topology.inseparable ⟷ Mathlib.Topology.Inseparable

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

@@ -369,7 +369,7 @@ theorem inseparable_iff_forall_closed : (x ~ y) ↔ ∀ s : Set X, IsClosed s 
 #print inseparable_iff_mem_closure /-
 theorem inseparable_iff_mem_closure :
     (x ~ y) ↔ x ∈ closure ({y} : Set X) ∧ y ∈ closure ({x} : Set X) :=
-  inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm']
+  inseparable_iff_specializes_and.trans <| by simp only [specializes_iff_mem_closure, and_comm]
 #align inseparable_iff_mem_closure inseparable_iff_mem_closure
 -/
 

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

@@ -3,9 +3,9 @@ Copyright (c) 2021 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang, Yury G. Kudryashov
 -/
-import Mathbin.Topology.ContinuousOn
-import Mathbin.Data.Setoid.Basic
-import Mathbin.Tactic.Tfae
+import Topology.ContinuousOn
+import Data.Setoid.Basic
+import Tactic.Tfae
 
 #align_import topology.inseparable from "leanprover-community/mathlib"@"e46da4e335b8671848ac711ccb34b42538c0d800"
 

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

@@ -109,10 +109,10 @@ theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
 #align specializes_iff_pure specializes_iff_pure
 -/
 
-alias specializes_iff_nhds ↔ Specializes.nhds_le_nhds _
+alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
 #align specializes.nhds_le_nhds Specializes.nhds_le_nhds
 
-alias specializes_iff_pure ↔ Specializes.pure_le_nhds _
+alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
 #align specializes.pure_le_nhds Specializes.pure_le_nhds
 
 #print specializes_iff_forall_open /-
@@ -157,7 +157,7 @@ theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
 #align specializes_iff_mem_closure specializes_iff_mem_closure
 -/
 
-alias specializes_iff_mem_closure ↔ Specializes.mem_closure _
+alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
 #align specializes.mem_closure Specializes.mem_closure
 
 #print specializes_iff_closure_subset /-
@@ -166,7 +166,7 @@ theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ c
 #align specializes_iff_closure_subset specializes_iff_closure_subset
 -/
 
-alias specializes_iff_closure_subset ↔ Specializes.closure_subset _
+alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
 #align specializes.closure_subset Specializes.closure_subset
 
 #print Filter.HasBasis.specializes_iff /-

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang, Yury G. Kudryashov
-
-! This file was ported from Lean 3 source module topology.inseparable
-! leanprover-community/mathlib commit e46da4e335b8671848ac711ccb34b42538c0d800
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.ContinuousOn
 import Mathbin.Data.Setoid.Basic
 import Mathbin.Tactic.Tfae
 
+#align_import topology.inseparable from "leanprover-community/mathlib"@"e46da4e335b8671848ac711ccb34b42538c0d800"
+
 /-!
 # Inseparable points in a topological space
 

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

@@ -74,9 +74,9 @@ def Specializes (x y : X) : Prop :=
 #align specializes Specializes
 -/
 
--- mathport name: «expr ⤳ »
 infixl:300 " ⤳ " => Specializes
 
+#print specializes_TFAE /-
 /-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
 below. -/
 theorem specializes_TFAE (x y : X) :
@@ -98,14 +98,19 @@ theorem specializes_TFAE (x y : X) :
     exact ho.mem_nhds hxs
   tfae_finish
 #align specializes_tfae specializes_TFAE
+-/
 
+#print specializes_iff_nhds /-
 theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
   Iff.rfl
 #align specializes_iff_nhds specializes_iff_nhds
+-/
 
+#print specializes_iff_pure /-
 theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
   (specializes_TFAE x y).out 0 1
 #align specializes_iff_pure specializes_iff_pure
+-/
 
 alias specializes_iff_nhds ↔ Specializes.nhds_le_nhds _
 #align specializes.nhds_le_nhds Specializes.nhds_le_nhds
@@ -200,6 +205,7 @@ theorem specializes_of_eq (e : x = y) : x ⤳ y :=
 #align specializes_of_eq specializes_of_eq
 -/
 
+#print specializes_of_nhdsWithin /-
 theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
   specializes_iff_pure.2 <|
     calc
@@ -207,20 +213,27 @@ theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈
       _ ≤ 𝓝[s] y := h₁
       _ ≤ 𝓝 y := inf_le_left
 #align specializes_of_nhds_within specializes_of_nhdsWithin
+-/
 
+#print Specializes.map_of_continuousAt /-
 theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
   specializes_iff_pure.2 fun s hs =>
     mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
 #align specializes.map_of_continuous_at Specializes.map_of_continuousAt
+-/
 
+#print Specializes.map /-
 theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
   h.map_of_continuousAt hf.ContinuousAt
 #align specializes.map Specializes.map
+-/
 
+#print Inducing.specializes_iff /-
 theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
   simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
     mem_preimage]
 #align inducing.specializes_iff Inducing.specializes_iff
+-/
 
 #print subtype_specializes_iff /-
 theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔ (x : X) ⤳ y :=
@@ -228,20 +241,26 @@ theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔
 #align subtype_specializes_iff subtype_specializes_iff
 -/
 
+#print specializes_prod /-
 @[simp]
 theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
   simp only [Specializes, nhds_prod_eq, prod_le_prod]
 #align specializes_prod specializes_prod
+-/
 
+#print Specializes.prod /-
 theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
     (x₁, y₁) ⤳ (x₂, y₂) :=
   specializes_prod.2 ⟨hx, hy⟩
 #align specializes.prod Specializes.prod
+-/
 
+#print specializes_pi /-
 @[simp]
 theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
   simp only [Specializes, nhds_pi, pi_le_pi]
 #align specializes_pi specializes_pi
+-/
 
 #print not_specializes_iff_exists_open /-
 theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
@@ -270,11 +289,13 @@ def specializationPreorder : Preorder X :=
 
 variable {X}
 
+#print Continuous.specialization_monotone /-
 /-- A continuous function is monotone with respect to the specialization preorders on the domain and
 the codomain. -/
 theorem Continuous.specialization_monotone (hf : Continuous f) :
     @Monotone _ _ (specializationPreorder X) (specializationPreorder Y) f := fun x y h => h.map hf
 #align continuous.specialization_monotone Continuous.specialization_monotone
+-/
 
 /-!
 ### `inseparable` relation
@@ -296,7 +317,6 @@ def Inseparable (x y : X) : Prop :=
 #align inseparable Inseparable
 -/
 
--- mathport name: «expr ~ »
 local infixl:0 " ~ " => Inseparable
 
 #print inseparable_def /-
@@ -369,9 +389,11 @@ theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s]
 #align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
 -/
 
+#print Inducing.inseparable_iff /-
 theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ f y) ↔ (x ~ y) := by
   simp only [inseparable_iff_specializes_and, hf.specializes_iff]
 #align inducing.inseparable_iff Inducing.inseparable_iff
+-/
 
 #print subtype_inseparable_iff /-
 theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ y) ↔ ((x : X) ~ y) :=
@@ -379,20 +401,26 @@ theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ y) ↔
 #align subtype_inseparable_iff subtype_inseparable_iff
 -/
 
+#print inseparable_prod /-
 @[simp]
 theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} : ((x₁, y₁) ~ (x₂, y₂)) ↔ (x₁ ~ x₂) ∧ (y₁ ~ y₂) :=
   by simp only [Inseparable, nhds_prod_eq, prod_inj]
 #align inseparable_prod inseparable_prod
+-/
 
+#print Inseparable.prod /-
 theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ x₂) (hy : y₁ ~ y₂) :
     (x₁, y₁) ~ (x₂, y₂) :=
   inseparable_prod.2 ⟨hx, hy⟩
 #align inseparable.prod Inseparable.prod
+-/
 
+#print inseparable_pi /-
 @[simp]
 theorem inseparable_pi {f g : ∀ i, π i} : (f ~ g) ↔ ∀ i, f i ~ g i := by
   simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
 #align inseparable_pi inseparable_pi
+-/
 
 namespace Inseparable
 
@@ -447,14 +475,18 @@ theorem mem_closed_iff (h : x ~ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
 #align inseparable.mem_closed_iff Inseparable.mem_closed_iff
 -/
 
+#print Inseparable.map_of_continuousAt /-
 theorem map_of_continuousAt (h : x ~ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
     f x ~ f y :=
   (h.Specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
 #align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
+-/
 
