢ t, mem_image2_of_mem hs ht, ⟨ha, hb⟩, hu⟩
+    IsTopologicalBasis (image2 (· ×ˢ ·) B₁ B₂) :=
+  h₁.inf_induced h₂ Prod.fst Prod.snd
 #align topological_space.is_topological_basis.prod TopologicalSpace.IsTopologicalBasis.prod
 
-protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α → β} {T : Set (Set β)}
-    (hf : Inducing f) (h : IsTopologicalBasis T) : IsTopologicalBasis ((preimage f) '' T) := by
-  refine' isTopologicalBasis_of_isOpen_of_nhds _ _
-  · rintro _ ⟨V, hV, rfl⟩
-    rw [hf.isOpen_iff]
-    exact ⟨V, h.isOpen hV, rfl⟩
-  · intro a U ha hU
-    rw [hf.isOpen_iff] at hU
-    obtain ⟨V, hV, rfl⟩ := hU
-    obtain ⟨S, hS, rfl⟩ := h.open_eq_sUnion hV
-    obtain ⟨W, hW, ha⟩ := ha
-    exact ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_sUnion_of_mem hW⟩
-#align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducing
-
 theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
     (Uc : ⋃ i, U i = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
     IsTopologicalBasis (⋃ i : ι, image ((↑) : U i → α) '' b i) := by
@@ -561,45 +569,44 @@ end TopologicalSpace
 
 open TopologicalSpace
 
+protected theorem IsTopologicalBasis.iInf {β : Type*} {ι : Type*} {t : ι → TopologicalSpace β}
+    {T : ι → Set (Set β)} (h_basis : ∀ i, IsTopologicalBasis (t := t i) (T i)) :
+    IsTopologicalBasis (t := ⨅ i, t i)
+      { S | ∃ (U : ι → Set β) (F : Finset ι), (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ i ∈ F, U i } := by
+  let _ := ⨅ i, t i
+  refine isTopologicalBasis_of_isOpen_of_nhds ?_ ?_
+  · rintro - ⟨U, F, hU, rfl⟩
+    refine isOpen_biInter_finset fun i hi ↦
+      (h_basis i).isOpen (t := t i) (hU i hi) |>.mono (iInf_le _ _)
+  · intro a u ha hu
+    rcases (nhds_iInf (t := t) (a := a)).symm ▸ hasBasis_iInf'
+      (fun i ↦ (h_basis i).nhds_hasBasis (t := t i)) |>.mem_iff.1 (hu.mem_nhds ha)
+      with ⟨⟨F, U⟩, ⟨hF, hU⟩, hUu⟩
+    refine ⟨_, ⟨U, hF.toFinset, ?_, rfl⟩, ?_, ?_⟩ <;> simp only [Finite.mem_toFinset, mem_iInter]
+    · exact fun i hi ↦ (hU i hi).1
+    · exact fun i hi ↦ (hU i hi).2
+    · exact hUu
+
+theorem IsTopologicalBasis.iInf_induced {β : Type*} {ι : Type*} {X : ι → Type*}
+    [t : Π i, TopologicalSpace (X i)] {T : Π i, Set (Set (X i))}
+    (cond : ∀ i, IsTopologicalBasis (T i)) (f : Π i, β → X i) :
+    IsTopologicalBasis (t := ⨅ i, induced (f i) (t i))
+      { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
+        (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (_ : i ∈ F), f i ⁻¹' U i } := by
+  convert IsTopologicalBasis.iInf (fun i ↦ (cond i).induced (f i)) with S
+  constructor <;> rintro ⟨U, F, hUT, hSU⟩
+  · exact ⟨fun i ↦ (f i) ⁻¹' (U i), F, fun i hi ↦ mem_image_of_mem _ (hUT i hi), hSU⟩
+  · choose! U' hU' hUU' using hUT
+    exact ⟨U', F, hU', hSU ▸ (.symm <| iInter₂_congr hUU')⟩
+#align is_topological_basis_infi IsTopologicalBasis.iInf_induced
+
 theorem isTopologicalBasis_pi {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
     {T : ∀ i, Set (Set (X i))} (cond : ∀ i, IsTopologicalBasis (T i)) :
     IsTopologicalBasis { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
       (∀ i, i ∈ F → U i ∈ T i) ∧ S = (F : Set ι).pi U } := by
-  refine' isTopologicalBasis_of_isOpen_of_nhds _ _
-  · rintro _ ⟨U, F, h1, rfl⟩
-    apply isOpen_set_pi F.finite_toSet
-    intro i hi
-    exact (cond i).isOpen (h1 i hi)
-  · intro a U ha hU
-    obtain ⟨I, t, hta, htU⟩ : ∃ (I : Finset ι) (t : ∀ i : ι, Set (X i)),
-        (∀ i, t i ∈ 𝓝 (a i)) ∧ Set.pi (↑I) t ⊆ U := by
-      rw [← Filter.mem_pi', ← nhds_pi]
-      exact hU.mem_nhds ha
-    have : ∀ i, ∃ V ∈ T i, a i ∈ V ∧ V ⊆ t i := fun i => (cond i).mem_nhds_iff.1 (hta i)
-    choose V hVT haV hVt using this
-    exact ⟨_, ⟨V, I, fun i _ => hVT i, rfl⟩, fun i _ => haV i, (pi_mono fun i _ => hVt i).trans htU⟩
+  simpa only [Set.pi_def] using IsTopologicalBasis.iInf_induced cond eval
 #align is_topological_basis_pi isTopologicalBasis_pi
 
-theorem isTopologicalBasis_iInf {β : Type*} {ι : Type*} {X : ι → Type*}
-    [t : ∀ i, TopologicalSpace (X i)] {T : ∀ i, Set (Set (X i))}
-    (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
-    @IsTopologicalBasis β (⨅ i, induced (f i) (t i))
-      { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
-        (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (_ : i ∈ F), f i ⁻¹' U i } := by
-  letI := ⨅ i, induced (f i) (t i)
-  convert (isTopologicalBasis_pi cond).inducing (inducing_iInf_to_pi f)
-  ext V
-  constructor
-  · rintro ⟨U, F, h1, rfl⟩
-    refine' ⟨(F : Set ι).pi U, ⟨U, F, h1, rfl⟩, _⟩
-    simp_rw [pi_def, Set.preimage_iInter]
-    rfl
-  · rintro ⟨U, ⟨U, F, h1, rfl⟩, rfl⟩
-    refine' ⟨U, F, h1, _⟩
-    simp_rw [pi_def, Set.preimage_iInter]
-    rfl
-#align is_topological_basis_infi isTopologicalBasis_iInf
-
 theorem isTopologicalBasis_singletons (α : Type*) [TopologicalSpace α] [DiscreteTopology α] :
     IsTopologicalBasis { s | ∃ x : α, (s : Set α) = {x} } :=
   isTopologicalBasis_of_isOpen_of_nhds (fun _ _ => isOpen_discrete _) fun x _ hx _ =>
@@ -886,18 +893,10 @@ topological bases on each of the parts of the space. -/
 theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
     (hs : ∀ i, IsTopologicalBasis (s i)) :
     IsTopologicalBasis (⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' s i) := by
-  apply isTopologicalBasis_of_isOpen_of_nhds
-  · intro u hu
-    obtain ⟨i, t, ts, rfl⟩ : ∃ (i : ι) (t : Set (E i)), t ∈ s i ∧ Sigma.mk i '' t = u := by
-      simpa only [mem_iUnion, mem_image] using hu
-    exact isOpenMap_sigmaMk _ ((hs i).isOpen ts)
-  · rintro ⟨i, x⟩ u hxu u_open
-    have hx : x ∈ Sigma.mk i ⁻¹' u := hxu
-    obtain ⟨v, vs, xv, hv⟩ : ∃ (v : Set (E i)), v ∈ s i ∧ x ∈ v ∧ v ⊆ Sigma.mk i ⁻¹' u :=
-      (hs i).exists_subset_of_mem_open hx (isOpen_sigma_iff.1 u_open i)
-    exact
-      ⟨Sigma.mk i '' v, mem_iUnion.2 ⟨i, mem_image_of_mem _ vs⟩, mem_image_of_mem _ xv,
-        image_subset_iff.2 hv⟩
+  refine .of_hasBasis_nhds fun a ↦ ?_
+  rw [Sigma.nhds_eq]
+  convert (((hs a.1).nhds_hasBasis).map _).to_image_id
+  aesop
 #align topological_space.is_topological_basis.sigma TopologicalSpace.IsTopologicalBasis.sigma
 
 /-- A countable disjoint union of second countable spaces is second countable. -/

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.

@@ -212,7 +212,7 @@ lemma IsTopologicalBasis.subset_of_forall_subset {t : Set α} (hB : IsTopologica
 lemma IsTopologicalBasis.eq_of_forall_subset_iff {t : Set α} (hB : IsTopologicalBasis B)
     (hs : IsOpen s) (ht : IsOpen t) (h : ∀ U ∈ B, U ⊆ s ↔ U ⊆ t) : s = t := by
   rw [hB.open_eq_sUnion' hs, hB.open_eq_sUnion' ht]
-  exact congr_arg _ (Set.ext λ U ↦ and_congr_right <| h _)
+  exact congr_arg _ (Set.ext fun U ↦ and_congr_right <| h _)
 
 /-- A point `a` is in the closure of `s` iff all basis sets containing `a` intersect `s`. -/
 theorem IsTopologicalBasis.mem_closure_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α}

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

@@ -311,8 +311,9 @@ latter should be used as a typeclass argument in theorems because Lean can autom
 `TopologicalSpace.SeparableSpace` from `SecondCountableTopology` but it can't
 deduce `SecondCountableTopology` from `TopologicalSpace.SeparableSpace`.
 
-Porting note: TODO: the previous paragraph describes the state of the art in Lean 3. We can have
-instance cycles in Lean 4 but we might want to postpone adding them till after the port. -/
+Porting note (#11215): TODO: the previous paragraph describes the state of the art in Lean 3.
+We can have instance cycles in Lean 4 but we might want to
+postpone adding them till after the port. -/
 @[mk_iff] class SeparableSpace : Prop where
   /-- There exists a countable dense set. -/
   exists_countable_dense : ∃ s : Set α, s.Countable ∧ Dense s

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".

@@ -186,13 +186,13 @@ theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologica
       fun ⟨_b, ⟨_, bu⟩, ab⟩ => bu ab⟩
 #align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 
--- porting note: use `∃ S, S ⊆ B ∧ _` instead of `∃ S (_ : S ⊆ B), _`
+-- Porting note: use `∃ S, S ⊆ B ∧ _` instead of `∃ S (_ : S ⊆ B), _`
 theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ S, S ⊆ B ∧ u = ⋃₀ S :=
   ⟨{ s ∈ B | s ⊆ u }, fun _ h => h.1, hB.open_eq_sUnion' ou⟩
 #align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 
--- porting note: use `∃ S, S ⊆ B ∧ _` instead of `∃ S (_ : S ⊆ B), _`
+-- Porting note: use `∃ S, S ⊆ B ∧ _` instead of `∃ S (_ : S ⊆ B), _`
 theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ S, S ⊆ B ∧ u = ⋃₀ S :=
   ⟨hB.open_eq_sUnion, fun ⟨_S, hSB, hu⟩ => hu.symm ▸ isOpen_sUnion fun _s hs => hB.isOpen (hSB hs)⟩
@@ -607,7 +607,7 @@ theorem isTopologicalBasis_singletons (α : Type*) [TopologicalSpace α] [Discre
 
 -- Porting note: moved `DenseRange.separableSpace` up
 
--- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
+-- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
 theorem Dense.exists_countable_dense_subset {α : Type*} [TopologicalSpace α] {s : Set α}
     [SeparableSpace s] (hs : Dense s) : ∃ t ⊆ s, t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
@@ -615,7 +615,7 @@ theorem Dense.exists_countable_dense_subset {α : Type*} [TopologicalSpace α] {
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
--- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
+-- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
 /-- Let `s` be a dense set in a topological space `α` with partial order structure. If `s` is a
 separable space (e.g., if `α` has a second countable topology), then there exists a countable
 dense subset `t ⊆ s` such that `t` contains bottom/top element of `α` when they exist and belong
@@ -866,7 +866,7 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
   exact fun x => ⟨x, mem_interior_iff_mem_nhds.2 (hf x)⟩
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds
 
--- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
+-- Porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
 theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set α} {s : Set α}
     (hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ ⋃ x ∈ t, f x := by
   have : ∀ x : s, (↑) ⁻¹' f x ∈ 𝓝 x := fun x => preimage_coe_mem_nhds_subtype.2 (hf x x.2)

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -397,7 +397,7 @@ instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i,
     choose y hyt hyu using this
     lift y to ∀ i : I, t i using hyt
     refine ⟨f ⟨I, y⟩, huU fun i (hi : i ∈ I) ↦ ?_, mem_range_self _⟩
-    simp only [dif_pos hi]
+    simp only [f, dif_pos hi]
     exact hyu _
 
 instance [SeparableSpace α] {r : α → α → Prop} : SeparableSpace (Quot r) :=
@@ -503,7 +503,7 @@ theorem IsSeparable.univ_pi {ι : Type*} [Countable ι] {X : ι → Type*} {s :
     suffices H : ∀ i ∈ I, (u i ∩ c i).Nonempty by
       choose f hfu hfc using H
       refine ⟨fun i ↦ ⟨f i i.2, hfc i i.2⟩, fun i (hi : i ∈ I) ↦ ?_⟩
-      simpa only [dif_pos hi] using hfu i hi
+      simpa only [g, dif_pos hi] using hfu i hi
     intro i hi
     exact mem_closure_iff.1 (hc i <| hf _ trivial) _ (huo i hi).1 (huo i hi).2
 

I replaced a few "terminal" refine/refine's with exact.

The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.

This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.

@@ -262,13 +262,13 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
   refine' isTopologicalBasis_of_isOpen_of_nhds _ _
   · rintro _ ⟨V, hV, rfl⟩
     rw [hf.isOpen_iff]
-    refine' ⟨V, h.isOpen hV, rfl⟩
+    exact ⟨V, h.isOpen hV, rfl⟩
   · intro a U ha hU
     rw [hf.isOpen_iff] at hU
     obtain ⟨V, hV, rfl⟩ := hU
     obtain ⟨S, hS, rfl⟩ := h.open_eq_sUnion hV
     obtain ⟨W, hW, ha⟩ := ha
-    refine' ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_sUnion_of_mem hW⟩
+    exact ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_sUnion_of_mem hW⟩
 #align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducing
 
 theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))

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>

@@ -860,8 +860,8 @@ theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (h
     ∃ s : Set α, s.Countable ∧ ⋃ x ∈ s, f x = univ := by
   rcases isOpen_iUnion_countable (fun x => interior (f x)) fun x => isOpen_interior with
     ⟨s, hsc, hsU⟩
-  suffices : ⋃ x ∈ s, interior (f x) = univ
-  exact ⟨s, hsc, flip eq_univ_of_subset this <| iUnion₂_mono fun _ _ => interior_subset⟩
+  suffices ⋃ x ∈ s, interior (f x) = univ from
+    ⟨s, hsc, flip eq_univ_of_subset this <| iUnion₂_mono fun _ _ => interior_subset⟩
   simp only [hsU, eq_univ_iff_forall, mem_iUnion]
   exact fun x => ⟨x, mem_interior_iff_mem_nhds.2 (hf x)⟩
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds

Add a version for Pairwise (Disjoint on _), not Set.PairwiseDisjoint _.

