topology.uniform_space.separation ⟷ Mathlib.Topology.UniformSpace.Separation

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

@@ -76,7 +76,7 @@ open scoped Classical Topology uniformity Filter
 
 noncomputable section
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:339:40: warning: unsupported option eqn_compiler.zeta -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:340:40: warning: unsupported option eqn_compiler.zeta -/
 set_option eqn_compiler.zeta true
 
 universe u v w

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

@@ -98,18 +98,19 @@ instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
 #align uniform_space.to_regular_space UniformSpace.to_regularSpace
 -/
 
-#print separationRel /-
+/- warning: separation_rel clashes with inseparable -> Inseparable
+Case conversion may be inaccurate. Consider using '#align separation_rel Inseparableₓ'. -/
+#print Inseparable /-
 /-- The separation relation is the intersection of all entourages.
   Two points which are related by the separation relation are "indistinguishable"
   according to the uniform structure. -/
-protected def separationRel (α : Type u) [u : UniformSpace α] :=
+protected def Inseparable (α : Type u) [u : UniformSpace α] :=
   ⋂₀ (𝓤 α).sets
-#align separation_rel separationRel
+#align separation_rel Inseparable
 -/
 
-scoped[uniformity] notation "𝓢" => separationRel
+scoped[uniformity] notation "𝓢" => Inseparable
 
-#print separated_equiv /-
 theorem separated_equiv : Equivalence fun x y => (x, y) ∈ 𝓢 α :=
   ⟨fun x => fun s => refl_mem_uniformity, fun x y => fun h (s : Set (α × α)) hs =>
     have : preimage Prod.swap s ∈ 𝓤 α := symm_le_uniformity hs
@@ -118,69 +119,70 @@ theorem separated_equiv : Equivalence fun x y => (x, y) ∈ 𝓢 α :=
     let ⟨t, ht, (h_ts : compRel t t ⊆ s)⟩ := comp_mem_uniformity_sets hs
     h_ts <| show (x, z) ∈ compRel t t from ⟨y, hxy t ht, hyz t ht⟩⟩
 #align separated_equiv separated_equiv
--/
 
-#print Filter.HasBasis.mem_separationRel /-
-theorem Filter.HasBasis.mem_separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
+#print Filter.HasBasis.inseparable_iff_uniformity /-
+theorem Filter.HasBasis.inseparable_iff_uniformity {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : (𝓤 α).HasBasis p s) {a : α × α} : a ∈ 𝓢 α ↔ ∀ i, p i → a ∈ s i :=
   h.forall_mem_mem
-#align filter.has_basis.mem_separation_rel Filter.HasBasis.mem_separationRel
+#align filter.has_basis.mem_separation_rel Filter.HasBasis.inseparable_iff_uniformity
 -/
 
-#print separationRel_iff_specializes /-
-theorem separationRel_iff_specializes {a b : α} : (a, b) ∈ 𝓢 α ↔ a ⤳ b := by
-  simp only [(𝓤 α).basis_sets.mem_separationRel, id, mem_set_of_eq,
+/- warning: separation_rel_iff_specializes clashes with specializes_iff_inseparable -> specializes_iff_inseparable
+Case conversion may be inaccurate. Consider using '#align separation_rel_iff_specializes specializes_iff_inseparableₓ'. -/
+#print specializes_iff_inseparable /-
+theorem specializes_iff_inseparable {a b : α} : (a, b) ∈ 𝓢 α ↔ a ⤳ b := by
+  simp only [(𝓤 α).basis_sets.inseparable_iff_uniformity, id, mem_set_of_eq,
     (nhds_basis_uniformity (𝓤 α).basis_sets).specializes_iff]
-#align separation_rel_iff_specializes separationRel_iff_specializes
+#align separation_rel_iff_specializes specializes_iff_inseparable
 -/
 
-#print separationRel_iff_inseparable /-
-theorem separationRel_iff_inseparable {a b : α} : (a, b) ∈ 𝓢 α ↔ Inseparable a b :=
-  separationRel_iff_specializes.trans specializes_iff_inseparable
-#align separation_rel_iff_inseparable separationRel_iff_inseparable
--/
+theorem inseparable_iff_inseparable {a b : α} : (a, b) ∈ 𝓢 α ↔ Inseparable a b :=
+  specializes_iff_inseparable.trans specializes_iff_inseparable
+#align separation_rel_iff_inseparable inseparable_iff_inseparable
 
-#print SeparatedSpace /-
+/- warning: separated_space clashes with t0_space -> T0Space
+Case conversion may be inaccurate. Consider using '#align separated_space T0Spaceₓ'. -/
+#print T0Space /-
 /-- A uniform space is separated if its separation relation is trivial (each point
 is related only to itself). -/
-class SeparatedSpace (α : Type u) [UniformSpace α] : Prop where
+class T0Space (α : Type u) [UniformSpace α] : Prop where
   out : 𝓢 α = idRel
-#align separated_space SeparatedSpace
+#align separated_space T0Space
 -/
 
