topology.metric_space.polish ⟷ Mathlib.Topology.MetricSpace.Polish

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

@@ -85,13 +85,13 @@ class UpgradedPolishSpace (α : Type _) extends MetricSpace α, SecondCountableT
 #align upgraded_polish_space UpgradedPolishSpace
 -/
 
-#print polishSpace_of_complete_second_countable /-
-instance (priority := 100) polishSpace_of_complete_second_countable [m : MetricSpace α]
-    [h : SecondCountableTopology α] [h' : CompleteSpace α] : PolishSpace α
+#print PolishSpace.of_separableSpace_completeSpace_metrizable /-
+instance (priority := 100) PolishSpace.of_separableSpace_completeSpace_metrizable
+    [m : MetricSpace α] [h : SecondCountableTopology α] [h' : CompleteSpace α] : PolishSpace α
     where
   second_countable := h
   complete := ⟨m, rfl, h'⟩
-#align polish_space_of_complete_second_countable polishSpace_of_complete_second_countable
+#align polish_space_of_complete_second_countable PolishSpace.of_separableSpace_completeSpace_metrizable
 -/
 
 #print polishSpaceMetric /-
@@ -210,14 +210,12 @@ theorem IsClosed.polishSpace {α : Type _} [TopologicalSpace α] [PolishSpace α
 #align is_closed.polish_space IsClosed.polishSpace
 -/
 
-#print PolishSpace.AuxCopy /-
 /-- A sequence of type synonyms of a given type `α`, useful in the proof of
 `exists_polish_space_forall_le` to endow each copy with a different topology. -/
 @[nolint unused_arguments has_nonempty_instance]
 def AuxCopy (α : Type _) {ι : Type _} (i : ι) : Type _ :=
   α
 #align polish_space.aux_copy PolishSpace.AuxCopy
--/
 
 #print PolishSpace.exists_polishSpace_forall_le /-
 /-- Given a Polish space, and countably many finer Polish topologies, there exists another Polish

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

@@ -453,7 +453,7 @@ theorem IsOpen.polishSpace {α : Type _} [TopologicalSpace α] [PolishSpace α]
     (hs : IsOpen s) : PolishSpace s :=
   by
   rcases eq_empty_or_nonempty (sᶜ) with (h's | h's)
-  · simp at h's 
+  · simp at h's
     apply IsClosed.polishSpace
     rw [h's]
     exact isClosed_univ

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

@@ -61,7 +61,7 @@ variable {α : Type _} {β : Type _}
 
 
 #print PolishSpace /-
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
 /-- A Polish space is a topological space with second countable topology, that can be endowed
 with a metric for which it is complete.
 We register an instance from complete second countable metric space to polish space, and not the

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

@@ -400,7 +400,7 @@ theorem TopologicalSpace.Opens.CompleteCopy.instCompleteSpace [CompleteSpace α]
     by
     apply cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => _) _
     · exact (dist_le_dist_complete_copy (u n) (u m)).trans (hu N n m hNn hNm).le
-    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
+    · exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
   obtain ⟨x, xlim⟩ : ∃ x, tendsto (fun n => (u n).1) at_top (𝓝 x) :=
     haveI : Nonempty α := ⟨(u 0).1⟩
     ⟨_, A.tendsto_lim⟩

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

@@ -61,7 +61,7 @@ variable {α : Type _} {β : Type _}
 
 
 #print PolishSpace /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
 /-- A Polish space is a topological space with second countable topology, that can be endowed
 with a metric for which it is complete.
 We register an instance from complete second countable metric space to polish space, and not the

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

@@ -3,10 +3,10 @@ Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Topology.MetricSpace.PiNat
-import Mathbin.Topology.MetricSpace.Isometry
-import Mathbin.Topology.MetricSpace.Gluing
-import Mathbin.Analysis.Normed.Field.Basic
+import Topology.MetricSpace.PiNat
+import Topology.MetricSpace.Isometry
+import Topology.MetricSpace.Gluing
+import Analysis.Normed.Field.Basic
 
 #align_import topology.metric_space.polish from "leanprover-community/mathlib"@"86d1873c01a723aba6788f0b9051ae3d23b4c1c3"
 
@@ -61,7 +61,7 @@ variable {α : Type _} {β : Type _}
 
 
 #print PolishSpace /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
 /-- A Polish space is a topological space with second countable topology, that can be endowed
 with a metric for which it is complete.
 We register an instance from complete second countable metric space to polish space, and not the

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

@@ -140,11 +140,9 @@ instance pi_countable {ι : Type _} [Countable ι] {E : ι → Type _} [∀ i, T
 #align polish_space.pi_countable PolishSpace.pi_countable
 -/
 
-#print PolishSpace.nat_fun /-
 /-- Without this instance, `polish_space (ℕ → ℕ)` is not found by typeclass inference. -/
 instance nat_fun [TopologicalSpace α] [PolishSpace α] : PolishSpace (ℕ → α) := by infer_instance
 #align polish_space.nat_fun PolishSpace.nat_fun
--/
 
 #print PolishSpace.sigma /-
 /-- A countable disjoint union of Polish spaces is Polish. -/

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

@@ -2,17 +2,14 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.polish
-! leanprover-community/mathlib commit 86d1873c01a723aba6788f0b9051ae3d23b4c1c3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.MetricSpace.PiNat
 import Mathbin.Topology.MetricSpace.Isometry
 import Mathbin.Topology.MetricSpace.Gluing
 import Mathbin.Analysis.Normed.Field.Basic
 
+#align_import topology.metric_space.polish from "leanprover-community/mathlib"@"86d1873c01a723aba6788f0b9051ae3d23b4c1c3"
+
 /-!
 # Polish spaces
 

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

@@ -64,7 +64,7 @@ variable {α : Type _} {β : Type _}
 
 
 #print PolishSpace /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
 /-- A Polish space is a topological space with second countable topology, that can be endowed
 with a metric for which it is complete.
 We register an instance from complete second countable metric space to polish space, and not the
@@ -149,6 +149,7 @@ instance nat_fun [TopologicalSpace α] [PolishSpace α] : PolishSpace (ℕ → 
 #align polish_space.nat_fun PolishSpace.nat_fun
 -/
 
+#print PolishSpace.sigma /-
 /-- A countable disjoint union of Polish spaces is Polish. -/
 instance sigma {ι : Type _} [Countable ι] {E : ι → Type _} [∀ n, TopologicalSpace (E n)]
     [∀ n, PolishSpace (E n)] : PolishSpace (Σ n, E n) :=
@@ -158,7 +159,9 @@ instance sigma {ι : Type _} [Countable ι] {E : ι → Type _} [∀ n, Topologi
   haveI : CompleteSpace (Σ n, E n) := sigma.complete_space
   infer_instance
 #align polish_space.sigma PolishSpace.sigma
+-/
 
+#print PolishSpace.sum /-
 /-- The disjoint union of two Polish spaces is Polish. -/
 instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [PolishSpace β] :
     PolishSpace (Sum α β) := by
@@ -167,14 +170,18 @@ instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [Polis
   letI : MetricSpace (Sum α β) := metric_space_sum
   infer_instance
 #align polish_space.sum PolishSpace.sum
+-/
 
+#print PolishSpace.exists_nat_nat_continuous_surjective /-
 /-- Any nonempty Polish space is the continuous image of the fundamental space `ℕ → ℕ`. -/
 theorem exists_nat_nat_continuous_surjective (α : Type _) [TopologicalSpace α] [PolishSpace α]
     [Nonempty α] : ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f :=
   letI := upgradePolishSpace α
   exists_nat_nat_continuous_surjective_of_completeSpace α
 #align polish_space.exists_nat_nat_continuous_surjective PolishSpace.exists_nat_nat_continuous_surjective
+-/
 
+#print ClosedEmbedding.polishSpace /-
 /-- Given a closed embedding into a Polish space, the source space is also Polish. -/
 theorem ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β] [PolishSpace β]
     {f : α → β} (hf : ClosedEmbedding f) : PolishSpace α :=
@@ -189,6 +196,7 @@ theorem ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β]
     exact hf.closed_range
   infer_instance
 #align closed_embedding.polish_space ClosedEmbedding.polishSpace
+-/
 
 #print Equiv.polishSpace_induced /-
 /-- Pulling back a Polish topology under an equiv gives again a Polish topology. -/
@@ -216,6 +224,7 @@ def AuxCopy (α : Type _) {ι : Type _} (i : ι) : Type _ :=
 #align polish_space.aux_copy PolishSpace.AuxCopy
 -/
 
+#print PolishSpace.exists_polishSpace_forall_le /-
 /-- Given a Polish space, and countably many finer Polish topologies, there exists another Polish
 topology which is finer than all of them. -/
 theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : TopologicalSpace α]
@@ -280,6 +289,7 @@ theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : Topologic
     · exact f_closed
   exact @ClosedEmbedding.polishSpace _ _ (T.induced f) T (by infer_instance) _ L
 #align polish_space.exists_polish_space_forall_le PolishSpace.exists_polishSpace_forall_le
+-/
 
 /-!
 ### An open subset of a Polish space is Polish
@@ -530,11 +540,13 @@ theorem IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α
 #align is_closed.is_clopenable IsClosed.isClopenable
 -/
 
+#print PolishSpace.IsClopenable.compl /-
 theorem IsClopenable.compl [TopologicalSpace α] {s : Set α} (hs : IsClopenable s) :
     IsClopenable (sᶜ) := by
   rcases hs with ⟨t, t_le, t_polish, h, h'⟩
   exact ⟨t, t_le, t_polish, @IsOpen.isClosed_compl α t s h', @IsClosed.isOpen_compl α t s h⟩
 #align polish_space.is_clopenable.compl PolishSpace.IsClopenable.compl
+-/
 
 #print IsOpen.isClopenable /-
 theorem IsOpen.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α} (hs : IsOpen s) :

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

@@ -343,7 +343,6 @@ def TopologicalSpace.Opens.CompleteCopy.instMetricSpace (s : Set α) :
         rw [← Real.dist_eq, ← Real.dist_eq, ← Real.dist_eq]
         exact add_le_add (dist_triangle _ _ _) (dist_triangle _ _ _)
       _ = dist x y + dist y z := by rw [dist_complete_copy_eq, dist_complete_copy_eq]; abel
-      
   eq_of_dist_eq_zero := by
     intro x y hxy
     apply Subtype.coe_injective
@@ -424,7 +423,6 @@ theorem TopologicalSpace.Opens.CompleteCopy.instCompleteSpace [CompleteSpace α]
       _ = dist (u n) (u 0) + dist (1 / inf_dist (u 0).1 (sᶜ)) 0 := rfl
       _ < (1 / 2) ^ 0 + dist (1 / inf_dist (u 0).1 (sᶜ)) 0 :=
         add_lt_add_right (hu 0 n 0 (zero_le _) le_rfl) _
-      
   have Cpos : 0 < C := by
     apply lt_of_le_of_lt _ (hC 0)
     simp [inf_dist_nonneg]

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

@@ -64,7 +64,7 @@ variable {α : Type _} {β : Type _}
 
 
 #print PolishSpace /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
 /-- A Polish space is a topological space with second countable topology, that can be endowed
 with a metric for which it is complete.
 We register an instance from complete second countable metric space to polish space, and not the
@@ -246,7 +246,7 @@ theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : Topologic
   refine' ⟨T.induced f, fun n => T_le_m n, (T_le_m default).trans (hm default), _⟩
   -- show that the new topology is Polish, as the pullback of a Polish topology under a closed
   -- embedding.
-  have A : range f = ⋂ n, { x | x n = x default } :=
+  have A : range f = ⋂ n, {x | x n = x default} :=
     by
     ext x
     constructor
@@ -404,7 +404,7 @@ theorem TopologicalSpace.Opens.CompleteCopy.instCompleteSpace [CompleteSpace α]
   · refine' ⟨⟨x, xs⟩, _⟩
     have L : tendsto (fun n => (id ⟨(u n).1, (u n).2⟩ : s)) at_top (𝓝 ⟨x, xs⟩) := by
       apply embedding_subtype_coe.tendsto_nhds_iff.2; exact xlim
-    convert((complete_copy_id_homeo hs h's).symm.Continuous.Tendsto _).comp L
+    convert ((complete_copy_id_homeo hs h's).symm.Continuous.Tendsto _).comp L
     ext1 n
     simp [complete_copy_id_homeo]
   obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / inf_dist (u n).1 (sᶜ) < C :=

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

@@ -84,7 +84,7 @@ class PolishSpace (α : Type _) [h : TopologicalSpace α] : Prop where
 class should be registered: It should be used as `letI := upgrade_polish_space α` to endow a Polish
 space with a complete metric. -/
 class UpgradedPolishSpace (α : Type _) extends MetricSpace α, SecondCountableTopology α,
-  CompleteSpace α
+    CompleteSpace α
 #align upgraded_polish_space UpgradedPolishSpace
 -/
 
@@ -151,11 +151,11 @@ instance nat_fun [TopologicalSpace α] [PolishSpace α] : PolishSpace (ℕ → 
 
 /-- A countable disjoint union of Polish spaces is Polish. -/
 instance sigma {ι : Type _} [Countable ι] {E : ι → Type _} [∀ n, TopologicalSpace (E n)]
-    [∀ n, PolishSpace (E n)] : PolishSpace (Σn, E n) :=
+    [∀ n, PolishSpace (E n)] : PolishSpace (Σ n, E n) :=
   by
   letI := fun n => upgradePolishSpace (E n)
-  letI : MetricSpace (Σn, E n) := sigma.metric_space
-  haveI : CompleteSpace (Σn, E n) := sigma.complete_space
+  letI : MetricSpace (Σ n, E n) := sigma.metric_space
+  haveI : CompleteSpace (Σ n, E n) := sigma.complete_space
   infer_instance
 #align polish_space.sigma PolishSpace.sigma
 
@@ -450,7 +450,7 @@ theorem IsOpen.polishSpace {α : Type _} [TopologicalSpace α] [PolishSpace α]
     (hs : IsOpen s) : PolishSpace s :=
   by
   rcases eq_empty_or_nonempty (sᶜ) with (h's | h's)
-  · simp at h's
+  · simp at h's 
     apply IsClosed.polishSpace
     rw [h's]
     exact isClosed_univ

