topology.G_delta ⟷ Mathlib.Topology.GDelta

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

@@ -283,7 +283,7 @@ theorem residual_of_dense_Gδ {s : Set α} (ho : IsGδ s) (hd : Dense s) : s ∈
 theorem mem_residual_iff {s : Set α} :
     s ∈ residual α ↔
       ∃ S : Set (Set α), (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s :=
-  mem_countableGenerate_iff.trans <| by simp_rw [subset_def, mem_set_of, forall_and, and_assoc']
+  mem_countableGenerate_iff.trans <| by simp_rw [subset_def, mem_set_of, forall_and, and_assoc]
 #align mem_residual_iff mem_residual_iff
 -/
 

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

@@ -62,65 +62,65 @@ theorem IsOpen.isGδ {s : Set α} (h : IsOpen s) : IsGδ s :=
 #align is_open.is_Gδ IsOpen.isGδ
 -/
 
-#print isGδ_empty /-
+#print IsGδ.empty /-
 @[simp]
-theorem isGδ_empty : IsGδ (∅ : Set α) :=
+theorem IsGδ.empty : IsGδ (∅ : Set α) :=
   isOpen_empty.IsGδ
-#align is_Gδ_empty isGδ_empty
+#align is_Gδ_empty IsGδ.empty
 -/
 
-#print isGδ_univ /-
+#print IsGδ.univ /-
 @[simp]
-theorem isGδ_univ : IsGδ (univ : Set α) :=
+theorem IsGδ.univ : IsGδ (univ : Set α) :=
   isOpen_univ.IsGδ
-#align is_Gδ_univ isGδ_univ
+#align is_Gδ_univ IsGδ.univ
 -/
 
-#print isGδ_biInter_of_isOpen /-
-theorem isGδ_biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set α}
+#print IsGδ.biInter_of_isOpen /-
+theorem IsGδ.biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set α}
     (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
   ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
-#align is_Gδ_bInter_of_open isGδ_biInter_of_isOpen
+#align is_Gδ_bInter_of_open IsGδ.biInter_of_isOpen
 -/
 
-#print isGδ_iInter_of_isOpen /-
-theorem isGδ_iInter_of_isOpen [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
+#print IsGδ.iInter_of_isOpen /-
+theorem IsGδ.iInter_of_isOpen [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
   ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
-#align is_Gδ_Inter_of_open isGδ_iInter_of_isOpen
+#align is_Gδ_Inter_of_open IsGδ.iInter_of_isOpen
 -/
 
-#print isGδ_iInter /-
+#print IsGδ.iInter /-
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
-theorem isGδ_iInter [Encodable ι] {s : ι → Set α} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) :=
+theorem IsGδ.iInter [Encodable ι] {s : ι → Set α} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) :=
   by
   choose T hTo hTc hTs using hs
   obtain rfl : s = fun i => ⋂₀ T i := funext hTs
   refine' ⟨⋃ i, T i, _, countable_Union hTc, (sInter_Union _).symm⟩
   simpa [@forall_swap ι] using hTo
-#align is_Gδ_Inter isGδ_iInter
+#align is_Gδ_Inter IsGδ.iInter
 -/
 
-#print isGδ_biInter /-
-theorem isGδ_biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set α}
+#print IsGδ.biInter /-
+theorem IsGδ.biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set α}
     (ht : ∀ i ∈ s, IsGδ (t i ‹_›)) : IsGδ (⋂ i ∈ s, t i ‹_›) :=
   by
   rw [bInter_eq_Inter]
   haveI := hs.to_encodable
-  exact isGδ_iInter fun x => ht x x.2
-#align is_Gδ_bInter isGδ_biInter
+  exact IsGδ.iInter fun x => ht x x.2
+#align is_Gδ_bInter IsGδ.biInter
 -/
 
-#print isGδ_sInter /-
+#print IsGδ.sInter /-
 /-- A countable intersection of Gδ sets is a Gδ set. -/
-theorem isGδ_sInter {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
-  simpa only [sInter_eq_bInter] using isGδ_biInter hS h
-#align is_Gδ_sInter isGδ_sInter
+theorem IsGδ.sInter {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
+  simpa only [sInter_eq_bInter] using IsGδ.biInter hS h
+#align is_Gδ_sInter IsGδ.sInter
 -/
 
 #print IsGδ.inter /-
 theorem IsGδ.inter {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
-  rw [inter_eq_Inter]; exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
+  rw [inter_eq_Inter]; exact IsGδ.iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
 #align is_Gδ.inter IsGδ.inter
 -/
 
@@ -131,20 +131,20 @@ theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪
   rcases hs with ⟨S, Sopen, Scount, rfl⟩
   rcases ht with ⟨T, Topen, Tcount, rfl⟩
   rw [sInter_union_sInter]
-  apply isGδ_biInter_of_isOpen (Scount.prod Tcount)
+  apply IsGδ.biInter_of_isOpen (Scount.prod Tcount)
   rintro ⟨a, b⟩ ⟨ha, hb⟩
   exact (Sopen a ha).union (Topen b hb)
 #align is_Gδ.union IsGδ.union
 -/
 
-#print isGδ_biUnion /-
+#print IsGδ.biUnion /-
 /-- The union of finitely many Gδ sets is a Gδ set. -/
-theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set α} (h : ∀ i ∈ s, IsGδ (f i)) :
+theorem IsGδ.biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set α} (h : ∀ i ∈ s, IsGδ (f i)) :
     IsGδ (⋃ i ∈ s, f i) := by
   refine' finite.induction_on hs (by simp) _ h
   simp only [ball_insert_iff, bUnion_insert]
   exact fun a s _ _ ihs H => H.1.union (ihs H.2)
-#align is_Gδ_bUnion isGδ_biUnion
+#align is_Gδ_bUnion IsGδ.biUnion
 -/
 
 #print IsClosed.isGδ /-
@@ -153,7 +153,7 @@ theorem IsClosed.isGδ {α} [UniformSpace α] [IsCountablyGenerated (𝓤 α)] {
   by
   rcases(@uniformity_hasBasis_open α _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
   rw [← hs.closure_eq, ← hU.bInter_bUnion_ball]
-  refine' isGδ_biInter (to_countable _) fun n hn => IsOpen.isGδ _
+  refine' IsGδ.biInter (to_countable _) fun n hn => IsOpen.isGδ _
   exact isOpen_biUnion fun x hx => UniformSpace.isOpen_ball _ (hUo _).2
 #align is_closed.is_Gδ IsClosed.isGδ
 -/
@@ -162,17 +162,17 @@ section T1Space
 
 variable [T1Space α]
 
-#print isGδ_compl_singleton /-
-theorem isGδ_compl_singleton (a : α) : IsGδ ({a}ᶜ : Set α) :=
+#print IsGδ.compl_singleton /-
+theorem IsGδ.compl_singleton (a : α) : IsGδ ({a}ᶜ : Set α) :=
   isOpen_compl_singleton.IsGδ
-#align is_Gδ_compl_singleton isGδ_compl_singleton
+#align is_Gδ_compl_singleton IsGδ.compl_singleton
 -/
 
 #print Set.Countable.isGδ_compl /-
 theorem Set.Countable.isGδ_compl {s : Set α} (hs : s.Countable) : IsGδ (sᶜ) :=
   by
   rw [← bUnion_of_singleton s, compl_Union₂]
-  exact isGδ_biInter hs fun x _ => isGδ_compl_singleton x
+  exact IsGδ.biInter hs fun x _ => IsGδ.compl_singleton x
 #align set.countable.is_Gδ_compl Set.Countable.isGδ_compl
 -/
 
@@ -198,18 +198,18 @@ open TopologicalSpace
 
 variable [FirstCountableTopology α]
 
-#print isGδ_singleton /-
-theorem isGδ_singleton (a : α) : IsGδ ({a} : Set α) :=
+#print IsGδ.singleton /-
+theorem IsGδ.singleton (a : α) : IsGδ ({a} : Set α) :=
   by
   rcases(nhds_basis_opens a).exists_antitone_subbasis with ⟨U, hU, h_basis⟩
   rw [← biInter_basis_nhds h_basis.to_has_basis]
-  exact isGδ_biInter (to_countable _) fun n hn => (hU n).2.IsGδ
-#align is_Gδ_singleton isGδ_singleton
+  exact IsGδ.biInter (to_countable _) fun n hn => (hU n).2.IsGδ
+#align is_Gδ_singleton IsGδ.singleton
 -/
 
 #print Set.Finite.isGδ /-
 theorem Set.Finite.isGδ {s : Set α} (hs : s.Finite) : IsGδ s :=
-  Finite.induction_on hs isGδ_empty fun a s _ _ hs => (isGδ_singleton a).union hs
+  Finite.induction_on hs IsGδ.empty fun a s _ _ hs => (IsGδ.singleton a).union hs
 #align set.finite.is_Gδ Set.Finite.isGδ
 -/
 
@@ -225,19 +225,19 @@ open scoped uniformity
 
 variable [TopologicalSpace α]
 
-#print isGδ_setOf_continuousAt /-
+#print IsGδ.setOf_continuousAt /-
 /-- The set of points where a function is continuous is a Gδ set. -/
-theorem isGδ_setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (𝓤 β)] (f : α → β) :
+theorem IsGδ.setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (𝓤 β)] (f : α → β) :
     IsGδ {x | ContinuousAt f x} :=
   by
   obtain ⟨U, hUo, hU⟩ := (@uniformity_hasBasis_open_symmetric β _).exists_antitone_subbasis
   simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
   simp only [(nhds_basis_opens _).prod_self.tendsto_iffₓ hU.to_has_basis, forall_prop_of_true,
     set_of_forall, id]
-  refine' isGδ_iInter fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
+  refine' IsGδ.iInter fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
   rintro ⟨s, ⟨hsx, hso⟩, hsU⟩
   filter_upwards [IsOpen.mem_nhds hso hsx] with _ hy using ⟨s, ⟨hy, hso⟩, hsU⟩
-#align is_Gδ_set_of_continuous_at isGδ_setOf_continuousAt
+#align is_Gδ_set_of_continuous_at IsGδ.setOf_continuousAt
 -/
 
 end ContinuousAt

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

@@ -76,18 +76,18 @@ theorem isGδ_univ : IsGδ (univ : Set α) :=
 #align is_Gδ_univ isGδ_univ
 -/
 
-#print isGδ_biInter_of_open /-
-theorem isGδ_biInter_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set α}
+#print isGδ_biInter_of_isOpen /-
+theorem isGδ_biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set α}
     (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
   ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
-#align is_Gδ_bInter_of_open isGδ_biInter_of_open
+#align is_Gδ_bInter_of_open isGδ_biInter_of_isOpen
 -/
 
-#print isGδ_iInter_of_open /-
-theorem isGδ_iInter_of_open [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
+#print isGδ_iInter_of_isOpen /-
+theorem isGδ_iInter_of_isOpen [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
   ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
-#align is_Gδ_Inter_of_open isGδ_iInter_of_open
+#align is_Gδ_Inter_of_open isGδ_iInter_of_isOpen
 -/
 
 #print isGδ_iInter /-
@@ -131,7 +131,7 @@ theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪
   rcases hs with ⟨S, Sopen, Scount, rfl⟩
   rcases ht with ⟨T, Topen, Tcount, rfl⟩
   rw [sInter_union_sInter]
-  apply isGδ_biInter_of_open (Scount.prod Tcount)
+  apply isGδ_biInter_of_isOpen (Scount.prod Tcount)
   rintro ⟨a, b⟩ ⟨ha, hb⟩
   exact (Sopen a ha).union (Topen b hb)
 #align is_Gδ.union IsGδ.union

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

@@ -3,9 +3,9 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 -/
-import Mathbin.Topology.UniformSpace.Basic
-import Mathbin.Topology.Separation
-import Mathbin.Order.Filter.CountableInter
+import Topology.UniformSpace.Basic
+import Topology.Separation
+import Order.Filter.CountableInter
 
 #align_import topology.G_delta from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.G_delta
-! leanprover-community/mathlib commit b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.UniformSpace.Basic
 import Mathbin.Topology.Separation
 import Mathbin.Order.Filter.CountableInter
 
+#align_import topology.G_delta from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
+
 /-!
 # `Gδ` sets
 

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

@@ -121,10 +121,13 @@ theorem isGδ_sInter {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Count
 #align is_Gδ_sInter isGδ_sInter
 -/
 
+#print IsGδ.inter /-
 theorem IsGδ.inter {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
   rw [inter_eq_Inter]; exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
 #align is_Gδ.inter IsGδ.inter
+-/
 
+#print IsGδ.union /-
 /-- The union of two Gδ sets is a Gδ set. -/
 theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t) :=
   by
@@ -135,6 +138,7 @@ theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪
   rintro ⟨a, b⟩ ⟨ha, hb⟩
   exact (Sopen a ha).union (Topen b hb)
 #align is_Gδ.union IsGδ.union
+-/
 
 #print isGδ_biUnion /-
 /-- The union of finitely many Gδ sets is a Gδ set. -/
@@ -161,27 +165,37 @@ section T1Space
 
 variable [T1Space α]
 
+#print isGδ_compl_singleton /-
 theorem isGδ_compl_singleton (a : α) : IsGδ ({a}ᶜ : Set α) :=
   isOpen_compl_singleton.IsGδ
 #align is_Gδ_compl_singleton isGδ_compl_singleton
+-/
 