-#print separatedSpace_iff /-
-theorem separatedSpace_iff {α : Type u} [UniformSpace α] : SeparatedSpace α ↔ 𝓢 α = idRel :=
+#print t0Space_iff_ker_uniformity /-
+theorem t0Space_iff_ker_uniformity {α : Type u} [UniformSpace α] : T0Space α ↔ 𝓢 α = idRel :=
   ⟨fun h => h.1, fun h => ⟨h⟩⟩
-#align separated_space_iff separatedSpace_iff
+#align separated_space_iff t0Space_iff_ker_uniformity
 -/
 
-#print separated_def /-
-theorem separated_def {α : Type u} [UniformSpace α] :
-    SeparatedSpace α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by
-  simp [separatedSpace_iff, idRel_subset.2 separated_equiv.1, subset.antisymm_iff] <;>
-    simp [subset_def, separationRel]
-#align separated_def separated_def
+#print t0Space_iff_uniformity /-
+theorem t0Space_iff_uniformity {α : Type u} [UniformSpace α] :
+    T0Space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by
+  simp [t0Space_iff_ker_uniformity, idRel_subset.2 separated_equiv.1, subset.antisymm_iff] <;>
+    simp [subset_def, Inseparable]
+#align separated_def t0Space_iff_uniformity
 -/
 
-#print separated_def' /-
-theorem separated_def' {α : Type u} [UniformSpace α] :
-    SeparatedSpace α ↔ ∀ x y, x ≠ y → ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
-  separated_def.trans <|
+#print t0Space_iff_uniformity' /-
+theorem t0Space_iff_uniformity' {α : Type u} [UniformSpace α] :
+    T0Space α ↔ ∀ x y, x ≠ y → ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
+  t0Space_iff_uniformity.trans <|
     forall₂_congr fun x y => by rw [← not_imp_not] <;> simp [Classical.not_forall]
-#align separated_def' separated_def'
+#align separated_def' t0Space_iff_uniformity'
 -/
 
 #print eq_of_uniformity /-
-theorem eq_of_uniformity {α : Type _} [UniformSpace α] [SeparatedSpace α] {x y : α}
+theorem eq_of_uniformity {α : Type _} [UniformSpace α] [T0Space α] {x y : α}
     (h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y :=
-  separated_def.mp ‹SeparatedSpace α› x y fun _ => h
+  t0Space_iff_uniformity.mp ‹T0Space α› x y fun _ => h
 #align eq_of_uniformity eq_of_uniformity
 -/
 
 #print eq_of_uniformity_basis /-
-theorem eq_of_uniformity_basis {α : Type _} [UniformSpace α] [SeparatedSpace α] {ι : Type _}
-    {p : ι → Prop} {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α}
-    (h : ∀ {i}, p i → (x, y) ∈ s i) : x = y :=
+theorem eq_of_uniformity_basis {α : Type _} [UniformSpace α] [T0Space α] {ι : Type _} {p : ι → Prop}
+    {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α} (h : ∀ {i}, p i → (x, y) ∈ s i) :
+    x = y :=
   eq_of_uniformity fun V V_in =>
     let ⟨i, hi, H⟩ := hs.mem_iff.mp V_in
     H (h hi)
@@ -188,102 +190,109 @@ theorem eq_of_uniformity_basis {α : Type _} [UniformSpace α] [SeparatedSpace 
 -/
 
 #print eq_of_forall_symmetric /-
-theorem eq_of_forall_symmetric {α : Type _} [UniformSpace α] [SeparatedSpace α] {x y : α}
+theorem eq_of_forall_symmetric {α : Type _} [UniformSpace α] [T0Space α] {x y : α}
     (h : ∀ {V}, V ∈ 𝓤 α → SymmetricRel V → (x, y) ∈ V) : x = y :=
   eq_of_uniformity_basis hasBasis_symmetric (by simpa [and_imp] using fun _ => h)
 #align eq_of_forall_symmetric eq_of_forall_symmetric
 -/
 
 #print eq_of_clusterPt_uniformity /-
-theorem eq_of_clusterPt_uniformity [SeparatedSpace α] {x y : α} (h : ClusterPt (x, y) (𝓤 α)) :
-    x = y :=
+theorem eq_of_clusterPt_uniformity [T0Space α] {x y : α} (h : ClusterPt (x, y) (𝓤 α)) : x = y :=
   eq_of_uniformity_basis uniformity_hasBasis_closed fun V ⟨hV, hVc⟩ =>
     isClosed_iff_clusterPt.1 hVc _ <| h.mono <| le_principal_iff.2 hV
 #align eq_of_cluster_pt_uniformity eq_of_clusterPt_uniformity
 -/
 