+#print Inseparable.map /-
 theorem map (h : x ~ y) (hf : Continuous f) : f x ~ f y :=
   h.map_of_continuousAt hf.ContinuousAt hf.ContinuousAt
 #align inseparable.map Inseparable.map
+-/
 
 end Inseparable
 
@@ -641,9 +673,11 @@ theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
 #align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
 -/
 
+#print SeparationQuotient.map_prod_map_mk_nhds /-
 theorem map_prod_map_mk_nhds (x : X) (y : Y) : map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) :=
   by rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
 #align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
+-/
 
 #print SeparationQuotient.map_mk_nhdsWithin_preimage /-
 theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
@@ -660,53 +694,69 @@ def lift (f : X → α) (hf : ∀ x y, (x ~ y) → f x = f y) : SeparationQuotie
 #align separation_quotient.lift SeparationQuotient.lift
 -/
 
+#print SeparationQuotient.lift_mk /-
 @[simp]
 theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
   rfl
 #align separation_quotient.lift_mk SeparationQuotient.lift_mk
+-/
 
+#print SeparationQuotient.lift_comp_mk /-
 @[simp]
 theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ y) → f x = f y) : lift f hf ∘ mk = f :=
   rfl
 #align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
+-/
 
+#print SeparationQuotient.tendsto_lift_nhds_mk /-
 @[simp]
 theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ y) → f x = f y} {x : X} {l : Filter α} :
     Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
   simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
 #align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
+-/
 
+#print SeparationQuotient.tendsto_lift_nhdsWithin_mk /-
 @[simp]
 theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ y) → f x = f y} {x : X}
     {s : Set (SeparationQuotient X)} {l : Filter α} :
     Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
   simp only [← map_mk_nhds_within_preimage, tendsto_map'_iff, lift_comp_mk]
 #align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
+-/
 
+#print SeparationQuotient.continuousAt_lift /-
 @[simp]
 theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f y} {x : X} :
     ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
   tendsto_lift_nhds_mk
 #align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
+-/
 
+#print SeparationQuotient.continuousWithinAt_lift /-
 @[simp]
 theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f y}
     {s : Set (SeparationQuotient X)} {x : X} :
     ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
   tendsto_lift_nhdsWithin_mk
 #align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
+-/
 
+#print SeparationQuotient.continuousOn_lift /-
 @[simp]
 theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f y}
     {s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
   simp only [ContinuousOn, surjective_mk.forall, continuous_within_at_lift, mem_preimage]
 #align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
+-/
 
+#print SeparationQuotient.continuous_lift /-
 @[simp]
 theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f y} :
     Continuous (lift f hf) ↔ Continuous f := by
   simp only [continuous_iff_continuousOn_univ, continuous_on_lift, preimage_univ]
 #align separation_quotient.continuous_lift SeparationQuotient.continuous_lift
+-/
 
 #print SeparationQuotient.lift₂ /-
 /-- Lift a map `f : X → Y → α` such that `inseparable a b → inseparable c d → f a c = f b d` to a
@@ -716,19 +766,24 @@ def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a
 #align separation_quotient.lift₂ SeparationQuotient.lift₂
 -/
 
+#print SeparationQuotient.lift₂_mk /-
 @[simp]
 theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d) (x : X)
     (y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
   rfl
 #align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
+-/
 
+#print SeparationQuotient.tendsto_lift₂_nhds /-
 @[simp]
 theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {x : X} {y : Y} {l : Filter α} :
     Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
   rw [← map_prod_map_mk_nhds, tendsto_map'_iff]; rfl
 #align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
+-/
 
+#print SeparationQuotient.tendsto_lift₂_nhdsWithin /-
 @[simp]
 theorem tendsto_lift₂_nhdsWithin {f : X → Y → α} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {x : X} {y : Y} {s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
@@ -736,14 +791,18 @@ theorem tendsto_lift₂_nhdsWithin {f : X → Y → α} {hf : ∀ a b c d, (a ~
       Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l :=
   by rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]; rfl
 #align separation_quotient.tendsto_lift₂_nhds_within SeparationQuotient.tendsto_lift₂_nhdsWithin
+-/
 
+#print SeparationQuotient.continuousAt_lift₂ /-
 @[simp]
 theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {x : X} {y : Y} :
     ContinuousAt (uncurry <| lift₂ f hf) (mk x, mk y) ↔ ContinuousAt (uncurry f) (x, y) :=
   tendsto_lift₂_nhds
 #align separation_quotient.continuous_at_lift₂ SeparationQuotient.continuousAt_lift₂
+-/
 
+#print SeparationQuotient.continuousWithinAt_lift₂ /-
 @[simp]
 theorem continuousWithinAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {s : Set (SeparationQuotient X × SeparationQuotient Y)} {x : X} {y : Y} :
@@ -751,7 +810,9 @@ theorem continuousWithinAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c
       ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (x, y) :=
   tendsto_lift₂_nhdsWithin
 #align separation_quotient.continuous_within_at_lift₂ SeparationQuotient.continuousWithinAt_lift₂
+-/
 
+#print SeparationQuotient.continuousOn_lift₂ /-
 @[simp]
 theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {s : Set (SeparationQuotient X × SeparationQuotient Y)} :
@@ -761,12 +822,15 @@ theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) →
     continuous_within_at_lift₂]
   rfl
 #align separation_quotient.continuous_on_lift₂ SeparationQuotient.continuousOn_lift₂
+-/
 
+#print SeparationQuotient.continuous_lift₂ /-
 @[simp]
 theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d} :
     Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) := by
   simp only [continuous_iff_continuousOn_univ, continuous_on_lift₂, preimage_univ]
 #align separation_quotient.continuous_lift₂ SeparationQuotient.continuous_lift₂
+-/
 
 end SeparationQuotient
 

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

@@ -206,7 +206,6 @@ theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈
       pure x ≤ 𝓝[s] x := le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
       _ ≤ 𝓝[s] y := h₁
       _ ≤ 𝓝 y := inf_le_left
-      
 #align specializes_of_nhds_within specializes_of_nhdsWithin
 
 theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=

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

@@ -491,7 +491,8 @@ def inseparableSetoid : Setoid X :=
 /-- The quotient of a topological space by its `inseparable_setoid`. This quotient is guaranteed to
 be a T₀ space. -/
 def SeparationQuotient :=
-  Quotient (inseparableSetoid X)deriving TopologicalSpace
+  Quotient (inseparableSetoid X)
+deriving TopologicalSpace
 #align separation_quotient SeparationQuotient
 -/
 

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

@@ -45,7 +45,7 @@ topological space, separation setoid
 
 open Set Filter Function
 
-open Topology Filter
+open scoped Topology Filter
 
 variable {X Y Z α ι : Type _} {π : ι → Type _} [TopologicalSpace X] [TopologicalSpace Y]
   [TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f : X → Y}

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

@@ -77,12 +77,6 @@ def Specializes (x y : X) : Prop :=
 -- mathport name: «expr ⤳ »
 infixl:300 " ⤳ " => Specializes
 
-/- warning: specializes_tfae -> specializes_TFAE is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] (x : X) (y : X), List.TFAE (List.cons.{0} Prop (Specializes.{u1} X _inst_1 x y) (List.cons.{0} Prop (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsOpen.{u1} X _inst_1 s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y s)) (List.cons.{0} Prop (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) x))) (List.cons.{0} Prop (HasSubset.Subset.{u1} (Set.{u1} X) (Set.hasSubset.{u1} X) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) y)) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) x))) (List.cons.{0} Prop (ClusterPt.{u1} X _inst_1 y (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x)) (List.nil.{0} Prop))))))))
-but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] (x : X) (y : X), List.TFAE (List.cons.{0} Prop (Specializes.{u1} X _inst_1 x y) (List.cons.{0} Prop (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} X x) (nhds.{u1} X _inst_1 y)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsOpen.{u1} X _inst_1 s) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y s) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x s)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x s) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y s)) (List.cons.{0} Prop (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.instSingletonSet.{u1} X) x))) (List.cons.{0} Prop (HasSubset.Subset.{u1} (Set.{u1} X) (Set.instHasSubsetSet.{u1} X) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.instSingletonSet.{u1} X) y)) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.instSingletonSet.{u1} X) x))) (List.cons.{0} Prop (ClusterPt.{u1} X _inst_1 y (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} X x)) (List.nil.{0} Prop))))))))
-Case conversion may be inaccurate. Consider using '#align specializes_tfae specializes_TFAEₓ'. -/
 /-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
 below. -/
 theorem specializes_TFAE (x y : X) :
@@ -105,41 +99,17 @@ theorem specializes_TFAE (x y : X) :
   tfae_finish
 #align specializes_tfae specializes_TFAE
 