+#print Set.Countable.isGδ_compl /-
 theorem Set.Countable.isGδ_compl {s : Set α} (hs : s.Countable) : IsGδ (sᶜ) :=
   by
   rw [← bUnion_of_singleton s, compl_Union₂]
   exact isGδ_biInter hs fun x _ => isGδ_compl_singleton x
 #align set.countable.is_Gδ_compl Set.Countable.isGδ_compl
+-/
 
+#print Set.Finite.isGδ_compl /-
 theorem Set.Finite.isGδ_compl {s : Set α} (hs : s.Finite) : IsGδ (sᶜ) :=
   hs.Countable.isGδ_compl
 #align set.finite.is_Gδ_compl Set.Finite.isGδ_compl
+-/
 
+#print Set.Subsingleton.isGδ_compl /-
 theorem Set.Subsingleton.isGδ_compl {s : Set α} (hs : s.Subsingleton) : IsGδ (sᶜ) :=
   hs.Finite.isGδ_compl
 #align set.subsingleton.is_Gδ_compl Set.Subsingleton.isGδ_compl
+-/
 
+#print Finset.isGδ_compl /-
 theorem Finset.isGδ_compl (s : Finset α) : IsGδ (sᶜ : Set α) :=
   s.finite_toSet.isGδ_compl
 #align finset.is_Gδ_compl Finset.isGδ_compl
+-/
 
 open TopologicalSpace
 

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

@@ -217,7 +217,7 @@ variable [TopologicalSpace α]
 #print isGδ_setOf_continuousAt /-
 /-- The set of points where a function is continuous is a Gδ set. -/
 theorem isGδ_setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (𝓤 β)] (f : α → β) :
-    IsGδ { x | ContinuousAt f x } :=
+    IsGδ {x | ContinuousAt f x} :=
   by
   obtain ⟨U, hUo, hU⟩ := (@uniformity_hasBasis_open_symmetric β _).exists_antitone_subbasis
   simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
@@ -225,7 +225,7 @@ theorem isGδ_setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (𝓤 
     set_of_forall, id]
   refine' isGδ_iInter fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
   rintro ⟨s, ⟨hsx, hso⟩, hsU⟩
-  filter_upwards [IsOpen.mem_nhds hso hsx]with _ hy using⟨s, ⟨hy, hso⟩, hsU⟩
+  filter_upwards [IsOpen.mem_nhds hso hsx] with _ hy using ⟨s, ⟨hy, hso⟩, hsU⟩
 #align is_Gδ_set_of_continuous_at isGδ_setOf_continuousAt
 -/
 
@@ -238,7 +238,7 @@ variable [TopologicalSpace α]
 #print residual /-
 /-- A set `s` is called *residual* if it includes a countable intersection of dense open sets. -/
 def residual (α : Type _) [TopologicalSpace α] : Filter α :=
-  Filter.countableGenerate { t | IsOpen t ∧ Dense t }
+  Filter.countableGenerate {t | IsOpen t ∧ Dense t}
 deriving CountableInterFilter
 #align residual residual
 -/

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

@@ -238,7 +238,8 @@ variable [TopologicalSpace α]
 #print residual /-
 /-- A set `s` is called *residual* if it includes a countable intersection of dense open sets. -/
 def residual (α : Type _) [TopologicalSpace α] : Filter α :=
-  Filter.countableGenerate { t | IsOpen t ∧ Dense t }deriving CountableInterFilter
+  Filter.countableGenerate { t | IsOpen t ∧ Dense t }
+deriving CountableInterFilter
 #align residual residual
 -/
 

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

@@ -41,7 +41,7 @@ Gδ set, residual set
 
 noncomputable section
 
-open Classical Topology Filter uniformity
+open scoped Classical Topology Filter uniformity
 
 open Filter Encodable Set
 
@@ -210,7 +210,7 @@ section ContinuousAt
 
 open TopologicalSpace
 
-open uniformity
+open scoped uniformity
 
 variable [TopologicalSpace α]
 

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

@@ -121,22 +121,10 @@ theorem isGδ_sInter {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Count
 #align is_Gδ_sInter isGδ_sInter
 -/
 
-/- warning: is_Gδ.inter -> IsGδ.inter is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (IsGδ.{u1} α _inst_1 s) -> (IsGδ.{u1} α _inst_1 t) -> (IsGδ.{u1} α _inst_1 (Inter.inter.{u1} (Set.{u1} α) (Set.hasInter.{u1} α) s t))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (IsGδ.{u1} α _inst_1 s) -> (IsGδ.{u1} α _inst_1 t) -> (IsGδ.{u1} α _inst_1 (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t))
-Case conversion may be inaccurate. Consider using '#align is_Gδ.inter IsGδ.interₓ'. -/
 theorem IsGδ.inter {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
   rw [inter_eq_Inter]; exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
 #align is_Gδ.inter IsGδ.inter
 
-/- warning: is_Gδ.union -> IsGδ.union is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (IsGδ.{u1} α _inst_1 s) -> (IsGδ.{u1} α _inst_1 t) -> (IsGδ.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (IsGδ.{u1} α _inst_1 s) -> (IsGδ.{u1} α _inst_1 t) -> (IsGδ.{u1} α _inst_1 (Union.union.{u1} (Set.{u1} α) (Set.instUnionSet.{u1} α) s t))
-Case conversion may be inaccurate. Consider using '#align is_Gδ.union IsGδ.unionₓ'. -/
 /-- The union of two Gδ sets is a Gδ set. -/
 theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t) :=
   by
@@ -173,54 +161,24 @@ section T1Space
 
 variable [T1Space α]
 
-/- warning: is_Gδ_compl_singleton -> isGδ_compl_singleton is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] (a : α), IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.hasSingleton.{u1} α) a))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] (a : α), IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Singleton.singleton.{u1, u1} α (Set.{u1} α) (Set.instSingletonSet.{u1} α) a))
-Case conversion may be inaccurate. Consider using '#align is_Gδ_compl_singleton isGδ_compl_singletonₓ'. -/
 theorem isGδ_compl_singleton (a : α) : IsGδ ({a}ᶜ : Set α) :=
   isOpen_compl_singleton.IsGδ
 #align is_Gδ_compl_singleton isGδ_compl_singleton
 
-/- warning: set.countable.is_Gδ_compl -> Set.Countable.isGδ_compl is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Countable.{u1} α s) -> (IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Countable.{u1} α s) -> (IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) s))
-Case conversion may be inaccurate. Consider using '#align set.countable.is_Gδ_compl Set.Countable.isGδ_complₓ'. -/
 theorem Set.Countable.isGδ_compl {s : Set α} (hs : s.Countable) : IsGδ (sᶜ) :=
   by
   rw [← bUnion_of_singleton s, compl_Union₂]
   exact isGδ_biInter hs fun x _ => isGδ_compl_singleton x
 #align set.countable.is_Gδ_compl Set.Countable.isGδ_compl
 
-/- warning: set.finite.is_Gδ_compl -> Set.Finite.isGδ_compl is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Finite.{u1} α s) -> (IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Finite.{u1} α s) -> (IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) s))
-Case conversion may be inaccurate. Consider using '#align set.finite.is_Gδ_compl Set.Finite.isGδ_complₓ'. -/
 theorem Set.Finite.isGδ_compl {s : Set α} (hs : s.Finite) : IsGδ (sᶜ) :=
   hs.Countable.isGδ_compl
 #align set.finite.is_Gδ_compl Set.Finite.isGδ_compl
 
-/- warning: set.subsingleton.is_Gδ_compl -> Set.Subsingleton.isGδ_compl is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Subsingleton.{u1} α s) -> (IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) s))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] {s : Set.{u1} α}, (Set.Subsingleton.{u1} α s) -> (IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) s))
-Case conversion may be inaccurate. Consider using '#align set.subsingleton.is_Gδ_compl Set.Subsingleton.isGδ_complₓ'. -/
 theorem Set.Subsingleton.isGδ_compl {s : Set α} (hs : s.Subsingleton) : IsGδ (sᶜ) :=
   hs.Finite.isGδ_compl
 #align set.subsingleton.is_Gδ_compl Set.Subsingleton.isGδ_compl
 
-/- warning: finset.is_Gδ_compl -> Finset.isGδ_compl is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] (s : Finset.{u1} α), IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.booleanAlgebra.{u1} α)) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} α) (Set.{u1} α) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} α) (Set.{u1} α) (CoeTCₓ.coe.{succ u1, succ u1} (Finset.{u1} α) (Set.{u1} α) (Finset.Set.hasCoeT.{u1} α))) s))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : T1Space.{u1} α _inst_1] (s : Finset.{u1} α), IsGδ.{u1} α _inst_1 (HasCompl.compl.{u1} (Set.{u1} α) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} α) (Set.instBooleanAlgebraSet.{u1} α)) (Finset.toSet.{u1} α s))
-Case conversion may be inaccurate. Consider using '#align finset.is_Gδ_compl Finset.isGδ_complₓ'. -/
 theorem Finset.isGδ_compl (s : Finset α) : IsGδ (sᶜ : Set α) :=
   s.finite_toSet.isGδ_compl
 #align finset.is_Gδ_compl Finset.isGδ_compl

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

@@ -127,10 +127,8 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α} {t : Set.{u1} α}, (IsGδ.{u1} α _inst_1 s) -> (IsGδ.{u1} α _inst_1 t) -> (IsGδ.{u1} α _inst_1 (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) s t))
 Case conversion may be inaccurate. Consider using '#align is_Gδ.inter IsGδ.interₓ'. -/
-theorem IsGδ.inter {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) :=
-  by
-  rw [inter_eq_Inter]
-  exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
+theorem IsGδ.inter {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
+  rw [inter_eq_Inter]; exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
 #align is_Gδ.inter IsGδ.inter
 
 /- warning: is_Gδ.union -> IsGδ.union is a dubious translation:

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

@@ -286,15 +286,20 @@ def residual (α : Type _) [TopologicalSpace α] : Filter α :=
 #align residual residual
 -/
 
+#print countableInterFilter_residual /-
 instance countableInterFilter_residual : CountableInterFilter (residual α) := by
   rw [residual] <;> infer_instance
 #align countable_Inter_filter_residual countableInterFilter_residual
+-/
 
+#print residual_of_dense_open /-
 /-- Dense open sets are residual. -/
 theorem residual_of_dense_open {s : Set α} (ho : IsOpen s) (hd : Dense s) : s ∈ residual α :=
   CountableGenerateSets.basic ⟨ho, hd⟩
 #align residual_of_dense_open residual_of_dense_open
+-/
 
+#print residual_of_dense_Gδ /-
 /-- Dense Gδ sets are residual. -/
 theorem residual_of_dense_Gδ {s : Set α} (ho : IsGδ s) (hd : Dense s) : s ∈ residual α :=
   by
@@ -303,13 +308,16 @@ theorem residual_of_dense_Gδ {s : Set α} (ho : IsGδ s) (hd : Dense s) : s ∈
     (countable_sInter_mem Tct).mpr fun t tT =>
       residual_of_dense_open (To t tT) (hd.mono (sInter_subset_of_mem tT))
 #align residual_of_dense_Gδ residual_of_dense_Gδ
+-/
 
+#print mem_residual_iff /-
 /-- A set is residual iff it includes a countable intersection of dense open sets. -/
 theorem mem_residual_iff {s : Set α} :
     s ∈ residual α ↔
       ∃ S : Set (Set α), (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s :=
   mem_countableGenerate_iff.trans <| by simp_rw [subset_def, mem_set_of, forall_and, and_assoc']
 #align mem_residual_iff mem_residual_iff
+-/
 
 end residual
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952

@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.G_delta
-! leanprover-community/mathlib commit 50832daea47b195a48b5b33b1c8b2162c48c3afc
+! leanprover-community/mathlib commit b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Topology.UniformSpace.Basic
 import Mathbin.Topology.Separation
+import Mathbin.Order.Filter.CountableInter
 
 /-!
 # `Gδ` sets
@@ -24,11 +25,8 @@ In this file we define `Gδ` sets and prove their basic properties.
 * `is_Gδ`: a set `s` is a `Gδ` set if it can be represented as an intersection
   of countably many open sets;
 
-* `residual`: the filter of residual sets. A set `s` is called *residual* if it includes a dense
-  `Gδ` set. In a Baire space (e.g., in a complete (e)metric space), residual sets form a filter.
-
-  For technical reasons, we define `residual` in any topological space but the definition agrees
-  with the description above only in Baire spaces.
+* `residual`: the σ-filter of residual sets. A set `s` is called *residual* if it includes a
+  countable intersection of dense open sets.
 