-#print idRel_sub_separationRel /-
-theorem idRel_sub_separationRel (α : Type _) [UniformSpace α] : idRel ⊆ 𝓢 α :=
+/- warning: id_rel_sub_separation_relation clashes with inseparable.rfl -> Inseparable.rfl
+Case conversion may be inaccurate. Consider using '#align id_rel_sub_separation_relation Inseparable.rflₓ'. -/
+#print Inseparable.rfl /-
+theorem Inseparable.rfl (α : Type _) [UniformSpace α] : idRel ⊆ 𝓢 α :=
   by
-  unfold separationRel
+  unfold Inseparable
   rw [idRel_subset]
   intro x
   suffices ∀ t ∈ 𝓤 α, (x, x) ∈ t by simpa only [refl_mem_uniformity]
   exact fun t => refl_mem_uniformity
-#align id_rel_sub_separation_relation idRel_sub_separationRel
+#align id_rel_sub_separation_relation Inseparable.rfl
 -/
 
-#print separationRel_comap /-
-theorem separationRel_comap {f : α → β}
+/- warning: separation_rel_comap clashes with inducing.inseparable_iff -> Inducing.inseparable_iff
+Case conversion may be inaccurate. Consider using '#align separation_rel_comap Inducing.inseparable_iffₓ'. -/
+#print Inducing.inseparable_iff /-
+theorem Inducing.inseparable_iff {f : α → β}
     (h : ‹UniformSpace α› = UniformSpace.comap f ‹UniformSpace β›) : 𝓢 α = Prod.map f f ⁻¹' 𝓢 β :=
   by
   subst h
-  dsimp [separationRel]
+  dsimp [Inseparable]
   simp_rw [uniformity_comap, (Filter.comap_hasBasis (Prod.map f f) (𝓤 β)).ker, ← preimage_Inter,
     sInter_eq_bInter]
   rfl
-#align separation_rel_comap separationRel_comap
+#align separation_rel_comap Inducing.inseparable_iff
 -/
 
-#print Filter.HasBasis.separationRel /-
-protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
-    (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (hi : p i), s i := by unfold separationRel;
+/- warning: filter.has_basis.separation_rel clashes with filter.has_basis.sInter_sets -> Filter.HasBasis.ker
+Case conversion may be inaccurate. Consider using '#align filter.has_basis.separation_rel Filter.HasBasis.kerₓ'. -/
+#print Filter.HasBasis.ker /-
+protected theorem Filter.HasBasis.ker {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
+    (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (hi : p i), s i := by unfold Inseparable;
   rw [h.sInter_sets]
-#align filter.has_basis.separation_rel Filter.HasBasis.separationRel
+#align filter.has_basis.separation_rel Filter.HasBasis.ker
 -/
 
-#print separationRel_eq_inter_closure /-
-theorem separationRel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) := by
+theorem inseparable_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) := by
   simp [uniformity_has_basis_closure.separation_rel]
-#align separation_rel_eq_inter_closure separationRel_eq_inter_closure
--/
+#align separation_rel_eq_inter_closure inseparable_eq_inter_closure
 
-#print isClosed_separationRel /-
-theorem isClosed_separationRel : IsClosed (𝓢 α) :=
+/- warning: is_closed_separation_rel clashes with is_closed_set_of_inseparable -> isClosed_setOf_inseparable
+Case conversion may be inaccurate. Consider using '#align is_closed_separation_rel isClosed_setOf_inseparableₓ'. -/
+#print isClosed_setOf_inseparable /-
+theorem isClosed_setOf_inseparable : IsClosed (𝓢 α) :=
   by
-  rw [separationRel_eq_inter_closure]
+  rw [inseparable_eq_inter_closure]
   apply isClosed_sInter
   rintro _ ⟨t, t_in, rfl⟩
   exact isClosed_closure
-#align is_closed_separation_rel isClosed_separationRel
+#align is_closed_separation_rel isClosed_setOf_inseparable
 -/
 
-#print separated_iff_t2 /-
-theorem separated_iff_t2 : SeparatedSpace α ↔ T2Space α := by
+#print R1Space.t2Space_iff_t0Space /-
+theorem R1Space.t2Space_iff_t0Space : T0Space α ↔ T2Space α := by
   classical
   constructor <;> intro h
   · rw [t2_iff_isClosed_diagonal, ← show 𝓢 α = diagonal α from h.1]
-    exact isClosed_separationRel
-  · rw [separated_def']
+    exact isClosed_setOf_inseparable
+  · rw [t0Space_iff_uniformity']
     intro x y hxy
     rcases t2_separation hxy with ⟨u, v, uo, vo, hx, hy, h⟩
     rcases isOpen_iff_ball_subset.1 uo x hx with ⟨r, hrU, hr⟩
     exact ⟨r, hrU, fun H => h.le_bot ⟨hr H, hy⟩⟩
-#align separated_iff_t2 separated_iff_t2
+#align separated_iff_t2 R1Space.t2Space_iff_t0Space
 -/
 