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

@@ -354,11 +354,10 @@ def TopologicalSpace.Opens.CompleteCopy.instMetricSpace (s : Set α) :
 
 attribute [local instance] complete_copy_metric_space
 
-/- warning: polish_space.complete_copy_id_homeo clashes with [anonymous] -> [anonymous]
-Case conversion may be inaccurate. Consider using '#align polish_space.complete_copy_id_homeo [anonymous]ₓ'. -/
 /-- The identity between the type synonym `complete_copy s` (with its modified metric) and the
 original subtype `s` is a homeomorphism. -/
-def [anonymous] (hs : IsOpen s) (h's : sᶜ.Nonempty) : TopologicalSpace.Opens.CompleteCopy s ≃ₜ s
+def completeCopyIdHomeo (hs : IsOpen s) (h's : sᶜ.Nonempty) :
+    TopologicalSpace.Opens.CompleteCopy s ≃ₜ s
     where
   toFun := id
   invFun := id
@@ -386,7 +385,7 @@ def [anonymous] (hs : IsOpen s) (h's : sᶜ.Nonempty) : TopologicalSpace.Opens.C
     · refine' (tendsto.sub_const _ _).abs
       refine' tendsto.div tendsto_const_nhds _ I.ne'
       exact ((continuous_inf_dist_pt _).comp continuous_subtype_val).Tendsto _
-#align polish_space.complete_copy_id_homeo [anonymous]
+#align polish_space.complete_copy_id_homeo PolishSpace.completeCopyIdHomeo
 
 theorem TopologicalSpace.Opens.CompleteCopy.instCompleteSpace [CompleteSpace α] (hs : IsOpen s)
     (h's : sᶜ.Nonempty) : CompleteSpace (TopologicalSpace.Opens.CompleteCopy s) :=

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

@@ -54,7 +54,7 @@ with additional properties:
 
 noncomputable section
 
-open Classical Topology Filter
+open scoped Classical Topology Filter
 
 open TopologicalSpace Set Metric Filter Function
 

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

@@ -149,12 +149,6 @@ instance nat_fun [TopologicalSpace α] [PolishSpace α] : PolishSpace (ℕ → 
 #align polish_space.nat_fun PolishSpace.nat_fun
 -/
 
-/- warning: polish_space.sigma -> PolishSpace.sigma is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {E : ι -> Type.{u2}} [_inst_2 : forall (n : ι), TopologicalSpace.{u2} (E n)] [_inst_3 : forall (n : ι), PolishSpace.{u2} (E n) (_inst_2 n)], PolishSpace.{max u1 u2} (Sigma.{u1, u2} ι (fun (n : ι) => E n)) (Sigma.topologicalSpace.{u1, u2} ι (fun (n : ι) => E n) (fun (a : ι) => _inst_2 a))
-but is expected to have type
-  forall {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {E : ι -> Type.{u2}} [_inst_2 : forall (n : ι), TopologicalSpace.{u2} (E n)] [_inst_3 : forall (n : ι), PolishSpace.{u2} (E n) (_inst_2 n)], PolishSpace.{max u2 u1} (Sigma.{u1, u2} ι (fun (n : ι) => E n)) (instTopologicalSpaceSigma.{u1, u2} ι (fun (n : ι) => E n) (fun (a : ι) => _inst_2 a))
-Case conversion may be inaccurate. Consider using '#align polish_space.sigma PolishSpace.sigmaₓ'. -/
 /-- A countable disjoint union of Polish spaces is Polish. -/
 instance sigma {ι : Type _} [Countable ι] {E : ι → Type _} [∀ n, TopologicalSpace (E n)]
     [∀ n, PolishSpace (E n)] : PolishSpace (Σn, E n) :=
@@ -165,12 +159,6 @@ instance sigma {ι : Type _} [Countable ι] {E : ι → Type _} [∀ n, Topologi
   infer_instance
 #align polish_space.sigma PolishSpace.sigma
 
-/- warning: polish_space.sum -> PolishSpace.sum is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : PolishSpace.{u2} β _inst_3], PolishSpace.{max u1 u2} (Sum.{u1, u2} α β) (Sum.topologicalSpace.{u1, u2} α β _inst_1 _inst_3)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : PolishSpace.{u2} β _inst_3], PolishSpace.{max u2 u1} (Sum.{u1, u2} α β) (instTopologicalSpaceSum.{u1, u2} α β _inst_1 _inst_3)
-Case conversion may be inaccurate. Consider using '#align polish_space.sum PolishSpace.sumₓ'. -/
 /-- The disjoint union of two Polish spaces is Polish. -/
 instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [PolishSpace β] :
     PolishSpace (Sum α β) := by
@@ -180,12 +168,6 @@ instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [Polis
   infer_instance
 #align polish_space.sum PolishSpace.sum
 
-/- warning: polish_space.exists_nat_nat_continuous_surjective -> PolishSpace.exists_nat_nat_continuous_surjective is a dubious translation:
-lean 3 declaration is
-  forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α _inst_1] [_inst_3 : Nonempty.{succ u1} α], Exists.{succ u1} ((Nat -> Nat) -> α) (fun (f : (Nat -> Nat) -> α) => And (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => Nat.topologicalSpace)) _inst_1 f) (Function.Surjective.{1, succ u1} (Nat -> Nat) α f))
-but is expected to have type
-  forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α _inst_1] [_inst_3 : Nonempty.{succ u1} α], Exists.{succ u1} ((Nat -> Nat) -> α) (fun (f : (Nat -> Nat) -> α) => And (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => instTopologicalSpaceNat)) _inst_1 f) (Function.Surjective.{1, succ u1} (Nat -> Nat) α f))
-Case conversion may be inaccurate. Consider using '#align polish_space.exists_nat_nat_continuous_surjective PolishSpace.exists_nat_nat_continuous_surjectiveₓ'. -/
 /-- Any nonempty Polish space is the continuous image of the fundamental space `ℕ → ℕ`. -/
 theorem exists_nat_nat_continuous_surjective (α : Type _) [TopologicalSpace α] [PolishSpace α]
     [Nonempty α] : ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f :=
@@ -193,12 +175,6 @@ theorem exists_nat_nat_continuous_surjective (α : Type _) [TopologicalSpace α]
   exists_nat_nat_continuous_surjective_of_completeSpace α
 #align polish_space.exists_nat_nat_continuous_surjective PolishSpace.exists_nat_nat_continuous_surjective
 
-/- warning: closed_embedding.polish_space -> ClosedEmbedding.polishSpace is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : PolishSpace.{u2} β _inst_2] {f : α -> β}, (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (PolishSpace.{u1} α _inst_1)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : PolishSpace.{u1} β _inst_2] {f : α -> β}, (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) -> (PolishSpace.{u2} α _inst_1)
-Case conversion may be inaccurate. Consider using '#align closed_embedding.polish_space ClosedEmbedding.polishSpaceₓ'. -/
 /-- Given a closed embedding into a Polish space, the source space is also Polish. -/
 theorem ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β] [PolishSpace β]
     {f : α → β} (hf : ClosedEmbedding f) : PolishSpace α :=
@@ -240,12 +216,6 @@ def AuxCopy (α : Type _) {ι : Type _} (i : ι) : Type _ :=
 #align polish_space.aux_copy PolishSpace.AuxCopy
 -/
 
-/- warning: polish_space.exists_polish_space_forall_le -> PolishSpace.exists_polishSpace_forall_le is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] [t : TopologicalSpace.{u1} α] [p : PolishSpace.{u1} α t] (m : ι -> (TopologicalSpace.{u1} α)), (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toHasLe.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) (m n) t) -> (forall (n : ι), PolishSpace.{u1} α (m n)) -> (Exists.{succ u1} (TopologicalSpace.{u1} α) (fun (t' : TopologicalSpace.{u1} α) => And (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toHasLe.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' (m n)) (And (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toHasLe.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' t) (PolishSpace.{u1} α t'))))
-but is expected to have type
-  forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] [t : TopologicalSpace.{u1} α] [p : PolishSpace.{u1} α t] (m : ι -> (TopologicalSpace.{u1} α)), (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) (m n) t) -> (forall (n : ι), PolishSpace.{u1} α (m n)) -> (Exists.{succ u1} (TopologicalSpace.{u1} α) (fun (t' : TopologicalSpace.{u1} α) => And (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) t' (m n)) (And (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) t' t) (PolishSpace.{u1} α t'))))
-Case conversion may be inaccurate. Consider using '#align polish_space.exists_polish_space_forall_le PolishSpace.exists_polishSpace_forall_leₓ'. -/
 /-- Given a Polish space, and countably many finer Polish topologies, there exists another Polish
 topology which is finer than all of them. -/
 theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : TopologicalSpace α]
@@ -385,11 +355,6 @@ def TopologicalSpace.Opens.CompleteCopy.instMetricSpace (s : Set α) :
 attribute [local instance] complete_copy_metric_space
 
 /- warning: polish_space.complete_copy_id_homeo clashes with [anonymous] -> [anonymous]
-warning: polish_space.complete_copy_id_homeo -> [anonymous] is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u_1}} [_inst_1 : MetricSpace.{u_1} α] {s : Set.{u_1} α}, (IsOpen.{u_1} α (UniformSpace.toTopologicalSpace.{u_1} α (PseudoMetricSpace.toUniformSpace.{u_1} α (MetricSpace.toPseudoMetricSpace.{u_1} α _inst_1))) s) -> (Set.Nonempty.{u_1} α (HasCompl.compl.{u_1} (Set.{u_1} α) (BooleanAlgebra.toHasCompl.{u_1} (Set.{u_1} α) (Set.booleanAlgebra.{u_1} α)) s)) -> (Homeomorph.{u_1, u_1} (TopologicalSpace.Opens.CompleteCopyₓ.{u_1} α s) (coeSort.{max (succ u_1) 1, succ (succ u_1)} (Set.{u_1} α) Type.{u_1} (Set.hasCoeToSort.{u_1} α) s) (UniformSpace.toTopologicalSpace.{u_1} (TopologicalSpace.Opens.CompleteCopyₓ.{u_1} α s) (PseudoMetricSpace.toUniformSpace.{u_1} (TopologicalSpace.Opens.CompleteCopyₓ.{u_1} α s) (MetricSpace.toPseudoMetricSpace.{u_1} (TopologicalSpace.Opens.CompleteCopyₓ.{u_1} α s) (TopologicalSpace.Opens.CompleteCopy.instMetricSpaceₓ.{u_1} α _inst_1 s)))) (Subtype.topologicalSpace.{u_1} α (fun (x : α) => Membership.Mem.{u_1, u_1} α (Set.{u_1} α) (Set.hasMem.{u_1} α) x s) (UniformSpace.toTopologicalSpace.{u_1} α (PseudoMetricSpace.toUniformSpace.{u_1} α (MetricSpace.toPseudoMetricSpace.{u_1} α _inst_1)))))
-but is expected to have type
-  forall {α : Type.{u}} {_inst_1 : Type.{v}}, (Nat -> α -> _inst_1) -> Nat -> (List.{u} α) -> (List.{v} _inst_1)
 Case conversion may be inaccurate. Consider using '#align polish_space.complete_copy_id_homeo [anonymous]ₓ'. -/
 /-- The identity between the type synonym `complete_copy s` (with its modified metric) and the
 original subtype `s` is a homeomorphism. -/
@@ -568,12 +533,6 @@ theorem IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α
 #align is_closed.is_clopenable IsClosed.isClopenable
 -/
 
-/- warning: polish_space.is_clopenable.compl -> PolishSpace.IsClopenable.compl is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, (PolishSpace.IsClopenable.{u1} α _inst_1 s) -> (PolishSpace.IsClopenable.{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] {s : Set.{u1} α}, (PolishSpace.IsClopenable.{u1} α _inst_1 s) -> (PolishSpace.IsClopenable.{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 polish_space.is_clopenable.compl PolishSpace.IsClopenable.complₓ'. -/
 theorem IsClopenable.compl [TopologicalSpace α] {s : Set α} (hs : IsClopenable s) :
     IsClopenable (sᶜ) := by
   rcases hs with ⟨t, t_le, t_polish, h, h'⟩

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

@@ -127,9 +127,7 @@ namespace PolishSpace
 
 #print PolishSpace.t2Space /-
 instance (priority := 100) t2Space (α : Type _) [TopologicalSpace α] [PolishSpace α] : T2Space α :=
-  by
-  letI := upgradePolishSpace α
-  infer_instance
+  by letI := upgradePolishSpace α; infer_instance
 #align polish_space.t2_space PolishSpace.t2Space
 -/
 