@@ -411,20 +411,21 @@ theorem separableSpace_iff_countable [DiscreteTopology α] : SeparableSpace α 
   simp [separableSpace_iff, countable_univ_iff]
 
 /-- In a separable space, a family of nonempty disjoint open sets is countable. -/
-theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type*}
-    {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ha : ∀ i ∈ a, IsOpen (s i))
-    (h'a : ∀ i ∈ a, (s i).Nonempty) : a.Countable := by
+theorem _root_.Pairwise.countable_of_isOpen_disjoint [SeparableSpace α] {ι : Type*}
+    {s : ι → Set α} (hd : Pairwise (Disjoint on s)) (ho : ∀ i, IsOpen (s i))
+    (hne : ∀ i, (s i).Nonempty) : Countable ι := by
   rcases exists_countable_dense α with ⟨u, u_countable, u_dense⟩
-  have : ∀ i : a, (s i ∩ u).Nonempty := fun i =>
-    dense_iff_inter_open.1 u_dense (s i) (ha i i.2) (h'a i i.2)
-  choose f hfs hfu using this
-  lift f to a → u using hfu
-  have f_inj : Injective f := by
-    refine' injective_iff_pairwise_ne.mpr
-      ((h.subtype _ _).mono fun i j hij hfij => hij.le_bot ⟨hfs i, _⟩)
-    simp only [congr_arg Subtype.val hfij, hfs j]
+  choose f hfu hfs using fun i ↦ u_dense.exists_mem_open (ho i) (hne i)
+  have f_inj : Injective f := fun i j hij ↦
+    hd.eq <| not_disjoint_iff.2 ⟨f i, hfs i, hij.symm ▸ hfs j⟩
   have := u_countable.to_subtype
-  exact f_inj.countable
+  exact (f_inj.codRestrict hfu).countable
+
+/-- In a separable space, a family of nonempty disjoint open sets is countable. -/
+theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type*}
+    {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ho : ∀ i ∈ a, IsOpen (s i))
+    (hne : ∀ i ∈ a, (s i).Nonempty) : a.Countable :=
+  (h.subtype _ _).countable_of_isOpen_disjoint (Subtype.forall.2 ho) (Subtype.forall.2 hne)
 #align set.pairwise_disjoint.countable_of_is_open Set.PairwiseDisjoint.countable_of_isOpen
 
 /-- In a separable space, a family of disjoint sets with nonempty interiors is countable. -/

• upgrade isSeparable_iUnion to an Iff lemma, restore the original version as IsSeparable.iUnion;
• add isSeparable_union and isSeparable_closure;
• upgrade isSeparable_pi from [Finite ι] to [Countable ι], add IsSeparable.univ_pi version;
• add Dense.isSeparable_iff and isSeparable_range;
• rename isSeparable_of_separableSpace_subtype to IsSeparable.of_subtype;
• rename isSeparable_of_separableSpace to IsSeparable.of_separableSpace.

@@ -448,37 +448,68 @@ theorem IsSeparable.mono {s u : Set α} (hs : IsSeparable s) (hu : u ⊆ s) : Is
   exact ⟨c, c_count, hu.trans hs⟩
 #align topological_space.is_separable.mono TopologicalSpace.IsSeparable.mono
 
+theorem IsSeparable.iUnion {ι : Sort*} [Countable ι] {s : ι → Set α}
+    (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) := by
+  choose c hc h'c using hs
+  refine' ⟨⋃ i, c i, countable_iUnion hc, iUnion_subset_iff.2 fun i => _⟩
+  exact (h'c i).trans (closure_mono (subset_iUnion _ i))
+#align topological_space.is_separable_Union TopologicalSpace.IsSeparable.iUnion
+
+@[simp]
+theorem isSeparable_iUnion {ι : Sort*} [Countable ι] {s : ι → Set α} :
+    IsSeparable (⋃ i, s i) ↔ ∀ i, IsSeparable (s i) :=
+  ⟨fun h i ↦ h.mono <| subset_iUnion s i, .iUnion⟩
+
+@[simp]
+theorem isSeparable_union {s t : Set α} : IsSeparable (s ∪ t) ↔ IsSeparable s ∧ IsSeparable t := by
+  simp [union_eq_iUnion, and_comm]
+
 theorem IsSeparable.union {s u : Set α} (hs : IsSeparable s) (hu : IsSeparable u) :
-    IsSeparable (s ∪ u) := by
-  rcases hs with ⟨cs, cs_count, hcs⟩
-  rcases hu with ⟨cu, cu_count, hcu⟩
-  refine' ⟨cs ∪ cu, cs_count.union cu_count, _⟩
-  exact
-    union_subset (hcs.trans (closure_mono (subset_union_left _ _)))
-      (hcu.trans (closure_mono (subset_union_right _ _)))
+    IsSeparable (s ∪ u) :=
+  isSeparable_union.2 ⟨hs, hu⟩
 #align topological_space.is_separable.union TopologicalSpace.IsSeparable.union
 
-theorem IsSeparable.closure {s : Set α} (hs : IsSeparable s) : IsSeparable (closure s) := by
-  rcases hs with ⟨c, c_count, hs⟩
-  exact ⟨c, c_count, by simpa using closure_mono hs⟩
+@[simp]
+theorem isSeparable_closure : IsSeparable (closure s) ↔ IsSeparable s := by
+  simp only [IsSeparable, isClosed_closure.closure_subset_iff]
+
+protected alias ⟨_, IsSeparable.closure⟩ := isSeparable_closure
 #align topological_space.is_separable.closure TopologicalSpace.IsSeparable.closure
 
-theorem isSeparable_iUnion {ι : Type*} [Countable ι] {s : ι → Set α}
-    (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) := by
-  choose c hc h'c using hs
-  refine' ⟨⋃ i, c i, countable_iUnion hc, iUnion_subset_iff.2 fun i => _⟩
-  exact (h'c i).trans (closure_mono (subset_iUnion _ i))
-#align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnion
+theorem _root_.Set.Countable.isSeparable {s : Set α} (hs : s.Countable) : IsSeparable s :=
+  ⟨s, hs, subset_closure⟩
+#align set.countable.is_separable Set.Countable.isSeparable
+
+theorem _root_.Set.Finite.isSeparable {s : Set α} (hs : s.Finite) : IsSeparable s :=
+  hs.countable.isSeparable
+#align set.finite.is_separable Set.Finite.isSeparable
 
-lemma isSeparable_pi {ι : Type*} [Finite ι] {α : ι → Type*} {s : ∀ i, Set (α i)}
+theorem IsSeparable.univ_pi {ι : Type*} [Countable ι] {X : ι → Type*} {s : ∀ i, Set (X i)}
+    [∀ i, TopologicalSpace (X i)] (h : ∀ i, IsSeparable (s i)) :
+    IsSeparable (univ.pi s) := by
+  classical
+  rcases eq_empty_or_nonempty (univ.pi s) with he | ⟨f₀, -⟩
+  · rw [he]
+    exact countable_empty.isSeparable
+  · choose c c_count hc using h
+    haveI := fun i ↦ (c_count i).to_subtype
+    set g : (I : Finset ι) × ((i : I) → c i) → (i : ι) → X i := fun ⟨I, f⟩ i ↦
+      if hi : i ∈ I then f ⟨i, hi⟩ else f₀ i
+    refine ⟨range g, countable_range g, fun f hf ↦ mem_closure_iff.2 fun o ho hfo ↦ ?_⟩
+    rcases isOpen_pi_iff.1 ho f hfo with ⟨I, u, huo, hI⟩
+    rsuffices ⟨f, hf⟩ : ∃ f : (i : I) → c i, g ⟨I, f⟩ ∈ Set.pi I u
+    · exact ⟨g ⟨I, f⟩, hI hf, mem_range_self _⟩
+    suffices H : ∀ i ∈ I, (u i ∩ c i).Nonempty by
+      choose f hfu hfc using H
+      refine ⟨fun i ↦ ⟨f i i.2, hfc i i.2⟩, fun i (hi : i ∈ I) ↦ ?_⟩
+      simpa only [dif_pos hi] using hfu i hi
+    intro i hi
+    exact mem_closure_iff.1 (hc i <| hf _ trivial) _ (huo i hi).1 (huo i hi).2
+
+lemma isSeparable_pi {ι : Type*} [Countable ι] {α : ι → Type*} {s : ∀ i, Set (α i)}
     [∀ i, TopologicalSpace (α i)] (h : ∀ i, IsSeparable (s i)) :
     IsSeparable {f : ∀ i, α i | ∀ i, f i ∈ s i} := by
-  choose c c_count hc using h
-  refine ⟨{f | ∀ i, f i ∈ c i}, countable_pi c_count, ?_⟩
-  simp_rw [← mem_univ_pi]
-  dsimp
-  rw [closure_pi_set]
-  exact Set.pi_mono (fun i _ ↦ hc i)
+  simpa only [← mem_univ_pi] using IsSeparable.univ_pi h
 
 lemma IsSeparable.prod {β : Type*} [TopologicalSpace β]
     {s : Set α} {t : Set β} (hs : IsSeparable s) (ht : IsSeparable t) :
@@ -489,27 +520,6 @@ lemma IsSeparable.prod {β : Type*} [TopologicalSpace β]
   rw [closure_prod_eq]
   exact Set.prod_mono hcs hct
 
-theorem _root_.Set.Countable.isSeparable {s : Set α} (hs : s.Countable) : IsSeparable s :=
-  ⟨s, hs, subset_closure⟩
-#align set.countable.is_separable Set.Countable.isSeparable
-
-theorem _root_.Set.Finite.isSeparable {s : Set α} (hs : s.Finite) : IsSeparable s :=
-  hs.countable.isSeparable
-#align set.finite.is_separable Set.Finite.isSeparable
-
-theorem isSeparable_univ_iff : IsSeparable (univ : Set α) ↔ SeparableSpace α := by
-  constructor
-  · rintro ⟨c, c_count, hc⟩
-    refine' ⟨⟨c, c_count, by rwa [dense_iff_closure_eq, ← univ_subset_iff]⟩⟩
-  · intro h
-    rcases exists_countable_dense α with ⟨c, c_count, hc⟩
-    exact ⟨c, c_count, by rwa [univ_subset_iff, ← dense_iff_closure_eq]⟩
-#align topological_space.is_separable_univ_iff TopologicalSpace.isSeparable_univ_iff
-
-theorem isSeparable_of_separableSpace [h : SeparableSpace α] (s : Set α) : IsSeparable s :=
-  IsSeparable.mono (isSeparable_univ_iff.2 h) (subset_univ _)
-#align topological_space.is_separable_of_separable_space TopologicalSpace.isSeparable_of_separableSpace
-
 theorem IsSeparable.image {β : Type*} [TopologicalSpace β] {s : Set α} (hs : IsSeparable s)
     {f : α → β} (hf : Continuous f) : IsSeparable (f '' s) := by
   rcases hs with ⟨c, c_count, hc⟩
@@ -518,11 +528,32 @@ theorem IsSeparable.image {β : Type*} [TopologicalSpace β] {s : Set α} (hs :
   exact hc.trans (closure_subset_preimage_closure_image hf)
 #align topological_space.is_separable.image TopologicalSpace.IsSeparable.image
 
-theorem isSeparable_of_separableSpace_subtype (s : Set α) [SeparableSpace s] : IsSeparable s := by
-  have : IsSeparable (((↑) : s → α) '' (univ : Set s)) :=
-    (isSeparable_of_separableSpace _).image continuous_subtype_val
-  simpa only [image_univ, Subtype.range_val_subtype] using this
-#align topological_space.is_separable_of_separable_space_subtype TopologicalSpace.isSeparable_of_separableSpace_subtype
+theorem _root_.Dense.isSeparable_iff (hs : Dense s) :
+    IsSeparable s ↔ SeparableSpace α := by
+  simp_rw [IsSeparable, separableSpace_iff, dense_iff_closure_eq, ← univ_subset_iff,
+    ← hs.closure_eq, isClosed_closure.closure_subset_iff]
+
+theorem isSeparable_univ_iff : IsSeparable (univ : Set α) ↔ SeparableSpace α :=
+  dense_univ.isSeparable_iff
+#align topological_space.is_separable_univ_iff TopologicalSpace.isSeparable_univ_iff
+
+theorem isSeparable_range [TopologicalSpace β] [SeparableSpace α] {f : α → β} (hf : Continuous f) :
+    IsSeparable (range f) :=
+  image_univ (f := f) ▸ (isSeparable_univ_iff.2 ‹_›).image hf
+
+theorem IsSeparable.of_subtype (s : Set α) [SeparableSpace s] : IsSeparable s := by
+  simpa using isSeparable_range (continuous_subtype_val (p := (· ∈ s)))
+#align topological_space.is_separable_of_separable_space_subtype TopologicalSpace.IsSeparable.of_subtype
+
+@[deprecated] -- Since 2024/02/05
+alias isSeparable_of_separableSpace_subtype := IsSeparable.of_subtype
+
+theorem IsSeparable.of_separableSpace [h : SeparableSpace α] (s : Set α) : IsSeparable s :=
+  IsSeparable.mono (isSeparable_univ_iff.2 h) (subset_univ _)
+#align topological_space.is_separable_of_separable_space TopologicalSpace.IsSeparable.of_separableSpace
+
+@[deprecated] -- Since 2024/02/05
+alias isSeparable_of_separableSpace := IsSeparable.of_separableSpace
 
 end TopologicalSpace
 

@@ -470,7 +470,7 @@ theorem isSeparable_iUnion {ι : Type*} [Countable ι] {s : ι → Set α}
   exact (h'c i).trans (closure_mono (subset_iUnion _ i))
 #align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnion
 
-lemma isSeparable_pi {ι : Type*} [Fintype ι] {α : ∀ (_ : ι), Type*} {s : ∀ i, Set (α i)}
+lemma isSeparable_pi {ι : Type*} [Finite ι] {α : ι → Type*} {s : ∀ i, Set (α i)}
     [∀ i, TopologicalSpace (α i)] (h : ∀ i, IsSeparable (s i)) :
     IsSeparable {f : ∀ i, α i | ∀ i, f i ∈ s i} := by
   choose c c_count hc using h

Redefine Set.Countable s as _root_.Countable s. Fix compile, golf some of the broken proofs.