 ## Main results
 
@@ -277,14 +275,41 @@ theorem isGδ_setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (𝓤 
 
 end ContinuousAt
 
-#print residual /-
-/-- A set `s` is called *residual* if it includes a dense `Gδ` set. If `α` is a Baire space
-(e.g., a complete metric space), then residual sets form a filter, see `mem_residual`.
+section residual
 
-For technical reasons we define the filter `residual` in any topological space but in a non-Baire
-space it is not useful because it may contain some non-residual sets. -/
+variable [TopologicalSpace α]
+
+#print residual /-
+/-- A set `s` is called *residual* if it includes a countable intersection of dense open sets. -/
 def residual (α : Type _) [TopologicalSpace α] : Filter α :=
-  ⨅ (t) (ht : IsGδ t) (ht' : Dense t), 𝓟 t
+  Filter.countableGenerate { t | IsOpen t ∧ Dense t }deriving CountableInterFilter
 #align residual residual
 -/
 
+instance countableInterFilter_residual : CountableInterFilter (residual α) := by
+  rw [residual] <;> infer_instance
+#align countable_Inter_filter_residual countableInterFilter_residual
+
+/-- Dense open sets are residual. -/
+theorem residual_of_dense_open {s : Set α} (ho : IsOpen s) (hd : Dense s) : s ∈ residual α :=
+  CountableGenerateSets.basic ⟨ho, hd⟩
+#align residual_of_dense_open residual_of_dense_open
+
+/-- Dense Gδ sets are residual. -/
+theorem residual_of_dense_Gδ {s : Set α} (ho : IsGδ s) (hd : Dense s) : s ∈ residual α :=
+  by
+  rcases ho with ⟨T, To, Tct, rfl⟩
+  exact
+    (countable_sInter_mem Tct).mpr fun t tT =>
+      residual_of_dense_open (To t tT) (hd.mono (sInter_subset_of_mem tT))
+#align residual_of_dense_Gδ residual_of_dense_Gδ
+
+/-- A set is residual iff it includes a countable intersection of dense open sets. -/
+theorem mem_residual_iff {s : Set α} :
+    s ∈ residual α ↔
+      ∃ S : Set (Set α), (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s :=
+  mem_countableGenerate_iff.trans <| by simp_rw [subset_def, mem_set_of, forall_and, and_assoc']
+#align mem_residual_iff mem_residual_iff
+
+end residual
+

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

@@ -63,7 +63,7 @@ def IsGδ (s : Set α) : Prop :=
 #print IsOpen.isGδ /-
 /-- An open set is a Gδ set. -/
 theorem IsOpen.isGδ {s : Set α} (h : IsOpen s) : IsGδ s :=
-  ⟨{s}, by simp [h], countable_singleton _, (Set.interₛ_singleton _).symm⟩
+  ⟨{s}, by simp [h], countable_singleton _, (Set.sInter_singleton _).symm⟩
 #align is_open.is_Gδ IsOpen.isGδ
 -/
 
@@ -81,46 +81,46 @@ theorem isGδ_univ : IsGδ (univ : Set α) :=
 #align is_Gδ_univ isGδ_univ
 -/
 
-#print isGδ_binterᵢ_of_open /-
-theorem isGδ_binterᵢ_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set α}
+#print isGδ_biInter_of_open /-
+theorem isGδ_biInter_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set α}
     (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
   ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
-#align is_Gδ_bInter_of_open isGδ_binterᵢ_of_open
+#align is_Gδ_bInter_of_open isGδ_biInter_of_open
 -/
 
-#print isGδ_interᵢ_of_open /-
-theorem isGδ_interᵢ_of_open [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
+#print isGδ_iInter_of_open /-
+theorem isGδ_iInter_of_open [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
   ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
-#align is_Gδ_Inter_of_open isGδ_interᵢ_of_open
+#align is_Gδ_Inter_of_open isGδ_iInter_of_open
 -/
 
-#print isGδ_interᵢ /-
+#print isGδ_iInter /-
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
-theorem isGδ_interᵢ [Encodable ι] {s : ι → Set α} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) :=
+theorem isGδ_iInter [Encodable ι] {s : ι → Set α} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) :=
   by
   choose T hTo hTc hTs using hs
   obtain rfl : s = fun i => ⋂₀ T i := funext hTs
   refine' ⟨⋃ i, T i, _, countable_Union hTc, (sInter_Union _).symm⟩
   simpa [@forall_swap ι] using hTo
-#align is_Gδ_Inter isGδ_interᵢ
+#align is_Gδ_Inter isGδ_iInter
 -/
 
-#print isGδ_binterᵢ /-
-theorem isGδ_binterᵢ {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set α}
+#print isGδ_biInter /-
+theorem isGδ_biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set α}
     (ht : ∀ i ∈ s, IsGδ (t i ‹_›)) : IsGδ (⋂ i ∈ s, t i ‹_›) :=
   by
   rw [bInter_eq_Inter]
   haveI := hs.to_encodable
-  exact isGδ_interᵢ fun x => ht x x.2
-#align is_Gδ_bInter isGδ_binterᵢ
+  exact isGδ_iInter fun x => ht x x.2
+#align is_Gδ_bInter isGδ_biInter
 -/
 
-#print isGδ_interₛ /-
+#print isGδ_sInter /-
 /-- A countable intersection of Gδ sets is a Gδ set. -/
-theorem isGδ_interₛ {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
-  simpa only [sInter_eq_bInter] using isGδ_binterᵢ hS h
-#align is_Gδ_sInter isGδ_interₛ
+theorem isGδ_sInter {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
+  simpa only [sInter_eq_bInter] using isGδ_biInter hS h
+#align is_Gδ_sInter isGδ_sInter
 -/
 
 /- warning: is_Gδ.inter -> IsGδ.inter is a dubious translation:
@@ -132,7 +132,7 @@ Case conversion may be inaccurate. Consider using '#align is_Gδ.inter IsGδ.int
 theorem IsGδ.inter {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) :=
   by
   rw [inter_eq_Inter]
-  exact isGδ_interᵢ (Bool.forall_bool.2 ⟨ht, hs⟩)
+  exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
 #align is_Gδ.inter IsGδ.inter
 
 /- warning: is_Gδ.union -> IsGδ.union is a dubious translation:
@@ -147,19 +147,19 @@ theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪
   rcases hs with ⟨S, Sopen, Scount, rfl⟩
   rcases ht with ⟨T, Topen, Tcount, rfl⟩
   rw [sInter_union_sInter]
-  apply isGδ_binterᵢ_of_open (Scount.prod Tcount)
+  apply isGδ_biInter_of_open (Scount.prod Tcount)
   rintro ⟨a, b⟩ ⟨ha, hb⟩
   exact (Sopen a ha).union (Topen b hb)
 #align is_Gδ.union IsGδ.union
 
-#print isGδ_bunionᵢ /-
+#print isGδ_biUnion /-
 /-- The union of finitely many Gδ sets is a Gδ set. -/
-theorem isGδ_bunionᵢ {s : Set ι} (hs : s.Finite) {f : ι → Set α} (h : ∀ i ∈ s, IsGδ (f i)) :
+theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set α} (h : ∀ i ∈ s, IsGδ (f i)) :
     IsGδ (⋃ i ∈ s, f i) := by
   refine' finite.induction_on hs (by simp) _ h
   simp only [ball_insert_iff, bUnion_insert]
   exact fun a s _ _ ihs H => H.1.union (ihs H.2)
-#align is_Gδ_bUnion isGδ_bunionᵢ
+#align is_Gδ_bUnion isGδ_biUnion
 -/
 
 #print IsClosed.isGδ /-
@@ -168,8 +168,8 @@ theorem IsClosed.isGδ {α} [UniformSpace α] [IsCountablyGenerated (𝓤 α)] {
   by
   rcases(@uniformity_hasBasis_open α _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
   rw [← hs.closure_eq, ← hU.bInter_bUnion_ball]
-  refine' isGδ_binterᵢ (to_countable _) fun n hn => IsOpen.isGδ _
-  exact isOpen_bunionᵢ fun x hx => UniformSpace.isOpen_ball _ (hUo _).2
+  refine' isGδ_biInter (to_countable _) fun n hn => IsOpen.isGδ _
+  exact isOpen_biUnion fun x hx => UniformSpace.isOpen_ball _ (hUo _).2
 #align is_closed.is_Gδ IsClosed.isGδ
 -/
 
@@ -196,7 +196,7 @@ Case conversion may be inaccurate. Consider using '#align set.countable.is_Gδ_c
 theorem Set.Countable.isGδ_compl {s : Set α} (hs : s.Countable) : IsGδ (sᶜ) :=
   by
   rw [← bUnion_of_singleton s, compl_Union₂]
-  exact isGδ_binterᵢ hs fun x _ => isGδ_compl_singleton x
+  exact isGδ_biInter hs fun x _ => isGδ_compl_singleton x
 #align set.countable.is_Gδ_compl Set.Countable.isGδ_compl
 
 /- warning: set.finite.is_Gδ_compl -> Set.Finite.isGδ_compl is a dubious translation:
@@ -237,8 +237,8 @@ variable [FirstCountableTopology α]
 theorem isGδ_singleton (a : α) : IsGδ ({a} : Set α) :=
   by
   rcases(nhds_basis_opens a).exists_antitone_subbasis with ⟨U, hU, h_basis⟩
-  rw [← binterᵢ_basis_nhds h_basis.to_has_basis]
-  exact isGδ_binterᵢ (to_countable _) fun n hn => (hU n).2.IsGδ
+  rw [← biInter_basis_nhds h_basis.to_has_basis]
+  exact isGδ_biInter (to_countable _) fun n hn => (hU n).2.IsGδ
 #align is_Gδ_singleton isGδ_singleton
 -/
 
@@ -269,7 +269,7 @@ theorem isGδ_setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (𝓤 
   simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
   simp only [(nhds_basis_opens _).prod_self.tendsto_iffₓ hU.to_has_basis, forall_prop_of_true,
     set_of_forall, id]
-  refine' isGδ_interᵢ fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
+  refine' isGδ_iInter fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
   rintro ⟨s, ⟨hsx, hso⟩, hsU⟩
   filter_upwards [IsOpen.mem_nhds hso hsx]with _ hy using⟨s, ⟨hy, hso⟩, hsU⟩
 #align is_Gδ_set_of_continuous_at isGδ_setOf_continuousAt

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

Follow-up to #10816.

Remaining places containing such lemmas are

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

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

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

@@ -154,7 +154,7 @@ theorem IsGδ.sUnion {S : Set (Set X)} (hS : S.Finite) (h : ∀ s ∈ S, IsGδ s
   induction S, hS using Set.Finite.dinduction_on with
   | H0 => simp
   | H1 _ _ ih =>
-    simp only [ball_insert_iff, sUnion_insert] at *
+    simp only [forall_mem_insert, sUnion_insert] at *
     exact h.1.union (ih h.2)
 
 /-- The union of finitely many Gδ sets is a Gδ set, bounded indexed union version. -/

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

@@ -296,7 +296,7 @@ lemma IsClosed.isNowhereDense_iff {s : Set X} (hs : IsClosed s) :
     IsNowhereDense s ↔ interior s = ∅ := by
   rw [IsNowhereDense, IsClosed.closure_eq hs]
 
-/-- If a set `s` is nowhere dense, so is its closure.-/
+/-- If a set `s` is nowhere dense, so is its closure. -/
 protected lemma IsNowhereDense.closure {s : Set X} (hs : IsNowhereDense s) :
     IsNowhereDense (closure s) := by
   rwa [IsNowhereDense, closure_closure]

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

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

@@ -81,14 +81,14 @@ protected theorem IsGδ.univ : IsGδ (univ : Set X) :=
 
 theorem IsGδ.biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set X}
     (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
-  ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
+  ⟨f '' I, by rwa [forall_mem_image], hI.image _, by rw [sInter_image]⟩
 #align is_Gδ_bInter_of_open IsGδ.biInter_of_isOpen
 
 @[deprecated] alias isGδ_biInter_of_isOpen := IsGδ.biInter_of_isOpen -- 2024-02-15
 
 theorem IsGδ.iInter_of_isOpen [Countable ι'] {f : ι' → Set X} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
-  ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
+  ⟨range f, by rwa [forall_mem_range], countable_range _, by rw [sInter_range]⟩
 #align is_Gδ_Inter_of_open IsGδ.iInter_of_isOpen
 
 @[deprecated] alias isGδ_iInter_of_isOpen := IsGδ.iInter_of_isOpen -- 2024-02-15
@@ -161,7 +161,7 @@ theorem IsGδ.sUnion {S : Set (Set X)} (hS : S.Finite) (h : ∀ s ∈ S, IsGδ s
 theorem IsGδ.biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i)) :
     IsGδ (⋃ i ∈ s, f i) := by
   rw [← sUnion_image]
-  exact .sUnion (hs.image _) (ball_image_iff.2 h)
+  exact .sUnion (hs.image _) (forall_mem_image.2 h)
 #align is_Gδ_bUnion IsGδ.biUnion
 
 @[deprecated] -- 2024-02-15
@@ -169,7 +169,7 @@ alias isGδ_biUnion := IsGδ.biUnion
 
 /-- The union of finitely many Gδ sets is a Gδ set, bounded indexed union version. -/
 theorem IsGδ.iUnion [Finite ι'] {f : ι' → Set X} (h : ∀ i, IsGδ (f i)) : IsGδ (⋃ i, f i) :=
-  .sUnion (finite_range _) <| forall_range_iff.2 h
+  .sUnion (finite_range _) <| forall_mem_range.2 h
 
 theorem IsClosed.isGδ {X : Type*} [UniformSpace X] [IsCountablyGenerated (𝓤 X)] {s : Set X}
     (hs : IsClosed s) : IsGδ s := by
@@ -337,13 +337,13 @@ lemma isMeagre_iUnion {s : ℕ → Set X} (hs : ∀ n, IsMeagre (s n)) : IsMeagr
 lemma isMeagre_iff_countable_union_isNowhereDense {s : Set X} :
     IsMeagre s ↔ ∃ S : Set (Set X), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S := by
   rw [IsMeagre, mem_residual_iff, compl_bijective.surjective.image_surjective.exists]
-  simp_rw [← and_assoc, ← forall_and, ball_image_iff, ← isClosed_isNowhereDense_iff_compl,
+  simp_rw [← and_assoc, ← forall_and, forall_mem_image, ← isClosed_isNowhereDense_iff_compl,
     sInter_image, ← compl_iUnion₂, compl_subset_compl, ← sUnion_eq_biUnion, and_assoc]
-  refine ⟨fun ⟨S, hS, hc, hsub⟩ ↦ ⟨S, fun s hs ↦ (hS s hs).2, ?_, hsub⟩, ?_⟩
+  refine ⟨fun ⟨S, hS, hc, hsub⟩ ↦ ⟨S, fun s hs ↦ (hS hs).2, ?_, hsub⟩, ?_⟩
   · rw [← compl_compl_image S]; exact hc.image _
   · intro ⟨S, hS, hc, hsub⟩
     use closure '' S
-    rw [ball_image_iff]
+    rw [forall_mem_image]
     exact ⟨fun s hs ↦ ⟨isClosed_closure, (hS s hs).closure⟩,
       (hc.image _).image _, hsub.trans (sUnion_mono_subsets fun s ↦ subset_closure)⟩
 

Rename many isGδ_some lemmas to IsGδ.some. Also resolve a TODO.