-#print separated_t3 /-
+#print RegularSpace.t3Space_iff_t0Space /-
 -- see Note [lower instance priority]
-instance (priority := 100) separated_t3 [SeparatedSpace α] : T3Space α :=
+instance (priority := 100) RegularSpace.t3Space_iff_t0Space [T0Space α] : T3Space α :=
   haveI := separated_iff_t2.mp ‹_›
   ⟨⟩
-#align separated_t3 separated_t3
+#align separated_t3 RegularSpace.t3Space_iff_t0Space
 -/
 
-#print Subtype.separatedSpace /-
-instance Subtype.separatedSpace [SeparatedSpace α] (s : Set α) : SeparatedSpace s :=
-  separated_iff_t2.mpr Subtype.t2Space
-#align subtype.separated_space Subtype.separatedSpace
+/- warning: subtype.separated_space clashes with subtype.t0_space -> Subtype.t0Space
+Case conversion may be inaccurate. Consider using '#align subtype.separated_space Subtype.t0Spaceₓ'. -/
+#print Subtype.t0Space /-
+instance Subtype.t0Space [T0Space α] (s : Set α) : T0Space s :=
+  R1Space.t2Space_iff_t0Space.mpr Subtype.t2Space
+#align subtype.separated_space Subtype.t0Space
 -/
 
 #print isClosed_of_spaced_out /-
-theorem isClosed_of_spaced_out [SeparatedSpace α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α) {s : Set α}
+theorem isClosed_of_spaced_out [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α) {s : Set α}
     (hs : s.Pairwise fun x y => (x, y) ∉ V₀) : IsClosed s :=
   by
   rcases comp_symm_mem_uniformity_sets V₀_in with ⟨V₁, V₁_in, V₁_symm, h_comp⟩
   apply isClosed_of_closure_subset
   intro x hx
-  rw [mem_closure_iff_ball] at hx 
+  rw [mem_closure_iff_ball] at hx
   rcases hx V₁_in with ⟨y, hy, hy'⟩
   suffices x = y by rwa [this]
   apply eq_of_forall_symmetric
@@ -297,7 +306,7 @@ theorem isClosed_of_spaced_out [SeparatedSpace α] {V₀ : Set (α × α)} (V₀
 -/
 
 #print isClosed_range_of_spaced_out /-
-theorem isClosed_range_of_spaced_out {ι} [SeparatedSpace α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
+theorem isClosed_range_of_spaced_out {ι} [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
     {f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (range f) :=
   isClosed_of_spaced_out V₀_in <| by rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h; exact hf (ne_of_apply_ne f h)
 #align is_closed_range_of_spaced_out isClosed_range_of_spaced_out
@@ -310,18 +319,19 @@ theorem isClosed_range_of_spaced_out {ι} [SeparatedSpace α] {V₀ : Set (α ×
 
 namespace UniformSpace
 
-#print UniformSpace.separationSetoid /-
+/- warning: uniform_space.separation_setoid clashes with inseparable_setoid -> inseparableSetoid
+Case conversion may be inaccurate. Consider using '#align uniform_space.separation_setoid inseparableSetoidₓ'. -/
+#print inseparableSetoid /-
 /-- The separation relation of a uniform space seen as a setoid. -/
-def separationSetoid (α : Type u) [UniformSpace α] : Setoid α :=
+def inseparableSetoid (α : Type u) [UniformSpace α] : Setoid α :=
   ⟨fun x y => (x, y) ∈ 𝓢 α, separated_equiv⟩
-#align uniform_space.separation_setoid UniformSpace.separationSetoid
+#align uniform_space.separation_setoid inseparableSetoid
 -/
 
 attribute [local instance] separation_setoid
 
-#print UniformSpace.separationSetoid.uniformSpace /-
-instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
-    UniformSpace (Quotient (separationSetoid α))
+instance inseparableSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
+    UniformSpace (Quotient (inseparableSetoid α))
     where
   toTopologicalSpace := u.toTopologicalSpace.coinduced fun x => ⟦x⟧
   uniformity := map (fun p : α × α => (⟦p.1⟧, ⟦p.2⟧)) u.uniformity
@@ -339,8 +349,8 @@ instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
               compRel s (compRel s s)) :=
         (lift'_mono' fun s hs ⟨a, b⟩ ⟨c, ⟨⟨a₁, a₂⟩, ha, a_eq⟩, ⟨⟨b₁, b₂⟩, hb, b_eq⟩⟩ =>
           by
-          simp at a_eq 
-          simp at b_eq 
+          simp at a_eq
+          simp at b_eq
           have h : ⟦a₂⟧ = ⟦b₁⟧ := by rw [a_eq.right, b_eq.left]
           have h : (a₂, b₁) ∈ 𝓢 α := Quotient.exact h
           simp [Function.comp, Set.image, compRel, and_comm, and_left_comm, and_assoc]
@@ -367,52 +377,50 @@ instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
     simp only [isOpen_coinduced, isOpen_uniformity, uniformity, Quotient.forall, mem_preimage,
       mem_map, preimage_set_of_eq, Quotient.eq']
     exact ⟨fun h a ha => (this a ha).mp <| h a ha, fun h a ha => (this a ha).mpr <| h a ha⟩
-#align uniform_space.separation_setoid.uniform_space UniformSpace.separationSetoid.uniformSpace
--/
+#align uniform_space.separation_setoid.uniform_space inseparableSetoid.uniformSpace
 