@@ -374,10 +372,7 @@ def TopologicalSpace.Opens.CompleteCopy.instMetricSpace (s : Set α) :
         by
         rw [← Real.dist_eq, ← Real.dist_eq, ← Real.dist_eq]
         exact add_le_add (dist_triangle _ _ _) (dist_triangle _ _ _)
-      _ = dist x y + dist y z :=
-        by
-        rw [dist_complete_copy_eq, dist_complete_copy_eq]
-        abel
+      _ = dist x y + dist y z := by rw [dist_complete_copy_eq, dist_complete_copy_eq]; abel
       
   eq_of_dist_eq_zero := by
     intro x y hxy
@@ -422,8 +417,7 @@ def [anonymous] (hs : IsOpen s) (h's : sᶜ.Nonempty) : TopologicalSpace.Opens.C
       rw [← hs.is_closed_compl.not_mem_iff_inf_dist_pos h's]
       simp
     apply tendsto.add
-    · apply Continuous.tendsto
-      exact continuous_subtype_coe.dist continuous_const
+    · apply Continuous.tendsto; exact continuous_subtype_coe.dist continuous_const
     · refine' (tendsto.sub_const _ _).abs
       refine' tendsto.div tendsto_const_nhds _ I.ne'
       exact ((continuous_inf_dist_pt _).comp continuous_subtype_val).Tendsto _
@@ -444,10 +438,8 @@ theorem TopologicalSpace.Opens.CompleteCopy.instCompleteSpace [CompleteSpace α]
     ⟨_, A.tendsto_lim⟩
   suffices xs : x ∈ s
   · refine' ⟨⟨x, xs⟩, _⟩
-    have L : tendsto (fun n => (id ⟨(u n).1, (u n).2⟩ : s)) at_top (𝓝 ⟨x, xs⟩) :=
-      by
-      apply embedding_subtype_coe.tendsto_nhds_iff.2
-      exact xlim
+    have L : tendsto (fun n => (id ⟨(u n).1, (u n).2⟩ : s)) at_top (𝓝 ⟨x, xs⟩) := by
+      apply embedding_subtype_coe.tendsto_nhds_iff.2; exact xlim
     convert((complete_copy_id_homeo hs h's).symm.Continuous.Tendsto _).comp L
     ext1 n
     simp [complete_copy_id_homeo]
@@ -455,10 +447,8 @@ theorem TopologicalSpace.Opens.CompleteCopy.instCompleteSpace [CompleteSpace α]
     by
     refine' ⟨(1 / 2) ^ 0 + dist (1 / inf_dist (u 0).1 (sᶜ)) 0, fun n => _⟩
     calc
-      1 / inf_dist (u n).val (sᶜ) ≤ dist (1 / inf_dist (u n).val (sᶜ)) 0 :=
-        by
-        rw [Real.dist_0_eq_abs]
-        exact le_abs_self _
+      1 / inf_dist (u n).val (sᶜ) ≤ dist (1 / inf_dist (u n).val (sᶜ)) 0 := by
+        rw [Real.dist_0_eq_abs]; exact le_abs_self _
       _ ≤
           dist (1 / inf_dist (u n).1 (sᶜ)) (1 / inf_dist (u 0).1 (sᶜ)) +
             dist (1 / inf_dist (u 0).1 (sᶜ)) 0 :=
@@ -477,10 +467,8 @@ theorem TopologicalSpace.Opens.CompleteCopy.instCompleteSpace [CompleteSpace α]
   have I : ∀ n, 1 / C ≤ inf_dist (u n).1 (sᶜ) :=
     by
     intro n
-    have : 0 < inf_dist (u n).val (sᶜ) :=
-      by
-      apply (hs.is_closed_compl.not_mem_iff_inf_dist_pos h's).1
-      simp
+    have : 0 < inf_dist (u n).val (sᶜ) := by
+      apply (hs.is_closed_compl.not_mem_iff_inf_dist_pos h's).1; simp
     rw [div_le_iff' Cpos]
     exact (div_le_iff this).1 (hC n).le
   have I' : 1 / C ≤ inf_dist x (sᶜ) :=
@@ -556,15 +544,11 @@ theorem IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α
     refine' ⟨f.symm ⁻¹' u, _, by simp only [Equiv.symm_symm, Equiv.symm_preimage_preimage]⟩
     refine' isOpen_sum_iff.2 ⟨_, _⟩
     · have : IsOpen ((coe : s → α) ⁻¹' u) := IsOpen.preimage continuous_subtype_val hu
-      have : Sum.inl ⁻¹' (⇑f.symm ⁻¹' u) = (coe : s → α) ⁻¹' u :=
-        by
-        ext x
+      have : Sum.inl ⁻¹' (⇑f.symm ⁻¹' u) = (coe : s → α) ⁻¹' u := by ext x;
         simp only [Equiv.symm_symm, mem_preimage, Equiv.Set.sumCompl_apply_inl]
       rwa [this]
     · have : IsOpen ((coe : t → α) ⁻¹' u) := IsOpen.preimage continuous_subtype_val hu
-      have : Sum.inr ⁻¹' (⇑f.symm ⁻¹' u) = (coe : t → α) ⁻¹' u :=
-        by
-        ext x
+      have : Sum.inr ⁻¹' (⇑f.symm ⁻¹' u) = (coe : t → α) ⁻¹' u := by ext x;
         simp only [Equiv.symm_symm, mem_preimage, Equiv.Set.sumCompl_apply_inr]
       rwa [this]
   · have : is_closed[t'] (g ⁻¹' range (Sum.inl : s → Sum s t)) :=

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

@@ -244,7 +244,7 @@ def AuxCopy (α : Type _) {ι : Type _} (i : ι) : Type _ :=
 
 /- warning: polish_space.exists_polish_space_forall_le -> PolishSpace.exists_polishSpace_forall_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] [t : TopologicalSpace.{u1} α] [p : PolishSpace.{u1} α t] (m : ι -> (TopologicalSpace.{u1} α)), (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) (m n) t) -> (forall (n : ι), PolishSpace.{u1} α (m n)) -> (Exists.{succ u1} (TopologicalSpace.{u1} α) (fun (t' : TopologicalSpace.{u1} α) => And (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' (m n)) (And (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' t) (PolishSpace.{u1} α t'))))
+  forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] [t : TopologicalSpace.{u1} α] [p : PolishSpace.{u1} α t] (m : ι -> (TopologicalSpace.{u1} α)), (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toHasLe.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) (m n) t) -> (forall (n : ι), PolishSpace.{u1} α (m n)) -> (Exists.{succ u1} (TopologicalSpace.{u1} α) (fun (t' : TopologicalSpace.{u1} α) => And (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toHasLe.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' (m n)) (And (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toHasLe.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' t) (PolishSpace.{u1} α t'))))
 but is expected to have type
   forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] [t : TopologicalSpace.{u1} α] [p : PolishSpace.{u1} α t] (m : ι -> (TopologicalSpace.{u1} α)), (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) (m n) t) -> (forall (n : ι), PolishSpace.{u1} α (m n)) -> (Exists.{succ u1} (TopologicalSpace.{u1} α) (fun (t' : TopologicalSpace.{u1} α) => And (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) t' (m n)) (And (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) t' t) (PolishSpace.{u1} α t'))))
 Case conversion may be inaccurate. Consider using '#align polish_space.exists_polish_space_forall_le PolishSpace.exists_polishSpace_forall_leₓ'. -/

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

@@ -291,7 +291,7 @@ theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : Topologic
       exact (mem_Inter.1 hx n : _)
   have f_closed : IsClosed (range f) := by
     rw [A]
-    apply isClosed_interᵢ fun n => _
+    apply isClosed_iInter fun n => _
     have C : ∀ i : ι, Continuous fun x : ∀ n, aux_copy α n => (id (x i) : α) :=
       by
       intro i
@@ -602,8 +602,8 @@ theorem IsOpen.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α}
 #align is_open.is_clopenable IsOpen.isClopenable
 -/
 
-#print PolishSpace.IsClopenable.unionᵢ /-
-theorem IsClopenable.unionᵢ [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α}
+#print PolishSpace.IsClopenable.iUnion /-
+theorem IsClopenable.iUnion [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClopenable (s n)) : IsClopenable (⋃ n, s n) :=
   by
   choose m mt m_polish m_closed m_open using hs
@@ -611,7 +611,7 @@ theorem IsClopenable.unionᵢ [t : TopologicalSpace α] [PolishSpace α] {s : 
     ∃ t' : TopologicalSpace α, (∀ n : ℕ, t' ≤ m n) ∧ t' ≤ t ∧ @PolishSpace α t' :=
     exists_polish_space_forall_le m mt m_polish
   have A : is_open[t'] (⋃ n, s n) := by
-    apply isOpen_unionᵢ
+    apply isOpen_iUnion
     intro n
     apply t'm n
     exact m_open n
@@ -620,7 +620,7 @@ theorem IsClopenable.unionᵢ [t : TopologicalSpace α] [PolishSpace α] {s : 
       t'' ≤ t' ∧ @PolishSpace α t'' ∧ is_closed[t''] (⋃ n, s n) ∧ is_open[t''] (⋃ n, s n) :=
     @IsOpen.isClopenable α t' t'_polish _ A
   exact ⟨t'', t''_le.trans ((t'm 0).trans (mt 0)), t''_polish, h1, h2⟩
-#align polish_space.is_clopenable.Union PolishSpace.IsClopenable.unionᵢ
+#align polish_space.is_clopenable.Union PolishSpace.IsClopenable.iUnion
 -/
 
 end PolishSpace

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

@@ -64,7 +64,7 @@ variable {α : Type _} {β : Type _}
 
 
 #print PolishSpace /-
-/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
 /-- A Polish space is a topological space with second countable topology, that can be endowed
 with a metric for which it is complete.
 We register an instance from complete second countable metric space to polish space, and not the

mathlib commit https://github.com/leanprover-community/mathlib/commit/52932b3a083d4142e78a15dc928084a22fea9ba0

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.polish
-! leanprover-community/mathlib commit bcfa726826abd57587355b4b5b7e78ad6527b7e4
+! leanprover-community/mathlib commit 86d1873c01a723aba6788f0b9051ae3d23b4c1c3
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.Analysis.Normed.Field.Basic
 /-!
 # Polish spaces
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A topological space is Polish if its topology is second-countable and there exists a compatible
 complete metric. This is the class of spaces that is well-behaved with respect to measure theory.
 In this file, we establish the basic properties of Polish spaces.

mathlib commit https://github.com/leanprover-community/mathlib/commit/730c6d4cab72b9d84fcfb9e95e8796e9cd8f40ba

@@ -60,6 +60,7 @@ variable {α : Type _} {β : Type _}
 /-! ### Basic properties of Polish spaces -/
 
 
+#print PolishSpace /-
 /- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`second_countable] [] -/
 /-- A Polish space is a topological space with second countable topology, that can be endowed
 with a metric for which it is complete.
@@ -73,33 +74,43 @@ class PolishSpace (α : Type _) [h : TopologicalSpace α] : Prop where
   complete :
     ∃ m : MetricSpace α, m.toUniformSpace.toTopologicalSpace = h ∧ @CompleteSpace α m.toUniformSpace
 #align polish_space PolishSpace
+-/
 
+#print UpgradedPolishSpace /-
 /-- A convenience class, for a Polish space endowed with a complete metric. No instance of this
 class should be registered: It should be used as `letI := upgrade_polish_space α` to endow a Polish
 space with a complete metric. -/
 class UpgradedPolishSpace (α : Type _) extends MetricSpace α, SecondCountableTopology α,
   CompleteSpace α
 #align upgraded_polish_space UpgradedPolishSpace
+-/
 
+#print polishSpace_of_complete_second_countable /-
 instance (priority := 100) polishSpace_of_complete_second_countable [m : MetricSpace α]
     [h : SecondCountableTopology α] [h' : CompleteSpace α] : PolishSpace α
     where
   second_countable := h
   complete := ⟨m, rfl, h'⟩
 #align polish_space_of_complete_second_countable polishSpace_of_complete_second_countable
+-/
 
+#print polishSpaceMetric /-
 /-- Construct on a Polish space a metric (compatible with the topology) which is complete. -/
 def polishSpaceMetric (α : Type _) [ht : TopologicalSpace α] [h : PolishSpace α] : MetricSpace α :=
   h.complete.some.replaceTopology h.complete.choose_spec.1.symm
 #align polish_space_metric polishSpaceMetric
+-/
 
+#print complete_polishSpaceMetric /-
 theorem complete_polishSpaceMetric (α : Type _) [ht : TopologicalSpace α] [h : PolishSpace α] :
     @CompleteSpace α (polishSpaceMetric α).toUniformSpace :=
   by
   convert h.complete.some_spec.2
   exact MetricSpace.replaceTopology_eq _ _
 #align complete_polish_space_metric complete_polishSpaceMetric
+-/
 
+#print upgradePolishSpace /-
 /-- This definition endows a Polish space with a complete metric. Use it as:
 `letI := upgrade_polish_space α`. -/
 def upgradePolishSpace (α : Type _) [ht : TopologicalSpace α] [h : PolishSpace α] :
@@ -107,15 +118,19 @@ def upgradePolishSpace (α : Type _) [ht : TopologicalSpace α] [h : PolishSpace
   letI := polishSpaceMetric α
   { complete_polishSpaceMetric α, PolishSpace.second_countable α with }
 #align upgrade_polish_space upgradePolishSpace
+-/
 
 namespace PolishSpace
 
+#print PolishSpace.t2Space /-
 instance (priority := 100) t2Space (α : Type _) [TopologicalSpace α] [PolishSpace α] : T2Space α :=
   by
   letI := upgradePolishSpace α
   infer_instance
 #align polish_space.t2_space PolishSpace.t2Space
+-/
 
+#print PolishSpace.pi_countable /-
 /-- A countable product of Polish spaces is Polish. -/
 instance pi_countable {ι : Type _} [Countable ι] {E : ι → Type _} [∀ i, TopologicalSpace (E i)]
     [∀ i, PolishSpace (E i)] : PolishSpace (∀ i, E i) :=
@@ -125,11 +140,20 @@ instance pi_countable {ι : Type _} [Countable ι] {E : ι → Type _} [∀ i, T
   letI : MetricSpace (∀ i, E i) := PiCountable.metricSpace
   infer_instance
 #align polish_space.pi_countable PolishSpace.pi_countable
+-/
 
+#print PolishSpace.nat_fun /-
 /-- Without this instance, `polish_space (ℕ → ℕ)` is not found by typeclass inference. -/
 instance nat_fun [TopologicalSpace α] [PolishSpace α] : PolishSpace (ℕ → α) := by infer_instance
 #align polish_space.nat_fun PolishSpace.nat_fun
+-/
 
+/- warning: polish_space.sigma -> PolishSpace.sigma is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {E : ι -> Type.{u2}} [_inst_2 : forall (n : ι), TopologicalSpace.{u2} (E n)] [_inst_3 : forall (n : ι), PolishSpace.{u2} (E n) (_inst_2 n)], PolishSpace.{max u1 u2} (Sigma.{u1, u2} ι (fun (n : ι) => E n)) (Sigma.topologicalSpace.{u1, u2} ι (fun (n : ι) => E n) (fun (a : ι) => _inst_2 a))
+but is expected to have type
+  forall {ι : Type.{u1}} [_inst_1 : Countable.{succ u1} ι] {E : ι -> Type.{u2}} [_inst_2 : forall (n : ι), TopologicalSpace.{u2} (E n)] [_inst_3 : forall (n : ι), PolishSpace.{u2} (E n) (_inst_2 n)], PolishSpace.{max u2 u1} (Sigma.{u1, u2} ι (fun (n : ι) => E n)) (instTopologicalSpaceSigma.{u1, u2} ι (fun (n : ι) => E n) (fun (a : ι) => _inst_2 a))
+Case conversion may be inaccurate. Consider using '#align polish_space.sigma PolishSpace.sigmaₓ'. -/
 /-- A countable disjoint union of Polish spaces is Polish. -/
 instance sigma {ι : Type _} [Countable ι] {E : ι → Type _} [∀ n, TopologicalSpace (E n)]
     [∀ n, PolishSpace (E n)] : PolishSpace (Σn, E n) :=
@@ -140,6 +164,12 @@ instance sigma {ι : Type _} [Countable ι] {E : ι → Type _} [∀ n, Topologi
   infer_instance
 #align polish_space.sigma PolishSpace.sigma
 