@@ -66,24 +66,32 @@ theorem IsOpen.isGδ {s : Set X} (h : IsOpen s) : IsGδ s :=
 #align is_open.is_Gδ IsOpen.isGδ
 
 @[simp]
-theorem isGδ_empty : IsGδ (∅ : Set X) :=
+protected theorem IsGδ.empty : IsGδ (∅ : Set X) :=
   isOpen_empty.isGδ
-#align is_Gδ_empty isGδ_empty
+#align is_Gδ_empty IsGδ.empty
+
+@[deprecated] alias isGδ_empty := IsGδ.empty -- 2024-02-15
 
 @[simp]
-theorem isGδ_univ : IsGδ (univ : Set X) :=
+protected theorem IsGδ.univ : IsGδ (univ : Set X) :=
   isOpen_univ.isGδ
-#align is_Gδ_univ isGδ_univ
+#align is_Gδ_univ IsGδ.univ
+
+@[deprecated] alias isGδ_univ := IsGδ.univ -- 2024-02-15
 
-theorem isGδ_biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set X}
+theorem IsGδ.biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set X}
     (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
   ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
-#align is_Gδ_bInter_of_open isGδ_biInter_of_isOpen
+#align is_Gδ_bInter_of_open IsGδ.biInter_of_isOpen
 
-theorem isGδ_iInter_of_isOpen [Countable ι'] {f : ι' → Set X} (hf : ∀ i, IsOpen (f i)) :
+@[deprecated] alias isGδ_biInter_of_isOpen := IsGδ.biInter_of_isOpen -- 2024-02-15
+
+theorem IsGδ.iInter_of_isOpen [Countable ι'] {f : ι' → Set X} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
   ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
-#align is_Gδ_Inter_of_open isGδ_iInter_of_isOpen
+#align is_Gδ_Inter_of_open IsGδ.iInter_of_isOpen
+
+@[deprecated] alias isGδ_iInter_of_isOpen := IsGδ.iInter_of_isOpen -- 2024-02-15
 
 lemma isGδ_iff_eq_iInter_nat {s : Set X} :
     IsGδ s ↔ ∃ (f : ℕ → Set X), (∀ n, IsOpen (f n)) ∧ s = ⋂ n, f n := by
@@ -94,33 +102,41 @@ lemma isGδ_iff_eq_iInter_nat {s : Set X} :
     · obtain ⟨f, hf⟩ : ∃ (f : ℕ → Set X), T = range f := Countable.exists_eq_range T_count hT
       exact ⟨f, by aesop, by simp [hf]⟩
   · rintro ⟨f, hf, rfl⟩
-    apply isGδ_iInter_of_isOpen hf
+    exact .iInter_of_isOpen hf
 
 alias ⟨IsGδ.eq_iInter_nat, _⟩ := isGδ_iff_eq_iInter_nat
 
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
-theorem isGδ_iInter [Countable ι'] {s : ι' → Set X} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
+protected theorem IsGδ.iInter [Countable ι'] {s : ι' → Set X} (hs : ∀ i, IsGδ (s i)) :
+    IsGδ (⋂ i, s i) := by
   choose T hTo hTc hTs using hs
   obtain rfl : s = fun i => ⋂₀ T i := funext hTs
   refine' ⟨⋃ i, T i, _, countable_iUnion hTc, (sInter_iUnion _).symm⟩
   simpa [@forall_swap ι'] using hTo
-#align is_Gδ_Inter isGδ_iInter
+#align is_Gδ_Inter IsGδ.iInter
 
-theorem isGδ_biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set X}
+@[deprecated] alias isGδ_iInter := IsGδ.iInter
+
+theorem IsGδ.biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set X}
     (ht : ∀ (i) (hi : i ∈ s), IsGδ (t i hi)) : IsGδ (⋂ i ∈ s, t i ‹_›) := by
   rw [biInter_eq_iInter]
-  haveI := hs.toEncodable
-  exact isGδ_iInter fun x => ht x x.2
-#align is_Gδ_bInter isGδ_biInter
+  haveI := hs.to_subtype
+  exact .iInter fun x => ht x x.2
+#align is_Gδ_bInter IsGδ.biInter
+
+@[deprecated] alias isGδ_biInter := IsGδ.biInter -- 2024-02-15
 
 /-- A countable intersection of Gδ sets is a Gδ set. -/
-theorem isGδ_sInter {S : Set (Set X)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
-  simpa only [sInter_eq_biInter] using isGδ_biInter hS h
-#align is_Gδ_sInter isGδ_sInter
+theorem IsGδ.sInter {S : Set (Set X)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
+  simpa only [sInter_eq_biInter] using IsGδ.biInter hS h
+#align is_Gδ_sInter IsGδ.sInter
+
+@[deprecated] -- 2024-02-15
+alias isGδ_sInter := IsGδ.sInter
 
 theorem IsGδ.inter {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
   rw [inter_eq_iInter]
-  exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
+  exact .iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
 #align is_Gδ.inter IsGδ.inter
 
 /-- The union of two Gδ sets is a Gδ set. -/
@@ -128,25 +144,38 @@ theorem IsGδ.union {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t
   rcases hs with ⟨S, Sopen, Scount, rfl⟩
   rcases ht with ⟨T, Topen, Tcount, rfl⟩
   rw [sInter_union_sInter]
-  apply isGδ_biInter_of_isOpen (Scount.prod Tcount)
+  refine .biInter_of_isOpen (Scount.prod Tcount) ?_
   rintro ⟨a, b⟩ ⟨ha, hb⟩
   exact (Sopen a ha).union (Topen b hb)
 #align is_Gδ.union IsGδ.union
 
--- TODO: add `iUnion` and `sUnion` versions
-/-- The union of finitely many Gδ sets is a Gδ set. -/
-theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i)) :
+/-- The union of finitely many Gδ sets is a Gδ set, `Set.sUnion` version. -/
+theorem IsGδ.sUnion {S : Set (Set X)} (hS : S.Finite) (h : ∀ s ∈ S, IsGδ s) : IsGδ (⋃₀ S) := by
+  induction S, hS using Set.Finite.dinduction_on with
+  | H0 => simp
+  | H1 _ _ ih =>
+    simp only [ball_insert_iff, sUnion_insert] at *
+    exact h.1.union (ih h.2)
+
+/-- The union of finitely many Gδ sets is a Gδ set, bounded indexed union version. -/
+theorem IsGδ.biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i)) :
     IsGδ (⋃ i ∈ s, f i) := by
-  refine' Finite.induction_on hs (by simp) _ h
-  simp only [ball_insert_iff, biUnion_insert]
-  exact fun _ _ ihs H => H.1.union (ihs H.2)
-#align is_Gδ_bUnion isGδ_biUnion
+  rw [← sUnion_image]
+  exact .sUnion (hs.image _) (ball_image_iff.2 h)
+#align is_Gδ_bUnion IsGδ.biUnion
+
+@[deprecated] -- 2024-02-15
+alias isGδ_biUnion := IsGδ.biUnion
+
+/-- The union of finitely many Gδ sets is a Gδ set, bounded indexed union version. -/
+theorem IsGδ.iUnion [Finite ι'] {f : ι' → Set X} (h : ∀ i, IsGδ (f i)) : IsGδ (⋃ i, f i) :=
+  .sUnion (finite_range _) <| forall_range_iff.2 h
 
-theorem IsClosed.isGδ {X} [UniformSpace X] [IsCountablyGenerated (𝓤 X)] {s : Set X}
+theorem IsClosed.isGδ {X : Type*} [UniformSpace X] [IsCountablyGenerated (𝓤 X)] {s : Set X}
     (hs : IsClosed s) : IsGδ s := by
   rcases (@uniformity_hasBasis_open X _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
   rw [← hs.closure_eq, ← hU.biInter_biUnion_ball]
-  refine' isGδ_biInter (to_countable _) fun n _ => IsOpen.isGδ _
+  refine .biInter (to_countable _) fun n _ => IsOpen.isGδ ?_
   exact isOpen_biUnion fun x _ => UniformSpace.isOpen_ball _ (hUo _).2
 #align is_closed.is_Gδ IsClosed.isGδ
 
@@ -154,13 +183,15 @@ section T1Space
 
 variable [T1Space X]
 
-theorem isGδ_compl_singleton (x : X) : IsGδ ({x}ᶜ : Set X) :=
+theorem IsGδ.compl_singleton (x : X) : IsGδ ({x}ᶜ : Set X) :=
   isOpen_compl_singleton.isGδ
-#align is_Gδ_compl_singleton isGδ_compl_singleton
+#align is_Gδ_compl_singleton IsGδ.compl_singleton
+
+@[deprecated] alias isGδ_compl_singleton := IsGδ.compl_singleton -- 2024-02-15
 
 theorem Set.Countable.isGδ_compl {s : Set X} (hs : s.Countable) : IsGδ sᶜ := by
   rw [← biUnion_of_singleton s, compl_iUnion₂]
-  exact isGδ_biInter hs fun x _ => isGδ_compl_singleton x
+  exact .biInter hs fun x _ => .compl_singleton x
 #align set.countable.is_Gδ_compl Set.Countable.isGδ_compl
 
 theorem Set.Finite.isGδ_compl {s : Set X} (hs : s.Finite) : IsGδ sᶜ :=
@@ -177,14 +208,16 @@ theorem Finset.isGδ_compl (s : Finset X) : IsGδ (sᶜ : Set X) :=
 
 variable [FirstCountableTopology X]
 
-theorem isGδ_singleton (x : X) : IsGδ ({x} : Set X) := by
+protected theorem IsGδ.singleton (x : X) : IsGδ ({x} : Set X) := by
   rcases (nhds_basis_opens x).exists_antitone_subbasis with ⟨U, hU, h_basis⟩
   rw [← biInter_basis_nhds h_basis.toHasBasis]
-  exact isGδ_biInter (to_countable _) fun n _ => (hU n).2.isGδ
-#align is_Gδ_singleton isGδ_singleton
+  exact .biInter (to_countable _) fun n _ => (hU n).2.isGδ
+#align is_Gδ_singleton IsGδ.singleton
+
+@[deprecated] alias isGδ_singleton := IsGδ.singleton -- 2024-02-15
 
 theorem Set.Finite.isGδ {s : Set X} (hs : s.Finite) : IsGδ s :=
-  Finite.induction_on hs isGδ_empty fun _ _ hs => (isGδ_singleton _).union hs
+  Finite.induction_on hs .empty fun _ _ ↦ .union (.singleton _)
 #align set.finite.is_Gδ Set.Finite.isGδ
 
 end T1Space
@@ -196,16 +229,18 @@ section ContinuousAt
 variable [TopologicalSpace X]
 
 /-- The set of points where a function is continuous is a Gδ set. -/
-theorem isGδ_setOf_continuousAt [UniformSpace Y] [IsCountablyGenerated (𝓤 Y)] (f : X → Y) :
+theorem IsGδ.setOf_continuousAt [UniformSpace Y] [IsCountablyGenerated (𝓤 Y)] (f : X → Y) :
     IsGδ { x | ContinuousAt f x } := by
   obtain ⟨U, _, hU⟩ := (@uniformity_hasBasis_open_symmetric Y _).exists_antitone_subbasis
   simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
   simp only [(nhds_basis_opens _).prod_self.tendsto_iff hU.toHasBasis, forall_prop_of_true,
     setOf_forall, id]
-  refine' isGδ_iInter fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
+  refine .iInter fun k ↦ IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x ↦ ?_
   rintro ⟨s, ⟨hsx, hso⟩, hsU⟩
   filter_upwards [IsOpen.mem_nhds hso hsx] with _ hy using ⟨s, ⟨hy, hso⟩, hsU⟩
-#align is_Gδ_set_of_continuous_at isGδ_setOf_continuousAt
+#align is_Gδ_set_of_continuous_at IsGδ.setOf_continuousAt
+
+@[deprecated] alias isGδ_setOf_continuousAt := IsGδ.setOf_continuousAt -- 2024-02-15
 
 end ContinuousAt
 

A TODO was added inside a porting note. We change it to a TODO.