-#print UniformSpace.uniformity_quotient /-
-theorem uniformity_quotient :
-    𝓤 (Quotient (separationSetoid α)) = (𝓤 α).map fun p : α × α => (⟦p.1⟧, ⟦p.2⟧) :=
+#print SeparationQuotient.uniformity_eq /-
+theorem uniformity_eq :
+    𝓤 (Quotient (inseparableSetoid α)) = (𝓤 α).map fun p : α × α => (⟦p.1⟧, ⟦p.2⟧) :=
   rfl
-#align uniform_space.uniformity_quotient UniformSpace.uniformity_quotient
+#align uniform_space.uniformity_quotient SeparationQuotient.uniformity_eq
 -/
 
-#print UniformSpace.uniformContinuous_quotient_mk' /-
-theorem uniformContinuous_quotient_mk' :
-    UniformContinuous (Quotient.mk' : α → Quotient (separationSetoid α)) :=
+#print SeparationQuotient.uniformContinuous_mk /-
+theorem uniformContinuous_mk :
+    UniformContinuous (Quotient.mk' : α → Quotient (inseparableSetoid α)) :=
   le_rfl
-#align uniform_space.uniform_continuous_quotient_mk UniformSpace.uniformContinuous_quotient_mk'
+#align uniform_space.uniform_continuous_quotient_mk SeparationQuotient.uniformContinuous_mk
 -/
 
-#print UniformSpace.uniformContinuous_quotient /-
-theorem uniformContinuous_quotient {f : Quotient (separationSetoid α) → β}
+#print SeparationQuotient.uniformContinuous_dom /-
+theorem uniformContinuous_dom {f : Quotient (inseparableSetoid α) → β}
     (hf : UniformContinuous fun x => f ⟦x⟧) : UniformContinuous f :=
   hf
-#align uniform_space.uniform_continuous_quotient UniformSpace.uniformContinuous_quotient
+#align uniform_space.uniform_continuous_quotient SeparationQuotient.uniformContinuous_dom
 -/
 
-#print UniformSpace.uniformContinuous_quotient_lift /-
-theorem uniformContinuous_quotient_lift {f : α → β} {h : ∀ a b, (a, b) ∈ 𝓢 α → f a = f b}
+#print SeparationQuotient.uniformContinuous_lift /-
+theorem uniformContinuous_lift {f : α → β} {h : ∀ a b, (a, b) ∈ 𝓢 α → f a = f b}
     (hf : UniformContinuous f) : UniformContinuous fun a => Quotient.lift f h a :=
-  uniformContinuous_quotient hf
-#align uniform_space.uniform_continuous_quotient_lift UniformSpace.uniformContinuous_quotient_lift
+  uniformContinuous_dom hf
+#align uniform_space.uniform_continuous_quotient_lift SeparationQuotient.uniformContinuous_lift
 -/
 
-#print UniformSpace.uniformContinuous_quotient_lift₂ /-
-theorem uniformContinuous_quotient_lift₂ {f : α → β → γ}
+#print SeparationQuotient.uniformContinuous_uncurry_lift₂ /-
+theorem uniformContinuous_uncurry_lift₂ {f : α → β → γ}
     {h : ∀ a c b d, (a, b) ∈ 𝓢 α → (c, d) ∈ 𝓢 β → f a c = f b d}
     (hf : UniformContinuous fun p : α × β => f p.1 p.2) :
     UniformContinuous fun p : _ × _ => Quotient.lift₂ f h p.1 p.2 :=
   by
   rw [UniformContinuous, uniformity_prod_eq_prod, uniformity_quotient, uniformity_quotient,
     Filter.prod_map_map_eq, Filter.tendsto_map'_iff, Filter.tendsto_map'_iff]
-  rwa [UniformContinuous, uniformity_prod_eq_prod, Filter.tendsto_map'_iff] at hf 
-#align uniform_space.uniform_continuous_quotient_lift₂ UniformSpace.uniformContinuous_quotient_lift₂
+  rwa [UniformContinuous, uniformity_prod_eq_prod, Filter.tendsto_map'_iff] at hf
+#align uniform_space.uniform_continuous_quotient_lift₂ SeparationQuotient.uniformContinuous_uncurry_lift₂
 -/
 
-#print UniformSpace.comap_quotient_le_uniformity /-
 theorem comap_quotient_le_uniformity :
-    ((𝓤 <| Quotient <| separationSetoid α).comap fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) ≤ 𝓤 α :=
+    ((𝓤 <| Quotient <| inseparableSetoid α).comap fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) ≤ 𝓤 α :=
   fun t' ht' =>
   let ⟨t, ht, tt_t'⟩ := comp_mem_uniformity_sets ht'
   let ⟨s, hs, ss_t⟩ := comp_mem_uniformity_sets ht
@@ -425,138 +433,148 @@ theorem comap_quotient_le_uniformity :
     tt_t'
       ⟨b₁, show ((a₁, a₂).1, b₁) ∈ t from ab₁, ss_t ⟨b₂, show ((b₁, a₂).1, b₂) ∈ s from hb, ba₂⟩⟩⟩
 #align uniform_space.comap_quotient_le_uniformity UniformSpace.comap_quotient_le_uniformity
--/
 