+/- warning: polish_space.sum -> PolishSpace.sum is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : PolishSpace.{u2} β _inst_3], PolishSpace.{max u1 u2} (Sum.{u1, u2} α β) (Sum.topologicalSpace.{u1, u2} α β _inst_1 _inst_3)
+but is expected to have type
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α _inst_1] [_inst_3 : TopologicalSpace.{u2} β] [_inst_4 : PolishSpace.{u2} β _inst_3], PolishSpace.{max u2 u1} (Sum.{u1, u2} α β) (instTopologicalSpaceSum.{u1, u2} α β _inst_1 _inst_3)
+Case conversion may be inaccurate. Consider using '#align polish_space.sum PolishSpace.sumₓ'. -/
 /-- The disjoint union of two Polish spaces is Polish. -/
 instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [PolishSpace β] :
     PolishSpace (Sum α β) := by
@@ -149,6 +179,12 @@ instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [Polis
   infer_instance
 #align polish_space.sum PolishSpace.sum
 
+/- warning: polish_space.exists_nat_nat_continuous_surjective -> PolishSpace.exists_nat_nat_continuous_surjective is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α _inst_1] [_inst_3 : Nonempty.{succ u1} α], Exists.{succ u1} ((Nat -> Nat) -> α) (fun (f : (Nat -> Nat) -> α) => And (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => Nat.topologicalSpace)) _inst_1 f) (Function.Surjective.{1, succ u1} (Nat -> Nat) α f))
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : PolishSpace.{u1} α _inst_1] [_inst_3 : Nonempty.{succ u1} α], Exists.{succ u1} ((Nat -> Nat) -> α) (fun (f : (Nat -> Nat) -> α) => And (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => instTopologicalSpaceNat)) _inst_1 f) (Function.Surjective.{1, succ u1} (Nat -> Nat) α f))
+Case conversion may be inaccurate. Consider using '#align polish_space.exists_nat_nat_continuous_surjective PolishSpace.exists_nat_nat_continuous_surjectiveₓ'. -/
 /-- Any nonempty Polish space is the continuous image of the fundamental space `ℕ → ℕ`. -/
 theorem exists_nat_nat_continuous_surjective (α : Type _) [TopologicalSpace α] [PolishSpace α]
     [Nonempty α] : ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f :=
@@ -156,6 +192,12 @@ theorem exists_nat_nat_continuous_surjective (α : Type _) [TopologicalSpace α]
   exists_nat_nat_continuous_surjective_of_completeSpace α
 #align polish_space.exists_nat_nat_continuous_surjective PolishSpace.exists_nat_nat_continuous_surjective
 
+/- warning: closed_embedding.polish_space -> ClosedEmbedding.polishSpace is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : PolishSpace.{u2} β _inst_2] {f : α -> β}, (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (PolishSpace.{u1} α _inst_1)
+but is expected to have type
+  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] [_inst_3 : PolishSpace.{u1} β _inst_2] {f : α -> β}, (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) -> (PolishSpace.{u2} α _inst_1)
+Case conversion may be inaccurate. Consider using '#align closed_embedding.polish_space ClosedEmbedding.polishSpaceₓ'. -/
 /-- Given a closed embedding into a Polish space, the source space is also Polish. -/
 theorem ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β] [PolishSpace β]
     {f : α → β} (hf : ClosedEmbedding f) : PolishSpace α :=
@@ -171,26 +213,38 @@ theorem ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β]
   infer_instance
 #align closed_embedding.polish_space ClosedEmbedding.polishSpace
 
+#print Equiv.polishSpace_induced /-
 /-- Pulling back a Polish topology under an equiv gives again a Polish topology. -/
 theorem Equiv.polishSpace_induced [t : TopologicalSpace β] [PolishSpace β] (f : α ≃ β) :
     @PolishSpace α (t.induced f) :=
   letI : TopologicalSpace α := t.induced f
   (f.to_homeomorph_of_inducing ⟨rfl⟩).ClosedEmbedding.PolishSpace
 #align equiv.polish_space_induced Equiv.polishSpace_induced
+-/
 
+#print IsClosed.polishSpace /-
 /-- A closed subset of a Polish space is also Polish. -/
 theorem IsClosed.polishSpace {α : Type _} [TopologicalSpace α] [PolishSpace α] {s : Set α}
     (hs : IsClosed s) : PolishSpace s :=
   (IsClosed.closedEmbedding_subtype_val hs).PolishSpace
 #align is_closed.polish_space IsClosed.polishSpace
+-/
 
+#print PolishSpace.AuxCopy /-
 /-- A sequence of type synonyms of a given type `α`, useful in the proof of
 `exists_polish_space_forall_le` to endow each copy with a different topology. -/
 @[nolint unused_arguments has_nonempty_instance]
 def AuxCopy (α : Type _) {ι : Type _} (i : ι) : Type _ :=
   α
 #align polish_space.aux_copy PolishSpace.AuxCopy
+-/
 
+/- warning: polish_space.exists_polish_space_forall_le -> PolishSpace.exists_polishSpace_forall_le is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] [t : TopologicalSpace.{u1} α] [p : PolishSpace.{u1} α t] (m : ι -> (TopologicalSpace.{u1} α)), (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) (m n) t) -> (forall (n : ι), PolishSpace.{u1} α (m n)) -> (Exists.{succ u1} (TopologicalSpace.{u1} α) (fun (t' : TopologicalSpace.{u1} α) => And (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' (m n)) (And (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.partialOrder.{u1} α))) t' t) (PolishSpace.{u1} α t'))))
+but is expected to have type
+  forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : Countable.{succ u2} ι] [t : TopologicalSpace.{u1} α] [p : PolishSpace.{u1} α t] (m : ι -> (TopologicalSpace.{u1} α)), (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) (m n) t) -> (forall (n : ι), PolishSpace.{u1} α (m n)) -> (Exists.{succ u1} (TopologicalSpace.{u1} α) (fun (t' : TopologicalSpace.{u1} α) => And (forall (n : ι), LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) t' (m n)) (And (LE.le.{u1} (TopologicalSpace.{u1} α) (Preorder.toLE.{u1} (TopologicalSpace.{u1} α) (PartialOrder.toPreorder.{u1} (TopologicalSpace.{u1} α) (TopologicalSpace.instPartialOrderTopologicalSpace.{u1} α))) t' t) (PolishSpace.{u1} α t'))))
+Case conversion may be inaccurate. Consider using '#align polish_space.exists_polish_space_forall_le PolishSpace.exists_polishSpace_forall_leₓ'. -/
 /-- Given a Polish space, and countably many finer Polish topologies, there exists another Polish
 topology which is finer than all of them. -/
 theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : TopologicalSpace α]
@@ -273,34 +327,37 @@ variable [MetricSpace α] {s : Set α}
 /-- A type synonym for a subset `s` of a metric space, on which we will construct another metric
 for which it will be complete. -/
 @[nolint has_nonempty_instance]
-def CompleteCopy {α : Type _} (s : Set α) : Type _ :=
+def TopologicalSpace.Opens.CompleteCopy {α : Type _} (s : Set α) : Type _ :=
   s
-#align polish_space.complete_copy PolishSpace.CompleteCopy
+#align polish_space.complete_copy TopologicalSpace.Opens.CompleteCopyₓ
 
 /-- A distance on a subset `s` of a metric space, designed to make it complete if `s` is open.
 It is given by `dist' x y = dist x y + |1 / dist x sᶜ - 1 / dist y sᶜ|`, where the second term
 blows up close to the boundary to ensure that Cauchy sequences for `dist'` remain well
 inside `s`. -/
-def hasDistCompleteCopy (s : Set α) : Dist (CompleteCopy s) :=
+def TopologicalSpace.Opens.CompleteCopy.instDist (s : Set α) :
+    Dist (TopologicalSpace.Opens.CompleteCopy s) :=
   ⟨fun x y => dist x.1 y.1 + abs (1 / infDist x.1 (sᶜ) - 1 / infDist y.1 (sᶜ))⟩
-#align polish_space.has_dist_complete_copy PolishSpace.hasDistCompleteCopy
+#align polish_space.has_dist_complete_copy TopologicalSpace.Opens.CompleteCopy.instDistₓ
 
 attribute [local instance] has_dist_complete_copy
 
-theorem dist_completeCopy_eq (x y : CompleteCopy s) :
+theorem TopologicalSpace.Opens.CompleteCopy.dist_eq (x y : TopologicalSpace.Opens.CompleteCopy s) :
     dist x y = dist x.1 y.1 + abs (1 / infDist x.1 (sᶜ) - 1 / infDist y.1 (sᶜ)) :=
   rfl
-#align polish_space.dist_complete_copy_eq PolishSpace.dist_completeCopy_eq
+#align polish_space.dist_complete_copy_eq TopologicalSpace.Opens.CompleteCopy.dist_eqₓ
 
-theorem dist_le_dist_completeCopy (x y : CompleteCopy s) : dist x.1 y.1 ≤ dist x y :=
+theorem TopologicalSpace.Opens.CompleteCopy.dist_val_le_dist
+    (x y : TopologicalSpace.Opens.CompleteCopy s) : dist x.1 y.1 ≤ dist x y :=
   (le_add_iff_nonneg_right _).2 (abs_nonneg _)
-#align polish_space.dist_le_dist_complete_copy PolishSpace.dist_le_dist_completeCopy
+#align polish_space.dist_le_dist_complete_copy TopologicalSpace.Opens.CompleteCopy.dist_val_le_distₓ
 
 /-- A metric space structure on a subset `s` of a metric space, designed to make it complete
 if `s` is open. It is given by `dist' x y = dist x y + |1 / dist x sᶜ - 1 / dist y sᶜ|`, where the
 second term blows up close to the boundary to ensure that Cauchy sequences for `dist'` remain well
 inside `s`. -/
-def completeCopyMetricSpace (s : Set α) : MetricSpace (CompleteCopy s)
+def TopologicalSpace.Opens.CompleteCopy.instMetricSpace (s : Set α) :
+    MetricSpace (TopologicalSpace.Opens.CompleteCopy s)
     where
   dist_self x := by simp [dist_complete_copy_eq]
   dist_comm x y := by simp [dist_complete_copy_eq, dist_comm, abs_sub_comm]
@@ -325,13 +382,20 @@ def completeCopyMetricSpace (s : Set α) : MetricSpace (CompleteCopy s)
     refine' dist_le_zero.1 _
     rw [← hxy]
     exact dist_le_dist_complete_copy x y
-#align polish_space.complete_copy_metric_space PolishSpace.completeCopyMetricSpace
+#align polish_space.complete_copy_metric_space TopologicalSpace.Opens.CompleteCopy.instMetricSpaceₓ
 
 attribute [local instance] complete_copy_metric_space
 
+/- warning: polish_space.complete_copy_id_homeo clashes with [anonymous] -> [anonymous]
+warning: polish_space.complete_copy_id_homeo -> [anonymous] is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u_1}} [_inst_1 : MetricSpace.{u_1} α] {s : Set.{u_1} α}, (IsOpen.{u_1} α (UniformSpace.toTopologicalSpace.{u_1} α (PseudoMetricSpace.toUniformSpace.{u_1} α (MetricSpace.toPseudoMetricSpace.{u_1} α _inst_1))) s) -> (Set.Nonempty.{u_1} α (HasCompl.compl.{u_1} (Set.{u_1} α) (BooleanAlgebra.toHasCompl.{u_1} (Set.{u_1} α) (Set.booleanAlgebra.{u_1} α)) s)) -> (Homeomorph.{u_1, u_1} (TopologicalSpace.Opens.CompleteCopyₓ.{u_1} α s) (coeSort.{max (succ u_1) 1, succ (succ u_1)} (Set.{u_1} α) Type.{u_1} (Set.hasCoeToSort.{u_1} α) s) (UniformSpace.toTopologicalSpace.{u_1} (TopologicalSpace.Opens.CompleteCopyₓ.{u_1} α s) (PseudoMetricSpace.toUniformSpace.{u_1} (TopologicalSpace.Opens.CompleteCopyₓ.{u_1} α s) (MetricSpace.toPseudoMetricSpace.{u_1} (TopologicalSpace.Opens.CompleteCopyₓ.{u_1} α s) (TopologicalSpace.Opens.CompleteCopy.instMetricSpaceₓ.{u_1} α _inst_1 s)))) (Subtype.topologicalSpace.{u_1} α (fun (x : α) => Membership.Mem.{u_1, u_1} α (Set.{u_1} α) (Set.hasMem.{u_1} α) x s) (UniformSpace.toTopologicalSpace.{u_1} α (PseudoMetricSpace.toUniformSpace.{u_1} α (MetricSpace.toPseudoMetricSpace.{u_1} α _inst_1)))))
+but is expected to have type
+  forall {α : Type.{u}} {_inst_1 : Type.{v}}, (Nat -> α -> _inst_1) -> Nat -> (List.{u} α) -> (List.{v} _inst_1)
+Case conversion may be inaccurate. Consider using '#align polish_space.complete_copy_id_homeo [anonymous]ₓ'. -/
 /-- The identity between the type synonym `complete_copy s` (with its modified metric) and the
 original subtype `s` is a homeomorphism. -/
-def completeCopyIdHomeo (hs : IsOpen s) (h's : sᶜ.Nonempty) : CompleteCopy s ≃ₜ s
+def [anonymous] (hs : IsOpen s) (h's : sᶜ.Nonempty) : TopologicalSpace.Opens.CompleteCopy s ≃ₜ s
     where
   toFun := id
   invFun := id
@@ -360,10 +424,10 @@ def completeCopyIdHomeo (hs : IsOpen s) (h's : sᶜ.Nonempty) : CompleteCopy s 
     · refine' (tendsto.sub_const _ _).abs
       refine' tendsto.div tendsto_const_nhds _ I.ne'
       exact ((continuous_inf_dist_pt _).comp continuous_subtype_val).Tendsto _
-#align polish_space.complete_copy_id_homeo PolishSpace.completeCopyIdHomeo
+#align polish_space.complete_copy_id_homeo [anonymous]
 