@@ -414,8 +414,8 @@ theorem separableSpace_iff_countable [DiscreteTopology α] : SeparableSpace α 
 theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type*}
     {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ha : ∀ i ∈ a, IsOpen (s i))
     (h'a : ∀ i ∈ a, (s i).Nonempty) : a.Countable := by
-  rcases exists_countable_dense α with ⟨u, ⟨u_encodable⟩, u_dense⟩
-  have : ∀ i : a, ∃ y, y ∈ s i ∩ u := fun i =>
+  rcases exists_countable_dense α with ⟨u, u_countable, u_dense⟩
+  have : ∀ i : a, (s i ∩ u).Nonempty := fun i =>
     dense_iff_inter_open.1 u_dense (s i) (ha i i.2) (h'a i i.2)
   choose f hfs hfu using this
   lift f to a → u using hfu
@@ -423,7 +423,8 @@ theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι
     refine' injective_iff_pairwise_ne.mpr
       ((h.subtype _ _).mono fun i j hij hfij => hij.le_bot ⟨hfs i, _⟩)
     simp only [congr_arg Subtype.val hfij, hfs j]
-  exact ⟨@Encodable.ofInj _ _ u_encodable f f_inj⟩
+  have := u_countable.to_subtype
+  exact f_inj.countable
 #align set.pairwise_disjoint.countable_of_is_open Set.PairwiseDisjoint.countable_of_isOpen
 
 /-- In a separable space, a family of disjoint sets with nonempty interiors is countable. -/
@@ -576,9 +577,9 @@ theorem isTopologicalBasis_singletons (α : Type*) [TopologicalSpace α] [Discre
 
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
 theorem Dense.exists_countable_dense_subset {α : Type*} [TopologicalSpace α] {s : Set α}
-    [SeparableSpace s] (hs : Dense s) : ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t :=
+    [SeparableSpace s] (hs : Dense s) : ∃ t ⊆ s, t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
-  ⟨(↑) '' t, image_subset_iff.2 fun _ _ => mem_preimage.2 <| Subtype.coe_prop _, htc.image (↑),
+  ⟨(↑) '' t, Subtype.coe_image_subset s t, htc.image Subtype.val,
     hs.denseRange_val.dense_image continuous_subtype_val htd⟩
 #align dense.exists_countable_dense_subset Dense.exists_countable_dense_subsetₓ
 
@@ -590,7 +591,7 @@ to `s`. For a dense subset containing neither bot nor top elements, see
 `Dense.exists_countable_dense_subset_no_bot_top`. -/
 theorem Dense.exists_countable_dense_subset_bot_top {α : Type*} [TopologicalSpace α]
     [PartialOrder α] {s : Set α} [SeparableSpace s] (hs : Dense s) :
-    ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∈ s → x ∈ t) ∧
+    ∃ t ⊆ s, t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∈ s → x ∈ t) ∧
       ∀ x, IsTop x → x ∈ s → x ∈ t := by
   rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩
   refine' ⟨(t ∪ ({ x | IsBot x } ∪ { x | IsTop x })) ∩ s, _, _, _, _, _⟩
@@ -787,7 +788,7 @@ instance (priority := 100) SecondCountableTopology.to_separableSpace [SecondCoun
 
 /-- A countable open cover induces a second-countable topology if all open covers
 are themselves second countable. -/
-theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι → Set α}
+theorem secondCountableTopology_of_countable_cover {ι} [Countable ι] {U : ι → Set α}
     [∀ i, SecondCountableTopology (U i)] (Uo : ∀ i, IsOpen (U i)) (hc : ⋃ i, U i = univ) :
     SecondCountableTopology α :=
   haveI : IsTopologicalBasis (⋃ i, image ((↑) : U i → α) '' countableBasis (U i)) :=
@@ -802,18 +803,22 @@ theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Se
     (H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i, s i := by
   let B := { b ∈ countableBasis α | ∃ i, b ⊆ s i }
   choose f hf using fun b : B => b.2.2
-  haveI : Encodable B := ((countable_countableBasis α).mono (sep_subset _ _)).toEncodable
+  haveI : Countable B := ((countable_countableBasis α).mono (sep_subset _ _)).to_subtype
   refine' ⟨_, countable_range f, (iUnion₂_subset_iUnion _ _).antisymm (sUnion_subset _)⟩
   rintro _ ⟨i, rfl⟩ x xs
   rcases (isBasis_countableBasis α).exists_subset_of_mem_open xs (H _) with ⟨b, hb, xb, bs⟩
   exact ⟨_, ⟨_, rfl⟩, _, ⟨⟨⟨_, hb, _, bs⟩, rfl⟩, rfl⟩, hf _ xb⟩
 #align topological_space.is_open_Union_countable TopologicalSpace.isOpen_iUnion_countable
 
+theorem isOpen_biUnion_countable [SecondCountableTopology α] {ι : Type*} (I : Set ι) (s : ι → Set α)
+    (H : ∀ i ∈ I, IsOpen (s i)) : ∃ T ⊆ I, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i ∈ I, s i := by
+  simp_rw [← Subtype.exists_set_subtype, biUnion_image]
+  rcases isOpen_iUnion_countable (fun i : I ↦ s i) fun i ↦ H i i.2 with ⟨T, hTc, hU⟩
+  exact ⟨T, hTc.image _, hU.trans <| iUnion_subtype ..⟩
+
 theorem isOpen_sUnion_countable [SecondCountableTopology α] (S : Set (Set α))
-    (H : ∀ s ∈ S, IsOpen s) : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
-  let ⟨T, cT, hT⟩ := isOpen_iUnion_countable (fun s : S => s.1) fun s => H s.1 s.2
-  ⟨Subtype.val '' T, cT.image _, image_subset_iff.2 fun ⟨_, h⟩ _ => h, by
-    rwa [sUnion_image, sUnion_eq_iUnion]⟩
+    (H : ∀ s ∈ S, IsOpen s) : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S := by
+  simpa only [and_left_comm, sUnion_eq_biUnion] using isOpen_biUnion_countable S id H
 #align topological_space.is_open_sUnion_countable TopologicalSpace.isOpen_sUnion_countable
 
 /-- In a topological space with second countable topology, if `f` is a function that sends each

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

@@ -130,7 +130,7 @@ theorem isTopologicalBasis_of_isOpen_of_nhds {s : Set (Set α)} (h_open : ∀ u
     rcases h_nhds a univ trivial isOpen_univ with ⟨u, h₁, h₂, -⟩
     exact ⟨u, h₁, h₂⟩
   · refine' (le_generateFrom h_open).antisymm fun u hu => _
-    refine' (@isOpen_iff_nhds α (generateFrom s) u).mpr fun a ha => _
+    refine (@isOpen_iff_nhds α u (generateFrom s)).mpr fun a ha ↦ ?_
     rcases h_nhds a u ha hu with ⟨v, hvs, hav, hvu⟩
     rw [nhds_generateFrom]
     exact iInf₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)

... where this is easy to do.

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

@@ -52,9 +52,6 @@ More fine grained instances for `FirstCountableTopology`,
 `TopologicalSpace.SeparableSpace`, and more.
 -/
 
-set_option autoImplicit true
-
-
 open Set Filter Function Topology
 
 noncomputable section
@@ -63,7 +60,7 @@ namespace TopologicalSpace
 
 universe u
 
-variable {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} {s : Set α}
+variable {α : Type u} {β : Type*} [t : TopologicalSpace α] {B : Set (Set α)} {s : Set α}
 
 /-- A topological basis is one that satisfies the necessary conditions so that
   it suffices to take unions of the basis sets to get a topology (without taking

See Zulip thread for the discussion.

@@ -117,7 +117,7 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
   · rw [sUnion_image, iUnion₂_eq_univ_iff]
     exact fun x => ⟨∅, ⟨finite_empty, empty_subset _⟩, sInter_empty.substr <| mem_univ x⟩
   · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩
-    exact hft.isOpen_sInter fun s hs ↦ GenerateOpen.basic _ $ htb hs
+    exact hft.isOpen_sInter fun s hs ↦ GenerateOpen.basic _ <| htb hs
   · rw [← sInter_singleton t]
     exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩
 #align topological_space.is_topological_basis_of_subbasis TopologicalSpace.isTopologicalBasis_of_subbasis
@@ -215,7 +215,7 @@ lemma IsTopologicalBasis.subset_of_forall_subset {t : Set α} (hB : IsTopologica
 lemma IsTopologicalBasis.eq_of_forall_subset_iff {t : Set α} (hB : IsTopologicalBasis B)
     (hs : IsOpen s) (ht : IsOpen t) (h : ∀ U ∈ B, U ⊆ s ↔ U ⊆ t) : s = t := by
   rw [hB.open_eq_sUnion' hs, hB.open_eq_sUnion' ht]
-  exact congr_arg _ (Set.ext λ U ↦ and_congr_right $ h _)
+  exact congr_arg _ (Set.ext λ U ↦ and_congr_right <| h _)
 
 /-- A point `a` is in the closure of `s` iff all basis sets containing `a` intersect `s`. -/
 theorem IsTopologicalBasis.mem_closure_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α}

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

Partially addresses issue #9129

@@ -411,7 +411,7 @@ instance [SeparableSpace α] {s : Setoid α} : SeparableSpace (Quotient s) :=
 
 /-- A topological space with discrete topology is separable iff it is countable. -/
 theorem separableSpace_iff_countable [DiscreteTopology α] : SeparableSpace α ↔ Countable α := by
-  simp [SeparableSpace_iff, countable_univ_iff]
+  simp [separableSpace_iff, countable_univ_iff]
 
 /-- In a separable space, a family of nonempty disjoint open sets is countable. -/
 theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type*}

• Redefine Set.image2 to use ∃ a ∈ s, ∃ b ∈ t, f a b = c instead of ∃ a b, a ∈ s ∧ b ∈ t ∧ f a b = c.
• Redefine Set.seq as Set.image2. The new definition is equal to the old one but rw [Set.seq] gives a different result.
• Redefine Filter.map₂ to use ∃ u ∈ f, ∃ v ∈ g, image2 m u v ⊆ s instead of ∃ u v, u ∈ f ∧ v ∈ g ∧ ...
• Update lemmas like Set.mem_image2, Finset.mem_image₂, Set.mem_mul, Finset.mem_div etc

The two reasons to make the change are:

• ∃ a ∈ s, ∃ b ∈ t, _ is a simp-normal form, and
• it looks a bit nicer.

@@ -252,7 +252,7 @@ protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set
     {B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) :
     IsTopologicalBasis (image2 (· ×ˢ ·) B₁ B₂) := by
   refine' isTopologicalBasis_of_isOpen_of_nhds _ _
-  · rintro _ ⟨u₁, u₂, hu₁, hu₂, rfl⟩
+  · rintro _ ⟨u₁, hu₁, u₂, hu₂, rfl⟩
     exact (h₁.isOpen hu₁).prod (h₂.isOpen hu₂)
   · rintro ⟨a, b⟩ u hu uo
     rcases (h₁.nhds_hasBasis.prod_nhds h₂.nhds_hasBasis).mem_iff.1 (IsOpen.mem_nhds uo hu) with

Similarly, upgrade tendsto_nhds_generateFrom, IsTopologicalBasis.continuous, Topology.IsLower.continuous_of_Ici, and Topology.IsUpper.continuous_iff_Iic.

The old lemmas are now deprecated, and the new ones have _iff in their names. Once we remove the old lemmas, we can drop the _iff suffixes.

@@ -290,9 +290,15 @@ theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen
       image_subset_iff.2 hvu⟩
 #align topological_space.is_topological_basis_of_cover TopologicalSpace.isTopologicalBasis_of_cover
 
+protected theorem IsTopologicalBasis.continuous_iff {β : Type*} [TopologicalSpace β]
+    {B : Set (Set β)} (hB : IsTopologicalBasis B) {f : α → β} :
+    Continuous f ↔ ∀ s ∈ B, IsOpen (f ⁻¹' s) := by
+  rw [hB.eq_generateFrom, continuous_generateFrom_iff]
+
+@[deprecated]
 protected theorem IsTopologicalBasis.continuous {β : Type*} [TopologicalSpace β] {B : Set (Set β)}
-    (hB : IsTopologicalBasis B) (f : α → β) (hf : ∀ s ∈ B, IsOpen (f ⁻¹' s)) : Continuous f := by
-  rw [hB.eq_generateFrom]; exact continuous_generateFrom hf
+    (hB : IsTopologicalBasis B) (f : α → β) (hf : ∀ s ∈ B, IsOpen (f ⁻¹' s)) : Continuous f :=
+  hB.continuous_iff.2 hf
 #align topological_space.is_topological_basis.continuous TopologicalSpace.IsTopologicalBasis.continuous
 
 variable (α)

From PFR

@@ -678,6 +678,10 @@ lemma SecondCountableTopology.mk' {b : Set (Set α)} (hc : b.Countable) :
     @SecondCountableTopology α (generateFrom b) :=
   @SecondCountableTopology.mk α (generateFrom b) ⟨b, hc, rfl⟩
 
+instance _root_.Finite.toSecondCountableTopology [Finite α] : SecondCountableTopology α where
+  is_open_generated_countable :=
+    ⟨_, {U | IsOpen U}.to_countable, TopologicalSpace.isTopologicalBasis_opens.eq_generateFrom⟩
+
 variable (α)
 
 theorem exists_countable_basis [SecondCountableTopology α] :

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

@@ -124,7 +124,7 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
 
 /-- If a family of open sets `s` is such that every open neighbourhood contains some
 member of `s`, then `s` is a topological basis. -/
-theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
+theorem isTopologicalBasis_of_isOpen_of_nhds {s : Set (Set α)} (h_open : ∀ u ∈ s, IsOpen u)
     (h_nhds : ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u) :
     IsTopologicalBasis s := by
   refine'
@@ -137,7 +137,7 @@ theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u 
     rcases h_nhds a u ha hu with ⟨v, hvs, hav, hvu⟩
     rw [nhds_generateFrom]
     exact iInf₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
-#align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_open_of_nhds
+#align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
 