-/- warning: specializes_iff_nhds -> specializes_iff_nhds is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
-but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
-Case conversion may be inaccurate. Consider using '#align specializes_iff_nhds specializes_iff_nhdsₓ'. -/
 theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
   Iff.rfl
 #align specializes_iff_nhds specializes_iff_nhds
 
-/- warning: specializes_iff_pure -> specializes_iff_pure is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y))
-but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} X x) (nhds.{u1} X _inst_1 y))
-Case conversion may be inaccurate. Consider using '#align specializes_iff_pure specializes_iff_pureₓ'. -/
 theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
   (specializes_TFAE x y).out 0 1
 #align specializes_iff_pure specializes_iff_pure
 
-/- warning: specializes.nhds_le_nhds -> Specializes.nhds_le_nhds is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
-but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
-Case conversion may be inaccurate. Consider using '#align specializes.nhds_le_nhds Specializes.nhds_le_nhdsₓ'. -/
 alias specializes_iff_nhds ↔ Specializes.nhds_le_nhds _
 #align specializes.nhds_le_nhds Specializes.nhds_le_nhds
 
-/- warning: specializes.pure_le_nhds -> Specializes.pure_le_nhds is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y))
-but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} X x) (nhds.{u1} X _inst_1 y))
-Case conversion may be inaccurate. Consider using '#align specializes.pure_le_nhds Specializes.pure_le_nhdsₓ'. -/
 alias specializes_iff_pure ↔ Specializes.pure_le_nhds _
 #align specializes.pure_le_nhds Specializes.pure_le_nhds
 
@@ -230,12 +200,6 @@ theorem specializes_of_eq (e : x = y) : x ⤳ y :=
 #align specializes_of_eq specializes_of_eq
 -/
 
-/- warning: specializes_of_nhds_within -> specializes_of_nhdsWithin is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X} {s : Set.{u1} X}, (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhdsWithin.{u1} X _inst_1 x s) (nhdsWithin.{u1} X _inst_1 y s)) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s) -> (Specializes.{u1} X _inst_1 x y)
-but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X} {s : Set.{u1} X}, (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (nhdsWithin.{u1} X _inst_1 x s) (nhdsWithin.{u1} X _inst_1 y s)) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x s) -> (Specializes.{u1} X _inst_1 x y)
-Case conversion may be inaccurate. Consider using '#align specializes_of_nhds_within specializes_of_nhdsWithinₓ'. -/
 theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y :=
   specializes_iff_pure.2 <|
     calc
@@ -245,33 +209,15 @@ theorem specializes_of_nhdsWithin (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈
       
 #align specializes_of_nhds_within specializes_of_nhdsWithin
 
-/- warning: specializes.map_of_continuous_at -> Specializes.map_of_continuousAt is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {x : X} {y : X} {f : X -> Y}, (Specializes.{u1} X _inst_1 x y) -> (ContinuousAt.{u1, u2} X Y _inst_1 _inst_2 f y) -> (Specializes.{u2} Y _inst_2 (f x) (f y))
-but is expected to have type
-  forall {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {x : X} {y : X} {f : X -> Y}, (Specializes.{u2} X _inst_1 x y) -> (ContinuousAt.{u2, u1} X Y _inst_1 _inst_2 f y) -> (Specializes.{u1} Y _inst_2 (f x) (f y))
-Case conversion may be inaccurate. Consider using '#align specializes.map_of_continuous_at Specializes.map_of_continuousAtₓ'. -/
 theorem Specializes.map_of_continuousAt (h : x ⤳ y) (hy : ContinuousAt f y) : f x ⤳ f y :=
   specializes_iff_pure.2 fun s hs =>
     mem_pure.2 <| mem_preimage.1 <| mem_of_mem_nhds <| hy.mono_left h hs
 #align specializes.map_of_continuous_at Specializes.map_of_continuousAt
 
-/- warning: specializes.map -> Specializes.map is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {x : X} {y : X} {f : X -> Y}, (Specializes.{u1} X _inst_1 x y) -> (Continuous.{u1, u2} X Y _inst_1 _inst_2 f) -> (Specializes.{u2} Y _inst_2 (f x) (f y))
-but is expected to have type
-  forall {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {x : X} {y : X} {f : X -> Y}, (Specializes.{u2} X _inst_1 x y) -> (Continuous.{u2, u1} X Y _inst_1 _inst_2 f) -> (Specializes.{u1} Y _inst_2 (f x) (f y))
-Case conversion may be inaccurate. Consider using '#align specializes.map Specializes.mapₓ'. -/
 theorem Specializes.map (h : x ⤳ y) (hf : Continuous f) : f x ⤳ f y :=
   h.map_of_continuousAt hf.ContinuousAt
 #align specializes.map Specializes.map
 
-/- warning: inducing.specializes_iff -> Inducing.specializes_iff is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {x : X} {y : X} {f : X -> Y}, (Inducing.{u1, u2} X Y _inst_1 _inst_2 f) -> (Iff (Specializes.{u2} Y _inst_2 (f x) (f y)) (Specializes.{u1} X _inst_1 x y))
-but is expected to have type
-  forall {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {x : X} {y : X} {f : X -> Y}, (Inducing.{u2, u1} X Y _inst_1 _inst_2 f) -> (Iff (Specializes.{u1} Y _inst_2 (f x) (f y)) (Specializes.{u2} X _inst_1 x y))
-Case conversion may be inaccurate. Consider using '#align inducing.specializes_iff Inducing.specializes_iffₓ'. -/
 theorem Inducing.specializes_iff (hf : Inducing f) : f x ⤳ f y ↔ x ⤳ y := by
   simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
     mem_preimage]
@@ -283,34 +229,16 @@ theorem subtype_specializes_iff {p : X → Prop} (x y : Subtype p) : x ⤳ y ↔
 #align subtype_specializes_iff subtype_specializes_iff
 -/
 
-/- warning: specializes_prod -> specializes_prod is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {x₁ : X} {x₂ : X} {y₁ : Y} {y₂ : Y}, Iff (Specializes.{max u1 u2} (Prod.{u1, u2} X Y) (Prod.topologicalSpace.{u1, u2} X Y _inst_1 _inst_2) (Prod.mk.{u1, u2} X Y x₁ y₁) (Prod.mk.{u1, u2} X Y x₂ y₂)) (And (Specializes.{u1} X _inst_1 x₁ x₂) (Specializes.{u2} Y _inst_2 y₁ y₂))
-but is expected to have type
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {x₁ : X} {x₂ : X} {y₁ : Y} {y₂ : Y}, Iff (Specializes.{max u2 u1} (Prod.{u1, u2} X Y) (instTopologicalSpaceProd.{u1, u2} X Y _inst_1 _inst_2) (Prod.mk.{u1, u2} X Y x₁ y₁) (Prod.mk.{u1, u2} X Y x₂ y₂)) (And (Specializes.{u1} X _inst_1 x₁ x₂) (Specializes.{u2} Y _inst_2 y₁ y₂))
-Case conversion may be inaccurate. Consider using '#align specializes_prod specializes_prodₓ'. -/
 @[simp]
 theorem specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} : (x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ := by
   simp only [Specializes, nhds_prod_eq, prod_le_prod]
 #align specializes_prod specializes_prod
 
-/- warning: specializes.prod -> Specializes.prod is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {x₁ : X} {x₂ : X} {y₁ : Y} {y₂ : Y}, (Specializes.{u1} X _inst_1 x₁ x₂) -> (Specializes.{u2} Y _inst_2 y₁ y₂) -> (Specializes.{max u1 u2} (Prod.{u1, u2} X Y) (Prod.topologicalSpace.{u1, u2} X Y _inst_1 _inst_2) (Prod.mk.{u1, u2} X Y x₁ y₁) (Prod.mk.{u1, u2} X Y x₂ y₂))
-but is expected to have type
-  forall {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {x₁ : X} {x₂ : X} {y₁ : Y} {y₂ : Y}, (Specializes.{u2} X _inst_1 x₁ x₂) -> (Specializes.{u1} Y _inst_2 y₁ y₂) -> (Specializes.{max u1 u2} (Prod.{u2, u1} X Y) (instTopologicalSpaceProd.{u2, u1} X Y _inst_1 _inst_2) (Prod.mk.{u2, u1} X Y x₁ y₁) (Prod.mk.{u2, u1} X Y x₂ y₂))
-Case conversion may be inaccurate. Consider using '#align specializes.prod Specializes.prodₓ'. -/
 theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
     (x₁, y₁) ⤳ (x₂, y₂) :=
   specializes_prod.2 ⟨hx, hy⟩
 #align specializes.prod Specializes.prod
 