-theorem completeSpace_completeCopy [CompleteSpace α] (hs : IsOpen s) (h's : sᶜ.Nonempty) :
-    CompleteSpace (CompleteCopy s) :=
+theorem TopologicalSpace.Opens.CompleteCopy.instCompleteSpace [CompleteSpace α] (hs : IsOpen s)
+    (h's : sᶜ.Nonempty) : CompleteSpace (TopologicalSpace.Opens.CompleteCopy s) :=
   by
   refine' Metric.complete_of_convergent_controlled_sequences (fun n => (1 / 2) ^ n) (by simp) _
   intro u hu
@@ -423,8 +487,9 @@ theorem completeSpace_completeCopy [CompleteSpace α] (hs : IsOpen s) (h's : s
   suffices x ∉ sᶜ by simpa
   apply (hs.is_closed_compl.not_mem_iff_inf_dist_pos h's).2 (lt_of_lt_of_le _ I')
   simp [Cpos]
-#align polish_space.complete_space_complete_copy PolishSpace.completeSpace_completeCopy
+#align polish_space.complete_space_complete_copy TopologicalSpace.Opens.CompleteCopy.instCompleteSpaceₓ
 
+#print IsOpen.polishSpace /-
 /-- An open subset of a Polish space is also Polish. -/
 theorem IsOpen.polishSpace {α : Type _} [TopologicalSpace α] [PolishSpace α] {s : Set α}
     (hs : IsOpen s) : PolishSpace s :=
@@ -440,19 +505,23 @@ theorem IsOpen.polishSpace {α : Type _} [TopologicalSpace α] [PolishSpace α]
       (complete_copy_id_homeo hs h's).Embedding.SecondCountableTopology
     exact (complete_copy_id_homeo hs h's).symm.ClosedEmbedding.PolishSpace
 #align is_open.polish_space IsOpen.polishSpace
+-/
 
 end CompleteCopy
 
 /-! ### Clopenable sets in Polish spaces -/
 
 
+#print PolishSpace.IsClopenable /-
 /-- A set in a topological space is clopenable if there exists a finer Polish topology for which
 this set is open and closed. It turns out that this notion is equivalent to being Borel-measurable,
 but this is nontrivial (see `is_clopenable_iff_measurable_set`). -/
 def IsClopenable [t : TopologicalSpace α] (s : Set α) : Prop :=
   ∃ t' : TopologicalSpace α, t' ≤ t ∧ @PolishSpace α t' ∧ is_closed[t'] s ∧ is_open[t'] s
 #align polish_space.is_clopenable PolishSpace.IsClopenable
+-/
 
+#print IsClosed.isClopenable /-
 /-- Given a closed set `s` in a Polish space, one can construct a finer Polish topology for
 which `s` is both open and closed. -/
 theorem IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α} (hs : IsClosed s) :
@@ -510,17 +579,27 @@ theorem IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α
     convert this
     exact A.symm
 #align is_closed.is_clopenable IsClosed.isClopenable
+-/
 
+/- warning: polish_space.is_clopenable.compl -> PolishSpace.IsClopenable.compl is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} α] {s : Set.{u1} α}, (PolishSpace.IsClopenable.{u1} α _inst_1 s) -> (PolishSpace.IsClopenable.{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] {s : Set.{u1} α}, (PolishSpace.IsClopenable.{u1} α _inst_1 s) -> (PolishSpace.IsClopenable.{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 polish_space.is_clopenable.compl PolishSpace.IsClopenable.complₓ'. -/
 theorem IsClopenable.compl [TopologicalSpace α] {s : Set α} (hs : IsClopenable s) :
     IsClopenable (sᶜ) := by
   rcases hs with ⟨t, t_le, t_polish, h, h'⟩
   exact ⟨t, t_le, t_polish, @IsOpen.isClosed_compl α t s h', @IsClosed.isOpen_compl α t s h⟩
 #align polish_space.is_clopenable.compl PolishSpace.IsClopenable.compl
 
+#print IsOpen.isClopenable /-
 theorem IsOpen.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α} (hs : IsOpen s) :
     IsClopenable s := by simpa using hs.is_closed_compl.is_clopenable.compl
 #align is_open.is_clopenable IsOpen.isClopenable
+-/
 
+#print PolishSpace.IsClopenable.unionᵢ /-
 theorem IsClopenable.unionᵢ [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClopenable (s n)) : IsClopenable (⋃ n, s n) :=
   by
@@ -539,6 +618,7 @@ theorem IsClopenable.unionᵢ [t : TopologicalSpace α] [PolishSpace α] {s : 
     @IsOpen.isClopenable α t' t'_polish _ A
   exact ⟨t'', t''_le.trans ((t'm 0).trans (mt 0)), t''_polish, h1, h2⟩
 #align polish_space.is_clopenable.Union PolishSpace.IsClopenable.unionᵢ
+-/
 
 end PolishSpace
 

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

@@ -381,7 +381,7 @@ theorem completeSpace_completeCopy [CompleteSpace α] (hs : IsOpen s) (h's : s
       by
       apply embedding_subtype_coe.tendsto_nhds_iff.2
       exact xlim
-    convert ((complete_copy_id_homeo hs h's).symm.Continuous.Tendsto _).comp L
+    convert((complete_copy_id_homeo hs h's).symm.Continuous.Tendsto _).comp L
     ext1 n
     simp [complete_copy_id_homeo]
   obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / inf_dist (u n).1 (sᶜ) < C :=

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

@@ -281,7 +281,7 @@ def CompleteCopy {α : Type _} (s : Set α) : Type _ :=
 It is given by `dist' x y = dist x y + |1 / dist x sᶜ - 1 / dist y sᶜ|`, where the second term
 blows up close to the boundary to ensure that Cauchy sequences for `dist'` remain well
 inside `s`. -/
-def hasDistCompleteCopy (s : Set α) : HasDist (CompleteCopy s) :=
+def hasDistCompleteCopy (s : Set α) : Dist (CompleteCopy s) :=
   ⟨fun x y => dist x.1 y.1 + abs (1 / infDist x.1 (sᶜ) - 1 / infDist y.1 (sᶜ))⟩
 #align polish_space.has_dist_complete_copy PolishSpace.hasDistCompleteCopy
 

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

@@ -395,11 +395,11 @@ theorem completeSpace_completeCopy [CompleteSpace α] (hs : IsOpen s) (h's : s
       _ ≤
           dist (1 / inf_dist (u n).1 (sᶜ)) (1 / inf_dist (u 0).1 (sᶜ)) +
             dist (1 / inf_dist (u 0).1 (sᶜ)) 0 :=
-        dist_triangle _ _ _
+        (dist_triangle _ _ _)
       _ ≤
           dist (u n).1 (u 0).1 + dist (1 / inf_dist (u n).1 (sᶜ)) (1 / inf_dist (u 0).1 (sᶜ)) +
             dist (1 / inf_dist (u 0).1 (sᶜ)) 0 :=
-        add_le_add (le_add_of_nonneg_left dist_nonneg) le_rfl
+        (add_le_add (le_add_of_nonneg_left dist_nonneg) le_rfl)
       _ = dist (u n) (u 0) + dist (1 / inf_dist (u 0).1 (sᶜ)) 0 := rfl
       _ < (1 / 2) ^ 0 + dist (1 / inf_dist (u 0).1 (sᶜ)) 0 :=
         add_lt_add_right (hu 0 n 0 (zero_le _) le_rfl) _

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

Reference the newly created issues #12094 and #12096, as well as the pre-existing #5171. Change all references to #10927 to #5171. Some of these changes were not labelled as "porting note"; change this for good measure.

@@ -251,7 +251,7 @@ variable [MetricSpace α] {s : Opens α}
 
 /-- A type synonym for a subset `s` of a metric space, on which we will construct another metric
 for which it will be complete. -/
--- Porting note: was @[nolint has_nonempty_instance]
+-- Porting note(#5171): was @[nolint has_nonempty_instance]
 def CompleteCopy {α : Type*} [MetricSpace α] (s : Opens α) : Type _ := s
 #align polish_space.complete_copy TopologicalSpace.Opens.CompleteCopyₓ
 

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

@@ -159,7 +159,7 @@ theorem _root_.ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpa
   haveI : SecondCountableTopology α := hf.toEmbedding.secondCountableTopology
   have : CompleteSpace α := by
     rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing]
-    exact hf.closed_range.isComplete
+    exact hf.isClosed_range.isComplete
   infer_instance
 #align closed_embedding.polish_space ClosedEmbedding.polishSpace
 

• Make it work for a UniformSpace with extra properties.
• Use it to golf some instances.

Co-authored-by: @ADedecker

@@ -48,8 +48,8 @@ with additional properties:
 
 noncomputable section
 
-open scoped Classical
-open Topology Filter TopologicalSpace Set Metric Function
+open scoped Topology Uniformity
+open Filter TopologicalSpace Set Metric Function
 
 variable {α : Type*} {β : Type*}
 
@@ -76,10 +76,12 @@ class UpgradedPolishSpace (α : Type*) extends MetricSpace α, SecondCountableTo
   CompleteSpace α
 #align upgraded_polish_space UpgradedPolishSpace
 
-instance (priority := 100) polishSpace_of_complete_second_countable [m : MetricSpace α]
-    [SecondCountableTopology α] [h' : CompleteSpace α] : PolishSpace α where
-  complete := ⟨m, rfl, h'⟩
-#align polish_space_of_complete_second_countable polishSpace_of_complete_second_countable
+instance (priority := 100) PolishSpace.of_separableSpace_completeSpace_metrizable [UniformSpace α]
+    [SeparableSpace α] [CompleteSpace α] [(𝓤 α).IsCountablyGenerated] [T0Space α] :
+    PolishSpace α where
+  toSecondCountableTopology := UniformSpace.secondCountable_of_separable α
+  complete := ⟨UniformSpace.metricSpace α, rfl, ‹_›⟩
+#align polish_space_of_complete_second_countable PolishSpace.of_separableSpace_completeSpace_metrizable
 
 /-- Construct on a Polish space a metric (compatible with the topology) which is complete. -/
 def polishSpaceMetric (α : Type*) [TopologicalSpace α] [h : PolishSpace α] : MetricSpace α :=
@@ -107,18 +109,14 @@ instance (priority := 100) instMetrizableSpace (α : Type*) [TopologicalSpace α
   letI := upgradePolishSpace α
   infer_instance
 
-instance (priority := 100) t2Space (α : Type*) [TopologicalSpace α] [PolishSpace α] :
-    T2Space α := by
-  letI := upgradePolishSpace α
-  infer_instance
+@[deprecated] -- 2024-02-23
+theorem t2Space (α : Type*) [TopologicalSpace α] [PolishSpace α] : T2Space α := inferInstance
 #align polish_space.t2_space PolishSpace.t2Space
 
 /-- A countable product of Polish spaces is Polish. -/
 instance pi_countable {ι : Type*} [Countable ι] {E : ι → Type*} [∀ i, TopologicalSpace (E i)]
     [∀ i, PolishSpace (E i)] : PolishSpace (∀ i, E i) := by
-  cases nonempty_encodable ι
   letI := fun i => upgradePolishSpace (E i)
-  letI : MetricSpace (∀ i, E i) := PiCountable.metricSpace
   infer_instance
 #align polish_space.pi_countable PolishSpace.pi_countable
 
@@ -143,7 +141,6 @@ instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [Polis
     PolishSpace (α ⊕ β) :=
   letI := upgradePolishSpace α
   letI := upgradePolishSpace β
-  letI : MetricSpace (α ⊕ β) := metricSpaceSum
   inferInstance
 #align polish_space.sum PolishSpace.sum
 
@@ -194,62 +191,42 @@ instance instPolishSpaceUniv [TopologicalSpace α] [PolishSpace α] :
   isClosed_univ.polishSpace
 #align measure_theory.set.univ.polish_space PolishSpace.instPolishSpaceUniv
 
-/-- A sequence of type synonyms of a given type `α`, useful in the proof of
-`exists_polishSpace_forall_le` to endow each copy with a different topology. -/
-@[nolint unusedArguments]
-def AuxCopy (α : Type*) {ι : Type*} (_i : ι) : Type _ := α
-#align polish_space.aux_copy PolishSpace.AuxCopy
+protected theorem _root_.CompletePseudometrizable.iInf {ι : Type*} [Countable ι]
+    {t : ι → TopologicalSpace α} (ht₀ : ∃ t₀, @T2Space α t₀ ∧ ∀ i, t i ≤ t₀)
+    (ht : ∀ i, ∃ u : UniformSpace α, CompleteSpace α ∧ 𝓤[u].IsCountablyGenerated ∧
+      u.toTopologicalSpace = t i) :
+    ∃ u : UniformSpace α, CompleteSpace α ∧
+      𝓤[u].IsCountablyGenerated ∧ u.toTopologicalSpace = ⨅ i, t i := by
+  choose u hcomp hcount hut using ht
+  obtain rfl : t = fun i ↦ (u i).toTopologicalSpace := (funext hut).symm
+  refine ⟨⨅ i, u i, .iInf hcomp ht₀, ?_, UniformSpace.toTopologicalSpace_iInf⟩
+  rw [iInf_uniformity]
+  infer_instance
+
+protected theorem iInf {ι : Type*} [Countable ι] {t : ι → TopologicalSpace α}
+    (ht₀ : ∃ i₀, ∀ i, t i ≤ t i₀) (ht : ∀ i, @PolishSpace α (t i)) : @PolishSpace α (⨅ i, t i) := by
+  rcases ht₀ with ⟨i₀, hi₀⟩
+  rcases CompletePseudometrizable.iInf ⟨t i₀, letI := t i₀; haveI := ht i₀; inferInstance, hi₀⟩
+    fun i ↦
+      letI := t i; haveI := ht i; letI := upgradePolishSpace α
+      ⟨inferInstance, inferInstance, inferInstance, rfl⟩
+    with ⟨u, hcomp, hcount, htop⟩
+  rw [← htop]
+  have : @SecondCountableTopology α u.toTopologicalSpace :=
+    htop.symm ▸ secondCountableTopology_iInf fun i ↦ letI := t i; (ht i).toSecondCountableTopology
+  have : @T1Space α u.toTopologicalSpace :=
+    htop.symm ▸ t1Space_antitone (iInf_le _ i₀) (by letI := t i₀; haveI := ht i₀; infer_instance)
+  infer_instance
+#noalign polish_space.aux_copy
 