@@ -133,7 +133,7 @@ theorem IsGδ.union {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t
   exact (Sopen a ha).union (Topen b hb)
 #align is_Gδ.union IsGδ.union
 
--- porting note: TODO: add `iUnion` and `sUnion` versions
+-- TODO: add `iUnion` and `sUnion` versions
 /-- The union of finitely many Gδ sets is a Gδ set. -/
 theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i)) :
     IsGδ (⋃ i ∈ s, f i) := by

@@ -96,6 +96,8 @@ lemma isGδ_iff_eq_iInter_nat {s : Set X} :
   · rintro ⟨f, hf, rfl⟩
     apply isGδ_iInter_of_isOpen hf
 
+alias ⟨IsGδ.eq_iInter_nat, _⟩ := isGδ_iff_eq_iInter_nat
+
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
 theorem isGδ_iInter [Countable ι'] {s : ι' → Set X} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
   choose T hTo hTc hTs using hs

• Prove that a set is a Gdelta if and only if it is an intersection of open sets indexed by ℕ.
• Cleanup porting todos in the Gdelta file
• Rename cluster_point_of_compact to exists_clusterPt_of_compactSpace
• Generalize the class T2Space to T2OrLocallyCompactRegularSpace in the file Support.lean

@@ -45,8 +45,9 @@ Gδ set, residual set, nowhere dense set, meagre set
 noncomputable section
 
 open Topology TopologicalSpace Filter Encodable Set
+open scoped Uniformity
 
-variable {X Y ι : Type*}
+variable {X Y ι : Type*} {ι' : Sort*}
 
 set_option linter.uppercaseLean3 false
 
@@ -79,19 +80,28 @@ theorem isGδ_biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set
   ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
 #align is_Gδ_bInter_of_open isGδ_biInter_of_isOpen
 
--- porting note: TODO: generalize to `Sort*` + `Countable _`
-theorem isGδ_iInter_of_isOpen [Encodable ι] {f : ι → Set X} (hf : ∀ i, IsOpen (f i)) :
+theorem isGδ_iInter_of_isOpen [Countable ι'] {f : ι' → Set X} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
   ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
 #align is_Gδ_Inter_of_open isGδ_iInter_of_isOpen
 
--- porting note: TODO: generalize to `Sort*` + `Countable _`
+lemma isGδ_iff_eq_iInter_nat {s : Set X} :
+    IsGδ s ↔ ∃ (f : ℕ → Set X), (∀ n, IsOpen (f n)) ∧ s = ⋂ n, f n := by
+  refine ⟨?_, ?_⟩
+  · rintro ⟨T, hT, T_count, rfl⟩
+    rcases Set.eq_empty_or_nonempty T with rfl|hT
+    · exact ⟨fun _n ↦ univ, fun _n ↦ isOpen_univ, by simp⟩
+    · obtain ⟨f, hf⟩ : ∃ (f : ℕ → Set X), T = range f := Countable.exists_eq_range T_count hT
+      exact ⟨f, by aesop, by simp [hf]⟩
+  · rintro ⟨f, hf, rfl⟩
+    apply isGδ_iInter_of_isOpen hf
+
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
-theorem isGδ_iInter [Encodable ι] {s : ι → Set X} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
+theorem isGδ_iInter [Countable ι'] {s : ι' → Set X} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
   choose T hTo hTc hTs using hs
   obtain rfl : s = fun i => ⋂₀ T i := funext hTs
   refine' ⟨⋃ i, T i, _, countable_iUnion hTc, (sInter_iUnion _).symm⟩
-  simpa [@forall_swap ι] using hTo
+  simpa [@forall_swap ι'] using hTo
 #align is_Gδ_Inter isGδ_iInter
 
 theorem isGδ_biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set X}
@@ -130,8 +140,7 @@ theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i
   exact fun _ _ ihs H => H.1.union (ihs H.2)
 #align is_Gδ_bUnion isGδ_biUnion
 
--- Porting note: Did not recognize notation 𝓤 X, needed to replace with uniformity X
-theorem IsClosed.isGδ {X} [UniformSpace X] [IsCountablyGenerated (uniformity X)] {s : Set X}
+theorem IsClosed.isGδ {X} [UniformSpace X] [IsCountablyGenerated (𝓤 X)] {s : Set X}
     (hs : IsClosed s) : IsGδ s := by
   rcases (@uniformity_hasBasis_open X _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
   rw [← hs.closure_eq, ← hU.biInter_biUnion_ball]
@@ -185,7 +194,7 @@ section ContinuousAt
 variable [TopologicalSpace X]
 
 /-- The set of points where a function is continuous is a Gδ set. -/
-theorem isGδ_setOf_continuousAt [UniformSpace Y] [IsCountablyGenerated (uniformity Y)] (f : X → Y) :
+theorem isGδ_setOf_continuousAt [UniformSpace Y] [IsCountablyGenerated (𝓤 Y)] (f : X → Y) :
     IsGδ { x | ContinuousAt f x } := by
   obtain ⟨U, _, hU⟩ := (@uniformity_hasBasis_open_symmetric Y _).exists_antitone_subbasis
   simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]

The corresponding Prop is called IsMeagre.

@@ -32,7 +32,7 @@ continuity set of a function from a topological space to an (e)metric space is a
 
 - `isClosed_isNowhereDense_iff_compl`: a closed set is nowhere dense iff
 its complement is open and dense
-- `meagre_iff_countable_union_isNowhereDense`: a set is meagre iff it is contained in a countable
+- `isMeagre_iff_countable_union_isNowhereDense`: a set is meagre iff it is contained in a countable
 union of nowhere dense sets
 - subsets of meagre sets are meagre; countable unions of meagre sets are meagre
 
@@ -283,13 +283,13 @@ lemma IsMeagre.inter {s t : Set X} (hs : IsMeagre s) : IsMeagre (s ∩ t) :=
   hs.mono (inter_subset_left s t)
 
 /-- A countable union of meagre sets is meagre. -/
-lemma meagre_iUnion {s : ℕ → Set X} (hs : ∀ n, IsMeagre (s n)) : IsMeagre (⋃ n, s n) := by
+lemma isMeagre_iUnion {s : ℕ → Set X} (hs : ∀ n, IsMeagre (s n)) : IsMeagre (⋃ n, s n) := by
   rw [IsMeagre, compl_iUnion]
   exact countable_iInter_mem.mpr hs
 
 /-- A set is meagre iff it is contained in a countable union of nowhere dense sets. -/
-lemma meagre_iff_countable_union_isNowhereDense {s : Set X} : IsMeagre s ↔
-    ∃ S : Set (Set X), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S := by
+lemma isMeagre_iff_countable_union_isNowhereDense {s : Set X} :
+    IsMeagre s ↔ ∃ S : Set (Set X), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S := by
   rw [IsMeagre, mem_residual_iff, compl_bijective.surjective.image_surjective.exists]
   simp_rw [← and_assoc, ← forall_and, ball_image_iff, ← isClosed_isNowhereDense_iff_compl,
     sInter_image, ← compl_iUnion₂, compl_subset_compl, ← sUnion_eq_biUnion, and_assoc]

@@ -242,7 +242,7 @@ def IsNowhereDense (s : Set X) := interior (closure s) = ∅
 
 /-- The empty set is nowhere dense. -/
 @[simp]
-lemma isNowhereDense_of_empty : IsNowhereDense (∅ : Set X) := by
+lemma isNowhereDense_empty : IsNowhereDense (∅ : Set X) := by
   rw [IsNowhereDense, closure_empty, interior_empty]
 
 /-- A closed set is nowhere dense iff its interior is empty. -/

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

@@ -74,16 +74,16 @@ theorem isGδ_univ : IsGδ (univ : Set X) :=
   isOpen_univ.isGδ
 #align is_Gδ_univ isGδ_univ
 
-theorem isGδ_biInter_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set X}
+theorem isGδ_biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set X}
     (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
   ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
-#align is_Gδ_bInter_of_open isGδ_biInter_of_open
+#align is_Gδ_bInter_of_open isGδ_biInter_of_isOpen
 
 -- porting note: TODO: generalize to `Sort*` + `Countable _`
-theorem isGδ_iInter_of_open [Encodable ι] {f : ι → Set X} (hf : ∀ i, IsOpen (f i)) :
+theorem isGδ_iInter_of_isOpen [Encodable ι] {f : ι → Set X} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
   ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
-#align is_Gδ_Inter_of_open isGδ_iInter_of_open
+#align is_Gδ_Inter_of_open isGδ_iInter_of_isOpen
 
 -- porting note: TODO: generalize to `Sort*` + `Countable _`
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
@@ -116,7 +116,7 @@ theorem IsGδ.union {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t
   rcases hs with ⟨S, Sopen, Scount, rfl⟩
   rcases ht with ⟨T, Topen, Tcount, rfl⟩
   rw [sInter_union_sInter]
-  apply isGδ_biInter_of_open (Scount.prod Tcount)
+  apply isGδ_biInter_of_isOpen (Scount.prod Tcount)
   rintro ⟨a, b⟩ ⟨ha, hb⟩
   exact (Sopen a ha).union (Topen b hb)
 #align is_Gδ.union IsGδ.union

The corresponding definition is called IsNowhereDense; this matches rules 1 and 5 of the naming convention.

While at it, fix a doc comment (missing word), use a focusing dot (small oversight) and fix line length.

@@ -30,10 +30,10 @@ In this file we define `Gδ` sets and prove their basic properties.
 We prove that finite or countable intersections of Gδ sets are Gδ sets. We also prove that the
 continuity set of a function from a topological space to an (e)metric space is a Gδ set.
 
-- `closed_isNowhereDense_iff_compl`: a closed set is nowhere dense iff
+- `isClosed_isNowhereDense_iff_compl`: a closed set is nowhere dense iff
 its complement is open and dense
-- `meagre_iff_countable_union_nowhereDense`: a set is meagre iff it is contained in a countable
-union of nowhere dense
+- `meagre_iff_countable_union_isNowhereDense`: a set is meagre iff it is contained in a countable
+union of nowhere dense sets
 - subsets of meagre sets are meagre; countable unions of meagre sets are meagre
 
 ## Tags
@@ -256,12 +256,12 @@ protected lemma IsNowhereDense.closure {s : Set X} (hs : IsNowhereDense s) :
   rwa [IsNowhereDense, closure_closure]
 
 /-- A nowhere dense set `s` is contained in a closed nowhere dense set (namely, its closure). -/
-lemma IsNowhereDense.subset_of_closed_nowhereDense {s : Set X} (hs : IsNowhereDense s) :
+lemma IsNowhereDense.subset_of_closed_isNowhereDense {s : Set X} (hs : IsNowhereDense s) :
     ∃ t : Set X, s ⊆ t ∧ IsNowhereDense t ∧ IsClosed t :=
   ⟨closure s, subset_closure, ⟨hs.closure, isClosed_closure⟩⟩
 
 /-- A set `s` is closed and nowhere dense iff its complement `sᶜ` is open and dense. -/
-lemma closed_isNowhereDense_iff_compl {s : Set X} :
+lemma isClosed_isNowhereDense_iff_compl {s : Set X} :
     IsClosed s ∧ IsNowhereDense s ↔ IsOpen sᶜ ∧ Dense sᶜ := by
   rw [and_congr_right IsClosed.isNowhereDense_iff,
     isOpen_compl_iff, interior_eq_empty_iff_dense_compl]
@@ -288,16 +288,17 @@ lemma meagre_iUnion {s : ℕ → Set X} (hs : ∀ n, IsMeagre (s n)) : IsMeagre
   exact countable_iInter_mem.mpr hs
 
 /-- A set is meagre iff it is contained in a countable union of nowhere dense sets. -/
-lemma meagre_iff_countable_union_nowhereDense {s : Set X} : IsMeagre s ↔
+lemma meagre_iff_countable_union_isNowhereDense {s : Set X} : IsMeagre s ↔
     ∃ S : Set (Set X), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S := by
   rw [IsMeagre, mem_residual_iff, compl_bijective.surjective.image_surjective.exists]
-  simp_rw [← and_assoc, ← forall_and, ball_image_iff, ← closed_isNowhereDense_iff_compl,
+  simp_rw [← and_assoc, ← forall_and, ball_image_iff, ← isClosed_isNowhereDense_iff_compl,
     sInter_image, ← compl_iUnion₂, compl_subset_compl, ← sUnion_eq_biUnion, and_assoc]
-  refine ⟨fun ⟨S, hS, hc, hsub⟩ ↦ ⟨S, fun s hs ↦ (hS s hs).2, ?_, hsub⟩, fun ⟨S, hS, hc, hsub⟩ ↦ ?_⟩
+  refine ⟨fun ⟨S, hS, hc, hsub⟩ ↦ ⟨S, fun s hs ↦ (hS s hs).2, ?_, hsub⟩, ?_⟩
   · rw [← compl_compl_image S]; exact hc.image _
-  use closure '' S
-  rw [ball_image_iff]
-  exact ⟨fun s hs ↦ ⟨isClosed_closure, (hS s hs).closure⟩,
-    (hc.image _).image _, hsub.trans (sUnion_mono_subsets fun s ↦ subset_closure)⟩
+  · intro ⟨S, hS, hc, hsub⟩
+    use closure '' S
+    rw [ball_image_iff]
+    exact ⟨fun s hs ↦ ⟨isClosed_closure, (hS s hs).closure⟩,
+      (hc.image _).image _, hsub.trans (sUnion_mono_subsets fun s ↦ subset_closure)⟩
 
 end meagre

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

@@ -133,7 +133,7 @@ theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i
 -- Porting note: Did not recognize notation 𝓤 X, needed to replace with uniformity X
 theorem IsClosed.isGδ {X} [UniformSpace X] [IsCountablyGenerated (uniformity X)] {s : Set X}
     (hs : IsClosed s) : IsGδ s := by
-  rcases(@uniformity_hasBasis_open X _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
+  rcases (@uniformity_hasBasis_open X _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
   rw [← hs.closure_eq, ← hU.biInter_biUnion_ball]
   refine' isGδ_biInter (to_countable _) fun n _ => IsOpen.isGδ _
   exact isOpen_biUnion fun x _ => UniformSpace.isOpen_ball _ (hUo _).2

mathport was forgetting a space in filter_upwards [...]with instead of filter_upwards [...] with.