-/- warning: specializes_pi -> specializes_pi is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {π : ι -> Type.{u2}} [_inst_4 : forall (i : ι), TopologicalSpace.{u2} (π i)] {f : forall (i : ι), π i} {g : forall (i : ι), π i}, Iff (Specializes.{max u1 u2} (forall (i : ι), π i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_4 a)) f g) (forall (i : ι), Specializes.{u2} (π i) (_inst_4 i) (f i) (g i))
-but is expected to have type
-  forall {ι : Type.{u2}} {π : ι -> Type.{u1}} [_inst_4 : forall (i : ι), TopologicalSpace.{u1} (π i)] {f : forall (i : ι), π i} {g : forall (i : ι), π i}, Iff (Specializes.{max u2 u1} (forall (i : ι), π i) (Pi.topologicalSpace.{u2, u1} ι (fun (i : ι) => π i) (fun (a : ι) => _inst_4 a)) f g) (forall (i : ι), Specializes.{u1} (π i) (_inst_4 i) (f i) (g i))
-Case conversion may be inaccurate. Consider using '#align specializes_pi specializes_piₓ'. -/
 @[simp]
 theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
   simp only [Specializes, nhds_pi, pi_le_pi]
@@ -343,12 +271,6 @@ def specializationPreorder : Preorder X :=
 
 variable {X}
 
-/- warning: continuous.specialization_monotone -> Continuous.specialization_monotone is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {f : X -> Y}, (Continuous.{u1, u2} X Y _inst_1 _inst_2 f) -> (Monotone.{u1, u2} X Y (specializationPreorder.{u1} X _inst_1) (specializationPreorder.{u2} Y _inst_2) f)
-but is expected to have type
-  forall {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : X -> Y}, (Continuous.{u2, u1} X Y _inst_1 _inst_2 f) -> (Monotone.{u2, u1} X Y (specializationPreorder.{u2} X _inst_1) (specializationPreorder.{u1} Y _inst_2) f)
-Case conversion may be inaccurate. Consider using '#align continuous.specialization_monotone Continuous.specialization_monotoneₓ'. -/
 /-- A continuous function is monotone with respect to the specialization preorders on the domain and
 the codomain. -/
 theorem Continuous.specialization_monotone (hf : Continuous f) :
@@ -448,12 +370,6 @@ theorem inseparable_of_nhdsWithin_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s]
 #align inseparable_of_nhds_within_eq inseparable_of_nhdsWithin_eq
 -/
 
-/- warning: inducing.inseparable_iff -> Inducing.inseparable_iff is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : TopologicalSpace.{u2} Y] {x : X} {y : X} {f : X -> Y}, (Inducing.{u1, u2} X Y _inst_1 _inst_2 f) -> (Iff (Inseparable.{u2} Y _inst_2 (f x) (f y)) (Inseparable.{u1} X _inst_1 x y))
-but is expected to have type
-  forall {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {x : X} {y : X} {f : X -> Y}, (Inducing.{u2, u1} X Y _inst_1 _inst_2 f) -> (Iff (Inseparable.{u1} Y _inst_2 (f x) (f y)) (Inseparable.{u2} X _inst_1 x y))
-Case conversion may be inaccurate. Consider using '#align inducing.inseparable_iff Inducing.inseparable_iffₓ'. -/
 theorem Inducing.inseparable_iff (hf : Inducing f) : (f x ~ f y) ↔ (x ~ y) := by
   simp only [inseparable_iff_specializes_and, hf.specializes_iff]
 #align inducing.inseparable_iff Inducing.inseparable_iff
@@ -464,34 +380,16 @@ theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ y) ↔
 #align subtype_inseparable_iff subtype_inseparable_iff
 -/
 
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 @[simp]
 theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} : ((x₁, y₁) ~ (x₂, y₂)) ↔ (x₁ ~ x₂) ∧ (y₁ ~ y₂) :=
   by simp only [Inseparable, nhds_prod_eq, prod_inj]
 #align inseparable_prod inseparable_prod
 
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-Case conversion may be inaccurate. Consider using '#align inseparable.prod Inseparable.prodₓ'. -/
 theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ x₂) (hy : y₁ ~ y₂) :
     (x₁, y₁) ~ (x₂, y₂) :=
   inseparable_prod.2 ⟨hx, hy⟩
 #align inseparable.prod Inseparable.prod
 
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 @[simp]
 theorem inseparable_pi {f g : ∀ i, π i} : (f ~ g) ↔ ∀ i, f i ~ g i := by
   simp only [Inseparable, nhds_pi, funext_iff, pi_inj]
@@ -550,23 +448,11 @@ theorem mem_closed_iff (h : x ~ y) (hs : IsClosed s) : x ∈ s ↔ y ∈ s :=
 #align inseparable.mem_closed_iff Inseparable.mem_closed_iff
 -/
 
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 theorem map_of_continuousAt (h : x ~ y) (hx : ContinuousAt f x) (hy : ContinuousAt f y) :
     f x ~ f y :=
   (h.Specializes.map_of_continuousAt hy).antisymm (h.specializes'.map_of_continuousAt hx)
 #align inseparable.map_of_continuous_at Inseparable.map_of_continuousAt
 
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 theorem map (h : x ~ y) (hf : Continuous f) : f x ~ f y :=
   h.map_of_continuousAt hf.ContinuousAt hf.ContinuousAt
 #align inseparable.map Inseparable.map
@@ -755,12 +641,6 @@ theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
 #align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
 -/
 
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 theorem map_prod_map_mk_nhds (x : X) (y : Y) : map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) :=
   by rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
 #align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
@@ -780,46 +660,22 @@ def lift (f : X → α) (hf : ∀ x y, (x ~ y) → f x = f y) : SeparationQuotie
 #align separation_quotient.lift SeparationQuotient.lift
 -/
 
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 @[simp]
 theorem lift_mk {f : X → α} (hf : ∀ x y, (x ~ y) → f x = f y) (x : X) : lift f hf (mk x) = f x :=
   rfl
 #align separation_quotient.lift_mk SeparationQuotient.lift_mk
 
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 @[simp]
 theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ y) → f x = f y) : lift f hf ∘ mk = f :=
   rfl
 #align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
 
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 @[simp]
 theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ y) → f x = f y} {x : X} {l : Filter α} :
     Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
   simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
 #align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
 
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 @[simp]
 theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ y) → f x = f y} {x : X}
     {s : Set (SeparationQuotient X)} {l : Filter α} :
@@ -827,24 +683,12 @@ theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ y) → f x
   simp only [← map_mk_nhds_within_preimage, tendsto_map'_iff, lift_comp_mk]
 #align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
 
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 @[simp]
 theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f y} {x : X} :
     ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
   tendsto_lift_nhds_mk
 #align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
 
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-Case conversion may be inaccurate. Consider using '#align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_liftₓ'. -/
 @[simp]
 theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f y}
     {s : Set (SeparationQuotient X)} {x : X} :
@@ -852,24 +696,12 @@ theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f
   tendsto_lift_nhdsWithin_mk
 #align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
 
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 @[simp]
 theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f y}
     {s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
   simp only [ContinuousOn, surjective_mk.forall, continuous_within_at_lift, mem_preimage]
 #align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
 
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-Case conversion may be inaccurate. Consider using '#align separation_quotient.continuous_lift SeparationQuotient.continuous_liftₓ'. -/
 @[simp]
 theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ y) → f x = f y} :
     Continuous (lift f hf) ↔ Continuous f := by
@@ -884,24 +716,12 @@ def lift₂ (f : X → Y → α) (hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a
 #align separation_quotient.lift₂ SeparationQuotient.lift₂
 -/
 
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-Case conversion may be inaccurate. Consider using '#align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mkₓ'. -/
 @[simp]
 theorem lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d) (x : X)
     (y : Y) : lift₂ f hf (mk x) (mk y) = f x y :=
   rfl
 #align separation_quotient.lift₂_mk SeparationQuotient.lift₂_mk
 