 /-- Given a Polish space, and countably many finer Polish topologies, there exists another Polish
-topology which is finer than all of them.
-
-Porting note (#11215): TODO: the topology `t'` is `t ⊓ ⨅ i, m i`. -/
+topology which is finer than all of them. -/
 theorem exists_polishSpace_forall_le {ι : Type*} [Countable ι] [t : TopologicalSpace α]
     [p : PolishSpace α] (m : ι → TopologicalSpace α) (hm : ∀ n, m n ≤ t)
     (h'm : ∀ n, @PolishSpace α (m n)) :
-    ∃ t' : TopologicalSpace α, (∀ n, t' ≤ m n) ∧ t' ≤ t ∧ @PolishSpace α t' := by
-  rcases isEmpty_or_nonempty ι with (hι | hι)
-  · exact ⟨t, fun i => (IsEmpty.elim hι i : _), le_rfl, p⟩
-  inhabit ι
-  /- Consider the product of infinitely many copies of `α`, each endowed with the topology `m n`.
-    This is a Polish space, as a product of Polish spaces. Pulling back this topology under the
-    diagonal embedding of `α`, one gets a Polish topology which is finer than all the `m n`. -/
-  letI : ∀ n : ι, TopologicalSpace (AuxCopy α n) := fun n => m n
-  haveI : ∀ n : ι, PolishSpace (AuxCopy α n) := fun n => h'm n
-  letI T : TopologicalSpace (∀ n : ι, AuxCopy α n) := inferInstance
-  let f : α → ∀ n : ι, AuxCopy α n := fun x _ => x
-  -- show that the induced topology is finer than all the `m n`.
-  have T_le_m : ∀ n, T.induced f ≤ m n := fun n ↦ by
-    rw [induced_to_pi]
-    exact iInf_le_of_le n (@induced_id _ (m n)).le
-  refine' ⟨T.induced f, fun n => T_le_m n, (T_le_m default).trans (hm default), _⟩
-  -- show that the new topology is Polish, as the pullback of a Polish topology under a closed
-  -- embedding.
-  have A : range f = ⋂ n, { x | x n = x default } := by
-    ext x
-    constructor
-    · rintro ⟨y, rfl⟩
-      exact mem_iInter.2 fun n => by simp only [mem_setOf_eq]
-    · refine fun hx ↦ ⟨x default, ?_⟩
-      ext1 n
-      symm
-      exact mem_iInter.1 hx n
-  have f_closed : IsClosed (range f) := by
-    rw [A]
-    refine isClosed_iInter fun n => ?_
-    have C : ∀ i : ι, Continuous fun x : ∀ n, AuxCopy α n => (id (x i) : α) := fun i ↦
-      have : Continuous (show AuxCopy α i → α from id) := continuous_id_of_le (hm i)
-      this.comp (continuous_apply i)
-    apply isClosed_eq (C n) (C default)
-  have K : @_root_.Embedding _ _ (T.induced f) T f := by
-    refine Function.Injective.embedding_induced fun x y hxy ↦ ?_
-    have : f x default = f y default := by rw [hxy]
-    exact this
-  have L : @ClosedEmbedding _ _ (T.induced f) T f := by
-    refine @ClosedEmbedding.mk _ _ (T.induced f) T f ?_ ?_
-    · exact K
-    · exact f_closed
-  exact @ClosedEmbedding.polishSpace _ _ (T.induced f) T (by infer_instance) _ L
+    ∃ t' : TopologicalSpace α, (∀ n, t' ≤ m n) ∧ t' ≤ t ∧ @PolishSpace α t' :=
+  ⟨⨅ i : Option ι, i.elim t m, fun i ↦ iInf_le _ (some i), iInf_le _ none,
+    .iInf ⟨none, Option.forall.2 ⟨le_rfl, hm⟩⟩ <| Option.forall.2 ⟨p, h'm⟩⟩
 #align polish_space.exists_polish_space_forall_le PolishSpace.exists_polishSpace_forall_le
 
 end PolishSpace
@@ -400,6 +377,7 @@ theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s :
   /- Both sets `s` and `sᶜ` admit a Polish topology. So does their disjoint union `s ⊕ sᶜ`.
     Pulling back this topology by the canonical bijection with `α` gives the desired Polish
     topology in which `s` is both open and closed. -/
+  classical
   haveI : PolishSpace s := hs.polishSpace
   let t : Set α := sᶜ
   haveI : PolishSpace t := hs.isOpen_compl.polishSpace

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

@@ -203,7 +203,7 @@ def AuxCopy (α : Type*) {ι : Type*} (_i : ι) : Type _ := α
 /-- Given a Polish space, and countably many finer Polish topologies, there exists another Polish
 topology which is finer than all of them.
 
-Porting note: TODO: the topology `t'` is `t ⊓ ⨅ i, m i`. -/
+Porting note (#11215): TODO: the topology `t'` is `t ⊓ ⨅ i, m i`. -/
 theorem exists_polishSpace_forall_le {ι : Type*} [Countable ι] [t : TopologicalSpace α]
     [p : PolishSpace α] (m : ι → TopologicalSpace α) (hm : ∀ n, m n ≤ t)
     (h'm : ∀ n, @PolishSpace α (m n)) :
@@ -426,7 +426,7 @@ theorem _root_.IsOpen.isClopenable [TopologicalSpace α] [PolishSpace α] {s : S
   simpa using hs.isClosed_compl.isClopenable.compl
 #align is_open.is_clopenable IsOpen.isClopenable
 
--- Porting note: TODO: generalize for free to `[Countable ι] {s : ι → Set α}`
+-- Porting note (#11215): TODO: generalize for free to `[Countable ι] {s : ι → Set α}`
 theorem IsClopenable.iUnion [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClopenable (s n)) : IsClopenable (⋃ n, s n) := by
   choose m mt m_polish _ m_open using hs

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -48,7 +48,8 @@ with additional properties:
 
 noncomputable section
 
-open Classical Topology Filter TopologicalSpace Set Metric Function
+open scoped Classical
+open Topology Filter TopologicalSpace Set Metric Function
 
 variable {α : Type*} {β : Type*}
 

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

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

@@ -273,7 +273,7 @@ variable [MetricSpace α] {s : Opens α}
 
 /-- A type synonym for a subset `s` of a metric space, on which we will construct another metric
 for which it will be complete. -/
--- porting note: was @[nolint has_nonempty_instance]
+-- Porting note: was @[nolint has_nonempty_instance]
 def CompleteCopy {α : Type*} [MetricSpace α] (s : Opens α) : Type _ := s
 #align polish_space.complete_copy TopologicalSpace.Opens.CompleteCopyₓ
 
@@ -425,7 +425,7 @@ theorem _root_.IsOpen.isClopenable [TopologicalSpace α] [PolishSpace α] {s : S
   simpa using hs.isClosed_compl.isClopenable.compl
 #align is_open.is_clopenable IsOpen.isClopenable
 
--- porting note: TODO: generalize for free to `[Countable ι] {s : ι → Set α}`
+-- Porting note: TODO: generalize for free to `[Countable ι] {s : ι → Set α}`
 theorem IsClopenable.iUnion [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClopenable (s n)) : IsClopenable (⋃ n, s n) := by
   choose m mt m_polish _ m_open using hs

This covers many instances, but is not exhaustive.

Independently of whether that syntax should be avoided (similar to #10534), I think all these changes are small improvements.

@@ -346,8 +346,8 @@ instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) :
   obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
   by_cases xs : x ∈ s
   · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
-  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
-  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
+  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C := by
+    refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
     rw [← sub_lt_iff_lt_add]
     calc
       _ ≤ |1 / infDist (u n).1 sᶜ - 1 / infDist (u 0).1 sᶜ| := le_abs_self _
@@ -430,7 +430,7 @@ theorem IsClopenable.iUnion [t : TopologicalSpace α] [PolishSpace α] {s : ℕ
     (hs : ∀ n, IsClopenable (s n)) : IsClopenable (⋃ n, s n) := by
   choose m mt m_polish _ m_open using hs
   obtain ⟨t', t'm, -, t'_polish⟩ :
-    ∃ t' : TopologicalSpace α, (∀ n : ℕ, t' ≤ m n) ∧ t' ≤ t ∧ @PolishSpace α t' :=
+      ∃ t' : TopologicalSpace α, (∀ n : ℕ, t' ≤ m n) ∧ t' ≤ t ∧ @PolishSpace α t' :=
     exists_polishSpace_forall_le m mt m_polish
   have A : IsOpen[t'] (⋃ n, s n) := by
     apply isOpen_iUnion

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

@@ -411,7 +411,7 @@ theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s :
   · rw [← f.induced_symm]
     exact f.symm.polishSpace_induced
   · rw [isOpen_coinduced, isOpen_sum_iff]
-    simp [preimage_preimage]
+    simp [f, preimage_preimage]
 #align is_closed.is_clopenable IsClosed.isClopenable
 
 theorem IsClopenable.compl [TopologicalSpace α] {s : Set α} (hs : IsClopenable s) :

Add measurableSet_tendsto: the set of points for which a measurable sequence of functions converges to a given filter is measurable.

Co-authored-by: Rémy Degenne <remydegenne@gmail.com>

@@ -101,6 +101,11 @@ def upgradePolishSpace (α : Type*) [TopologicalSpace α] [PolishSpace α] :
 
 namespace PolishSpace
 
+instance (priority := 100) instMetrizableSpace (α : Type*) [TopologicalSpace α] [PolishSpace α] :
+    MetrizableSpace α := by
+  letI := upgradePolishSpace α
+  infer_instance
+
 instance (priority := 100) t2Space (α : Type*) [TopologicalSpace α] [PolishSpace α] :
     T2Space α := by
   letI := upgradePolishSpace α

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

This follows on from #6964.

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

@@ -319,8 +319,8 @@ instance instMetricSpace : MetricSpace (CompleteCopy s) := by
     · rcases h x hx with ⟨ε, ε0, hε⟩
       simp only [dist_eq, one_div] at hε
       have : Tendsto (fun y : s ↦ dist x.1 y.1 + |(infDist x.1 sᶜ)⁻¹ - (infDist y.1 sᶜ)⁻¹|)
-        (𝓝 x) (𝓝 (dist x.1 x.1 + |(infDist x.1 sᶜ)⁻¹ - (infDist x.1 sᶜ)⁻¹|))
-      · refine (tendsto_const_nhds.dist continuous_subtype_val.continuousAt).add
+          (𝓝 x) (𝓝 (dist x.1 x.1 + |(infDist x.1 sᶜ)⁻¹ - (infDist x.1 sᶜ)⁻¹|)) := by
+        refine (tendsto_const_nhds.dist continuous_subtype_val.continuousAt).add
           (tendsto_const_nhds.sub <| ?_).abs
         refine (continuousAt_inv_infDist_pt ?_).comp continuous_subtype_val.continuousAt
         rw [s.isOpen.isClosed_compl.closure_eq, mem_compl_iff, not_not]
@@ -334,8 +334,8 @@ instance instMetricSpace : MetricSpace (CompleteCopy s) := by
 
 instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
   refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
-  have A : CauchySeq fun n => (u n).1
-  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
+  have A : CauchySeq fun n => (u n).1 := by
+    refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
     · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
     · exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
   obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
@@ -398,8 +398,8 @@ theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s :
   let t : Set α := sᶜ
   haveI : PolishSpace t := hs.isOpen_compl.polishSpace
   let f : s ⊕ t ≃ α := Equiv.Set.sumCompl s
-  have hle : TopologicalSpace.coinduced f instTopologicalSpaceSum ≤ ‹_›
-  · simp only [instTopologicalSpaceSum, coinduced_sup, coinduced_compose, sup_le_iff,
+  have hle : TopologicalSpace.coinduced f instTopologicalSpaceSum ≤ ‹_› := by
+    simp only [instTopologicalSpaceSum, coinduced_sup, coinduced_compose, sup_le_iff,
       ← continuous_iff_coinduced_le]
     exact ⟨continuous_subtype_val, continuous_subtype_val⟩
   refine ⟨.coinduced f instTopologicalSpaceSum, hle, ?_, hs.mono hle, ?_⟩

See here on Zulip.

This PR changes a bunch of names containing nhds_0 or/and lt_1 to nhds_zero or/and lt_one.

@@ -337,7 +337,7 @@ instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) :
   have A : CauchySeq fun n => (u n).1
   · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
     · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
-    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
+    · exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
   obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
   by_cases xs : x ∈ s
   · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩

Use X, Y, Z for topological spaces.

@@ -406,8 +406,7 @@ theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s :
   · rw [← f.induced_symm]
     exact f.symm.polishSpace_induced
   · rw [isOpen_coinduced, isOpen_sum_iff]
-    convert And.intro (isOpen_univ (α := s)) (isOpen_empty (α := (sᶜ : Set α)))
-      <;> ext ⟨x, hx⟩ <;> simpa using hx
+    simp [preimage_preimage]
 #align is_closed.is_clopenable IsClosed.isClopenable
 
 theorem IsClopenable.compl [TopologicalSpace α] {s : Set α} (hs : IsClopenable s) :

add a class StandardBorelSpace, whose relationship to PolishSpace is analogous to the relationship between PolishSpace and complete metric space. The usefulness is shown by _root_.MeasurableSet.standardBorel: A measurable subset of a standard Borel space is Polish. Note that a borel measurable subset of a Polish space will not typically be Polish in the induced topology!

Specific things done in this PR:

• added some basic instances for StandardBorelSpace
• changed theorems with hypotheses [MeasurableSpace X] [TopologicalSpace X] [BorelSpace X] [PolishSpace X] but which do not refer to the topology on X to depend instead on [MeasurableSpace X][StandardBorelSpace X]. These automatically generalize the old theorems thanks to standardBorel_of_polish.

I believe this sort of refactoring can be pushed a lot farther in future PRs.

Co-authored-by: Felix-Weilacher <112423742+Felix-Weilacher@users.noreply.github.com>

@@ -125,6 +125,13 @@ instance sigma {ι : Type*} [Countable ι] {E : ι → Type*} [∀ n, Topologica
   inferInstance
 #align polish_space.sigma PolishSpace.sigma
 
+/-- The product of two Polish spaces is Polish. -/
+instance prod [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [PolishSpace β] :
+    PolishSpace (α × β) :=
+  letI := upgradePolishSpace α
+  letI := upgradePolishSpace β
+  inferInstance
+
 /-- The disjoint union of two Polish spaces is Polish. -/
 instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [PolishSpace β] :
     PolishSpace (α ⊕ β) :=

Make the following changes which were intentionally left till after the port:

• Move several instances of PolishSpace from MeasureTheory.Constructions.Polish to Topology.MetricSpace.Polish

• Remove instance of PolishSpace for nat -> nat, since lean4 can find this automatically

• Change PolishSpace to extend SecondCountableTopology, since lean4 now allows loops in TC inference.