 /-- A set `s` is in the neighbourhood of `a` iff there is some basis set `t`, which
 contains `a` and is itself contained in `s`. -/
@@ -245,13 +245,13 @@ theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTop
 #align topological_space.is_topological_basis.exists_nonempty_subset TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset
 
 theorem isTopologicalBasis_opens : IsTopologicalBasis { U : Set α | IsOpen U } :=
-  isTopologicalBasis_of_open_of_nhds (by tauto) (by tauto)
+  isTopologicalBasis_of_isOpen_of_nhds (by tauto) (by tauto)
 #align topological_space.is_topological_basis_opens TopologicalSpace.isTopologicalBasis_opens
 
 protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set (Set α)}
     {B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) :
     IsTopologicalBasis (image2 (· ×ˢ ·) B₁ B₂) := by
-  refine' isTopologicalBasis_of_open_of_nhds _ _
+  refine' isTopologicalBasis_of_isOpen_of_nhds _ _
   · rintro _ ⟨u₁, u₂, hu₁, hu₂, rfl⟩
     exact (h₁.isOpen hu₁).prod (h₂.isOpen hu₂)
   · rintro ⟨a, b⟩ u hu uo
@@ -262,7 +262,7 @@ protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set
 
 protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α → β} {T : Set (Set β)}
     (hf : Inducing f) (h : IsTopologicalBasis T) : IsTopologicalBasis ((preimage f) '' T) := by
-  refine' isTopologicalBasis_of_open_of_nhds _ _
+  refine' isTopologicalBasis_of_isOpen_of_nhds _ _
   · rintro _ ⟨V, hV, rfl⟩
     rw [hf.isOpen_iff]
     refine' ⟨V, h.isOpen hV, rfl⟩
@@ -277,7 +277,7 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
 theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
     (Uc : ⋃ i, U i = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
     IsTopologicalBasis (⋃ i : ι, image ((↑) : U i → α) '' b i) := by
-  refine' isTopologicalBasis_of_open_of_nhds (fun u hu => _) _
+  refine' isTopologicalBasis_of_isOpen_of_nhds (fun u hu => _) _
   · simp only [mem_iUnion, mem_image] at hu
     rcases hu with ⟨i, s, sb, rfl⟩
     exact (Uo i).isOpenMap_subtype_val _ ((hb i).isOpen sb)
@@ -528,7 +528,7 @@ theorem isTopologicalBasis_pi {ι : Type*} {X : ι → Type*} [∀ i, Topologica
     {T : ∀ i, Set (Set (X i))} (cond : ∀ i, IsTopologicalBasis (T i)) :
     IsTopologicalBasis { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
       (∀ i, i ∈ F → U i ∈ T i) ∧ S = (F : Set ι).pi U } := by
-  refine' isTopologicalBasis_of_open_of_nhds _ _
+  refine' isTopologicalBasis_of_isOpen_of_nhds _ _
   · rintro _ ⟨U, F, h1, rfl⟩
     apply isOpen_set_pi F.finite_toSet
     intro i hi
@@ -565,7 +565,7 @@ theorem isTopologicalBasis_iInf {β : Type*} {ι : Type*} {X : ι → Type*}
 
 theorem isTopologicalBasis_singletons (α : Type*) [TopologicalSpace α] [DiscreteTopology α] :
     IsTopologicalBasis { s | ∃ x : α, (s : Set α) = {x} } :=
-  isTopologicalBasis_of_open_of_nhds (fun _ _ => isOpen_discrete _) fun x _ hx _ =>
+  isTopologicalBasis_of_isOpen_of_nhds (fun _ _ => isOpen_discrete _) fun x _ hx _ =>
     ⟨{x}, ⟨x, rfl⟩, mem_singleton x, singleton_subset_iff.2 hx⟩
 #align is_topological_basis_singletons isTopologicalBasis_singletons
 
@@ -841,7 +841,7 @@ topological bases on each of the parts of the space. -/
 theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
     (hs : ∀ i, IsTopologicalBasis (s i)) :
     IsTopologicalBasis (⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' s i) := by
-  apply isTopologicalBasis_of_open_of_nhds
+  apply isTopologicalBasis_of_isOpen_of_nhds
   · intro u hu
     obtain ⟨i, t, ts, rfl⟩ : ∃ (i : ι) (t : Set (E i)), t ∈ s i ∧ Sigma.mk i '' t = u := by
       simpa only [mem_iUnion, mem_image] using hu
@@ -874,7 +874,7 @@ topological bases on each of the two components. -/
 theorem IsTopologicalBasis.sum {s : Set (Set α)} (hs : IsTopologicalBasis s) {t : Set (Set β)}
     (ht : IsTopologicalBasis t) :
     IsTopologicalBasis ((fun u => Sum.inl '' u) '' s ∪ (fun u => Sum.inr '' u) '' t) := by
-  apply isTopologicalBasis_of_open_of_nhds
+  apply isTopologicalBasis_of_isOpen_of_nhds
   · rintro u (⟨w, hw, rfl⟩ | ⟨w, hw, rfl⟩)
     · exact openEmbedding_inl.isOpenMap w (hs.isOpen hw)
     · exact openEmbedding_inr.isOpenMap w (ht.isOpen hw)
@@ -909,7 +909,7 @@ variable {X : Type*} [TopologicalSpace X] {Y : Type*} [TopologicalSpace Y] {π :
 /-- The image of a topological basis under an open quotient map is a topological basis. -/
 theorem IsTopologicalBasis.quotientMap {V : Set (Set X)} (hV : IsTopologicalBasis V)
     (h' : QuotientMap π) (h : IsOpenMap π) : IsTopologicalBasis (Set.image π '' V) := by
-  apply isTopologicalBasis_of_open_of_nhds
+  apply isTopologicalBasis_of_isOpen_of_nhds
   · rintro - ⟨U, U_in_V, rfl⟩
     apply h U (hV.isOpen U_in_V)
   · intro y U y_in_U U_open

All the other properties of topological spaces like T0Space or RegularSpace are in the root namespace. Many files were opening TopologicalSpace just for the sake of shortening TopologicalSpace.SecondCountableTopology...

@@ -26,9 +26,9 @@ conditions are equivalent in this case).
 * `TopologicalSpace.IsTopologicalBasis s`: The topological space `t` has basis `s`.
 * `TopologicalSpace.SeparableSpace α`: The topological space `t` has a countable, dense subset.
 * `TopologicalSpace.IsSeparable s`: The set `s` is contained in the closure of a countable set.
-* `TopologicalSpace.FirstCountableTopology α`: A topology in which `𝓝 x` is countably generated for
+* `FirstCountableTopology α`: A topology in which `𝓝 x` is countably generated for
   every `x`.
-* `TopologicalSpace.SecondCountableTopology α`: A topology which has a topological basis which is
+* `SecondCountableTopology α`: A topology which has a topological basis which is
   countable.
 
 ## Main results
@@ -39,8 +39,8 @@ conditions are equivalent in this case).
   the union of arbitrarily-many open sets is equal to a sub-union of only countably many of these
   sets.
 * `TopologicalSpace.SecondCountableTopology.countable_cover_nhds`: Consider `f : α → Set α` with the
-  property that `f x ∈ 𝓝 x` for all `x`. Then there is some countable set `s` whose image covers the
-  space.
+  property that `f x ∈ 𝓝 x` for all `x`. Then there is some countable set `s` whose image covers
+  the space.
 
 ## Implementation Notes
 For our applications we are interested that there exists a countable basis, but we do not need the
@@ -48,7 +48,7 @@ concrete basis itself. This allows us to declare these type classes as `Prop` to
 
 ### TODO:
 
-More fine grained instances for `TopologicalSpace.FirstCountableTopology`,
+More fine grained instances for `FirstCountableTopology`,
 `TopologicalSpace.SeparableSpace`, and more.
 -/
 
@@ -303,10 +303,10 @@ variable (α)
 `TopologicalSpace.denseRange_denseSeq`.
 
 If `α` is a uniform space with countably generated uniformity filter (e.g., an `EMetricSpace`), then
-this condition is equivalent to `TopologicalSpace.SecondCountableTopology α`. In this case the
+this condition is equivalent to `SecondCountableTopology α`. In this case the
 latter should be used as a typeclass argument in theorems because Lean can automatically deduce
-`TopologicalSpace.SeparableSpace` from `TopologicalSpace.SecondCountableTopology` but it can't
-deduce `TopologicalSpace.SecondCountableTopology` from `TopologicalSpace.SeparableSpace`.
+`TopologicalSpace.SeparableSpace` from `SecondCountableTopology` but it can't
+deduce `SecondCountableTopology` from `TopologicalSpace.SeparableSpace`.
 
 Porting note: TODO: the previous paragraph describes the state of the art in Lean 3. We can have
 instance cycles in Lean 4 but we might want to postpone adding them till after the port. -/
@@ -620,10 +620,10 @@ variable (α : Type u) [t : TopologicalSpace α]
 
 /-- A first-countable space is one in which every point has a
   countable neighborhood basis. -/
-class FirstCountableTopology : Prop where
+class _root_.FirstCountableTopology : Prop where
   /-- The filter `𝓝 a` is countably generated for all points `a`. -/
   nhds_generated_countable : ∀ a : α, (𝓝 a).IsCountablyGenerated
-#align topological_space.first_countable_topology TopologicalSpace.FirstCountableTopology
+#align topological_space.first_countable_topology FirstCountableTopology
 
 attribute [instance] FirstCountableTopology.nhds_generated_countable
 
@@ -662,10 +662,10 @@ instance isCountablyGenerated_nhdsWithin (x : α) [IsCountablyGenerated (𝓝 x)
 variable (α)
 
 /-- A second-countable space is one with a countable basis. -/
-class SecondCountableTopology : Prop where
+class _root_.SecondCountableTopology : Prop where
   /-- There exists a countable set of sets that generates the topology. -/
   is_open_generated_countable : ∃ b : Set (Set α), b.Countable ∧ t = TopologicalSpace.generateFrom b
-#align topological_space.second_countable_topology TopologicalSpace.SecondCountableTopology
+#align topological_space.second_countable_topology SecondCountableTopology
 
 variable {α}
 

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

@@ -255,7 +255,7 @@ protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set
   · rintro _ ⟨u₁, u₂, hu₁, hu₂, rfl⟩
     exact (h₁.isOpen hu₁).prod (h₂.isOpen hu₂)
   · rintro ⟨a, b⟩ u hu uo
-    rcases(h₁.nhds_hasBasis.prod_nhds h₂.nhds_hasBasis).mem_iff.1 (IsOpen.mem_nhds uo hu) with
+    rcases (h₁.nhds_hasBasis.prod_nhds h₂.nhds_hasBasis).mem_iff.1 (IsOpen.mem_nhds uo hu) with
       ⟨⟨s, t⟩, ⟨⟨hs, ha⟩, ht, hb⟩, hu⟩
     exact ⟨s ×ˢ t, mem_image2_of_mem hs ht, ⟨ha, hb⟩, hu⟩
 #align topological_space.is_topological_basis.prod TopologicalSpace.IsTopologicalBasis.prod
@@ -284,7 +284,7 @@ theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen
   · intro a u ha uo
     rcases iUnion_eq_univ_iff.1 Uc a with ⟨i, hi⟩
     lift a to ↥(U i) using hi
-    rcases(hb i).exists_subset_of_mem_open ha (uo.preimage continuous_subtype_val) with
+    rcases (hb i).exists_subset_of_mem_open ha (uo.preimage continuous_subtype_val) with
       ⟨v, hvb, hav, hvu⟩
     exact ⟨(↑) '' v, mem_iUnion.2 ⟨i, mem_image_of_mem _ hvb⟩, mem_image_of_mem _ hav,
       image_subset_iff.2 hvu⟩

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

@@ -209,11 +209,11 @@ theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopological
 #align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
 
 lemma IsTopologicalBasis.subset_of_forall_subset {t : Set α} (hB : IsTopologicalBasis B)
-  (hs : IsOpen s) (h : ∀ U ∈ B, U ⊆ s → U ⊆ t) : s ⊆ t := by
+    (hs : IsOpen s) (h : ∀ U ∈ B, U ⊆ s → U ⊆ t) : s ⊆ t := by
   rw [hB.open_eq_sUnion' hs]; simpa [sUnion_subset_iff]
 
 lemma IsTopologicalBasis.eq_of_forall_subset_iff {t : Set α} (hB : IsTopologicalBasis B)
-  (hs : IsOpen s) (ht : IsOpen t) (h : ∀ U ∈ B, U ⊆ s ↔ U ⊆ t) : s = t := by
+    (hs : IsOpen s) (ht : IsOpen t) (h : ∀ U ∈ B, U ⊆ s ↔ U ⊆ t) : s = t := by
   rw [hB.open_eq_sUnion' hs, hB.open_eq_sUnion' ht]
   exact congr_arg _ (Set.ext λ U ↦ and_congr_right $ h _)
 

@@ -789,7 +789,8 @@ theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι 
 #align topological_space.second_countable_topology_of_countable_cover TopologicalSpace.secondCountableTopology_of_countable_cover
 
 /-- In a second-countable space, an open set, given as a union of open sets,
-is equal to the union of countably many of those sets. -/
+is equal to the union of countably many of those sets.
+In particular, any open covering of `α` has a countable subcover: α is a Lindelöf space. -/
 theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
     (H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i, s i := by
   let B := { b ∈ countableBasis α | ∃ i, b ⊆ s i }

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.

@@ -63,7 +63,7 @@ namespace TopologicalSpace
 
 universe u
 
-variable {α : Type u} [t : TopologicalSpace α]
+variable {α : Type u} [t : TopologicalSpace α] {B : Set (Set α)} {s : Set α}
 
 /-- A topological basis is one that satisfies the necessary conditions so that
   it suffices to take unions of the basis sets to get a topology (without taking
@@ -208,6 +208,15 @@ theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopological
     apply hB.open_eq_sUnion' ou, fun s => And.left s.2⟩
 #align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
 
+lemma IsTopologicalBasis.subset_of_forall_subset {t : Set α} (hB : IsTopologicalBasis B)
+  (hs : IsOpen s) (h : ∀ U ∈ B, U ⊆ s → U ⊆ t) : s ⊆ t := by
+  rw [hB.open_eq_sUnion' hs]; simpa [sUnion_subset_iff]
+
+lemma IsTopologicalBasis.eq_of_forall_subset_iff {t : Set α} (hB : IsTopologicalBasis B)
+  (hs : IsOpen s) (ht : IsOpen t) (h : ∀ U ∈ B, U ⊆ s ↔ U ⊆ t) : s = t := by
+  rw [hB.open_eq_sUnion' hs, hB.open_eq_sUnion' ht]
+  exact congr_arg _ (Set.ext λ U ↦ and_congr_right $ h _)
+
 /-- A point `a` is in the closure of `s` iff all basis sets containing `a` intersect `s`. -/
 theorem IsTopologicalBasis.mem_closure_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α}
     {a : α} : a ∈ closure s ↔ ∀ o ∈ b, a ∈ o → (o ∩ s).Nonempty :=

@@ -425,7 +425,7 @@ theorem _root_.Set.PairwiseDisjoint.countable_of_nonempty_interior [SeparableSpa
 set `c`. Beware that this definition does not require that `c` is contained in `s` (to express the
 latter, use `TopologicalSpace.SeparableSpace s` or
 `TopologicalSpace.IsSeparable (univ : Set s))`. In metric spaces, the two definitions are
-equivalent, see `TopologicalSpace.IsSeparable.SeparableSpace`. -/
+equivalent, see `TopologicalSpace.IsSeparable.separableSpace`. -/
 def IsSeparable (s : Set α) :=
   ∃ c : Set α, c.Countable ∧ s ⊆ closure c
 #align topological_space.is_separable TopologicalSpace.IsSeparable
@@ -457,6 +457,25 @@ theorem isSeparable_iUnion {ι : Type*} [Countable ι] {s : ι → Set α}
   exact (h'c i).trans (closure_mono (subset_iUnion _ i))
 #align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnion
 
+lemma isSeparable_pi {ι : Type*} [Fintype ι] {α : ∀ (_ : ι), Type*} {s : ∀ i, Set (α i)}
+    [∀ i, TopologicalSpace (α i)] (h : ∀ i, IsSeparable (s i)) :
+    IsSeparable {f : ∀ i, α i | ∀ i, f i ∈ s i} := by
+  choose c c_count hc using h
+  refine ⟨{f | ∀ i, f i ∈ c i}, countable_pi c_count, ?_⟩
+  simp_rw [← mem_univ_pi]
+  dsimp
+  rw [closure_pi_set]
+  exact Set.pi_mono (fun i _ ↦ hc i)
+
+lemma IsSeparable.prod {β : Type*} [TopologicalSpace β]
+    {s : Set α} {t : Set β} (hs : IsSeparable s) (ht : IsSeparable t) :
+    IsSeparable (s ×ˢ t) := by
+  rcases hs with ⟨cs, cs_count, hcs⟩
+  rcases ht with ⟨ct, ct_count, hct⟩
+  refine ⟨cs ×ˢ ct, cs_count.prod ct_count, ?_⟩
+  rw [closure_prod_eq]
+  exact Set.prod_mono hcs hct
+
 theorem _root_.Set.Countable.isSeparable {s : Set α} (hs : s.Countable) : IsSeparable s :=
   ⟨s, hs, subset_closure⟩
 #align set.countable.is_separable Set.Countable.isSeparable

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.