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 @[simp]
 theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {x : X} {y : Y} {l : Filter α} :
@@ -909,12 +729,6 @@ theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ c) →
   rw [← map_prod_map_mk_nhds, tendsto_map'_iff]; rfl
 #align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
 
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 @[simp]
 theorem tendsto_lift₂_nhdsWithin {f : X → Y → α} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {x : X} {y : Y} {s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
@@ -923,12 +737,6 @@ theorem tendsto_lift₂_nhdsWithin {f : X → Y → α} {hf : ∀ a b c d, (a ~
   by rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]; rfl
 #align separation_quotient.tendsto_lift₂_nhds_within SeparationQuotient.tendsto_lift₂_nhdsWithin
 
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 @[simp]
 theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {x : X} {y : Y} :
@@ -936,12 +744,6 @@ theorem continuousAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) →
   tendsto_lift₂_nhds
 #align separation_quotient.continuous_at_lift₂ SeparationQuotient.continuousAt_lift₂
 
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 @[simp]
 theorem continuousWithinAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {s : Set (SeparationQuotient X × SeparationQuotient Y)} {x : X} {y : Y} :
@@ -950,12 +752,6 @@ theorem continuousWithinAt_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c
   tendsto_lift₂_nhdsWithin
 #align separation_quotient.continuous_within_at_lift₂ SeparationQuotient.continuousWithinAt_lift₂
 
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-Case conversion may be inaccurate. Consider using '#align separation_quotient.continuous_on_lift₂ SeparationQuotient.continuousOn_lift₂ₓ'. -/
 @[simp]
 theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {s : Set (SeparationQuotient X × SeparationQuotient Y)} :
@@ -966,12 +762,6 @@ theorem continuousOn_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) →
   rfl
 #align separation_quotient.continuous_on_lift₂ SeparationQuotient.continuousOn_lift₂
 
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 @[simp]
 theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d} :
     Continuous (uncurry <| lift₂ f hf) ↔ Continuous (uncurry f) := by

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

@@ -317,20 +317,14 @@ theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := b
 #align specializes_pi specializes_pi
 
 #print not_specializes_iff_exists_open /-
-theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S :=
-  by
-  rw [specializes_iff_forall_open]
-  push_neg
-  rfl
+theorem not_specializes_iff_exists_open : ¬x ⤳ y ↔ ∃ S : Set X, IsOpen S ∧ y ∈ S ∧ x ∉ S := by
+  rw [specializes_iff_forall_open]; push_neg; rfl
 #align not_specializes_iff_exists_open not_specializes_iff_exists_open
 -/
 
 #print not_specializes_iff_exists_closed /-
-theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S :=
-  by
-  rw [specializes_iff_forall_closed]
-  push_neg
-  rfl
+theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClosed S ∧ x ∈ S ∧ y ∉ S := by
+  rw [specializes_iff_forall_closed]; push_neg; rfl
 #align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
 -/
 
@@ -700,9 +694,7 @@ theorem inducing_mk : Inducing (mk : X → SeparationQuotient X) :=
 
 #print SeparationQuotient.isClosedMap_mk /-
 theorem isClosedMap_mk : IsClosedMap (mk : X → SeparationQuotient X) :=
-  inducing_mk.IsClosedMap <| by
-    rw [range_mk]
-    exact isClosed_univ
+  inducing_mk.IsClosedMap <| by rw [range_mk]; exact isClosed_univ
 #align separation_quotient.is_closed_map_mk SeparationQuotient.isClosedMap_mk
 -/
 
@@ -913,10 +905,8 @@ Case conversion may be inaccurate. Consider using '#align separation_quotient.te
 @[simp]
 theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ c) → (b ~ d) → f a b = f c d}
     {x : X} {y : Y} {l : Filter α} :
-    Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l :=
-  by
-  rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
-  rfl
+    Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by
+  rw [← map_prod_map_mk_nhds, tendsto_map'_iff]; rfl
 #align separation_quotient.tendsto_lift₂_nhds SeparationQuotient.tendsto_lift₂_nhds
 
 /- warning: separation_quotient.tendsto_lift₂_nhds_within -> SeparationQuotient.tendsto_lift₂_nhdsWithin is a dubious translation:
@@ -930,9 +920,7 @@ theorem tendsto_lift₂_nhdsWithin {f : X → Y → α} {hf : ∀ a b c d, (a ~
     {x : X} {y : Y} {s : Set (SeparationQuotient X × SeparationQuotient Y)} {l : Filter α} :
     Tendsto (uncurry <| lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
       Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l :=
-  by
-  rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]
-  rfl
+  by rw [nhdsWithin, ← map_prod_map_mk_nhds, ← Filter.push_pull, comap_principal]; rfl
 #align separation_quotient.tendsto_lift₂_nhds_within SeparationQuotient.tendsto_lift₂_nhdsWithin
 
 /- warning: separation_quotient.continuous_at_lift₂ -> SeparationQuotient.continuousAt_lift₂ is a dubious translation:

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

@@ -79,7 +79,7 @@ infixl:300 " ⤳ " => Specializes
 
 /- warning: specializes_tfae -> specializes_TFAE is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] (x : X) (y : X), List.TFAE (List.cons.{0} Prop (Specializes.{u1} X _inst_1 x y) (List.cons.{0} Prop (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsOpen.{u1} X _inst_1 s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y s)) (List.cons.{0} Prop (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) x))) (List.cons.{0} Prop (HasSubset.Subset.{u1} (Set.{u1} X) (Set.hasSubset.{u1} X) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) y)) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) x))) (List.cons.{0} Prop (ClusterPt.{u1} X _inst_1 y (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x)) (List.nil.{0} Prop))))))))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] (x : X) (y : X), List.TFAE (List.cons.{0} Prop (Specializes.{u1} X _inst_1 x y) (List.cons.{0} Prop (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsOpen.{u1} X _inst_1 s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y s)) (List.cons.{0} Prop (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) y (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) x))) (List.cons.{0} Prop (HasSubset.Subset.{u1} (Set.{u1} X) (Set.hasSubset.{u1} X) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) y)) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.hasSingleton.{u1} X) x))) (List.cons.{0} Prop (ClusterPt.{u1} X _inst_1 y (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x)) (List.nil.{0} Prop))))))))
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] (x : X) (y : X), List.TFAE (List.cons.{0} Prop (Specializes.{u1} X _inst_1 x y) (List.cons.{0} Prop (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} X x) (nhds.{u1} X _inst_1 y)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsOpen.{u1} X _inst_1 s) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y s) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x s)) (List.cons.{0} Prop (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x s) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y s)) (List.cons.{0} Prop (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) y (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.instSingletonSet.{u1} X) x))) (List.cons.{0} Prop (HasSubset.Subset.{u1} (Set.{u1} X) (Set.instHasSubsetSet.{u1} X) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.instSingletonSet.{u1} X) y)) (closure.{u1} X _inst_1 (Singleton.singleton.{u1, u1} X (Set.{u1} X) (Set.instSingletonSet.{u1} X) x))) (List.cons.{0} Prop (ClusterPt.{u1} X _inst_1 y (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} X x)) (List.nil.{0} Prop))))))))
 Case conversion may be inaccurate. Consider using '#align specializes_tfae specializes_TFAEₓ'. -/
@@ -107,7 +107,7 @@ theorem specializes_TFAE (x y : X) :
 
 /- warning: specializes_iff_nhds -> specializes_iff_nhds is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
 Case conversion may be inaccurate. Consider using '#align specializes_iff_nhds specializes_iff_nhdsₓ'. -/
@@ -117,7 +117,7 @@ theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
 
 /- warning: specializes_iff_pure -> specializes_iff_pure is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y))
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, Iff (Specializes.{u1} X _inst_1 x y) (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} X x) (nhds.{u1} X _inst_1 y))
 Case conversion may be inaccurate. Consider using '#align specializes_iff_pure specializes_iff_pureₓ'. -/
@@ -127,7 +127,7 @@ theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
 
 /- warning: specializes.nhds_le_nhds -> Specializes.nhds_le_nhds is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (nhds.{u1} X _inst_1 x) (nhds.{u1} X _inst_1 y))
 Case conversion may be inaccurate. Consider using '#align specializes.nhds_le_nhds Specializes.nhds_le_nhdsₓ'. -/
@@ -136,7 +136,7 @@ alias specializes_iff_nhds ↔ Specializes.nhds_le_nhds _
 