@@ -193,7 +193,7 @@ theorem isGδ_setOf_continuousAt [UniformSpace Y] [IsCountablyGenerated (uniform
     setOf_forall, id]
   refine' isGδ_iInter fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
   rintro ⟨s, ⟨hsx, hso⟩, hsU⟩
-  filter_upwards [IsOpen.mem_nhds hso hsx]with _ hy using⟨s, ⟨hy, hso⟩, hsU⟩
+  filter_upwards [IsOpen.mem_nhds hso hsx] with _ hy using ⟨s, ⟨hy, hso⟩, hsU⟩
 #align is_Gδ_set_of_continuous_at isGδ_setOf_continuousAt
 
 end ContinuousAt

Greek letters for topological spaces are outdated, use letters X, Y, Z instead. Zulip discussion.

@@ -46,55 +46,55 @@ noncomputable section
 
 open Topology TopologicalSpace Filter Encodable Set
 
-variable {α β γ ι : Type*}
+variable {X Y ι : Type*}
 
 set_option linter.uppercaseLean3 false
 
 section IsGδ
 
-variable [TopologicalSpace α]
+variable [TopologicalSpace X]
 
 /-- A Gδ set is a countable intersection of open sets. -/
-def IsGδ (s : Set α) : Prop :=
-  ∃ T : Set (Set α), (∀ t ∈ T, IsOpen t) ∧ T.Countable ∧ s = ⋂₀ T
+def IsGδ (s : Set X) : Prop :=
+  ∃ T : Set (Set X), (∀ t ∈ T, IsOpen t) ∧ T.Countable ∧ s = ⋂₀ T
 #align is_Gδ IsGδ
 
 /-- An open set is a Gδ set. -/
-theorem IsOpen.isGδ {s : Set α} (h : IsOpen s) : IsGδ s :=
+theorem IsOpen.isGδ {s : Set X} (h : IsOpen s) : IsGδ s :=
   ⟨{s}, by simp [h], countable_singleton _, (Set.sInter_singleton _).symm⟩
 #align is_open.is_Gδ IsOpen.isGδ
 
 @[simp]
-theorem isGδ_empty : IsGδ (∅ : Set α) :=
+theorem isGδ_empty : IsGδ (∅ : Set X) :=
   isOpen_empty.isGδ
 #align is_Gδ_empty isGδ_empty
 
 @[simp]
-theorem isGδ_univ : IsGδ (univ : Set α) :=
+theorem isGδ_univ : IsGδ (univ : Set X) :=
   isOpen_univ.isGδ
 #align is_Gδ_univ isGδ_univ
 
-theorem isGδ_biInter_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set α}
+theorem isGδ_biInter_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set X}
     (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
   ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
 #align is_Gδ_bInter_of_open isGδ_biInter_of_open
 
 -- porting note: TODO: generalize to `Sort*` + `Countable _`
-theorem isGδ_iInter_of_open [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
+theorem isGδ_iInter_of_open [Encodable ι] {f : ι → Set X} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
   ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
 #align is_Gδ_Inter_of_open isGδ_iInter_of_open
 
 -- porting note: TODO: generalize to `Sort*` + `Countable _`
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
-theorem isGδ_iInter [Encodable ι] {s : ι → Set α} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
+theorem isGδ_iInter [Encodable ι] {s : ι → Set X} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
   choose T hTo hTc hTs using hs
   obtain rfl : s = fun i => ⋂₀ T i := funext hTs
   refine' ⟨⋃ i, T i, _, countable_iUnion hTc, (sInter_iUnion _).symm⟩
   simpa [@forall_swap ι] using hTo
 #align is_Gδ_Inter isGδ_iInter
 
-theorem isGδ_biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set α}
+theorem isGδ_biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set X}
     (ht : ∀ (i) (hi : i ∈ s), IsGδ (t i hi)) : IsGδ (⋂ i ∈ s, t i ‹_›) := by
   rw [biInter_eq_iInter]
   haveI := hs.toEncodable
@@ -102,17 +102,17 @@ theorem isGδ_biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set α}
 #align is_Gδ_bInter isGδ_biInter
 
 /-- A countable intersection of Gδ sets is a Gδ set. -/
-theorem isGδ_sInter {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
+theorem isGδ_sInter {S : Set (Set X)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
   simpa only [sInter_eq_biInter] using isGδ_biInter hS h
 #align is_Gδ_sInter isGδ_sInter
 
-theorem IsGδ.inter {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
+theorem IsGδ.inter {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
   rw [inter_eq_iInter]
   exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
 #align is_Gδ.inter IsGδ.inter
 
 /-- The union of two Gδ sets is a Gδ set. -/
-theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t) := by
+theorem IsGδ.union {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t) := by
   rcases hs with ⟨S, Sopen, Scount, rfl⟩
   rcases ht with ⟨T, Topen, Tcount, rfl⟩
   rw [sInter_union_sInter]
@@ -123,17 +123,17 @@ theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪
 
 -- porting note: TODO: add `iUnion` and `sUnion` versions
 /-- The union of finitely many Gδ sets is a Gδ set. -/
-theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set α} (h : ∀ i ∈ s, IsGδ (f i)) :
+theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i)) :
     IsGδ (⋃ i ∈ s, f i) := by
   refine' Finite.induction_on hs (by simp) _ h
   simp only [ball_insert_iff, biUnion_insert]
   exact fun _ _ ihs H => H.1.union (ihs H.2)
 #align is_Gδ_bUnion isGδ_biUnion
 
--- Porting note: Did not recognize notation 𝓤 α, needed to replace with uniformity α
-theorem IsClosed.isGδ {α} [UniformSpace α] [IsCountablyGenerated (uniformity α)] {s : Set α}
+-- Porting note: Did not recognize notation 𝓤 X, needed to replace with uniformity X
+theorem IsClosed.isGδ {X} [UniformSpace X] [IsCountablyGenerated (uniformity X)] {s : Set X}
     (hs : IsClosed s) : IsGδ s := by
-  rcases(@uniformity_hasBasis_open α _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
+  rcases(@uniformity_hasBasis_open X _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
   rw [← hs.closure_eq, ← hU.biInter_biUnion_ball]
   refine' isGδ_biInter (to_countable _) fun n _ => IsOpen.isGδ _
   exact isOpen_biUnion fun x _ => UniformSpace.isOpen_ball _ (hUo _).2
@@ -141,38 +141,38 @@ theorem IsClosed.isGδ {α} [UniformSpace α] [IsCountablyGenerated (uniformity
 
 section T1Space
 
-variable [T1Space α]
+variable [T1Space X]
 
-theorem isGδ_compl_singleton (a : α) : IsGδ ({a}ᶜ : Set α) :=
+theorem isGδ_compl_singleton (x : X) : IsGδ ({x}ᶜ : Set X) :=
   isOpen_compl_singleton.isGδ
 #align is_Gδ_compl_singleton isGδ_compl_singleton
 
-theorem Set.Countable.isGδ_compl {s : Set α} (hs : s.Countable) : IsGδ sᶜ := by
+theorem Set.Countable.isGδ_compl {s : Set X} (hs : s.Countable) : IsGδ sᶜ := by
   rw [← biUnion_of_singleton s, compl_iUnion₂]
   exact isGδ_biInter hs fun x _ => isGδ_compl_singleton x
 #align set.countable.is_Gδ_compl Set.Countable.isGδ_compl
 
-theorem Set.Finite.isGδ_compl {s : Set α} (hs : s.Finite) : IsGδ sᶜ :=
+theorem Set.Finite.isGδ_compl {s : Set X} (hs : s.Finite) : IsGδ sᶜ :=
   hs.countable.isGδ_compl
 #align set.finite.is_Gδ_compl Set.Finite.isGδ_compl
 
-theorem Set.Subsingleton.isGδ_compl {s : Set α} (hs : s.Subsingleton) : IsGδ sᶜ :=
+theorem Set.Subsingleton.isGδ_compl {s : Set X} (hs : s.Subsingleton) : IsGδ sᶜ :=
   hs.finite.isGδ_compl
 #align set.subsingleton.is_Gδ_compl Set.Subsingleton.isGδ_compl
 
-theorem Finset.isGδ_compl (s : Finset α) : IsGδ (sᶜ : Set α) :=
+theorem Finset.isGδ_compl (s : Finset X) : IsGδ (sᶜ : Set X) :=
   s.finite_toSet.isGδ_compl
 #align finset.is_Gδ_compl Finset.isGδ_compl
 
-variable [FirstCountableTopology α]
+variable [FirstCountableTopology X]
 
-theorem isGδ_singleton (a : α) : IsGδ ({a} : Set α) := by
-  rcases (nhds_basis_opens a).exists_antitone_subbasis with ⟨U, hU, h_basis⟩
+theorem isGδ_singleton (x : X) : IsGδ ({x} : Set X) := by
+  rcases (nhds_basis_opens x).exists_antitone_subbasis with ⟨U, hU, h_basis⟩
   rw [← biInter_basis_nhds h_basis.toHasBasis]
   exact isGδ_biInter (to_countable _) fun n _ => (hU n).2.isGδ
 #align is_Gδ_singleton isGδ_singleton
 
-theorem Set.Finite.isGδ {s : Set α} (hs : s.Finite) : IsGδ s :=
+theorem Set.Finite.isGδ {s : Set X} (hs : s.Finite) : IsGδ s :=
   Finite.induction_on hs isGδ_empty fun _ _ hs => (isGδ_singleton _).union hs
 #align set.finite.is_Gδ Set.Finite.isGδ
 
@@ -182,12 +182,12 @@ end IsGδ
 
 section ContinuousAt
 
-variable [TopologicalSpace α]
+variable [TopologicalSpace X]
 
 /-- The set of points where a function is continuous is a Gδ set. -/
-theorem isGδ_setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (uniformity β)] (f : α → β) :
+theorem isGδ_setOf_continuousAt [UniformSpace Y] [IsCountablyGenerated (uniformity Y)] (f : X → Y) :
     IsGδ { x | ContinuousAt f x } := by
-  obtain ⟨U, _, hU⟩ := (@uniformity_hasBasis_open_symmetric β _).exists_antitone_subbasis
+  obtain ⟨U, _, hU⟩ := (@uniformity_hasBasis_open_symmetric Y _).exists_antitone_subbasis
   simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
   simp only [(nhds_basis_opens _).prod_self.tendsto_iff hU.toHasBasis, forall_prop_of_true,
     setOf_forall, id]
@@ -200,24 +200,24 @@ end ContinuousAt
 
 section residual
 
-variable [TopologicalSpace α]
+variable [TopologicalSpace X]
 
 /-- A set `s` is called *residual* if it includes a countable intersection of dense open sets. -/
-def residual (α : Type*) [TopologicalSpace α] : Filter α :=
+def residual (X : Type*) [TopologicalSpace X] : Filter X :=
   Filter.countableGenerate { t | IsOpen t ∧ Dense t }
 #align residual residual
 
-instance countableInterFilter_residual : CountableInterFilter (residual α) := by
+instance countableInterFilter_residual : CountableInterFilter (residual X) := by
   rw [residual]; infer_instance
 #align countable_Inter_filter_residual countableInterFilter_residual
 
 /-- Dense open sets are residual. -/
-theorem residual_of_dense_open {s : Set α} (ho : IsOpen s) (hd : Dense s) : s ∈ residual α :=
+theorem residual_of_dense_open {s : Set X} (ho : IsOpen s) (hd : Dense s) : s ∈ residual X :=
   CountableGenerateSets.basic ⟨ho, hd⟩
 #align residual_of_dense_open residual_of_dense_open
 
 /-- Dense Gδ sets are residual. -/
-theorem residual_of_dense_Gδ {s : Set α} (ho : IsGδ s) (hd : Dense s) : s ∈ residual α := by
+theorem residual_of_dense_Gδ {s : Set X} (ho : IsGδ s) (hd : Dense s) : s ∈ residual X := by
   rcases ho with ⟨T, To, Tct, rfl⟩
   exact
     (countable_sInter_mem Tct).mpr fun t tT =>
@@ -225,9 +225,9 @@ theorem residual_of_dense_Gδ {s : Set α} (ho : IsGδ s) (hd : Dense s) : s ∈
 #align residual_of_dense_Gδ residual_of_dense_Gδ
 
 /-- A set is residual iff it includes a countable intersection of dense open sets. -/
-theorem mem_residual_iff {s : Set α} :
-    s ∈ residual α ↔
-      ∃ S : Set (Set α), (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s :=
+theorem mem_residual_iff {s : Set X} :
+    s ∈ residual X ↔
+      ∃ S : Set (Set X), (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s :=
   mem_countableGenerate_iff.trans <| by simp_rw [subset_def, mem_setOf, forall_and, and_assoc]
 #align mem_residual_iff mem_residual_iff
 