@@ -117,8 +117,7 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
   · rw [sUnion_image, iUnion₂_eq_univ_iff]
     exact fun x => ⟨∅, ⟨finite_empty, empty_subset _⟩, sInter_empty.substr <| mem_univ x⟩
   · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩
-    apply isOpen_sInter
-    exacts [hft, fun s hs => GenerateOpen.basic _ <| htb hs]
+    exact hft.isOpen_sInter fun s hs ↦ GenerateOpen.basic _ $ htb hs
   · rw [← sInter_singleton t]
     exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩
 #align topological_space.is_topological_basis_of_subbasis TopologicalSpace.isTopologicalBasis_of_subbasis

@@ -350,7 +350,7 @@ theorem SeparableSpace.of_denseRange {ι : Sort _} [Countable ι] (u : ι → α
   ⟨⟨range u, countable_range u, hu⟩⟩
 #align topological_space.separable_space_of_dense_range TopologicalSpace.SeparableSpace.of_denseRange
 
-alias SeparableSpace.of_denseRange ← _root_.DenseRange.separableSpace'
+alias _root_.DenseRange.separableSpace' := SeparableSpace.of_denseRange
 
 /-- If `α` is a separable space and `f : α → β` is a continuous map with dense range, then `β` is
 a separable space as well. E.g., the completion of a separable uniform space is separable. -/

@@ -230,11 +230,10 @@ theorem IsTopologicalBasis.isOpenMap_iff {β} [TopologicalSpace β] {B : Set (Se
 #align topological_space.is_topological_basis.is_open_map_iff TopologicalSpace.IsTopologicalBasis.isOpenMap_iff
 
 theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTopologicalBasis B)
-    {u : Set α} (hu : u.Nonempty) (ou : IsOpen u) : ∃ v ∈ B, Set.Nonempty v ∧ v ⊆ u := by
-  cases' hu with x hx
-  rw [hb.open_eq_sUnion' ou, mem_sUnion] at hx
-  rcases hx with ⟨v, hv, hxv⟩
-  exact ⟨v, hv.1, ⟨x, hxv⟩, hv.2⟩
+    {u : Set α} (hu : u.Nonempty) (ou : IsOpen u) : ∃ v ∈ B, Set.Nonempty v ∧ v ⊆ u :=
+  let ⟨x, hx⟩ := hu
+  let ⟨v, vB, xv, vu⟩ := hb.exists_subset_of_mem_open hx ou
+  ⟨v, vB, ⟨x, xv⟩, vu⟩
 #align topological_space.is_topological_basis.exists_nonempty_subset TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset
 
 theorem isTopologicalBasis_opens : IsTopologicalBasis { U : Set α | IsOpen U } :=

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.

@@ -52,6 +52,8 @@ More fine grained instances for `TopologicalSpace.FirstCountableTopology`,
 `TopologicalSpace.SeparableSpace`, and more.
 -/
 
+set_option autoImplicit true
+
 
 open Set Filter Function Topology
 

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

This has nice performance benefits.

@@ -281,7 +281,7 @@ theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen
       image_subset_iff.2 hvu⟩
 #align topological_space.is_topological_basis_of_cover TopologicalSpace.isTopologicalBasis_of_cover
 
-protected theorem IsTopologicalBasis.continuous {β : Type _} [TopologicalSpace β] {B : Set (Set β)}
+protected theorem IsTopologicalBasis.continuous {β : Type*} [TopologicalSpace β] {B : Set (Set β)}
     (hB : IsTopologicalBasis B) (f : α → β) (hf : ∀ s ∈ B, IsOpen (f ⁻¹' s)) : Continuous f := by
   rw [hB.eq_generateFrom]; exact continuous_generateFrom hf
 #align topological_space.is_topological_basis.continuous TopologicalSpace.IsTopologicalBasis.continuous
@@ -370,7 +370,7 @@ instance [TopologicalSpace β] [SeparableSpace α] [SeparableSpace β] : Separab
   exact ⟨⟨s ×ˢ t, hsc.prod htc, hsd.prod htd⟩⟩
 
 /-- The product of a countable family of separable spaces is a separable space. -/
-instance {ι : Type _} {X : ι → Type _} [∀ i, TopologicalSpace (X i)] [∀ i, SeparableSpace (X i)]
+instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, SeparableSpace (X i)]
     [Countable ι] : SeparableSpace (∀ i, X i) := by
   choose t htc htd using (exists_countable_dense <| X ·)
   haveI := fun i ↦ (htc i).to_subtype
@@ -399,7 +399,7 @@ theorem separableSpace_iff_countable [DiscreteTopology α] : SeparableSpace α 
   simp [SeparableSpace_iff, countable_univ_iff]
 
 /-- In a separable space, a family of nonempty disjoint open sets is countable. -/
-theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type _}
+theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type*}
     {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s) (ha : ∀ i ∈ a, IsOpen (s i))
     (h'a : ∀ i ∈ a, (s i).Nonempty) : a.Countable := by
   rcases exists_countable_dense α with ⟨u, ⟨u_encodable⟩, u_dense⟩
@@ -415,7 +415,7 @@ theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι
 #align set.pairwise_disjoint.countable_of_is_open Set.PairwiseDisjoint.countable_of_isOpen
 
 /-- In a separable space, a family of disjoint sets with nonempty interiors is countable. -/
-theorem _root_.Set.PairwiseDisjoint.countable_of_nonempty_interior [SeparableSpace α] {ι : Type _}
+theorem _root_.Set.PairwiseDisjoint.countable_of_nonempty_interior [SeparableSpace α] {ι : Type*}
     {s : ι → Set α} {a : Set ι} (h : a.PairwiseDisjoint s)
     (ha : ∀ i ∈ a, (interior (s i)).Nonempty) : a.Countable :=
   (h.mono fun _ => interior_subset).countable_of_isOpen (fun _ _ => isOpen_interior) ha
@@ -450,7 +450,7 @@ theorem IsSeparable.closure {s : Set α} (hs : IsSeparable s) : IsSeparable (clo
   exact ⟨c, c_count, by simpa using closure_mono hs⟩
 #align topological_space.is_separable.closure TopologicalSpace.IsSeparable.closure
 
-theorem isSeparable_iUnion {ι : Type _} [Countable ι] {s : ι → Set α}
+theorem isSeparable_iUnion {ι : Type*} [Countable ι] {s : ι → Set α}
     (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) := by
   choose c hc h'c using hs
   refine' ⟨⋃ i, c i, countable_iUnion hc, iUnion_subset_iff.2 fun i => _⟩
@@ -478,7 +478,7 @@ theorem isSeparable_of_separableSpace [h : SeparableSpace α] (s : Set α) : IsS
   IsSeparable.mono (isSeparable_univ_iff.2 h) (subset_univ _)
 #align topological_space.is_separable_of_separable_space TopologicalSpace.isSeparable_of_separableSpace
 
-theorem IsSeparable.image {β : Type _} [TopologicalSpace β] {s : Set α} (hs : IsSeparable s)
+theorem IsSeparable.image {β : Type*} [TopologicalSpace β] {s : Set α} (hs : IsSeparable s)
     {f : α → β} (hf : Continuous f) : IsSeparable (f '' s) := by
   rcases hs with ⟨c, c_count, hc⟩
   refine' ⟨f '' c, c_count.image _, _⟩
@@ -496,7 +496,7 @@ end TopologicalSpace
 
 open TopologicalSpace
 
-theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, TopologicalSpace (X i)]
+theorem isTopologicalBasis_pi {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
     {T : ∀ i, Set (Set (X i))} (cond : ∀ i, IsTopologicalBasis (T i)) :
     IsTopologicalBasis { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
       (∀ i, i ∈ F → U i ∈ T i) ∧ S = (F : Set ι).pi U } := by
@@ -515,7 +515,7 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
     exact ⟨_, ⟨V, I, fun i _ => hVT i, rfl⟩, fun i _ => haV i, (pi_mono fun i _ => hVt i).trans htU⟩
 #align is_topological_basis_pi isTopologicalBasis_pi
 
-theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
+theorem isTopologicalBasis_iInf {β : Type*} {ι : Type*} {X : ι → Type*}
     [t : ∀ i, TopologicalSpace (X i)] {T : ∀ i, Set (Set (X i))}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
     @IsTopologicalBasis β (⨅ i, induced (f i) (t i))
@@ -535,7 +535,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     rfl
 #align is_topological_basis_infi isTopologicalBasis_iInf
 
-theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [DiscreteTopology α] :
+theorem isTopologicalBasis_singletons (α : Type*) [TopologicalSpace α] [DiscreteTopology α] :
     IsTopologicalBasis { s | ∃ x : α, (s : Set α) = {x} } :=
   isTopologicalBasis_of_open_of_nhds (fun _ _ => isOpen_discrete _) fun x _ hx _ =>
     ⟨{x}, ⟨x, rfl⟩, mem_singleton x, singleton_subset_iff.2 hx⟩
@@ -544,7 +544,7 @@ theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [Discr
 -- Porting note: moved `DenseRange.separableSpace` up
 
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
-theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}
+theorem Dense.exists_countable_dense_subset {α : Type*} [TopologicalSpace α] {s : Set α}
     [SeparableSpace s] (hs : Dense s) : ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t :=
   let ⟨t, htc, htd⟩ := exists_countable_dense s
   ⟨(↑) '' t, image_subset_iff.2 fun _ _ => mem_preimage.2 <| Subtype.coe_prop _, htc.image (↑),
@@ -557,7 +557,7 @@ separable space (e.g., if `α` has a second countable topology), then there exis
 dense subset `t ⊆ s` such that `t` contains bottom/top element of `α` when they exist and belong
 to `s`. For a dense subset containing neither bot nor top elements, see
 `Dense.exists_countable_dense_subset_no_bot_top`. -/
-theorem Dense.exists_countable_dense_subset_bot_top {α : Type _} [TopologicalSpace α]
+theorem Dense.exists_countable_dense_subset_bot_top {α : Type*} [TopologicalSpace α]
     [PartialOrder α] {s : Set α} [SeparableSpace s] (hs : Dense s) :
     ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∈ s → x ∈ t) ∧
       ∀ x, IsTop x → x ∈ s → x ∈ t := by
@@ -569,7 +569,7 @@ theorem Dense.exists_countable_dense_subset_bot_top {α : Type _} [TopologicalSp
     fun x hx hxs => ⟨Or.inr <| Or.inr hx, hxs⟩]
 #align dense.exists_countable_dense_subset_bot_top Dense.exists_countable_dense_subset_bot_top
 
-instance separableSpace_univ {α : Type _} [TopologicalSpace α] [SeparableSpace α] :
+instance separableSpace_univ {α : Type*} [TopologicalSpace α] [SeparableSpace α] :
     SeparableSpace (univ : Set α) :=
   (Equiv.Set.univ α).symm.surjective.denseRange.separableSpace (continuous_id.subtype_mk _)
 #align separable_space_univ separableSpace_univ
@@ -578,7 +578,7 @@ instance separableSpace_univ {α : Type _} [TopologicalSpace α] [SeparableSpace
 dense set `s : Set α` that contains those of both bottom and top elements of `α` that actually
 exist. For a dense set containing neither bot nor top elements, see
 `exists_countable_dense_no_bot_top`. -/
-theorem exists_countable_dense_bot_top (α : Type _) [TopologicalSpace α] [SeparableSpace α]
+theorem exists_countable_dense_bot_top (α : Type*) [TopologicalSpace α] [SeparableSpace α]
     [PartialOrder α] :
     ∃ s : Set α, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∈ s) ∧ ∀ x, IsTop x → x ∈ s := by
   simpa using dense_univ.exists_countable_dense_subset_bot_top
@@ -620,7 +620,7 @@ instance {β} [TopologicalSpace β] [FirstCountableTopology α] [FirstCountableT
 
 section Pi
 
-instance {ι : Type _} {π : ι → Type _} [Countable ι] [∀ i, TopologicalSpace (π i)]
+instance {ι : Type*} {π : ι → Type*} [Countable ι] [∀ i, TopologicalSpace (π i)]
     [∀ i, FirstCountableTopology (π i)] : FirstCountableTopology (∀ i, π i) :=
   ⟨fun f => by rw [nhds_pi]; infer_instance⟩
 
@@ -732,12 +732,12 @@ lemma secondCountableTopology_iInf {ι} [Countable ι] {t : ι → TopologicalSp
     countable_iUnion fun i => @countable_countableBasis _ (t i) (ht i)
 
 -- TODO: more fine grained instances for first_countable_topology, separable_space, t2_space, ...
-instance {β : Type _} [TopologicalSpace β] [SecondCountableTopology α] [SecondCountableTopology β] :
+instance {β : Type*} [TopologicalSpace β] [SecondCountableTopology α] [SecondCountableTopology β] :
     SecondCountableTopology (α × β) :=
   ((isBasis_countableBasis α).prod (isBasis_countableBasis β)).secondCountableTopology <|
     (countable_countableBasis α).image2 (countable_countableBasis β) _
 
-instance {ι : Type _} {π : ι → Type _} [Countable ι] [∀ a, TopologicalSpace (π a)]
+instance {ι : Type*} {π : ι → Type*} [Countable ι] [∀ a, TopologicalSpace (π a)]
     [∀ a, SecondCountableTopology (π a)] : SecondCountableTopology (∀ a, π a) :=
   secondCountableTopology_iInf fun _ => secondCountableTopology_induced _ _ _
 
@@ -805,7 +805,7 @@ theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set
 
 section Sigma
 
-variable {ι : Type _} {E : ι → Type _} [∀ i, TopologicalSpace (E i)]
+variable {ι : Type*} {E : ι → Type*} [∀ i, TopologicalSpace (E i)]
 
 /-- In a disjoint union space `Σ i, E i`, one can form a topological basis by taking the union of
 topological bases on each of the parts of the space. -/
@@ -838,7 +838,7 @@ end Sigma
 
 section Sum
 
-variable {β : Type _} [TopologicalSpace α] [TopologicalSpace β]
+variable {β : Type*} [TopologicalSpace α] [TopologicalSpace β]
 
 /-- In a sum space `α ⊕ β`, one can form a topological basis by taking the union of
 topological bases on each of the two components. -/
@@ -875,7 +875,7 @@ end Sum
 
 section Quotient
 
-variable {X : Type _} [TopologicalSpace X] {Y : Type _} [TopologicalSpace Y] {π : X → Y}
+variable {X : Type*} [TopologicalSpace X] {Y : Type*} [TopologicalSpace Y] {π : X → Y}
 
 /-- The image of a topological basis under an open quotient map is a topological basis. -/
 theorem IsTopologicalBasis.quotientMap {V : Set (Set X)} (hV : IsTopologicalBasis V)
@@ -923,7 +923,7 @@ end TopologicalSpace
 
 open TopologicalSpace
 
-variable {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
+variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
 
 protected theorem Inducing.secondCountableTopology [SecondCountableTopology β] (hf : Inducing f) :
     SecondCountableTopology α := by

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
-
-! This file was ported from Lean 3 source module topology.bases
-! leanprover-community/mathlib commit bcfa726826abd57587355b4b5b7e78ad6527b7e4
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Constructions
 import Mathlib.Topology.ContinuousOn
 
+#align_import topology.bases from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
+
 /-!
 # Bases of topologies. Countability axioms.
 