@@ -62,8 +62,8 @@ other way around as this is the most common use case.
 
 To endow a Polish space with a complete metric space structure, do `letI := upgradePolishSpace α`.
 -/
-class PolishSpace (α : Type*) [h : TopologicalSpace α] : Prop where
-  secondCountableTopology : SecondCountableTopology α
+class PolishSpace (α : Type*) [h : TopologicalSpace α]
+    extends SecondCountableTopology α : Prop where
   complete : ∃ m : MetricSpace α, m.toUniformSpace.toTopologicalSpace = h ∧
     @CompleteSpace α m.toUniformSpace
 #align polish_space PolishSpace
@@ -76,8 +76,7 @@ class UpgradedPolishSpace (α : Type*) extends MetricSpace α, SecondCountableTo
 #align upgraded_polish_space UpgradedPolishSpace
 
 instance (priority := 100) polishSpace_of_complete_second_countable [m : MetricSpace α]
-    [h : SecondCountableTopology α] [h' : CompleteSpace α] : PolishSpace α where
-  secondCountableTopology := h
+    [SecondCountableTopology α] [h' : CompleteSpace α] : PolishSpace α where
   complete := ⟨m, rfl, h'⟩
 #align polish_space_of_complete_second_countable polishSpace_of_complete_second_countable
 
@@ -97,7 +96,7 @@ theorem complete_polishSpaceMetric (α : Type*) [ht : TopologicalSpace α] [h :
 def upgradePolishSpace (α : Type*) [TopologicalSpace α] [PolishSpace α] :
     UpgradedPolishSpace α :=
   letI := polishSpaceMetric α
-  { complete_polishSpaceMetric α, PolishSpace.secondCountableTopology with }
+  { complete_polishSpaceMetric α with }
 #align upgrade_polish_space upgradePolishSpace
 
 namespace PolishSpace
@@ -117,11 +116,6 @@ instance pi_countable {ι : Type*} [Countable ι] {E : ι → Type*} [∀ i, Top
   infer_instance
 #align polish_space.pi_countable PolishSpace.pi_countable
 
-/-- Without this instance, Lean 3 was unable to find `PolishSpace (ℕ → ℕ)` by typeclass inference.
-Porting note: TODO: test with Lean 4. -/
-instance nat_fun [TopologicalSpace α] [PolishSpace α] : PolishSpace (ℕ → α) := inferInstance
-#align polish_space.nat_fun PolishSpace.nat_fun
-
 /-- A countable disjoint union of Polish spaces is Polish. -/
 instance sigma {ι : Type*} [Countable ι] {E : ι → Type*} [∀ n, TopologicalSpace (E n)]
     [∀ n, PolishSpace (E n)] : PolishSpace (Σn, E n) :=
@@ -159,6 +153,16 @@ theorem _root_.ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpa
   infer_instance
 #align closed_embedding.polish_space ClosedEmbedding.polishSpace
 
+/-- Any countable discrete space is Polish. -/
+instance (priority := 50) polish_of_countable [TopologicalSpace α]
+    [h : Countable α] [DiscreteTopology α] : PolishSpace α := by
+  obtain ⟨f, hf⟩ := h.exists_injective_nat
+  have : ClosedEmbedding f := by
+    apply closedEmbedding_of_continuous_injective_closed continuous_of_discreteTopology hf
+    exact fun t _ => isClosed_discrete _
+  exact this.polishSpace
+#align polish_of_countable PolishSpace.polish_of_countable
+
 /-- Pulling back a Polish topology under an equiv gives again a Polish topology. -/
 theorem _root_.Equiv.polishSpace_induced [t : TopologicalSpace β] [PolishSpace β] (f : α ≃ β) :
     @PolishSpace α (t.induced f) :=
@@ -167,11 +171,16 @@ theorem _root_.Equiv.polishSpace_induced [t : TopologicalSpace β] [PolishSpace
 #align equiv.polish_space_induced Equiv.polishSpace_induced
 
 /-- A closed subset of a Polish space is also Polish. -/
-theorem _root_.IsClosed.polishSpace {α : Type*} [TopologicalSpace α] [PolishSpace α] {s : Set α}
+theorem _root_.IsClosed.polishSpace [TopologicalSpace α] [PolishSpace α] {s : Set α}
     (hs : IsClosed s) : PolishSpace s :=
   (IsClosed.closedEmbedding_subtype_val hs).polishSpace
 #align is_closed.polish_space IsClosed.polishSpace
 
+instance instPolishSpaceUniv [TopologicalSpace α] [PolishSpace α] :
+    PolishSpace (univ : Set α) :=
+  isClosed_univ.polishSpace
+#align measure_theory.set.univ.polish_space PolishSpace.instPolishSpaceUniv
+
 /-- A sequence of type synonyms of a given type `α`, useful in the proof of
 `exists_polishSpace_forall_le` to endow each copy with a different topology. -/
 @[nolint unusedArguments]

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

This has nice performance benefits.

@@ -50,7 +50,7 @@ noncomputable section
 
 open Classical Topology Filter TopologicalSpace Set Metric Function
 
-variable {α : Type _} {β : Type _}
+variable {α : Type*} {β : Type*}
 
 /-! ### Basic properties of Polish spaces -/
 
@@ -62,7 +62,7 @@ other way around as this is the most common use case.
 
 To endow a Polish space with a complete metric space structure, do `letI := upgradePolishSpace α`.
 -/
-class PolishSpace (α : Type _) [h : TopologicalSpace α] : Prop where
+class PolishSpace (α : Type*) [h : TopologicalSpace α] : Prop where
   secondCountableTopology : SecondCountableTopology α
   complete : ∃ m : MetricSpace α, m.toUniformSpace.toTopologicalSpace = h ∧
     @CompleteSpace α m.toUniformSpace
@@ -71,7 +71,7 @@ class PolishSpace (α : Type _) [h : TopologicalSpace α] : Prop where
 /-- A convenience class, for a Polish space endowed with a complete metric. No instance of this
 class should be registered: It should be used as `letI := upgradePolishSpace α` to endow a Polish
 space with a complete metric. -/
-class UpgradedPolishSpace (α : Type _) extends MetricSpace α, SecondCountableTopology α,
+class UpgradedPolishSpace (α : Type*) extends MetricSpace α, SecondCountableTopology α,
   CompleteSpace α
 #align upgraded_polish_space UpgradedPolishSpace
 
@@ -82,11 +82,11 @@ instance (priority := 100) polishSpace_of_complete_second_countable [m : MetricS
 #align polish_space_of_complete_second_countable polishSpace_of_complete_second_countable
 
 /-- Construct on a Polish space a metric (compatible with the topology) which is complete. -/
-def polishSpaceMetric (α : Type _) [TopologicalSpace α] [h : PolishSpace α] : MetricSpace α :=
+def polishSpaceMetric (α : Type*) [TopologicalSpace α] [h : PolishSpace α] : MetricSpace α :=
   h.complete.choose.replaceTopology h.complete.choose_spec.1.symm
 #align polish_space_metric polishSpaceMetric
 
-theorem complete_polishSpaceMetric (α : Type _) [ht : TopologicalSpace α] [h : PolishSpace α] :
+theorem complete_polishSpaceMetric (α : Type*) [ht : TopologicalSpace α] [h : PolishSpace α] :
     @CompleteSpace α (polishSpaceMetric α).toUniformSpace := by
   convert h.complete.choose_spec.2
   exact MetricSpace.replaceTopology_eq _ _
@@ -94,7 +94,7 @@ theorem complete_polishSpaceMetric (α : Type _) [ht : TopologicalSpace α] [h :
 
 /-- This definition endows a Polish space with a complete metric. Use it as:
 `letI := upgradePolishSpace α`. -/
-def upgradePolishSpace (α : Type _) [TopologicalSpace α] [PolishSpace α] :
+def upgradePolishSpace (α : Type*) [TopologicalSpace α] [PolishSpace α] :
     UpgradedPolishSpace α :=
   letI := polishSpaceMetric α
   { complete_polishSpaceMetric α, PolishSpace.secondCountableTopology with }
@@ -102,14 +102,14 @@ def upgradePolishSpace (α : Type _) [TopologicalSpace α] [PolishSpace α] :
 
 namespace PolishSpace
 
-instance (priority := 100) t2Space (α : Type _) [TopologicalSpace α] [PolishSpace α] :
+instance (priority := 100) t2Space (α : Type*) [TopologicalSpace α] [PolishSpace α] :
     T2Space α := by
   letI := upgradePolishSpace α
   infer_instance
 #align polish_space.t2_space PolishSpace.t2Space
 
 /-- A countable product of Polish spaces is Polish. -/
-instance pi_countable {ι : Type _} [Countable ι] {E : ι → Type _} [∀ i, TopologicalSpace (E i)]
+instance pi_countable {ι : Type*} [Countable ι] {E : ι → Type*} [∀ i, TopologicalSpace (E i)]
     [∀ i, PolishSpace (E i)] : PolishSpace (∀ i, E i) := by
   cases nonempty_encodable ι
   letI := fun i => upgradePolishSpace (E i)
@@ -123,7 +123,7 @@ instance nat_fun [TopologicalSpace α] [PolishSpace α] : PolishSpace (ℕ → 
 #align polish_space.nat_fun PolishSpace.nat_fun
 
 /-- A countable disjoint union of Polish spaces is Polish. -/
-instance sigma {ι : Type _} [Countable ι] {E : ι → Type _} [∀ n, TopologicalSpace (E n)]
+instance sigma {ι : Type*} [Countable ι] {E : ι → Type*} [∀ n, TopologicalSpace (E n)]
     [∀ n, PolishSpace (E n)] : PolishSpace (Σn, E n) :=
   letI := fun n => upgradePolishSpace (E n)
   letI : MetricSpace (Σn, E n) := Sigma.metricSpace
@@ -141,7 +141,7 @@ instance sum [TopologicalSpace α] [PolishSpace α] [TopologicalSpace β] [Polis
 #align polish_space.sum PolishSpace.sum
 
 /-- Any nonempty Polish space is the continuous image of the fundamental space `ℕ → ℕ`. -/
-theorem exists_nat_nat_continuous_surjective (α : Type _) [TopologicalSpace α] [PolishSpace α]
+theorem exists_nat_nat_continuous_surjective (α : Type*) [TopologicalSpace α] [PolishSpace α]
     [Nonempty α] : ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f :=
   letI := upgradePolishSpace α
   exists_nat_nat_continuous_surjective_of_completeSpace α
@@ -167,7 +167,7 @@ theorem _root_.Equiv.polishSpace_induced [t : TopologicalSpace β] [PolishSpace
 #align equiv.polish_space_induced Equiv.polishSpace_induced
 
 /-- A closed subset of a Polish space is also Polish. -/
-theorem _root_.IsClosed.polishSpace {α : Type _} [TopologicalSpace α] [PolishSpace α] {s : Set α}
+theorem _root_.IsClosed.polishSpace {α : Type*} [TopologicalSpace α] [PolishSpace α] {s : Set α}
     (hs : IsClosed s) : PolishSpace s :=
   (IsClosed.closedEmbedding_subtype_val hs).polishSpace
 #align is_closed.polish_space IsClosed.polishSpace
@@ -175,14 +175,14 @@ theorem _root_.IsClosed.polishSpace {α : Type _} [TopologicalSpace α] [PolishS
 /-- A sequence of type synonyms of a given type `α`, useful in the proof of
 `exists_polishSpace_forall_le` to endow each copy with a different topology. -/
 @[nolint unusedArguments]
-def AuxCopy (α : Type _) {ι : Type _} (_i : ι) : Type _ := α
+def AuxCopy (α : Type*) {ι : Type*} (_i : ι) : Type _ := α
 #align polish_space.aux_copy PolishSpace.AuxCopy
 
 /-- Given a Polish space, and countably many finer Polish topologies, there exists another Polish
 topology which is finer than all of them.
 
 Porting note: TODO: the topology `t'` is `t ⊓ ⨅ i, m i`. -/
-theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : TopologicalSpace α]
+theorem exists_polishSpace_forall_le {ι : Type*} [Countable ι] [t : TopologicalSpace α]
     [p : PolishSpace α] (m : ι → TopologicalSpace α) (hm : ∀ n, m n ≤ t)
     (h'm : ∀ n, @PolishSpace α (m n)) :
     ∃ t' : TopologicalSpace α, (∀ n, t' ≤ m n) ∧ t' ≤ t ∧ @PolishSpace α t' := by
@@ -253,7 +253,7 @@ variable [MetricSpace α] {s : Opens α}
 /-- A type synonym for a subset `s` of a metric space, on which we will construct another metric
 for which it will be complete. -/
 -- porting note: was @[nolint has_nonempty_instance]
-def CompleteCopy {α : Type _} [MetricSpace α] (s : Opens α) : Type _ := s
+def CompleteCopy {α : Type*} [MetricSpace α] (s : Opens α) : Type _ := s
 #align polish_space.complete_copy TopologicalSpace.Opens.CompleteCopyₓ
 
 namespace CompleteCopy
@@ -348,7 +348,7 @@ instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) :
 #align polish_space.complete_space_complete_copy TopologicalSpace.Opens.CompleteCopy.instCompleteSpaceₓ
 
 /-- An open subset of a Polish space is also Polish. -/
-theorem _root_.IsOpen.polishSpace {α : Type _} [TopologicalSpace α] [PolishSpace α] {s : Set α}
+theorem _root_.IsOpen.polishSpace {α : Type*} [TopologicalSpace α] [PolishSpace α] {s : Set α}
     (hs : IsOpen s) : PolishSpace s := by
   letI := upgradePolishSpace α
   lift s to Opens α using hs

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.polish
-! leanprover-community/mathlib commit bcfa726826abd57587355b4b5b7e78ad6527b7e4
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.MetricSpace.PiNat
 import Mathlib.Topology.MetricSpace.Isometry
@@ -14,6 +9,8 @@ import Mathlib.Topology.MetricSpace.Gluing
 import Mathlib.Topology.Sets.Opens
 import Mathlib.Analysis.Normed.Field.Basic
 
+#align_import topology.metric_space.polish from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
+
 /-!
 # Polish spaces
 

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

@@ -245,7 +245,7 @@ boundary.
 