@@ -235,61 +235,61 @@ end residual
 
 section meagre
 open Function TopologicalSpace Set
-variable {α : Type*} [TopologicalSpace α]
+variable {X : Type*} [TopologicalSpace X]
 
 /-- A set is called **nowhere dense** iff its closure has empty interior. -/
-def IsNowhereDense (s : Set α) := interior (closure s) = ∅
+def IsNowhereDense (s : Set X) := interior (closure s) = ∅
 
 /-- The empty set is nowhere dense. -/
 @[simp]
-lemma isNowhereDense_of_empty : IsNowhereDense (∅ : Set α) := by
+lemma isNowhereDense_of_empty : IsNowhereDense (∅ : Set X) := by
   rw [IsNowhereDense, closure_empty, interior_empty]
 
 /-- A closed set is nowhere dense iff its interior is empty. -/
-lemma IsClosed.isNowhereDense_iff {s : Set α} (hs : IsClosed s) :
+lemma IsClosed.isNowhereDense_iff {s : Set X} (hs : IsClosed s) :
     IsNowhereDense s ↔ interior s = ∅ := by
   rw [IsNowhereDense, IsClosed.closure_eq hs]
 
 /-- If a set `s` is nowhere dense, so is its closure.-/
-protected lemma IsNowhereDense.closure {s : Set α} (hs : IsNowhereDense s) :
+protected lemma IsNowhereDense.closure {s : Set X} (hs : IsNowhereDense s) :
     IsNowhereDense (closure s) := by
   rwa [IsNowhereDense, closure_closure]
 
 /-- A nowhere dense set `s` is contained in a closed nowhere dense set (namely, its closure). -/
-lemma IsNowhereDense.subset_of_closed_nowhereDense {s : Set α} (hs : IsNowhereDense s) :
-    ∃ t : Set α, s ⊆ t ∧ IsNowhereDense t ∧ IsClosed t :=
+lemma IsNowhereDense.subset_of_closed_nowhereDense {s : Set X} (hs : IsNowhereDense s) :
+    ∃ t : Set X, s ⊆ t ∧ IsNowhereDense t ∧ IsClosed t :=
   ⟨closure s, subset_closure, ⟨hs.closure, isClosed_closure⟩⟩
 
 /-- A set `s` is closed and nowhere dense iff its complement `sᶜ` is open and dense. -/
-lemma closed_isNowhereDense_iff_compl {s : Set α} :
+lemma closed_isNowhereDense_iff_compl {s : Set X} :
     IsClosed s ∧ IsNowhereDense s ↔ IsOpen sᶜ ∧ Dense sᶜ := by
   rw [and_congr_right IsClosed.isNowhereDense_iff,
     isOpen_compl_iff, interior_eq_empty_iff_dense_compl]
 
 /-- A set is called **meagre** iff its complement is a residual (or comeagre) set. -/
-def IsMeagre (s : Set α) := sᶜ ∈ residual α
+def IsMeagre (s : Set X) := sᶜ ∈ residual X
 
 /-- The empty set is meagre. -/
-lemma meagre_empty : IsMeagre (∅ : Set α) := by
+lemma meagre_empty : IsMeagre (∅ : Set X) := by
   rw [IsMeagre, compl_empty]
   exact Filter.univ_mem
 
 /-- Subsets of meagre sets are meagre. -/
-lemma IsMeagre.mono {s t : Set α} (hs : IsMeagre s) (hts: t ⊆ s) : IsMeagre t :=
+lemma IsMeagre.mono {s t : Set X} (hs : IsMeagre s) (hts: t ⊆ s) : IsMeagre t :=
   Filter.mem_of_superset hs (compl_subset_compl.mpr hts)
 
 /-- An intersection with a meagre set is meagre. -/
-lemma IsMeagre.inter {s t : Set α} (hs : IsMeagre s) : IsMeagre (s ∩ t) :=
+lemma IsMeagre.inter {s t : Set X} (hs : IsMeagre s) : IsMeagre (s ∩ t) :=
   hs.mono (inter_subset_left s t)
 
 /-- A countable union of meagre sets is meagre. -/
-lemma meagre_iUnion {s : ℕ → Set α} (hs : ∀ n, IsMeagre (s n)) : IsMeagre (⋃ n, s n) := by
+lemma meagre_iUnion {s : ℕ → Set X} (hs : ∀ n, IsMeagre (s n)) : IsMeagre (⋃ n, s n) := by
   rw [IsMeagre, compl_iUnion]
   exact countable_iInter_mem.mpr hs
 
 /-- A set is meagre iff it is contained in a countable union of nowhere dense sets. -/
-lemma meagre_iff_countable_union_nowhereDense {s : Set α} : IsMeagre s ↔
-    ∃ S : Set (Set α), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S := by
+lemma meagre_iff_countable_union_nowhereDense {s : Set X} : IsMeagre s ↔
+    ∃ S : Set (Set X), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S := by
   rw [IsMeagre, mem_residual_iff, compl_bijective.surjective.image_surjective.exists]
   simp_rw [← and_assoc, ← forall_and, ball_image_iff, ← closed_isNowhereDense_iff_compl,
     sInter_image, ← compl_iUnion₂, compl_subset_compl, ← sUnion_eq_biUnion, and_assoc]

Define nowhere dense and meagre sets and show their basic properties. Meagre sets are defined as the complement of comeagre(=residual) sets (and shown to be equivalent to the standard definition); we deduce their API from the API for comeagre sets.

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

@@ -22,14 +22,23 @@ In this file we define `Gδ` sets and prove their basic properties.
 * `residual`: the σ-filter of residual sets. A set `s` is called *residual* if it includes a
   countable intersection of dense open sets.
 
+* `IsNowhereDense`: a set is called *nowhere dense* iff its closure has empty interior
+* `IsMeagre`: a set `s` is called *meagre* iff its complement is residual
+
 ## Main results
 
-We prove that finite or countable intersections of Gδ sets is a Gδ set. We also prove that the
+We prove that finite or countable intersections of Gδ sets are Gδ sets. We also prove that the
 continuity set of a function from a topological space to an (e)metric space is a Gδ set.
 
+- `closed_isNowhereDense_iff_compl`: a closed set is nowhere dense iff
+its complement is open and dense
+- `meagre_iff_countable_union_nowhereDense`: a set is meagre iff it is contained in a countable
+union of nowhere dense
+- subsets of meagre sets are meagre; countable unions of meagre sets are meagre
+
 ## Tags
 
-Gδ set, residual set
+Gδ set, residual set, nowhere dense set, meagre set
 -/
 
 
@@ -223,3 +232,72 @@ theorem mem_residual_iff {s : Set α} :
 #align mem_residual_iff mem_residual_iff
 
 end residual
+
+section meagre
+open Function TopologicalSpace Set
+variable {α : Type*} [TopologicalSpace α]
+
+/-- A set is called **nowhere dense** iff its closure has empty interior. -/
+def IsNowhereDense (s : Set α) := interior (closure s) = ∅
+
+/-- The empty set is nowhere dense. -/
+@[simp]
+lemma isNowhereDense_of_empty : IsNowhereDense (∅ : Set α) := by
+  rw [IsNowhereDense, closure_empty, interior_empty]
+
+/-- A closed set is nowhere dense iff its interior is empty. -/
+lemma IsClosed.isNowhereDense_iff {s : Set α} (hs : IsClosed s) :
+    IsNowhereDense s ↔ interior s = ∅ := by
+  rw [IsNowhereDense, IsClosed.closure_eq hs]
+
+/-- If a set `s` is nowhere dense, so is its closure.-/
+protected lemma IsNowhereDense.closure {s : Set α} (hs : IsNowhereDense s) :
+    IsNowhereDense (closure s) := by
+  rwa [IsNowhereDense, closure_closure]
+
+/-- A nowhere dense set `s` is contained in a closed nowhere dense set (namely, its closure). -/
+lemma IsNowhereDense.subset_of_closed_nowhereDense {s : Set α} (hs : IsNowhereDense s) :
+    ∃ t : Set α, s ⊆ t ∧ IsNowhereDense t ∧ IsClosed t :=
+  ⟨closure s, subset_closure, ⟨hs.closure, isClosed_closure⟩⟩
+
+/-- A set `s` is closed and nowhere dense iff its complement `sᶜ` is open and dense. -/
+lemma closed_isNowhereDense_iff_compl {s : Set α} :
+    IsClosed s ∧ IsNowhereDense s ↔ IsOpen sᶜ ∧ Dense sᶜ := by
+  rw [and_congr_right IsClosed.isNowhereDense_iff,
+    isOpen_compl_iff, interior_eq_empty_iff_dense_compl]
+
+/-- A set is called **meagre** iff its complement is a residual (or comeagre) set. -/
+def IsMeagre (s : Set α) := sᶜ ∈ residual α
+
+/-- The empty set is meagre. -/
+lemma meagre_empty : IsMeagre (∅ : Set α) := by
+  rw [IsMeagre, compl_empty]
+  exact Filter.univ_mem
+
+/-- Subsets of meagre sets are meagre. -/
+lemma IsMeagre.mono {s t : Set α} (hs : IsMeagre s) (hts: t ⊆ s) : IsMeagre t :=
+  Filter.mem_of_superset hs (compl_subset_compl.mpr hts)
+
+/-- An intersection with a meagre set is meagre. -/
+lemma IsMeagre.inter {s t : Set α} (hs : IsMeagre s) : IsMeagre (s ∩ t) :=
+  hs.mono (inter_subset_left s t)
+
+/-- A countable union of meagre sets is meagre. -/
+lemma meagre_iUnion {s : ℕ → Set α} (hs : ∀ n, IsMeagre (s n)) : IsMeagre (⋃ n, s n) := by
+  rw [IsMeagre, compl_iUnion]
+  exact countable_iInter_mem.mpr hs
+
+/-- A set is meagre iff it is contained in a countable union of nowhere dense sets. -/
+lemma meagre_iff_countable_union_nowhereDense {s : Set α} : IsMeagre s ↔
+    ∃ S : Set (Set α), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S := by
+  rw [IsMeagre, mem_residual_iff, compl_bijective.surjective.image_surjective.exists]
+  simp_rw [← and_assoc, ← forall_and, ball_image_iff, ← closed_isNowhereDense_iff_compl,
+    sInter_image, ← compl_iUnion₂, compl_subset_compl, ← sUnion_eq_biUnion, and_assoc]
+  refine ⟨fun ⟨S, hS, hc, hsub⟩ ↦ ⟨S, fun s hs ↦ (hS s hs).2, ?_, hsub⟩, fun ⟨S, hS, hc, hsub⟩ ↦ ?_⟩
+  · rw [← compl_compl_image S]; exact hc.image _
+  use closure '' S
+  rw [ball_image_iff]
+  exact ⟨fun s hs ↦ ⟨isClosed_closure, (hS s hs).closure⟩,
+    (hc.image _).image _, hsub.trans (sUnion_mono_subsets fun s ↦ subset_closure)⟩
+
+end meagre

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

This has nice performance benefits.

@@ -37,7 +37,7 @@ noncomputable section
 
 open Topology TopologicalSpace Filter Encodable Set
 
-variable {α β γ ι : Type _}
+variable {α β γ ι : Type*}
 
 set_option linter.uppercaseLean3 false
 
@@ -70,13 +70,13 @@ theorem isGδ_biInter_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set α
   ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
 #align is_Gδ_bInter_of_open isGδ_biInter_of_open
 
--- porting note: TODO: generalize to `Sort _` + `Countable _`
+-- porting note: TODO: generalize to `Sort*` + `Countable _`
 theorem isGδ_iInter_of_open [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
   ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
 #align is_Gδ_Inter_of_open isGδ_iInter_of_open
 
--- porting note: TODO: generalize to `Sort _` + `Countable _`
+-- porting note: TODO: generalize to `Sort*` + `Countable _`
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
 theorem isGδ_iInter [Encodable ι] {s : ι → Set α} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
   choose T hTo hTc hTs using hs
@@ -194,7 +194,7 @@ section residual
 variable [TopologicalSpace α]
 
 /-- A set `s` is called *residual* if it includes a countable intersection of dense open sets. -/
-def residual (α : Type _) [TopologicalSpace α] : Filter α :=
+def residual (α : Type*) [TopologicalSpace α] : Filter α :=
   Filter.countableGenerate { t | IsOpen t ∧ Dense t }
 #align residual residual
 

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.G_delta
-! leanprover-community/mathlib commit b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.UniformSpace.Basic
 import Mathlib.Topology.Separation
 import Mathlib.Order.Filter.CountableInter
 
+#align_import topology.G_delta from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
+
 /-!
 # `Gδ` sets
 

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

@@ -141,16 +141,16 @@ theorem isGδ_compl_singleton (a : α) : IsGδ ({a}ᶜ : Set α) :=
   isOpen_compl_singleton.isGδ
 #align is_Gδ_compl_singleton isGδ_compl_singleton
 
-theorem Set.Countable.isGδ_compl {s : Set α} (hs : s.Countable) : IsGδ (sᶜ) := by
+theorem Set.Countable.isGδ_compl {s : Set α} (hs : s.Countable) : IsGδ sᶜ := by
   rw [← biUnion_of_singleton s, compl_iUnion₂]
   exact isGδ_biInter hs fun x _ => isGδ_compl_singleton x
 #align set.countable.is_Gδ_compl Set.Countable.isGδ_compl
 