@@ -304,7 +304,7 @@ deduce `TopologicalSpace.SecondCountableTopology` from `TopologicalSpace.Separab
 
 Porting note: TODO: the previous paragraph describes the state of the art in Lean 3. We can have
 instance cycles in Lean 4 but we might want to postpone adding them till after the port. -/
-class SeparableSpace : Prop where
+@[mk_iff] class SeparableSpace : Prop where
   /-- There exists a countable dense set. -/
   exists_countable_dense : ∃ s : Set α, s.Countable ∧ Dense s
 #align topological_space.separable_space TopologicalSpace.SeparableSpace
@@ -341,14 +341,65 @@ theorem denseRange_denseSeq [SeparableSpace α] [Nonempty α] : DenseRange (dens
 
 variable {α}
 
-instance (priority := 100) Countable.to_separableSpace [Countable α] : SeparableSpace α
-    where exists_countable_dense := ⟨Set.univ, Set.countable_univ, dense_univ⟩
+instance (priority := 100) Countable.to_separableSpace [Countable α] : SeparableSpace α where
+  exists_countable_dense := ⟨Set.univ, Set.countable_univ, dense_univ⟩
 #align topological_space.countable.to_separable_space TopologicalSpace.Countable.to_separableSpace
 
-theorem separableSpace_of_denseRange {ι : Type _} [Countable ι] (u : ι → α) (hu : DenseRange u) :
+/-- If `f` has a dense range and its domain is countable, then its codomain is a separable space.
+See also `DenseRange.separableSpace`. -/
+theorem SeparableSpace.of_denseRange {ι : Sort _} [Countable ι] (u : ι → α) (hu : DenseRange u) :
     SeparableSpace α :=
   ⟨⟨range u, countable_range u, hu⟩⟩
-#align topological_space.separable_space_of_dense_range TopologicalSpace.separableSpace_of_denseRange
+#align topological_space.separable_space_of_dense_range TopologicalSpace.SeparableSpace.of_denseRange
+
+alias SeparableSpace.of_denseRange ← _root_.DenseRange.separableSpace'
+
+/-- If `α` is a separable space and `f : α → β` is a continuous map with dense range, then `β` is
+a separable space as well. E.g., the completion of a separable uniform space is separable. -/
+protected theorem _root_.DenseRange.separableSpace [SeparableSpace α] [TopologicalSpace β]
+    {f : α → β} (h : DenseRange f) (h' : Continuous f) : SeparableSpace β :=
+  let ⟨s, s_cnt, s_dense⟩ := exists_countable_dense α
+  ⟨⟨f '' s, Countable.image s_cnt f, h.dense_image h' s_dense⟩⟩
+#align dense_range.separable_space DenseRange.separableSpace
+
+theorem _root_.QuotientMap.separableSpace [SeparableSpace α] [TopologicalSpace β] {f : α → β}
+    (hf : QuotientMap f) : SeparableSpace β :=
+  hf.surjective.denseRange.separableSpace hf.continuous
+
+/-- The product of two separable spaces is a separable space. -/
+instance [TopologicalSpace β] [SeparableSpace α] [SeparableSpace β] : SeparableSpace (α × β) := by
+  rcases exists_countable_dense α with ⟨s, hsc, hsd⟩
+  rcases exists_countable_dense β with ⟨t, htc, htd⟩
+  exact ⟨⟨s ×ˢ t, hsc.prod htc, hsd.prod htd⟩⟩
+
+/-- The product of a countable family of separable spaces is a separable space. -/
+instance {ι : Type _} {X : ι → Type _} [∀ i, TopologicalSpace (X i)] [∀ i, SeparableSpace (X i)]
+    [Countable ι] : SeparableSpace (∀ i, X i) := by
+  choose t htc htd using (exists_countable_dense <| X ·)
+  haveI := fun i ↦ (htc i).to_subtype
+  nontriviality ∀ i, X i; inhabit ∀ i, X i
+  classical
+    set f : (Σ I : Finset ι, ∀ i : I, t i) → ∀ i, X i := fun ⟨I, g⟩ i ↦
+      if hi : i ∈ I then g ⟨i, hi⟩ else (default : ∀ i, X i) i
+    refine ⟨⟨range f, countable_range f, dense_iff_inter_open.2 fun U hU ⟨g, hg⟩ ↦ ?_⟩⟩
+    rcases isOpen_pi_iff.1 hU g hg with ⟨I, u, huo, huU⟩
+    have : ∀ i : I, ∃ y ∈ t i, y ∈ u i := fun i ↦
+      (htd i).exists_mem_open (huo i i.2).1 ⟨_, (huo i i.2).2⟩
+    choose y hyt hyu using this
+    lift y to ∀ i : I, t i using hyt
+    refine ⟨f ⟨I, y⟩, huU fun i (hi : i ∈ I) ↦ ?_, mem_range_self _⟩
+    simp only [dif_pos hi]
+    exact hyu _
+
+instance [SeparableSpace α] {r : α → α → Prop} : SeparableSpace (Quot r) :=
+  quotientMap_quot_mk.separableSpace
+
+instance [SeparableSpace α] {s : Setoid α} : SeparableSpace (Quotient s) :=
+  quotientMap_quot_mk.separableSpace
+
+/-- A topological space with discrete topology is separable iff it is countable. -/
+theorem separableSpace_iff_countable [DiscreteTopology α] : SeparableSpace α ↔ Countable α := by
+  simp [SeparableSpace_iff, countable_univ_iff]
 
 /-- In a separable space, a family of nonempty disjoint open sets is countable. -/
 theorem _root_.Set.PairwiseDisjoint.countable_of_isOpen [SeparableSpace α] {ι : Type _}
@@ -493,13 +544,7 @@ theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [Discr
     ⟨{x}, ⟨x, rfl⟩, mem_singleton x, singleton_subset_iff.2 hx⟩
 #align is_topological_basis_singletons isTopologicalBasis_singletons
 
-/-- If `α` is a separable space and `f : α → β` is a continuous map with dense range, then `β` is
-a separable space as well. E.g., the completion of a separable uniform space is separable. -/
-protected theorem DenseRange.separableSpace {α β : Type _} [TopologicalSpace α] [SeparableSpace α]
-    [TopologicalSpace β] {f : α → β} (h : DenseRange f) (h' : Continuous f) : SeparableSpace β :=
-  let ⟨s, s_cnt, s_dense⟩ := exists_countable_dense α
-  ⟨⟨f '' s, Countable.image s_cnt f, h.dense_image h' s_dense⟩⟩
-#align dense_range.separable_space DenseRange.separableSpace
+-- Porting note: moved `DenseRange.separableSpace` up
 
 -- porting note: use `∃ t, t ⊆ s ∧ _` instead of `∃ t (_ : t ⊆ s), _`
 theorem Dense.exists_countable_dense_subset {α : Type _} [TopologicalSpace α] {s : Set α}

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

@@ -162,7 +162,7 @@ theorem IsTopologicalBasis.isOpen_iff {s : Set α} {b : Set (Set α)} (hb : IsTo
 
 theorem IsTopologicalBasis.nhds_hasBasis {b : Set (Set α)} (hb : IsTopologicalBasis b) {a : α} :
     (𝓝 a).HasBasis (fun t : Set α => t ∈ b ∧ a ∈ t) fun t => t :=
-  ⟨fun s => hb.mem_nhds_iff.trans <| by simp only [exists_prop, and_assoc]⟩
+  ⟨fun s => hb.mem_nhds_iff.trans <| by simp only [and_assoc]⟩
 #align topological_space.is_topological_basis.nhds_has_basis TopologicalSpace.IsTopologicalBasis.nhds_hasBasis
 
 protected theorem IsTopologicalBasis.isOpen {s : Set α} {b : Set (Set α)}

@@ -269,7 +269,7 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
 #align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducing
 
 theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
-    (Uc : (⋃ i, U i) = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
+    (Uc : ⋃ i, U i = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
     IsTopologicalBasis (⋃ i : ι, image ((↑) : U i → α) '' b i) := by
   refine' isTopologicalBasis_of_open_of_nhds (fun u hu => _) _
   · simp only [mem_iUnion, mem_image] at hu
@@ -711,7 +711,7 @@ instance (priority := 100) SecondCountableTopology.to_separableSpace [SecondCoun
 /-- A countable open cover induces a second-countable topology if all open covers
 are themselves second countable. -/
 theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι → Set α}
-    [∀ i, SecondCountableTopology (U i)] (Uo : ∀ i, IsOpen (U i)) (hc : (⋃ i, U i) = univ) :
+    [∀ i, SecondCountableTopology (U i)] (Uo : ∀ i, IsOpen (U i)) (hc : ⋃ i, U i = univ) :
     SecondCountableTopology α :=
   haveI : IsTopologicalBasis (⋃ i, image ((↑) : U i → α) '' countableBasis (U i)) :=
     isTopologicalBasis_of_cover Uo hc fun i => isBasis_countableBasis (U i)
@@ -721,7 +721,7 @@ theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι 
 /-- In a second-countable space, an open set, given as a union of open sets,
 is equal to the union of countably many of those sets. -/
 theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
-    (H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ (⋃ i ∈ T, s i) = ⋃ i, s i := by
+    (H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i, s i := by
   let B := { b ∈ countableBasis α | ∃ i, b ⊆ s i }
   choose f hf using fun b : B => b.2.2
   haveI : Encodable B := ((countable_countableBasis α).mono (sep_subset _ _)).toEncodable
@@ -742,10 +742,10 @@ theorem isOpen_sUnion_countable [SecondCountableTopology α] (S : Set (Set α))
 point `x` to a neighborhood of `x`, then for some countable set `s`, the neighborhoods `f x`,
 `x ∈ s`, cover the whole space. -/
 theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (hf : ∀ x, f x ∈ 𝓝 x) :
-    ∃ s : Set α, s.Countable ∧ (⋃ x ∈ s, f x) = univ := by
+    ∃ s : Set α, s.Countable ∧ ⋃ x ∈ s, f x = univ := by
   rcases isOpen_iUnion_countable (fun x => interior (f x)) fun x => isOpen_interior with
     ⟨s, hsc, hsU⟩
-  suffices : (⋃ x ∈ s, interior (f x)) = univ
+  suffices : ⋃ x ∈ s, interior (f x) = univ
   exact ⟨s, hsc, flip eq_univ_of_subset this <| iUnion₂_mono fun _ _ => interior_subset⟩
   simp only [hsU, eq_univ_iff_forall, mem_iUnion]
   exact fun x => ⟨x, mem_interior_iff_mem_nhds.2 (hf x)⟩

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -757,7 +757,7 @@ theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set
   have : ∀ x : s, (↑) ⁻¹' f x ∈ 𝓝 x := fun x => preimage_coe_mem_nhds_subtype.2 (hf x x.2)
   rcases countable_cover_nhds this with ⟨t, htc, htU⟩
   refine' ⟨(↑) '' t, Subtype.coe_image_subset _ _, htc.image _, fun x hx => _⟩
-  simp only [biUnion_image, eq_univ_iff_forall, ← preimage_iUnion, mem_preimage] at htU⊢
+  simp only [biUnion_image, eq_univ_iff_forall, ← preimage_iUnion, mem_preimage] at htU ⊢
   exact htU ⟨x, hx⟩
 #align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithin
 

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

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

@@ -521,7 +521,7 @@ theorem Dense.exists_countable_dense_subset_bot_top {α : Type _} [TopologicalSp
       ∀ x, IsTop x → x ∈ s → x ∈ t := by
   rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩
   refine' ⟨(t ∪ ({ x | IsBot x } ∪ { x | IsTop x })) ∩ s, _, _, _, _, _⟩
-  exacts[inter_subset_right _ _,
+  exacts [inter_subset_right _ _,
     (htc.union ((countable_isBot α).union (countable_isTop α))).mono (inter_subset_left _ _),
     htd.mono (subset_inter (subset_union_left _ _) hts), fun x hx hxs => ⟨Or.inr <| Or.inl hx, hxs⟩,
     fun x hx hxs => ⟨Or.inr <| Or.inr hx, hxs⟩]

@@ -472,7 +472,7 @@ theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
     @IsTopologicalBasis β (⨅ i, induced (f i) (t i))
       { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
-        (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (_hi : i ∈ F), f i ⁻¹' U i } := by
+        (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (_ : i ∈ F), f i ⁻¹' U i } := by
   letI := ⨅ i, induced (f i) (t i)
   convert (isTopologicalBasis_pi cond).inducing (inducing_iInf_to_pi f)
   ext V

Currently, the following notations are changed from · ×ˢ · because Lean 4 can't deal with ambiguous notations. | Definition | Notation | | :

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -242,10 +242,9 @@ theorem isTopologicalBasis_opens : IsTopologicalBasis { U : Set α | IsOpen U }
   isTopologicalBasis_of_open_of_nhds (by tauto) (by tauto)
 #align topological_space.is_topological_basis_opens TopologicalSpace.isTopologicalBasis_opens
 
--- porting note: Lean 4 can't handle ambigous notation here
 protected theorem IsTopologicalBasis.prod {β} [TopologicalSpace β] {B₁ : Set (Set α)}
     {B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) :
-    IsTopologicalBasis (image2 Set.prod B₁ B₂) := by
+    IsTopologicalBasis (image2 (· ×ˢ ·) B₁ B₂) := by
   refine' isTopologicalBasis_of_open_of_nhds _ _
   · rintro _ ⟨u₁, u₂, hu₁, hu₂, rfl⟩
     exact (h₁.isOpen hu₁).prod (h₂.isOpen hu₂)

As discussed on Zulip

Renames

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

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -38,7 +38,7 @@ conditions are equivalent in this case).
 