 Porting note: definitions and lemmas in this section now take `(s : Opens α)` instead of
 `{s : Set α} (hs : IsOpen s)` so that we can turn various definitions and lemmas into instances.
-Also, some lemmas used to assume `Set.Nonempty (sᶜ)` in Lean 3. In fact, this assumption is not
+Also, some lemmas used to assume `Set.Nonempty sᶜ` in Lean 3. In fact, this assumption is not
 needed, so it was dropped.
 -/
 
@@ -266,11 +266,11 @@ by `dist' x y = dist x y + |1 / dist x sᶜ - 1 / dist y sᶜ|`, where the secon
 the boundary to ensure that Cauchy sequences for `dist'` remain well inside `s`. -/
 -- Porting note: in mathlib3 this was only a local instance.
 instance instDist : Dist (CompleteCopy s) where
-  dist x y := dist x.1 y.1 + abs (1 / infDist x.1 (sᶜ) - 1 / infDist y.1 (sᶜ))
+  dist x y := dist x.1 y.1 + abs (1 / infDist x.1 sᶜ - 1 / infDist y.1 sᶜ)
 #align polish_space.has_dist_complete_copy TopologicalSpace.Opens.CompleteCopy.instDistₓ
 
 theorem dist_eq (x y : CompleteCopy s) :
-    dist x y = dist x.1 y.1 + abs (1 / infDist x.1 (sᶜ) - 1 / infDist y.1 (sᶜ)) :=
+    dist x y = dist x.1 y.1 + abs (1 / infDist x.1 sᶜ - 1 / infDist y.1 sᶜ) :=
   rfl
 #align polish_space.dist_complete_copy_eq TopologicalSpace.Opens.CompleteCopy.dist_eqₓ
 
@@ -295,9 +295,9 @@ instance instMetricSpace : MetricSpace (CompleteCopy s) := by
   · simp only [dist_eq, dist_self, one_div, sub_self, abs_zero, add_zero]
   · simp only [dist_eq, dist_comm, abs_sub_comm]
   · calc
-      dist x z = dist x.1 z.1 + |1 / infDist x.1 (sᶜ) - 1 / infDist z.1 (sᶜ)| := rfl
-      _ ≤ dist x.1 y.1 + dist y.1 z.1 + (|1 / infDist x.1 (sᶜ) - 1 / infDist y.1 (sᶜ)| +
-            |1 / infDist y.1 (sᶜ) - 1 / infDist z.1 (sᶜ)|) :=
+      dist x z = dist x.1 z.1 + |1 / infDist x.1 sᶜ - 1 / infDist z.1 sᶜ| := rfl
+      _ ≤ dist x.1 y.1 + dist y.1 z.1 + (|1 / infDist x.1 sᶜ - 1 / infDist y.1 sᶜ| +
+            |1 / infDist y.1 sᶜ - 1 / infDist z.1 sᶜ|) :=
         add_le_add (dist_triangle _ _ _) (dist_triangle (1 / infDist _ _) _ _)
       _ = dist x y + dist y z := add_add_add_comm ..
   · refine ⟨fun h x hx ↦ ?_, fun h ↦ isOpen_iff_mem_nhds.2 fun x hx ↦ ?_⟩
@@ -305,8 +305,8 @@ instance instMetricSpace : MetricSpace (CompleteCopy s) := by
       exact ⟨ε, ε0, fun y hy ↦ hε <| (dist_comm _ _).trans_lt <| (dist_val_le_dist _ _).trans_lt hy⟩
     · rcases h x hx with ⟨ε, ε0, hε⟩
       simp only [dist_eq, one_div] at hε
-      have : Tendsto (fun y : s ↦ dist x.1 y.1 + |(infDist x.1 (sᶜ))⁻¹ - (infDist y.1 (sᶜ))⁻¹|)
-        (𝓝 x) (𝓝 (dist x.1 x.1 + |(infDist x.1 (sᶜ))⁻¹ - (infDist x.1 (sᶜ))⁻¹|))
+      have : Tendsto (fun y : s ↦ dist x.1 y.1 + |(infDist x.1 sᶜ)⁻¹ - (infDist y.1 sᶜ)⁻¹|)
+        (𝓝 x) (𝓝 (dist x.1 x.1 + |(infDist x.1 sᶜ)⁻¹ - (infDist x.1 sᶜ)⁻¹|))
       · refine (tendsto_const_nhds.dist continuous_subtype_val.continuousAt).add
           (tendsto_const_nhds.sub <| ?_).abs
         refine (continuousAt_inv_infDist_pt ?_).comp continuous_subtype_val.continuousAt
@@ -328,23 +328,23 @@ instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) :
   obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
   by_cases xs : x ∈ s
   · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
-  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 (sᶜ) < C
-  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 (sᶜ), fun n ↦ ?_⟩
+  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
+  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
     rw [← sub_lt_iff_lt_add]
     calc
-      _ ≤ |1 / infDist (u n).1 (sᶜ) - 1 / infDist (u 0).1 (sᶜ)| := le_abs_self _
-      _ = |1 / infDist (u 0).1 (sᶜ) - 1 / infDist (u n).1 (sᶜ)| := abs_sub_comm _ _
+      _ ≤ |1 / infDist (u n).1 sᶜ - 1 / infDist (u 0).1 sᶜ| := le_abs_self _
+      _ = |1 / infDist (u 0).1 sᶜ - 1 / infDist (u n).1 sᶜ| := abs_sub_comm _ _
       _ ≤ dist (u 0) (u n) := le_add_of_nonneg_left dist_nonneg
       _ < (1 / 2) ^ 0 := hu 0 0 n le_rfl n.zero_le
   have Cpos : 0 < C := lt_of_le_of_lt (div_nonneg zero_le_one infDist_nonneg) (hC 0)
-  have Hmem : ∀ {y}, y ∈ s ↔ 0 < infDist y (sᶜ) := fun {y} ↦ by
+  have Hmem : ∀ {y}, y ∈ s ↔ 0 < infDist y sᶜ := fun {y} ↦ by
     rw [← s.isOpen.isClosed_compl.not_mem_iff_infDist_pos ⟨x, xs⟩]; exact not_not.symm
-  have I : ∀ n, 1 / C ≤ infDist (u n).1 (sᶜ) := fun n ↦ by
-    have : 0 < infDist (u n).1 (sᶜ) := Hmem.1 (u n).2
+  have I : ∀ n, 1 / C ≤ infDist (u n).1 sᶜ := fun n ↦ by
+    have : 0 < infDist (u n).1 sᶜ := Hmem.1 (u n).2
     rw [div_le_iff' Cpos]
     exact (div_le_iff this).1 (hC n).le
-  have I' : 1 / C ≤ infDist x (sᶜ) :=
-    have : Tendsto (fun n => infDist (u n).1 (sᶜ)) atTop (𝓝 (infDist x (sᶜ))) :=
+  have I' : 1 / C ≤ infDist x sᶜ :=
+    have : Tendsto (fun n => infDist (u n).1 sᶜ) atTop (𝓝 (infDist x sᶜ)) :=
       ((continuous_infDist_pt (sᶜ : Set α)).tendsto x).comp xlim
     ge_of_tendsto' this I
   exact absurd (Hmem.2 <| lt_of_lt_of_le (div_pos one_pos Cpos) I') xs
@@ -398,7 +398,7 @@ theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s :
 #align is_closed.is_clopenable IsClosed.isClopenable
 
 theorem IsClopenable.compl [TopologicalSpace α] {s : Set α} (hs : IsClopenable s) :
-    IsClopenable (sᶜ) := by
+    IsClopenable sᶜ := by
   rcases hs with ⟨t, t_le, t_polish, h, h'⟩
   exact ⟨t, t_le, t_polish, @IsOpen.isClosed_compl α t s h', @IsClosed.isOpen_compl α t s h⟩
 #align polish_space.is_clopenable.compl PolishSpace.IsClopenable.compl

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>

@@ -202,7 +202,7 @@ theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : Topologic
   -- show that the induced topology is finer than all the `m n`.
   have T_le_m : ∀ n, T.induced f ≤ m n := fun n ↦ by
     rw [induced_to_pi]
-    exact infᵢ_le_of_le n (@induced_id _ (m n)).le
+    exact iInf_le_of_le n (@induced_id _ (m n)).le
   refine' ⟨T.induced f, fun n => T_le_m n, (T_le_m default).trans (hm default), _⟩
   -- show that the new topology is Polish, as the pullback of a Polish topology under a closed
   -- embedding.
@@ -210,14 +210,14 @@ theorem exists_polishSpace_forall_le {ι : Type _} [Countable ι] [t : Topologic
     ext x
     constructor
     · rintro ⟨y, rfl⟩
-      exact mem_interᵢ.2 fun n => by simp only [mem_setOf_eq]
+      exact mem_iInter.2 fun n => by simp only [mem_setOf_eq]
     · refine fun hx ↦ ⟨x default, ?_⟩
       ext1 n
       symm
-      exact mem_interᵢ.1 hx n
+      exact mem_iInter.1 hx n
   have f_closed : IsClosed (range f) := by
     rw [A]
-    refine isClosed_interᵢ fun n => ?_
+    refine isClosed_iInter fun n => ?_
     have C : ∀ i : ι, Continuous fun x : ∀ n, AuxCopy α n => (id (x i) : α) := fun i ↦
       have : Continuous (show AuxCopy α i → α from id) := continuous_id_of_le (hm i)
       this.comp (continuous_apply i)
@@ -409,14 +409,14 @@ theorem _root_.IsOpen.isClopenable [TopologicalSpace α] [PolishSpace α] {s : S
 #align is_open.is_clopenable IsOpen.isClopenable
 
 -- porting note: TODO: generalize for free to `[Countable ι] {s : ι → Set α}`
-theorem IsClopenable.unionᵢ [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α}
+theorem IsClopenable.iUnion [t : TopologicalSpace α] [PolishSpace α] {s : ℕ → Set α}
     (hs : ∀ n, IsClopenable (s n)) : IsClopenable (⋃ n, s n) := by
   choose m mt m_polish _ m_open using hs
   obtain ⟨t', t'm, -, t'_polish⟩ :
     ∃ t' : TopologicalSpace α, (∀ n : ℕ, t' ≤ m n) ∧ t' ≤ t ∧ @PolishSpace α t' :=
     exists_polishSpace_forall_le m mt m_polish
   have A : IsOpen[t'] (⋃ n, s n) := by
-    apply isOpen_unionᵢ
+    apply isOpen_iUnion
     intro n
     apply t'm n
     exact m_open n
@@ -424,6 +424,6 @@ theorem IsClopenable.unionᵢ [t : TopologicalSpace α] [PolishSpace α] {s : 
       t'' ≤ t' ∧ @PolishSpace α t'' ∧ IsClosed[t''] (⋃ n, s n) ∧ IsOpen[t''] (⋃ n, s n) :=
     @IsOpen.isClopenable α t' t'_polish _ A
   exact ⟨t'', t''_le.trans ((t'm 0).trans (mt 0)), t''_polish, h1, h2⟩
-#align polish_space.is_clopenable.Union PolishSpace.IsClopenable.unionᵢ
+#align polish_space.is_clopenable.Union PolishSpace.IsClopenable.iUnion
 
 end PolishSpace

@@ -37,16 +37,16 @@ In this file, we establish the basic properties of Polish spaces.
 A fundamental property of Polish spaces is that one can put finer topologies, still Polish,
 with additional properties:
 
-* `exists_polish_space_forall_le`: on a topological space, consider countably many topologies
+* `exists_polishSpace_forall_le`: on a topological space, consider countably many topologies
   `t n`, all Polish and finer than the original topology. Then there exists another Polish
   topology which is finer than all the `t n`.
-* `is_clopenable s` is a property of a subset `s` of a topological space, requiring that there
+* `IsClopenable s` is a property of a subset `s` of a topological space, requiring that there
   exists a finer topology, which is Polish, for which `s` becomes open and closed. We show that
   this property is satisfied for open sets, closed sets, for complements, and for countable unions.
   Once Borel-measurable sets are defined in later files, it will follow that any Borel-measurable
   set is clopenable. Once the Lusin-Souslin theorem is proved using analytic sets, we will even
   show that a set is clopenable if and only if it is Borel-measurable, see
-  `is_clopenable_iff_measurable_set`.
+  `isClopenable_iff_measurableSet`.
 -/
 
 noncomputable section
@@ -176,7 +176,7 @@ theorem _root_.IsClosed.polishSpace {α : Type _} [TopologicalSpace α] [PolishS
 #align is_closed.polish_space IsClosed.polishSpace
 
 /-- A sequence of type synonyms of a given type `α`, useful in the proof of
-`exists_polish_space_forall_le` to endow each copy with a different topology. -/
+`exists_polishSpace_forall_le` to endow each copy with a different topology. -/
 @[nolint unusedArguments]
 def AuxCopy (α : Type _) {ι : Type _} (_i : ι) : Type _ := α
 #align polish_space.aux_copy PolishSpace.AuxCopy
@@ -369,7 +369,7 @@ namespace PolishSpace
 
 /-- A set in a topological space is clopenable if there exists a finer Polish topology for which
 this set is open and closed. It turns out that this notion is equivalent to being Borel-measurable,
-but this is nontrivial (see `is_clopenable_iff_measurable_set`). -/
+but this is nontrivial (see `isClopenable_iff_measurableSet`). -/
 def IsClopenable [t : TopologicalSpace α] (s : Set α) : Prop :=
   ∃ t' : TopologicalSpace α, t' ≤ t ∧ @PolishSpace α t' ∧ IsClosed[t'] s ∧ IsOpen[t'] s
 #align polish_space.is_clopenable PolishSpace.IsClopenable
@@ -427,4 +427,3 @@ theorem IsClopenable.unionᵢ [t : TopologicalSpace α] [PolishSpace α] {s : 
 #align polish_space.is_clopenable.Union PolishSpace.IsClopenable.unionᵢ
 
 end PolishSpace
-

Co-authored-by: Scott Morrison <scott@tqft.net>