-theorem Set.Finite.isGδ_compl {s : Set α} (hs : s.Finite) : IsGδ (sᶜ) :=
+theorem Set.Finite.isGδ_compl {s : Set α} (hs : s.Finite) : IsGδ sᶜ :=
   hs.countable.isGδ_compl
 #align set.finite.is_Gδ_compl Set.Finite.isGδ_compl
 
-theorem Set.Subsingleton.isGδ_compl {s : Set α} (hs : s.Subsingleton) : IsGδ (sᶜ) :=
+theorem Set.Subsingleton.isGδ_compl {s : Set α} (hs : s.Subsingleton) : IsGδ sᶜ :=
   hs.finite.isGδ_compl
 #align set.subsingleton.is_Gδ_compl Set.Subsingleton.isGδ_compl
 

Co-authored-by: Chris Hughes <33847686+ChrisHughes24@users.noreply.github.com>

@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.G_delta
-! leanprover-community/mathlib commit b363547b3113d350d053abdf2884e9850a56b205
+! leanprover-community/mathlib commit b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathlib.Topology.UniformSpace.Basic
 import Mathlib.Topology.Separation
+import Mathlib.Order.Filter.CountableInter
 
 /-!
 # `Gδ` sets
@@ -21,11 +22,8 @@ In this file we define `Gδ` sets and prove their basic properties.
 * `IsGδ`: a set `s` is a `Gδ` set if it can be represented as an intersection
   of countably many open sets;
 
-* `residual`: the filter of residual sets. A set `s` is called *residual* if it includes a dense
-  `Gδ` set. In a Baire space (e.g., in a complete (e)metric space), residual sets form a filter.
-
-  For technical reasons, we define `residual` in any topological space but the definition agrees
-  with the description above only in Baire spaces.
+* `residual`: the σ-filter of residual sets. A set `s` is called *residual* if it includes a
+  countable intersection of dense open sets.
 
 ## Main results
 
@@ -194,11 +192,37 @@ theorem isGδ_setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (unifor
 
 end ContinuousAt
 
-/-- A set `s` is called *residual* if it includes a dense `Gδ` set. If `α` is a Baire space
-(e.g., a complete metric space), then residual sets form a filter, see `mem_residual`.
+section residual
+
+variable [TopologicalSpace α]
 
-For technical reasons we define the filter `residual` in any topological space but in a non-Baire
-space it is not useful because it may contain some non-residual sets. -/
+/-- A set `s` is called *residual* if it includes a countable intersection of dense open sets. -/
 def residual (α : Type _) [TopologicalSpace α] : Filter α :=
-  ⨅ (t) (_ht : IsGδ t) (_ht' : Dense t), 𝓟 t
+  Filter.countableGenerate { t | IsOpen t ∧ Dense t }
 #align residual residual
+
+instance countableInterFilter_residual : CountableInterFilter (residual α) := by
+  rw [residual]; infer_instance
+#align countable_Inter_filter_residual countableInterFilter_residual
+
+/-- Dense open sets are residual. -/
+theorem residual_of_dense_open {s : Set α} (ho : IsOpen s) (hd : Dense s) : s ∈ residual α :=
+  CountableGenerateSets.basic ⟨ho, hd⟩
+#align residual_of_dense_open residual_of_dense_open
+
+/-- Dense Gδ sets are residual. -/
+theorem residual_of_dense_Gδ {s : Set α} (ho : IsGδ s) (hd : Dense s) : s ∈ residual α := by
+  rcases ho with ⟨T, To, Tct, rfl⟩
+  exact
+    (countable_sInter_mem Tct).mpr fun t tT =>
+      residual_of_dense_open (To t tT) (hd.mono (sInter_subset_of_mem tT))
+#align residual_of_dense_Gδ residual_of_dense_Gδ
+
+/-- A set is residual iff it includes a countable intersection of dense open sets. -/
+theorem mem_residual_iff {s : Set α} :
+    s ∈ residual α ↔
+      ∃ S : Set (Set α), (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s :=
+  mem_countableGenerate_iff.trans <| by simp_rw [subset_def, mem_setOf, forall_and, and_assoc]
+#align mem_residual_iff mem_residual_iff
+
+end residual

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>

@@ -57,7 +57,7 @@ def IsGδ (s : Set α) : Prop :=
 
 /-- An open set is a Gδ set. -/
 theorem IsOpen.isGδ {s : Set α} (h : IsOpen s) : IsGδ s :=
-  ⟨{s}, by simp [h], countable_singleton _, (Set.interₛ_singleton _).symm⟩
+  ⟨{s}, by simp [h], countable_singleton _, (Set.sInter_singleton _).symm⟩
 #align is_open.is_Gδ IsOpen.isGδ
 
 @[simp]
@@ -70,69 +70,69 @@ theorem isGδ_univ : IsGδ (univ : Set α) :=
   isOpen_univ.isGδ
 #align is_Gδ_univ isGδ_univ
 
-theorem isGδ_binterᵢ_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set α}
+theorem isGδ_biInter_of_open {I : Set ι} (hI : I.Countable) {f : ι → Set α}
     (hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
-  ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [interₛ_image]⟩
-#align is_Gδ_bInter_of_open isGδ_binterᵢ_of_open
+  ⟨f '' I, by rwa [ball_image_iff], hI.image _, by rw [sInter_image]⟩
+#align is_Gδ_bInter_of_open isGδ_biInter_of_open
 
 -- porting note: TODO: generalize to `Sort _` + `Countable _`
-theorem isGδ_interᵢ_of_open [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
+theorem isGδ_iInter_of_open [Encodable ι] {f : ι → Set α} (hf : ∀ i, IsOpen (f i)) :
     IsGδ (⋂ i, f i) :=
-  ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [interₛ_range]⟩
-#align is_Gδ_Inter_of_open isGδ_interᵢ_of_open
+  ⟨range f, by rwa [forall_range_iff], countable_range _, by rw [sInter_range]⟩
+#align is_Gδ_Inter_of_open isGδ_iInter_of_open
 
 -- porting note: TODO: generalize to `Sort _` + `Countable _`
 /-- The intersection of an encodable family of Gδ sets is a Gδ set. -/
-theorem isGδ_interᵢ [Encodable ι] {s : ι → Set α} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
+theorem isGδ_iInter [Encodable ι] {s : ι → Set α} (hs : ∀ i, IsGδ (s i)) : IsGδ (⋂ i, s i) := by
   choose T hTo hTc hTs using hs
   obtain rfl : s = fun i => ⋂₀ T i := funext hTs
-  refine' ⟨⋃ i, T i, _, countable_unionᵢ hTc, (interₛ_unionᵢ _).symm⟩
+  refine' ⟨⋃ i, T i, _, countable_iUnion hTc, (sInter_iUnion _).symm⟩
   simpa [@forall_swap ι] using hTo
-#align is_Gδ_Inter isGδ_interᵢ
+#align is_Gδ_Inter isGδ_iInter
 
-theorem isGδ_binterᵢ {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set α}
+theorem isGδ_biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set α}
     (ht : ∀ (i) (hi : i ∈ s), IsGδ (t i hi)) : IsGδ (⋂ i ∈ s, t i ‹_›) := by
-  rw [binterᵢ_eq_interᵢ]
+  rw [biInter_eq_iInter]
   haveI := hs.toEncodable
-  exact isGδ_interᵢ fun x => ht x x.2
-#align is_Gδ_bInter isGδ_binterᵢ
+  exact isGδ_iInter fun x => ht x x.2
+#align is_Gδ_bInter isGδ_biInter
 
 /-- A countable intersection of Gδ sets is a Gδ set. -/
-theorem isGδ_interₛ {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
-  simpa only [interₛ_eq_binterᵢ] using isGδ_binterᵢ hS h
-#align is_Gδ_sInter isGδ_interₛ
+theorem isGδ_sInter {S : Set (Set α)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
+  simpa only [sInter_eq_biInter] using isGδ_biInter hS h
+#align is_Gδ_sInter isGδ_sInter
 
 theorem IsGδ.inter {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
-  rw [inter_eq_interᵢ]
-  exact isGδ_interᵢ (Bool.forall_bool.2 ⟨ht, hs⟩)
+  rw [inter_eq_iInter]
+  exact isGδ_iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
 #align is_Gδ.inter IsGδ.inter
 
 /-- The union of two Gδ sets is a Gδ set. -/
 theorem IsGδ.union {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t) := by
   rcases hs with ⟨S, Sopen, Scount, rfl⟩
   rcases ht with ⟨T, Topen, Tcount, rfl⟩
-  rw [interₛ_union_interₛ]
-  apply isGδ_binterᵢ_of_open (Scount.prod Tcount)
+  rw [sInter_union_sInter]
+  apply isGδ_biInter_of_open (Scount.prod Tcount)
   rintro ⟨a, b⟩ ⟨ha, hb⟩
   exact (Sopen a ha).union (Topen b hb)
 #align is_Gδ.union IsGδ.union
 
--- porting note: TODO: add `unionᵢ` and `unionₛ` versions
+-- porting note: TODO: add `iUnion` and `sUnion` versions
 /-- The union of finitely many Gδ sets is a Gδ set. -/
-theorem isGδ_bunionᵢ {s : Set ι} (hs : s.Finite) {f : ι → Set α} (h : ∀ i ∈ s, IsGδ (f i)) :
+theorem isGδ_biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set α} (h : ∀ i ∈ s, IsGδ (f i)) :
     IsGδ (⋃ i ∈ s, f i) := by
   refine' Finite.induction_on hs (by simp) _ h
-  simp only [ball_insert_iff, bunionᵢ_insert]
+  simp only [ball_insert_iff, biUnion_insert]
   exact fun _ _ ihs H => H.1.union (ihs H.2)
-#align is_Gδ_bUnion isGδ_bunionᵢ
+#align is_Gδ_bUnion isGδ_biUnion
 
 -- Porting note: Did not recognize notation 𝓤 α, needed to replace with uniformity α
 theorem IsClosed.isGδ {α} [UniformSpace α] [IsCountablyGenerated (uniformity α)] {s : Set α}
     (hs : IsClosed s) : IsGδ s := by
   rcases(@uniformity_hasBasis_open α _).exists_antitone_subbasis with ⟨U, hUo, hU, -⟩
-  rw [← hs.closure_eq, ← hU.binterᵢ_bunionᵢ_ball]
-  refine' isGδ_binterᵢ (to_countable _) fun n _ => IsOpen.isGδ _
-  exact isOpen_bunionᵢ fun x _ => UniformSpace.isOpen_ball _ (hUo _).2
+  rw [← hs.closure_eq, ← hU.biInter_biUnion_ball]
+  refine' isGδ_biInter (to_countable _) fun n _ => IsOpen.isGδ _
+  exact isOpen_biUnion fun x _ => UniformSpace.isOpen_ball _ (hUo _).2
 #align is_closed.is_Gδ IsClosed.isGδ
 
 section T1Space
@@ -144,8 +144,8 @@ theorem isGδ_compl_singleton (a : α) : IsGδ ({a}ᶜ : Set α) :=
 #align is_Gδ_compl_singleton isGδ_compl_singleton
 
 theorem Set.Countable.isGδ_compl {s : Set α} (hs : s.Countable) : IsGδ (sᶜ) := by
-  rw [← bunionᵢ_of_singleton s, compl_unionᵢ₂]
-  exact isGδ_binterᵢ hs fun x _ => isGδ_compl_singleton x
+  rw [← biUnion_of_singleton s, compl_iUnion₂]
+  exact isGδ_biInter hs fun x _ => isGδ_compl_singleton x
 #align set.countable.is_Gδ_compl Set.Countable.isGδ_compl
 
 theorem Set.Finite.isGδ_compl {s : Set α} (hs : s.Finite) : IsGδ (sᶜ) :=
@@ -164,8 +164,8 @@ variable [FirstCountableTopology α]
 
 theorem isGδ_singleton (a : α) : IsGδ ({a} : Set α) := by
   rcases (nhds_basis_opens a).exists_antitone_subbasis with ⟨U, hU, h_basis⟩
-  rw [← binterᵢ_basis_nhds h_basis.toHasBasis]
-  exact isGδ_binterᵢ (to_countable _) fun n _ => (hU n).2.isGδ
+  rw [← biInter_basis_nhds h_basis.toHasBasis]
+  exact isGδ_biInter (to_countable _) fun n _ => (hU n).2.isGδ
 #align is_Gδ_singleton isGδ_singleton
 
 theorem Set.Finite.isGδ {s : Set α} (hs : s.Finite) : IsGδ s :=
@@ -187,7 +187,7 @@ theorem isGδ_setOf_continuousAt [UniformSpace β] [IsCountablyGenerated (unifor
   simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
   simp only [(nhds_basis_opens _).prod_self.tendsto_iff hU.toHasBasis, forall_prop_of_true,
     setOf_forall, id]
-  refine' isGδ_interᵢ fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
+  refine' isGδ_iInter fun k => IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x => _
   rintro ⟨s, ⟨hsx, hso⟩, hsU⟩
   filter_upwards [IsOpen.mem_nhds hso hsx]with _ hy using⟨s, ⟨hy, hso⟩, hsU⟩
 #align is_Gδ_set_of_continuous_at isGδ_setOf_continuousAt

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