 /- warning: specializes.pure_le_nhds -> Specializes.pure_le_nhds is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} X x) (nhds.{u1} X _inst_1 y))
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X}, (Specializes.{u1} X _inst_1 x y) -> (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} X x) (nhds.{u1} X _inst_1 y))
 Case conversion may be inaccurate. Consider using '#align specializes.pure_le_nhds Specializes.pure_le_nhdsₓ'. -/
@@ -232,7 +232,7 @@ theorem specializes_of_eq (e : x = y) : x ⤳ y :=
 
 /- warning: specializes_of_nhds_within -> specializes_of_nhdsWithin is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X} {s : Set.{u1} X}, (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhdsWithin.{u1} X _inst_1 x s) (nhdsWithin.{u1} X _inst_1 y s)) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s) -> (Specializes.{u1} X _inst_1 x y)
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X} {s : Set.{u1} X}, (LE.le.{u1} (Filter.{u1} X) (Preorder.toHasLe.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.partialOrder.{u1} X))) (nhdsWithin.{u1} X _inst_1 x s) (nhdsWithin.{u1} X _inst_1 y s)) -> (Membership.Mem.{u1, u1} X (Set.{u1} X) (Set.hasMem.{u1} X) x s) -> (Specializes.{u1} X _inst_1 x y)
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {x : X} {y : X} {s : Set.{u1} X}, (LE.le.{u1} (Filter.{u1} X) (Preorder.toLE.{u1} (Filter.{u1} X) (PartialOrder.toPreorder.{u1} (Filter.{u1} X) (Filter.instPartialOrderFilter.{u1} X))) (nhdsWithin.{u1} X _inst_1 x s) (nhdsWithin.{u1} X _inst_1 y s)) -> (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x s) -> (Specializes.{u1} X _inst_1 x y)
 Case conversion may be inaccurate. Consider using '#align specializes_of_nhds_within specializes_of_nhdsWithinₓ'. -/

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

This is needed to prove continuity of binary arithmetic operations on the separation quotient.

@@ -489,6 +489,16 @@ theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
   rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
 #align separation_quotient.map_mk_nhds_within_preimage SeparationQuotient.map_mk_nhdsWithin_preimage
 
+/-- The map `(x, y) ↦ (mk x, mk y)` is a quotient map. -/
+theorem quotientMap_prodMap_mk : QuotientMap (Prod.map mk mk : X × Y → _) := by
+  have hsurj : Surjective (Prod.map mk mk : X × Y → _) := surjective_mk.Prod_map surjective_mk
+  refine quotientMap_iff.2 ⟨hsurj, fun s ↦ ?_⟩
+  refine ⟨fun hs ↦ hs.preimage (continuous_mk.prod_map continuous_mk), fun hs ↦ ?_⟩
+  refine isOpen_iff_mem_nhds.2 <| hsurj.forall.2 fun (x, y) h ↦ ?_
+  rw [Prod.map_mk, nhds_prod_eq, ← map_mk_nhds, ← map_mk_nhds, Filter.prod_map_map_eq',
+    ← nhds_prod_eq, Filter.mem_map]
+  exact hs.mem_nhds h
+
 /-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
 `SeparationQuotient X → α`. -/
 def lift (f : X → α) (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : SeparationQuotient X → α := fun x =>

Generalize coclosedCompact_eq_cocompact and relativelyCompact.

@@ -195,6 +195,9 @@ theorem Specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (h
   specializes_prod.2 ⟨hx, hy⟩
 #align specializes.prod Specializes.prod
 
+theorem Specializes.fst {a b : X × Y} (h : a ⤳ b) : a.1 ⤳ b.1 := (specializes_prod.1 h).1
+theorem Specializes.snd {a b : X × Y} (h : a ⤳ b) : a.2 ⤳ b.2 := (specializes_prod.1 h).2
+
 @[simp]
 theorem specializes_pi {f g : ∀ i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i := by
   simp only [Specializes, nhds_pi, pi_le_pi]

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

• "new lemma"
• "added lemma"

@@ -133,7 +133,7 @@ theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ c
 alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
 #align specializes.closure_subset Specializes.closure_subset
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
   (specializes_TFAE x y).out 0 6
 

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".

@@ -133,7 +133,7 @@ theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ c
 alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
 #align specializes.closure_subset Specializes.closure_subset
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
   (specializes_TFAE x y).out 0 6
 

@@ -11,7 +11,7 @@ import Mathlib.Topology.ContinuousOn
 /-!
 # Inseparable points in a topological space
 
-In this file we define
+In this file we prove basic properties of the following notions defined elsewhere.
 
 * `Specializes` (notation: `x ⤳ y`) : a relation saying that `𝓝 x ≤ 𝓝 y`;
 
@@ -45,25 +45,6 @@ variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [Topolog
 ### `Specializes` relation
 -/
 
-/-- `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
-hold:
-
-* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
-* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
-* `y ∈ closure {x}`;
-* `closure {y} ⊆ closure {x}`;
-* for any closed set `s` we have `x ∈ s → y ∈ s`;
-* for any open set `s` we have `y ∈ s → x ∈ s`;
-* `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
-
-This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
-order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
-def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
-#align specializes Specializes
-
-@[inherit_doc]
-infixl:300 " ⤳ " => Specializes
-
 /-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
 below. -/
 theorem specializes_TFAE (x y : X) :
@@ -245,17 +226,6 @@ theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)]
     Continuous (s.piecewise f g) := by
   simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
 
-variable (X)
-
-/-- Specialization forms a preorder on the topological space. -/
-def specializationPreorder : Preorder X :=
-  { Preorder.lift (OrderDual.toDual ∘ 𝓝) with
-    le := fun x y => y ⤳ x
-    lt := fun x y => y ⤳ x ∧ ¬x ⤳ y }
-#align specialization_preorder specializationPreorder
-
-variable {X}
-
 /-- A continuous function is monotone with respect to the specialization preorders on the domain and
 the codomain. -/
 theorem Continuous.specialization_monotone (hf : Continuous f) :
@@ -266,19 +236,6 @@ theorem Continuous.specialization_monotone (hf : Continuous f) :
 ### `Inseparable` relation
 -/
 
-/-- Two points `x` and `y` in a topological space are `Inseparable` if any of the following
-equivalent properties hold:
-
-- `𝓝 x = 𝓝 y`; we use this property as the definition;
-- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
-- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
-- `x ∈ closure {y}` and `y ∈ closure {x}`, see `inseparable_iff_mem_closure`;
-- `closure {x} = closure {y}`, see `inseparable_iff_closure_eq`.
--/
-def Inseparable (x y : X) : Prop :=
-  𝓝 x = 𝓝 y
-#align inseparable Inseparable
-
 local infixl:0 " ~ᵢ " => Inseparable
 
 theorem inseparable_def : (x ~ᵢ y) ↔ 𝓝 x = 𝓝 y :=
@@ -413,15 +370,6 @@ In this section we define the quotient of a topological space by the `Inseparabl
 
 variable (X)
 
-/-- A `setoid` version of `Inseparable`, used to define the `SeparationQuotient`. -/
-def inseparableSetoid : Setoid X := { Setoid.comap 𝓝 ⊥ with r := Inseparable }
-#align inseparable_setoid inseparableSetoid
-
-/-- The quotient of a topological space by its `inseparableSetoid`. This quotient is guaranteed to
-be a T₀ space. -/
-def SeparationQuotient := Quotient (inseparableSetoid X)
-#align separation_quotient SeparationQuotient
-
 instance : TopologicalSpace (SeparationQuotient X) := instTopologicalSpaceQuotient
 
 variable {X}

Main API changes

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

New theorems

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

Deprecated theorems

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

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

Renamed/moved theorems

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

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

@@ -98,6 +98,9 @@ theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=
   Iff.rfl
 #align specializes_iff_nhds specializes_iff_nhds
 
+theorem Specializes.not_disjoint (h : x ⤳ y) : ¬Disjoint (𝓝 x) (𝓝 y) := fun hd ↦
+  absurd (hd.mono_right h) <| by simp [NeBot.ne']
+
 theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
   (specializes_TFAE x y).out 0 1
 #align specializes_iff_pure specializes_iff_pure

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

@@ -552,39 +552,39 @@ theorem lift_comp_mk {f : X → α} (hf : ∀ x y, (x ~ᵢ y) → f x = f y) : l
 #align separation_quotient.lift_comp_mk SeparationQuotient.lift_comp_mk
 