 * `TopologicalSpace.FirstCountableTopology.tendsto_subseq`: In a first-countable space,
   cluster points are limits of subsequences.
-* `TopologicalSpace.SecondCountableTopology.isOpen_unionᵢ_countable`: In a second-countable space,
+* `TopologicalSpace.SecondCountableTopology.isOpen_iUnion_countable`: In a second-countable space,
   the union of arbitrarily-many open sets is equal to a sub-union of only countably many of these
   sets.
 * `TopologicalSpace.SecondCountableTopology.countable_cover_nhds`: Consider `f : α → Set α` with the
@@ -73,14 +73,14 @@ structure IsTopologicalBasis (s : Set (Set α)) : Prop where
   /-- For every point `x`, the set of `t ∈ s` such that `x ∈ t` is directed downwards.  -/
   exists_subset_inter : ∀ t₁ ∈ s, ∀ t₂ ∈ s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
   /-- The sets from `s` cover the whole space. -/
-  unionₛ_eq : ⋃₀ s = univ
+  sUnion_eq : ⋃₀ s = univ
   /-- The topology is generated by sets from `s`. -/
   eq_generateFrom : t = generateFrom s
 #align topological_space.is_topological_basis TopologicalSpace.IsTopologicalBasis
 
 theorem IsTopologicalBasis.insert_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
     IsTopologicalBasis (insert ∅ s) := by
-  refine' ⟨_, by rw [unionₛ_insert, empty_union, h.unionₛ_eq], _⟩
+  refine' ⟨_, by rw [sUnion_insert, empty_union, h.sUnion_eq], _⟩
   · rintro t₁ (rfl | h₁) t₂ (rfl | h₂) x ⟨hx₁, hx₂⟩
     · cases hx₁
     · cases hx₁
@@ -96,7 +96,7 @@ theorem IsTopologicalBasis.insert_empty {s : Set (Set α)} (h : IsTopologicalBas
 
 theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
     IsTopologicalBasis (s \ {∅}) := by
-  refine' ⟨_, by rw [unionₛ_diff_singleton_empty, h.unionₛ_eq], _⟩
+  refine' ⟨_, by rw [sUnion_diff_singleton_empty, h.sUnion_eq], _⟩
   · rintro t₁ ⟨h₁, -⟩ t₂ ⟨h₂, -⟩ x hx
     obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x hx
     exact ⟨t₃, ⟨h₃, Nonempty.ne_empty ⟨x, hs.1⟩⟩, hs⟩
@@ -114,13 +114,13 @@ theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom
   subst t; letI := generateFrom s
   refine' ⟨_, _, le_antisymm (le_generateFrom _) <| generateFrom_anti fun t ht => _⟩
   · rintro _ ⟨t₁, ⟨hft₁, ht₁b⟩, rfl⟩ _ ⟨t₂, ⟨hft₂, ht₂b⟩, rfl⟩ x h
-    exact ⟨_, ⟨_, ⟨hft₁.union hft₂, union_subset ht₁b ht₂b⟩, interₛ_union t₁ t₂⟩, h, Subset.rfl⟩
-  · rw [unionₛ_image, unionᵢ₂_eq_univ_iff]
-    exact fun x => ⟨∅, ⟨finite_empty, empty_subset _⟩, interₛ_empty.substr <| mem_univ x⟩
+    exact ⟨_, ⟨_, ⟨hft₁.union hft₂, union_subset ht₁b ht₂b⟩, sInter_union t₁ t₂⟩, h, Subset.rfl⟩
+  · rw [sUnion_image, iUnion₂_eq_univ_iff]
+    exact fun x => ⟨∅, ⟨finite_empty, empty_subset _⟩, sInter_empty.substr <| mem_univ x⟩
   · rintro _ ⟨t, ⟨hft, htb⟩, rfl⟩
-    apply isOpen_interₛ
+    apply isOpen_sInter
     exacts [hft, fun s hs => GenerateOpen.basic _ <| htb hs]
-  · rw [← interₛ_singleton t]
+  · rw [← sInter_singleton t]
     exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩
 #align topological_space.is_topological_basis_of_subbasis TopologicalSpace.isTopologicalBasis_of_subbasis
 
@@ -131,14 +131,14 @@ theorem isTopologicalBasis_of_open_of_nhds {s : Set (Set α)} (h_open : ∀ u 
     IsTopologicalBasis s := by
   refine'
     ⟨fun t₁ ht₁ t₂ ht₂ x hx => h_nhds _ _ hx (IsOpen.inter (h_open _ ht₁) (h_open _ ht₂)), _, _⟩
-  · refine' unionₛ_eq_univ_iff.2 fun a => _
+  · refine' sUnion_eq_univ_iff.2 fun a => _
     rcases h_nhds a univ trivial isOpen_univ with ⟨u, h₁, h₂, -⟩
     exact ⟨u, h₁, h₂⟩
   · refine' (le_generateFrom h_open).antisymm fun u hu => _
     refine' (@isOpen_iff_nhds α (generateFrom s) u).mpr fun a ha => _
     rcases h_nhds a u ha hu with ⟨v, hvs, hav, hvu⟩
     rw [nhds_generateFrom]
-    exact infᵢ₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
+    exact iInf₂_le_of_le v ⟨hav, hvs⟩ (le_principal_iff.2 hvu)
 #align topological_space.is_topological_basis_of_open_of_nhds TopologicalSpace.isTopologicalBasis_of_open_of_nhds
 
 /-- A set `s` is in the neighbourhood of `a` iff there is some basis set `t`, which
@@ -146,13 +146,13 @@ contains `a` and is itself contained in `s`. -/
 theorem IsTopologicalBasis.mem_nhds_iff {a : α} {s : Set α} {b : Set (Set α)}
     (hb : IsTopologicalBasis b) : s ∈ 𝓝 a ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s := by
   change s ∈ (𝓝 a).sets ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s
-  rw [hb.eq_generateFrom, nhds_generateFrom, binfᵢ_sets_eq]
+  rw [hb.eq_generateFrom, nhds_generateFrom, biInf_sets_eq]
   · simp [and_assoc, and_left_comm]
   · rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩
     let ⟨u, hu₁, hu₂, hu₃⟩ := hb.1 _ hs₂ _ ht₂ _ ⟨hs₁, ht₁⟩
     exact ⟨u, ⟨hu₂, hu₁⟩, le_principal_iff.2 (hu₃.trans (inter_subset_left _ _)),
       le_principal_iff.2 (hu₃.trans (inter_subset_right _ _))⟩
-  · rcases eq_univ_iff_forall.1 hb.unionₛ_eq a with ⟨i, h1, h2⟩
+  · rcases eq_univ_iff_forall.1 hb.sUnion_eq a with ⟨i, h1, h2⟩
     exact ⟨i, h2, h1⟩
 #align topological_space.is_topological_basis.mem_nhds_iff TopologicalSpace.IsTopologicalBasis.mem_nhds_iff
 
@@ -182,33 +182,33 @@ theorem IsTopologicalBasis.exists_subset_of_mem_open {b : Set (Set α)} (hb : Is
 #align topological_space.is_topological_basis.exists_subset_of_mem_open TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_open
 
 /-- Any open set is the union of the basis sets contained in it. -/
-theorem IsTopologicalBasis.open_eq_unionₛ' {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
+theorem IsTopologicalBasis.open_eq_sUnion' {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : u = ⋃₀ { s ∈ B | s ⊆ u } :=
   ext fun _a =>
     ⟨fun ha =>
       let ⟨b, hb, ab, bu⟩ := hB.exists_subset_of_mem_open ha ou
       ⟨b, ⟨hb, bu⟩, ab⟩,
       fun ⟨_b, ⟨_, bu⟩, ab⟩ => bu ab⟩
-#align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_unionₛ'
+#align topological_space.is_topological_basis.open_eq_sUnion' TopologicalSpace.IsTopologicalBasis.open_eq_sUnion'
 
 -- porting note: use `∃ S, S ⊆ B ∧ _` instead of `∃ S (_ : S ⊆ B), _`
-theorem IsTopologicalBasis.open_eq_unionₛ {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
+theorem IsTopologicalBasis.open_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ S, S ⊆ B ∧ u = ⋃₀ S :=
-  ⟨{ s ∈ B | s ⊆ u }, fun _ h => h.1, hB.open_eq_unionₛ' ou⟩
-#align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_unionₛ
+  ⟨{ s ∈ B | s ⊆ u }, fun _ h => h.1, hB.open_eq_sUnion' ou⟩
+#align topological_space.is_topological_basis.open_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
 
 -- porting note: use `∃ S, S ⊆ B ∧ _` instead of `∃ S (_ : S ⊆ B), _`
-theorem IsTopologicalBasis.open_iff_eq_unionₛ {B : Set (Set α)} (hB : IsTopologicalBasis B)
+theorem IsTopologicalBasis.open_iff_eq_sUnion {B : Set (Set α)} (hB : IsTopologicalBasis B)
     {u : Set α} : IsOpen u ↔ ∃ S, S ⊆ B ∧ u = ⋃₀ S :=
-  ⟨hB.open_eq_unionₛ, fun ⟨_S, hSB, hu⟩ => hu.symm ▸ isOpen_unionₛ fun _s hs => hB.isOpen (hSB hs)⟩
-#align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_unionₛ
+  ⟨hB.open_eq_sUnion, fun ⟨_S, hSB, hu⟩ => hu.symm ▸ isOpen_sUnion fun _s hs => hB.isOpen (hSB hs)⟩
+#align topological_space.is_topological_basis.open_iff_eq_sUnion TopologicalSpace.IsTopologicalBasis.open_iff_eq_sUnion
 
-theorem IsTopologicalBasis.open_eq_unionᵢ {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
+theorem IsTopologicalBasis.open_eq_iUnion {B : Set (Set α)} (hB : IsTopologicalBasis B) {u : Set α}
     (ou : IsOpen u) : ∃ (β : Type u) (f : β → Set α), (u = ⋃ i, f i) ∧ ∀ i, f i ∈ B :=
   ⟨↥({ s ∈ B | s ⊆ u }), (↑), by
-    rw [← unionₛ_eq_unionᵢ]
-    apply hB.open_eq_unionₛ' ou, fun s => And.left s.2⟩
-#align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_unionᵢ
+    rw [← sUnion_eq_iUnion]
+    apply hB.open_eq_sUnion' ou, fun s => And.left s.2⟩
+#align topological_space.is_topological_basis.open_eq_Union TopologicalSpace.IsTopologicalBasis.open_eq_iUnion
 
 /-- A point `a` is in the closure of `s` iff all basis sets containing `a` intersect `s`. -/
 theorem IsTopologicalBasis.mem_closure_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α}
@@ -226,14 +226,14 @@ theorem IsTopologicalBasis.dense_iff {b : Set (Set α)} (hb : IsTopologicalBasis
 theorem IsTopologicalBasis.isOpenMap_iff {β} [TopologicalSpace β] {B : Set (Set α)}
     (hB : IsTopologicalBasis B) {f : α → β} : IsOpenMap f ↔ ∀ s ∈ B, IsOpen (f '' s) := by
   refine' ⟨fun H o ho => H _ (hB.isOpen ho), fun hf o ho => _⟩
-  rw [hB.open_eq_unionₛ' ho, unionₛ_eq_unionᵢ, image_unionᵢ]
-  exact isOpen_unionᵢ fun s => hf s s.2.1
+  rw [hB.open_eq_sUnion' ho, sUnion_eq_iUnion, image_iUnion]
+  exact isOpen_iUnion fun s => hf s s.2.1
 #align topological_space.is_topological_basis.is_open_map_iff TopologicalSpace.IsTopologicalBasis.isOpenMap_iff
 
 theorem IsTopologicalBasis.exists_nonempty_subset {B : Set (Set α)} (hb : IsTopologicalBasis B)
     {u : Set α} (hu : u.Nonempty) (ou : IsOpen u) : ∃ v ∈ B, Set.Nonempty v ∧ v ⊆ u := by
   cases' hu with x hx
-  rw [hb.open_eq_unionₛ' ou, mem_unionₛ] at hx
+  rw [hb.open_eq_sUnion' ou, mem_sUnion] at hx
   rcases hx with ⟨v, hv, hxv⟩
   exact ⟨v, hv.1, ⟨x, hxv⟩, hv.2⟩
 #align topological_space.is_topological_basis.exists_nonempty_subset TopologicalSpace.IsTopologicalBasis.exists_nonempty_subset
@@ -264,24 +264,24 @@ protected theorem IsTopologicalBasis.inducing {β} [TopologicalSpace β] {f : α
   · intro a U ha hU
     rw [hf.isOpen_iff] at hU
     obtain ⟨V, hV, rfl⟩ := hU
-    obtain ⟨S, hS, rfl⟩ := h.open_eq_unionₛ hV
+    obtain ⟨S, hS, rfl⟩ := h.open_eq_sUnion hV
     obtain ⟨W, hW, ha⟩ := ha
-    refine' ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_unionₛ_of_mem hW⟩
+    refine' ⟨f ⁻¹' W, ⟨_, hS hW, rfl⟩, ha, Set.preimage_mono <| Set.subset_sUnion_of_mem hW⟩
 #align topological_space.is_topological_basis.inducing TopologicalSpace.IsTopologicalBasis.inducing
 
 theorem isTopologicalBasis_of_cover {ι} {U : ι → Set α} (Uo : ∀ i, IsOpen (U i))
     (Uc : (⋃ i, U i) = univ) {b : ∀ i, Set (Set (U i))} (hb : ∀ i, IsTopologicalBasis (b i)) :
     IsTopologicalBasis (⋃ i : ι, image ((↑) : U i → α) '' b i) := by
   refine' isTopologicalBasis_of_open_of_nhds (fun u hu => _) _
-  · simp only [mem_unionᵢ, mem_image] at hu
+  · simp only [mem_iUnion, mem_image] at hu
     rcases hu with ⟨i, s, sb, rfl⟩
     exact (Uo i).isOpenMap_subtype_val _ ((hb i).isOpen sb)
   · intro a u ha uo
-    rcases unionᵢ_eq_univ_iff.1 Uc a with ⟨i, hi⟩
+    rcases iUnion_eq_univ_iff.1 Uc a with ⟨i, hi⟩
     lift a to ↥(U i) using hi
     rcases(hb i).exists_subset_of_mem_open ha (uo.preimage continuous_subtype_val) with
       ⟨v, hvb, hav, hvu⟩
-    exact ⟨(↑) '' v, mem_unionᵢ.2 ⟨i, mem_image_of_mem _ hvb⟩, mem_image_of_mem _ hav,
+    exact ⟨(↑) '' v, mem_iUnion.2 ⟨i, mem_image_of_mem _ hvb⟩, mem_image_of_mem _ hav,
       image_subset_iff.2 hvu⟩
 #align topological_space.is_topological_basis_of_cover TopologicalSpace.isTopologicalBasis_of_cover
 