-#print UniformSpace.comap_quotient_eq_uniformity /-
-theorem comap_quotient_eq_uniformity :
-    ((𝓤 <| Quotient <| separationSetoid α).comap fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) = 𝓤 α :=
+#print SeparationQuotient.comap_mk_uniformity /-
+theorem comap_mk_uniformity :
+    ((𝓤 <| Quotient <| inseparableSetoid α).comap fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) = 𝓤 α :=
   le_antisymm comap_quotient_le_uniformity le_comap_map
-#align uniform_space.comap_quotient_eq_uniformity UniformSpace.comap_quotient_eq_uniformity
+#align uniform_space.comap_quotient_eq_uniformity SeparationQuotient.comap_mk_uniformity
 -/
 
-#print UniformSpace.separated_separation /-
-instance separated_separation : SeparatedSpace (Quotient (separationSetoid α)) :=
+#print SeparationQuotient.instT0Space /-
+instance instT0Space : T0Space (Quotient (inseparableSetoid α)) :=
   ⟨Set.ext fun ⟨a, b⟩ =>
       Quotient.induction_on₂ a b fun a b =>
         ⟨fun h =>
           have : a ≈ b := fun s hs =>
             have :
-              s ∈ (𝓤 <| Quotient <| separationSetoid α).comap fun p : α × α => (⟦p.1⟧, ⟦p.2⟧) :=
+              s ∈ (𝓤 <| Quotient <| inseparableSetoid α).comap fun p : α × α => (⟦p.1⟧, ⟦p.2⟧) :=
               comap_quotient_le_uniformity hs
             let ⟨t, ht, hts⟩ := this
             hts (by dsimp [preimage]; exact h t ht)
           show ⟦a⟧ = ⟦b⟧ from Quotient.sound this,
           fun heq : ⟦a⟧ = ⟦b⟧ => fun h hs => HEq ▸ refl_mem_uniformity hs⟩⟩
-#align uniform_space.separated_separation UniformSpace.separated_separation
+#align uniform_space.separated_separation SeparationQuotient.instT0Space
 -/
 
-#print UniformSpace.separated_of_uniformContinuous /-
-theorem separated_of_uniformContinuous {f : α → β} {x y : α} (H : UniformContinuous f) (h : x ≈ y) :
-    f x ≈ f y := fun _ h' => h _ (H h')
-#align uniform_space.separated_of_uniform_continuous UniformSpace.separated_of_uniformContinuous
+/- warning: uniform_space.separated_of_uniform_continuous clashes with inseparable.map -> Inseparable.map
+Case conversion may be inaccurate. Consider using '#align uniform_space.separated_of_uniform_continuous Inseparable.mapₓ'. -/
+#print Inseparable.map /-
+theorem map {f : α → β} {x y : α} (H : UniformContinuous f) (h : x ≈ y) : f x ≈ f y := fun _ h' =>
+  h _ (H h')
+#align uniform_space.separated_of_uniform_continuous Inseparable.map
 -/
 
-#print UniformSpace.eq_of_separated_of_uniformContinuous /-
-theorem eq_of_separated_of_uniformContinuous [SeparatedSpace β] {f : α → β} {x y : α}
+theorem eq_of_separated_of_uniformContinuous [T0Space β] {f : α → β} {x y : α}
     (H : UniformContinuous f) (h : x ≈ y) : f x = f y :=
-  separated_def.1 (by infer_instance) _ _ <| separated_of_uniformContinuous H h
+  t0Space_iff_uniformity.1 (by infer_instance) _ _ <| map H h
 #align uniform_space.eq_of_separated_of_uniform_continuous UniformSpace.eq_of_separated_of_uniformContinuous
--/
 
-#print UniformSpace.SeparationQuotient /-
+/- warning: uniform_space.separation_quotient clashes with separation_quotient -> SeparationQuotient
+Case conversion may be inaccurate. Consider using '#align uniform_space.separation_quotient SeparationQuotientₓ'. -/
+#print SeparationQuotient /-
 /-- The maximal separated quotient of a uniform space `α`. -/
 def SeparationQuotient (α : Type _) [UniformSpace α] :=
-  Quotient (separationSetoid α)
-#align uniform_space.separation_quotient UniformSpace.SeparationQuotient
+  Quotient (inseparableSetoid α)
+#align uniform_space.separation_quotient SeparationQuotient
 -/
 
 namespace SeparationQuotient
 
 instance : UniformSpace (SeparationQuotient α) :=
-  separationSetoid.uniformSpace
+  inseparableSetoid.uniformSpace
 