 @[simp]
-theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} {l : Filter α} :
+theorem tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {l : Filter α} :
     Tendsto (lift f hf) (𝓝 <| mk x) l ↔ Tendsto f (𝓝 x) l := by
   simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk]
 #align separation_quotient.tendsto_lift_nhds_mk SeparationQuotient.tendsto_lift_nhds_mk
 
 @[simp]
-theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X}
+theorem tendsto_lift_nhdsWithin_mk {f : X → α} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
     {s : Set (SeparationQuotient X)} {l : Filter α} :
     Tendsto (lift f hf) (𝓝[s] mk x) l ↔ Tendsto f (𝓝[mk ⁻¹' s] x) l := by
   simp only [← map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]
 #align separation_quotient.tendsto_lift_nhds_within_mk SeparationQuotient.tendsto_lift_nhdsWithin_mk
 
 @[simp]
-theorem continuousAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {x : X} :
+theorem continuousAt_lift {hf : ∀ x y, (x ~ᵢ y) → f x = f y}:
     ContinuousAt (lift f hf) (mk x) ↔ ContinuousAt f x :=
   tendsto_lift_nhds_mk
 #align separation_quotient.continuous_at_lift SeparationQuotient.continuousAt_lift
 
 @[simp]
-theorem continuousWithinAt_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
-    {s : Set (SeparationQuotient X)} {x : X} :
+theorem continuousWithinAt_lift {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
+    {s : Set (SeparationQuotient X)}:
     ContinuousWithinAt (lift f hf) s (mk x) ↔ ContinuousWithinAt f (mk ⁻¹' s) x :=
   tendsto_lift_nhdsWithin_mk
 #align separation_quotient.continuous_within_at_lift SeparationQuotient.continuousWithinAt_lift
 
 @[simp]
-theorem continuousOn_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y}
-    {s : Set (SeparationQuotient X)} : ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
+theorem continuousOn_lift {hf : ∀ x y, (x ~ᵢ y) → f x = f y} {s : Set (SeparationQuotient X)} :
+    ContinuousOn (lift f hf) s ↔ ContinuousOn f (mk ⁻¹' s) := by
   simp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]
 #align separation_quotient.continuous_on_lift SeparationQuotient.continuousOn_lift
 
 @[simp]
-theorem continuous_lift {f : X → Y} {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
+theorem continuous_lift {hf : ∀ x y, (x ~ᵢ y) → f x = f y} :
     Continuous (lift f hf) ↔ Continuous f := by
   simp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]
 #align separation_quotient.continuous_lift SeparationQuotient.continuous_lift

This file was mostly using Greek letters, but used letters X, Y, Z in comments and one theorem. Switch to using the latter consistently, per Zulip discussion.

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

@@ -652,5 +652,5 @@ end SeparationQuotient
 
 theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
     Continuous f ↔ Continuous g := by
-  simp_rw [SeparationQuotient.inducing_mk.continuous_iff (β := Y)]
+  simp_rw [SeparationQuotient.inducing_mk.continuous_iff (Y := Y)]
   exact continuous_congr fun x ↦ SeparationQuotient.mk_eq_mk.mpr (h x)

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

@@ -302,8 +302,8 @@ theorem inseparable_iff_forall_open : (x ~ᵢ y) ↔ ∀ s : Set X, IsOpen s →
 #align inseparable_iff_forall_open inseparable_iff_forall_open
 
 theorem not_inseparable_iff_exists_open :
-    ¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) :=
-  by simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
+    ¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by
+  simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
 #align not_inseparable_iff_exists_open not_inseparable_iff_exists_open
 
 theorem inseparable_iff_forall_closed : (x ~ᵢ y) ↔ ∀ s : Set X, IsClosed s → (x ∈ s ↔ y ∈ s) := by
@@ -334,8 +334,8 @@ theorem subtype_inseparable_iff {p : X → Prop} (x y : Subtype p) : (x ~ᵢ y)
 #align subtype_inseparable_iff subtype_inseparable_iff
 
 @[simp] theorem inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
-    ((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) :=
-  by simp only [Inseparable, nhds_prod_eq, prod_inj]
+    ((x₁, y₁) ~ᵢ (x₂, y₂)) ↔ (x₁ ~ᵢ x₂) ∧ (y₁ ~ᵢ y₂) := by
+  simp only [Inseparable, nhds_prod_eq, prod_inj]
 #align inseparable_prod inseparable_prod
 
 theorem Inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ᵢ x₂) (hy : y₁ ~ᵢ y₂) :
@@ -525,8 +525,9 @@ theorem image_mk_closure : mk '' closure s = closure (mk '' s) :=
     isClosedMap_mk.closure_image_subset _
 #align separation_quotient.image_mk_closure SeparationQuotient.image_mk_closure
 
-theorem map_prod_map_mk_nhds (x : X) (y : Y) : map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) :=
-  by rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
+theorem map_prod_map_mk_nhds (x : X) (y : Y) :
+    map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by
+  rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
 #align separation_quotient.map_prod_map_mk_nhds SeparationQuotient.map_prod_map_mk_nhds
 
 theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :

Generalizing one of the proofs introduced in #7878

@@ -18,7 +18,7 @@ In this file we define
 * `Inseparable`: a relation saying that two points in a topological space have the same
   neighbourhoods; equivalently, they can't be separated by an open set;
 
-* `InseparableSetoid X`: same relation, as a `setoid`;
+* `InseparableSetoid X`: same relation, as a `Setoid`;
 
 * `SeparationQuotient X`: the quotient of `X` by its `InseparableSetoid`.
 
@@ -39,7 +39,7 @@ topological space, separation setoid
 open Set Filter Function Topology List
 
 variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
-  [TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f : X → Y}
+  [TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f g : X → Y}
 
 /-!
 ### `Specializes` relation
@@ -228,6 +228,20 @@ theorem not_specializes_iff_exists_closed : ¬x ⤳ y ↔ ∃ S : Set X, IsClose
   rfl
 #align not_specializes_iff_exists_closed not_specializes_iff_exists_closed
 
+theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
+    (hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
+    Continuous (s.piecewise f g) := by
+  have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
+  rw [continuous_def]
+  intro U hU
+  rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
+  exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
+
+theorem IsClosed.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsClosed s)
+    (hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, g x ⤳ f x) :
+    Continuous (s.piecewise f g) := by
+  simpa only [piecewise_compl] using hs.isOpen_compl.continuous_piecewise_of_specializes hg hf hspec
+
 variable (X)
 
 /-- Specialization forms a preorder on the topological space. -/
@@ -634,3 +648,8 @@ theorem continuous_lift₂ {f : X → Y → Z} {hf : ∀ a b c d, (a ~ᵢ c) →
 #align separation_quotient.continuous_lift₂ SeparationQuotient.continuous_lift₂
 
 end SeparationQuotient
+
+theorem continuous_congr_of_inseparable (h : ∀ x, f x ~ᵢ g x) :
+    Continuous f ↔ Continuous g := by
+  simp_rw [SeparationQuotient.inducing_mk.continuous_iff (β := Y)]
+  exact continuous_congr fun x ↦ SeparationQuotient.mk_eq_mk.mpr (h x)

For faster build times and clearer dependencies. No attempt at being exhaustive.

The new import in Clopen.lean had been transitively imported before.

@@ -3,7 +3,6 @@ Copyright (c) 2021 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang, Yury G. Kudryashov
 -/
-import Mathlib.Data.Setoid.Basic
 import Mathlib.Tactic.TFAE
 import Mathlib.Topology.ContinuousOn
 

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

@@ -109,8 +109,8 @@ alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
 alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
 #align specializes.pure_le_nhds Specializes.pure_le_nhds
 
-theorem sInter_nhds_sets_eq_specializes : ⋂₀ (𝓝 x).sets = {y | y ⤳ x} :=
-  Set.ext fun _ ↦ specializes_iff_pure.symm
+theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by
+  ext; simp [specializes_iff_pure, le_def]
 
 theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
   (specializes_TFAE x y).out 0 2

@@ -103,10 +103,10 @@ theorem specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y :=
   (specializes_TFAE x y).out 0 1
 #align specializes_iff_pure specializes_iff_pure
 
-alias specializes_iff_nhds ↔ Specializes.nhds_le_nhds _
+alias ⟨Specializes.nhds_le_nhds, _⟩ := specializes_iff_nhds
 #align specializes.nhds_le_nhds Specializes.nhds_le_nhds
 
-alias specializes_iff_pure ↔ Specializes.pure_le_nhds _
+alias ⟨Specializes.pure_le_nhds, _⟩ := specializes_iff_pure
 #align specializes.pure_le_nhds Specializes.pure_le_nhds
 
 theorem sInter_nhds_sets_eq_specializes : ⋂₀ (𝓝 x).sets = {y | y ⤳ x} :=
@@ -140,14 +140,14 @@ theorem specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : Set X) :=
   (specializes_TFAE x y).out 0 4
 #align specializes_iff_mem_closure specializes_iff_mem_closure
 
-alias specializes_iff_mem_closure ↔ Specializes.mem_closure _
+alias ⟨Specializes.mem_closure, _⟩ := specializes_iff_mem_closure
 #align specializes.mem_closure Specializes.mem_closure
 
 theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ closure {x} :=
   (specializes_TFAE x y).out 0 5
 #align specializes_iff_closure_subset specializes_iff_closure_subset
 
-alias specializes_iff_closure_subset ↔ Specializes.closure_subset _
+alias ⟨Specializes.closure_subset, _⟩ := specializes_iff_closure_subset
 #align specializes.closure_subset Specializes.closure_subset
 
 -- porting note: new lemma

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

This has nice performance benefits.