@@ -403,12 +403,12 @@ theorem IsSeparable.closure {s : Set α} (hs : IsSeparable s) : IsSeparable (clo
   exact ⟨c, c_count, by simpa using closure_mono hs⟩
 #align topological_space.is_separable.closure TopologicalSpace.IsSeparable.closure
 
-theorem isSeparable_unionᵢ {ι : Type _} [Countable ι] {s : ι → Set α}
+theorem isSeparable_iUnion {ι : Type _} [Countable ι] {s : ι → Set α}
     (hs : ∀ i, IsSeparable (s i)) : IsSeparable (⋃ i, s i) := by
   choose c hc h'c using hs
-  refine' ⟨⋃ i, c i, countable_unionᵢ hc, unionᵢ_subset_iff.2 fun i => _⟩
-  exact (h'c i).trans (closure_mono (subset_unionᵢ _ i))
-#align topological_space.is_separable_Union TopologicalSpace.isSeparable_unionᵢ
+  refine' ⟨⋃ i, c i, countable_iUnion hc, iUnion_subset_iff.2 fun i => _⟩
+  exact (h'c i).trans (closure_mono (subset_iUnion _ i))
+#align topological_space.is_separable_Union TopologicalSpace.isSeparable_iUnion
 
 theorem _root_.Set.Countable.isSeparable {s : Set α} (hs : s.Countable) : IsSeparable s :=
   ⟨s, hs, subset_closure⟩
@@ -468,25 +468,25 @@ theorem isTopologicalBasis_pi {ι : Type _} {X : ι → Type _} [∀ i, Topologi
     exact ⟨_, ⟨V, I, fun i _ => hVT i, rfl⟩, fun i _ => haV i, (pi_mono fun i _ => hVt i).trans htU⟩
 #align is_topological_basis_pi isTopologicalBasis_pi
 
-theorem isTopologicalBasis_infᵢ {β : Type _} {ι : Type _} {X : ι → Type _}
+theorem isTopologicalBasis_iInf {β : Type _} {ι : Type _} {X : ι → Type _}
     [t : ∀ i, TopologicalSpace (X i)] {T : ∀ i, Set (Set (X i))}
     (cond : ∀ i, IsTopologicalBasis (T i)) (f : ∀ i, β → X i) :
     @IsTopologicalBasis β (⨅ i, induced (f i) (t i))
       { S | ∃ (U : ∀ i, Set (X i)) (F : Finset ι),
         (∀ i, i ∈ F → U i ∈ T i) ∧ S = ⋂ (i) (_hi : i ∈ F), f i ⁻¹' U i } := by
   letI := ⨅ i, induced (f i) (t i)
-  convert (isTopologicalBasis_pi cond).inducing (inducing_infᵢ_to_pi f)
+  convert (isTopologicalBasis_pi cond).inducing (inducing_iInf_to_pi f)
   ext V
   constructor
   · rintro ⟨U, F, h1, rfl⟩
     refine' ⟨(F : Set ι).pi U, ⟨U, F, h1, rfl⟩, _⟩
-    simp_rw [pi_def, Set.preimage_interᵢ]
+    simp_rw [pi_def, Set.preimage_iInter]
     rfl
   · rintro ⟨U, ⟨U, F, h1, rfl⟩, rfl⟩
     refine' ⟨U, F, h1, _⟩
-    simp_rw [pi_def, Set.preimage_interᵢ]
+    simp_rw [pi_def, Set.preimage_iInter]
     rfl
-#align is_topological_basis_infi isTopologicalBasis_infᵢ
+#align is_topological_basis_infi isTopologicalBasis_iInf
 
 theorem isTopologicalBasis_singletons (α : Type _) [TopologicalSpace α] [DiscreteTopology α] :
     IsTopologicalBasis { s | ∃ x : α, (s : Set α) = {x} } :=
@@ -684,11 +684,11 @@ instance Subtype.secondCountableTopology (s : Set α) [SecondCountableTopology 
   secondCountableTopology_induced s α (↑)
 #align topological_space.subtype.second_countable_topology TopologicalSpace.Subtype.secondCountableTopology
 
-lemma secondCountableTopology_infᵢ {ι} [Countable ι] {t : ι → TopologicalSpace α}
+lemma secondCountableTopology_iInf {ι} [Countable ι] {t : ι → TopologicalSpace α}
     (ht : ∀ i, @SecondCountableTopology α (t i)) : @SecondCountableTopology α (⨅ i, t i) := by
-  rw [funext fun i => @eq_generateFrom_countableBasis α (t i) (ht i), ← generateFrom_unionᵢ]
+  rw [funext fun i => @eq_generateFrom_countableBasis α (t i) (ht i), ← generateFrom_iUnion]
   exact SecondCountableTopology.mk' <|
-    countable_unionᵢ fun i => @countable_countableBasis _ (t i) (ht i)
+    countable_iUnion fun i => @countable_countableBasis _ (t i) (ht i)
 
 -- TODO: more fine grained instances for first_countable_topology, separable_space, t2_space, ...
 instance {β : Type _} [TopologicalSpace β] [SecondCountableTopology α] [SecondCountableTopology β] :
@@ -698,7 +698,7 @@ instance {β : Type _} [TopologicalSpace β] [SecondCountableTopology α] [Secon
 
 instance {ι : Type _} {π : ι → Type _} [Countable ι] [∀ a, TopologicalSpace (π a)]
     [∀ a, SecondCountableTopology (π a)] : SecondCountableTopology (∀ a, π a) :=
-  secondCountableTopology_infᵢ fun _ => secondCountableTopology_induced _ _ _
+  secondCountableTopology_iInf fun _ => secondCountableTopology_induced _ _ _
 
 -- see Note [lower instance priority]
 instance (priority := 100) SecondCountableTopology.to_separableSpace [SecondCountableTopology α] :
@@ -716,39 +716,39 @@ theorem secondCountableTopology_of_countable_cover {ι} [Encodable ι] {U : ι 
     SecondCountableTopology α :=
   haveI : IsTopologicalBasis (⋃ i, image ((↑) : U i → α) '' countableBasis (U i)) :=
     isTopologicalBasis_of_cover Uo hc fun i => isBasis_countableBasis (U i)
-  this.secondCountableTopology (countable_unionᵢ fun _ => (countable_countableBasis _).image _)
+  this.secondCountableTopology (countable_iUnion fun _ => (countable_countableBasis _).image _)
 #align topological_space.second_countable_topology_of_countable_cover TopologicalSpace.secondCountableTopology_of_countable_cover
 
 /-- In a second-countable space, an open set, given as a union of open sets,
 is equal to the union of countably many of those sets. -/
-theorem isOpen_unionᵢ_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
+theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
     (H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ (⋃ i ∈ T, s i) = ⋃ i, s i := by
   let B := { b ∈ countableBasis α | ∃ i, b ⊆ s i }
   choose f hf using fun b : B => b.2.2
   haveI : Encodable B := ((countable_countableBasis α).mono (sep_subset _ _)).toEncodable
-  refine' ⟨_, countable_range f, (unionᵢ₂_subset_unionᵢ _ _).antisymm (unionₛ_subset _)⟩
+  refine' ⟨_, countable_range f, (iUnion₂_subset_iUnion _ _).antisymm (sUnion_subset _)⟩
   rintro _ ⟨i, rfl⟩ x xs
   rcases (isBasis_countableBasis α).exists_subset_of_mem_open xs (H _) with ⟨b, hb, xb, bs⟩
   exact ⟨_, ⟨_, rfl⟩, _, ⟨⟨⟨_, hb, _, bs⟩, rfl⟩, rfl⟩, hf _ xb⟩
-#align topological_space.is_open_Union_countable TopologicalSpace.isOpen_unionᵢ_countable
+#align topological_space.is_open_Union_countable TopologicalSpace.isOpen_iUnion_countable
 
-theorem isOpen_unionₛ_countable [SecondCountableTopology α] (S : Set (Set α))
+theorem isOpen_sUnion_countable [SecondCountableTopology α] (S : Set (Set α))
     (H : ∀ s ∈ S, IsOpen s) : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
-  let ⟨T, cT, hT⟩ := isOpen_unionᵢ_countable (fun s : S => s.1) fun s => H s.1 s.2
+  let ⟨T, cT, hT⟩ := isOpen_iUnion_countable (fun s : S => s.1) fun s => H s.1 s.2
   ⟨Subtype.val '' T, cT.image _, image_subset_iff.2 fun ⟨_, h⟩ _ => h, by
-    rwa [unionₛ_image, unionₛ_eq_unionᵢ]⟩
-#align topological_space.is_open_sUnion_countable TopologicalSpace.isOpen_unionₛ_countable
+    rwa [sUnion_image, sUnion_eq_iUnion]⟩
+#align topological_space.is_open_sUnion_countable TopologicalSpace.isOpen_sUnion_countable
 
 /-- In a topological space with second countable topology, if `f` is a function that sends each
 point `x` to a neighborhood of `x`, then for some countable set `s`, the neighborhoods `f x`,
 `x ∈ s`, cover the whole space. -/
 theorem countable_cover_nhds [SecondCountableTopology α] {f : α → Set α} (hf : ∀ x, f x ∈ 𝓝 x) :
     ∃ s : Set α, s.Countable ∧ (⋃ x ∈ s, f x) = univ := by
-  rcases isOpen_unionᵢ_countable (fun x => interior (f x)) fun x => isOpen_interior with
+  rcases isOpen_iUnion_countable (fun x => interior (f x)) fun x => isOpen_interior with
     ⟨s, hsc, hsU⟩
   suffices : (⋃ x ∈ s, interior (f x)) = univ
-  exact ⟨s, hsc, flip eq_univ_of_subset this <| unionᵢ₂_mono fun _ _ => interior_subset⟩
-  simp only [hsU, eq_univ_iff_forall, mem_unionᵢ]
+  exact ⟨s, hsc, flip eq_univ_of_subset this <| iUnion₂_mono fun _ _ => interior_subset⟩
+  simp only [hsU, eq_univ_iff_forall, mem_iUnion]
   exact fun x => ⟨x, mem_interior_iff_mem_nhds.2 (hf x)⟩
 #align topological_space.countable_cover_nhds TopologicalSpace.countable_cover_nhds
 
@@ -758,7 +758,7 @@ theorem countable_cover_nhdsWithin [SecondCountableTopology α] {f : α → Set
   have : ∀ x : s, (↑) ⁻¹' f x ∈ 𝓝 x := fun x => preimage_coe_mem_nhds_subtype.2 (hf x x.2)
   rcases countable_cover_nhds this with ⟨t, htc, htU⟩
   refine' ⟨(↑) '' t, Subtype.coe_image_subset _ _, htc.image _, fun x hx => _⟩
-  simp only [bunionᵢ_image, eq_univ_iff_forall, ← preimage_unionᵢ, mem_preimage] at htU⊢
+  simp only [biUnion_image, eq_univ_iff_forall, ← preimage_iUnion, mem_preimage] at htU⊢
   exact htU ⟨x, hx⟩
 #align topological_space.countable_cover_nhds_within TopologicalSpace.countable_cover_nhdsWithin
 
@@ -774,14 +774,14 @@ theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
   apply isTopologicalBasis_of_open_of_nhds
   · intro u hu
     obtain ⟨i, t, ts, rfl⟩ : ∃ (i : ι) (t : Set (E i)), t ∈ s i ∧ Sigma.mk i '' t = u := by
-      simpa only [mem_unionᵢ, mem_image] using hu
+      simpa only [mem_iUnion, mem_image] using hu
     exact isOpenMap_sigmaMk _ ((hs i).isOpen ts)
   · rintro ⟨i, x⟩ u hxu u_open
     have hx : x ∈ Sigma.mk i ⁻¹' u := hxu
     obtain ⟨v, vs, xv, hv⟩ : ∃ (v : Set (E i)), v ∈ s i ∧ x ∈ v ∧ v ⊆ Sigma.mk i ⁻¹' u :=
       (hs i).exists_subset_of_mem_open hx (isOpen_sigma_iff.1 u_open i)
     exact
-      ⟨Sigma.mk i '' v, mem_unionᵢ.2 ⟨i, mem_image_of_mem _ vs⟩, mem_image_of_mem _ xv,
+      ⟨Sigma.mk i '' v, mem_iUnion.2 ⟨i, mem_image_of_mem _ vs⟩, mem_image_of_mem _ xv,
         image_subset_iff.2 hv⟩
 #align topological_space.is_topological_basis.sigma TopologicalSpace.IsTopologicalBasis.sigma
 
@@ -790,7 +790,7 @@ instance [Countable ι] [∀ i, SecondCountableTopology (E i)] :
     SecondCountableTopology (Σi, E i) := by
   let b := ⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' countableBasis (E i)
   have A : IsTopologicalBasis b := IsTopologicalBasis.sigma fun i => isBasis_countableBasis _
-  have B : b.Countable := countable_unionᵢ fun i => (countable_countableBasis _).image _
+  have B : b.Countable := countable_iUnion fun i => (countable_countableBasis _).image _
   exact A.secondCountableTopology B
 
 end Sigma

This PR puts, with one exception, every single remaining by that lies all by itself on its own line to the previous line, thus matching the current behaviour of start-port.sh. The exception is when the by begins the second or later argument to a tuple or anonymous constructor; see https://github.com/leanprover-community/mathlib4/pull/3825#discussion_r1186702599.

Essentially this is s/\n *by$/ by/g, but with manual editing to satisfy the linter's max-100-char-line requirement. The Python style linter is also modified to catch these "isolated bys".

@@ -786,8 +786,8 @@ theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
 #align topological_space.is_topological_basis.sigma TopologicalSpace.IsTopologicalBasis.sigma
 
 /-- A countable disjoint union of second countable spaces is second countable. -/
-instance [Countable ι] [∀ i, SecondCountableTopology (E i)] : SecondCountableTopology (Σi, E i) :=
-  by
+instance [Countable ι] [∀ i, SecondCountableTopology (E i)] :
+    SecondCountableTopology (Σi, E i) := by
   let b := ⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' countableBasis (E i)
   have A : IsTopologicalBasis b := IsTopologicalBasis.sigma fun i => isBasis_countableBasis _
   have B : b.Countable := countable_unionᵢ fun i => (countable_countableBasis _).image _