-instance : SeparatedSpace (SeparationQuotient α) :=
-  UniformSpace.separated_separation
+instance : T0Space (SeparationQuotient α) :=
+  SeparationQuotient.instT0Space
 
 instance [Inhabited α] : Inhabited (SeparationQuotient α) :=
-  Quotient.inhabited (separationSetoid α)
+  Quotient.inhabited (inseparableSetoid α)
 
-#print UniformSpace.SeparationQuotient.mk_eq_mk /-
-theorem mk_eq_mk {x y : α} : (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=
-  Quotient.eq''.trans separationRel_iff_inseparable
-#align uniform_space.separation_quotient.mk_eq_mk UniformSpace.SeparationQuotient.mk_eq_mk
+/- warning: uniform_space.separation_quotient.mk_eq_mk clashes with separation_quotient.mk_eq_mk -> SeparationQuotient.mk_eq_mk
+Case conversion may be inaccurate. Consider using '#align uniform_space.separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mkₓ'. -/
+#print SeparationQuotient.mk_eq_mk /-
+theorem SeparationQuotient.mk_eq_mk {x y : α} :
+    (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=
+  Quotient.eq''.trans inseparable_iff_inseparable
+#align uniform_space.separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
 -/
 
-#print UniformSpace.SeparationQuotient.lift /-
+#print SeparationQuotient.lift' /-
 /-- Factoring functions to a separated space through the separation quotient. -/
-def lift [SeparatedSpace β] (f : α → β) : SeparationQuotient α → β :=
+def SeparationQuotient.lift' [T0Space β] (f : α → β) : SeparationQuotient α → β :=
   if h : UniformContinuous f then Quotient.lift f fun x y => eq_of_separated_of_uniformContinuous h
   else fun x => f (Nonempty.some ⟨x.out⟩)
-#align uniform_space.separation_quotient.lift UniformSpace.SeparationQuotient.lift
+#align uniform_space.separation_quotient.lift SeparationQuotient.lift'
 -/
 
-#print UniformSpace.SeparationQuotient.lift_mk /-
-theorem lift_mk [SeparatedSpace β] {f : α → β} (h : UniformContinuous f) (a : α) :
-    lift f ⟦a⟧ = f a := by rw [lift, dif_pos h] <;> rfl
-#align uniform_space.separation_quotient.lift_mk UniformSpace.SeparationQuotient.lift_mk
+#print SeparationQuotient.lift'_mk /-
+theorem SeparationQuotient.lift'_mk [T0Space β] {f : α → β} (h : UniformContinuous f) (a : α) :
+    SeparationQuotient.lift' f ⟦a⟧ = f a := by rw [lift, dif_pos h] <;> rfl
+#align uniform_space.separation_quotient.lift_mk SeparationQuotient.lift'_mk
 -/
 
-#print UniformSpace.SeparationQuotient.uniformContinuous_lift /-
-theorem uniformContinuous_lift [SeparatedSpace β] (f : α → β) : UniformContinuous (lift f) :=
+#print SeparationQuotient.uniformContinuous_lift' /-
+theorem SeparationQuotient.uniformContinuous_lift' [T0Space β] (f : α → β) :
+    UniformContinuous (SeparationQuotient.lift' f) :=
   by
   by_cases hf : UniformContinuous f
   · rw [lift, dif_pos hf]; exact uniform_continuous_quotient_lift hf
   · rw [lift, dif_neg hf]; exact uniformContinuous_of_const fun a b => rfl
-#align uniform_space.separation_quotient.uniform_continuous_lift UniformSpace.SeparationQuotient.uniformContinuous_lift
+#align uniform_space.separation_quotient.uniform_continuous_lift SeparationQuotient.uniformContinuous_lift'
 -/
 
-#print UniformSpace.SeparationQuotient.map /-
+#print SeparationQuotient.map /-
 /-- The separation quotient functor acting on functions. -/
-def map (f : α → β) : SeparationQuotient α → SeparationQuotient β :=
-  lift (Quotient.mk' ∘ f)
-#align uniform_space.separation_quotient.map UniformSpace.SeparationQuotient.map
+def SeparationQuotient.map (f : α → β) : SeparationQuotient α → SeparationQuotient β :=
+  SeparationQuotient.lift' (Quotient.mk' ∘ f)
+#align uniform_space.separation_quotient.map SeparationQuotient.map
 -/
 
-#print UniformSpace.SeparationQuotient.map_mk /-
-theorem map_mk {f : α → β} (h : UniformContinuous f) (a : α) : map f ⟦a⟧ = ⟦f a⟧ := by
+#print SeparationQuotient.map_mk /-
+theorem SeparationQuotient.map_mk {f : α → β} (h : UniformContinuous f) (a : α) :
+    SeparationQuotient.map f ⟦a⟧ = ⟦f a⟧ := by
   rw [map, lift_mk (uniform_continuous_quotient_mk.comp h)]
-#align uniform_space.separation_quotient.map_mk UniformSpace.SeparationQuotient.map_mk
+#align uniform_space.separation_quotient.map_mk SeparationQuotient.map_mk
 -/
 