@@ -39,7 +39,7 @@ topological space, separation setoid
 
 open Set Filter Function Topology List
 
-variable {X Y Z α ι : Type _} {π : ι → Type _} [TopologicalSpace X] [TopologicalSpace Y]
+variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
   [TopologicalSpace Z] [∀ i, TopologicalSpace (π i)] {x y z : X} {s : Set X} {f : X → Y}
 
 /-!

A step towards showing that Pmfs have countable support.

Thanks Eric Rodriguez and Kevin Buzzard for helping on zulip.

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

@@ -109,6 +109,9 @@ alias specializes_iff_nhds ↔ Specializes.nhds_le_nhds _
 alias specializes_iff_pure ↔ Specializes.pure_le_nhds _
 #align specializes.pure_le_nhds Specializes.pure_le_nhds
 
+theorem sInter_nhds_sets_eq_specializes : ⋂₀ (𝓝 x).sets = {y | y ⤳ x} :=
+  Set.ext fun _ ↦ specializes_iff_pure.symm
+
 theorem specializes_iff_forall_open : x ⤳ y ↔ ∀ s : Set X, IsOpen s → y ∈ s → x ∈ s :=
   (specializes_TFAE x y).out 0 2
 #align specializes_iff_forall_open specializes_iff_forall_open

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang, Yury G. Kudryashov
-
-! This file was ported from Lean 3 source module topology.inseparable
-! 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.Data.Setoid.Basic
 import Mathlib.Tactic.TFAE
 import Mathlib.Topology.ContinuousOn
 
+#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
+
 /-!
 # Inseparable points in a topological space
 

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

@@ -83,7 +83,7 @@ theorem specializes_TFAE (x y : X) :
   tfae_have 2 → 3
   · exact fun h s hso hy => h (hso.mem_nhds hy)
   tfae_have 3 → 4
-  · exact fun h s hsc hx => of_not_not fun hy => h (sᶜ) hsc.isOpen_compl hy hx
+  · exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
   tfae_have 4 → 5
   · exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
   tfae_have 6 ↔ 5

@@ -8,9 +8,9 @@ Authors: Andrew Yang, Yury G. Kudryashov
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathlib.Topology.ContinuousOn
 import Mathlib.Data.Setoid.Basic
-import Mathlib.Data.List.TFAE
+import Mathlib.Tactic.TFAE
+import Mathlib.Topology.ContinuousOn
 
 /-!
 # Inseparable points in a topological space
@@ -70,7 +70,7 @@ infixl:300 " ⤳ " => Specializes
 
 /-- A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
 below. -/
-theorem specializes_TFAE ( x y : X ) :
+theorem specializes_TFAE (x y : X) :
     TFAE [x ⤳ y,
       pure x ≤ 𝓝 y,
       ∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
@@ -78,30 +78,24 @@ theorem specializes_TFAE ( x y : X ) :
       y ∈ closure ({ x } : Set X),
       closure ({ y } : Set X) ⊆ closure { x },
       ClusterPt y (pure x)] := by
-  -- todo: rewrite using `tfae_have` etc
-  apply_rules [tfae_of_cycle, Chain.cons, Chain.nil] <;> dsimp only [ilast']
-  · exact le_trans (pure_le_nhds _)
+  tfae_have 1 → 2
+  · exact (pure_le_nhds _).trans
+  tfae_have 2 → 3
   · exact fun h s hso hy => h (hso.mem_nhds hy)
+  tfae_have 3 → 4
   · exact fun h s hsc hx => of_not_not fun hy => h (sᶜ) hsc.isOpen_compl hy hx
-  · exact fun h => h _ isClosed_closure (subset_closure rfl)
-  · exact fun h => closure_minimal (singleton_subset_iff.2 h) isClosed_closure
-  · rw [← principal_singleton, ← mem_closure_iff_clusterPt]
-    exact fun h => h (subset_closure rfl)
-  · refine fun h => (nhds_basis_opens _).ge_iff.2 fun U ⟨hyU, hUo⟩ => ?_
-    rw [← Ultrafilter.coe_pure, Ultrafilter.clusterPt_iff] at h
-    exact hUo.mem_nhds (h <| hUo.mem_nhds hyU)
-  -- tfae_have 1 → 2; exact pure_le_nhds _ . trans
-  -- tfae_have 2 → 3; exact fun h s hso hy => h hso . mem_nhds hy
-  -- tfae_have 3 → 4; exact fun h s hsc hx => of_not_not fun hy => h s ᶜ hsc . is_open_compl hy hx
-  -- tfae_have 4 → 5; exact fun h => h _ isClosed_closure subset_closure <| mem_singleton _
-  -- tfae_have 6 ↔ 5; exact is_closed_closure.closure_subset_iff.trans singleton_subset_iff
-  -- tfae_have 5 ↔ 7; rw [ mem_closure_iff_clusterPt, principal_singleton ]
-  -- tfae_have 5 → 1
-  -- · refine' fun h => nhds_basis_opens _ . ge_iff . 2 _
-  --   rintro s ⟨ hy , ho ⟩
-  --   rcases mem_closure_iff . 1 h s ho hy with ⟨ z , hxs , rfl : z = x ⟩
-  --   exact ho.mem_nhds hxs
-  -- tfae_finish
+  tfae_have 4 → 5
+  · exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
+  tfae_have 6 ↔ 5
+  · exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
+  tfae_have 5 ↔ 7
+  · rw [mem_closure_iff_clusterPt, principal_singleton]
+  tfae_have 5 → 1
+  · refine' fun h => (nhds_basis_opens _).ge_iff.2 _
+    rintro s ⟨hy, ho⟩
+    rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
+    exact ho.mem_nhds hxs
+  tfae_finish
 #align specializes_tfae specializes_TFAE
 
 theorem specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y :=

• Run a non-interactive version of fix-comments.py on all files.
• Go through the diff and manually add/discard/edit chunks.

@@ -60,7 +60,7 @@ hold:
 * for any open set `s` we have `y ∈ s → x ∈ s`;
 * `y` is a cluster point of the filter `pure x = 𝓟 {x}`.
 
-This relation defines a `preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
+This relation defines a `Preorder` on `X`. If `X` is a T₀ space, then this preorder is a partial
 order. If `X` is a T₁ space, then this partial order is trivial : `x ⤳ y ↔ x = y`. -/
 def Specializes (x y : X) : Prop := 𝓝 x ≤ 𝓝 y
 #align specializes Specializes

Co-authored-by: Johan Commelin <johan@commelin.net>

@@ -156,6 +156,10 @@ theorem specializes_iff_closure_subset : x ⤳ y ↔ closure ({y} : Set X) ⊆ c
 alias specializes_iff_closure_subset ↔ Specializes.closure_subset _
 #align specializes.closure_subset Specializes.closure_subset
 
+-- porting note: new lemma
+theorem specializes_iff_clusterPt : x ⤳ y ↔ ClusterPt y (pure x) :=
+  (specializes_TFAE x y).out 0 6
+
 theorem Filter.HasBasis.specializes_iff {ι} {p : ι → Prop} {s : ι → Set X}
     (h : (𝓝 y).HasBasis p s) : x ⤳ y ↔ ∀ i, p i → x ∈ s i :=
   specializes_iff_pure.trans h.ge_iff