-#print UniformSpace.SeparationQuotient.uniformContinuous_map /-
-theorem uniformContinuous_map (f : α → β) : UniformContinuous (map f) :=
-  uniformContinuous_lift (Quotient.mk' ∘ f)
-#align uniform_space.separation_quotient.uniform_continuous_map UniformSpace.SeparationQuotient.uniformContinuous_map
+#print SeparationQuotient.uniformContinuous_map /-
+theorem SeparationQuotient.uniformContinuous_map (f : α → β) :
+    UniformContinuous (SeparationQuotient.map f) :=
+  SeparationQuotient.uniformContinuous_lift' (Quotient.mk' ∘ f)
+#align uniform_space.separation_quotient.uniform_continuous_map SeparationQuotient.uniformContinuous_map
 -/
 
-#print UniformSpace.SeparationQuotient.map_unique /-
-theorem map_unique {f : α → β} (hf : UniformContinuous f)
+#print SeparationQuotient.map_unique /-
+theorem SeparationQuotient.map_unique {f : α → β} (hf : UniformContinuous f)
     {g : SeparationQuotient α → SeparationQuotient β} (comm : Quotient.mk' ∘ f = g ∘ Quotient.mk') :
-    map f = g := by
+    SeparationQuotient.map f = g := by
   ext ⟨a⟩ <;>
     calc
       map f ⟦a⟧ = ⟦f a⟧ := map_mk hf a
       _ = g ⟦a⟧ := congr_fun comm a
-#align uniform_space.separation_quotient.map_unique UniformSpace.SeparationQuotient.map_unique
+#align uniform_space.separation_quotient.map_unique SeparationQuotient.map_unique
 -/
 
-#print UniformSpace.SeparationQuotient.map_id /-
-theorem map_id : map (@id α) = id :=
-  map_unique uniformContinuous_id rfl
-#align uniform_space.separation_quotient.map_id UniformSpace.SeparationQuotient.map_id
+#print SeparationQuotient.map_id /-
+theorem SeparationQuotient.map_id : SeparationQuotient.map (@id α) = id :=
+  SeparationQuotient.map_unique uniformContinuous_id rfl
+#align uniform_space.separation_quotient.map_id SeparationQuotient.map_id
 -/
 
-#print UniformSpace.SeparationQuotient.map_comp /-
-theorem map_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) :
-    map g ∘ map f = map (g ∘ f) :=
-  (map_unique (hg.comp hf) <| by simp only [(· ∘ ·), map_mk, hf, hg]).symm
-#align uniform_space.separation_quotient.map_comp UniformSpace.SeparationQuotient.map_comp
+#print SeparationQuotient.map_comp /-
+theorem SeparationQuotient.map_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f)
+    (hg : UniformContinuous g) :
+    SeparationQuotient.map g ∘ SeparationQuotient.map f = SeparationQuotient.map (g ∘ f) :=
+  (SeparationQuotient.map_unique (hg.comp hf) <| by simp only [(· ∘ ·), map_mk, hf, hg]).symm
+#align uniform_space.separation_quotient.map_comp SeparationQuotient.map_comp
 -/
 
 end SeparationQuotient
 
-#print UniformSpace.separation_prod /-
-theorem separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a₂, b₂) ↔ a₁ ≈ a₂ ∧ b₁ ≈ b₂ :=
+/- warning: uniform_space.separation_prod clashes with inseparable_prod -> inseparable_prod
+Case conversion may be inaccurate. Consider using '#align uniform_space.separation_prod inseparable_prodₓ'. -/
+#print inseparable_prod /-
+theorem inseparable_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a₂, b₂) ↔ a₁ ≈ a₂ ∧ b₁ ≈ b₂ :=
   by
   constructor
   · intro h
@@ -564,22 +582,22 @@ theorem separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a
       ⟨separated_of_uniform_continuous uniformContinuous_fst h,
         separated_of_uniform_continuous uniformContinuous_snd h⟩
   · rintro ⟨eqv_α, eqv_β⟩ r r_in
-    rw [uniformity_prod] at r_in 
+    rw [uniformity_prod] at r_in
     rcases r_in with ⟨t_α, ⟨r_α, r_α_in, h_α⟩, t_β, ⟨r_β, r_β_in, h_β⟩, rfl⟩
     let p_α := fun p : (α × β) × α × β => (p.1.1, p.2.1)
     let p_β := fun p : (α × β) × α × β => (p.1.2, p.2.2)
     have key_α : p_α ((a₁, b₁), (a₂, b₂)) ∈ r_α := by simp [p_α, eqv_α r_α r_α_in]
     have key_β : p_β ((a₁, b₁), (a₂, b₂)) ∈ r_β := by simp [p_β, eqv_β r_β r_β_in]
     exact ⟨h_α key_α, h_β key_β⟩
-#align uniform_space.separation_prod UniformSpace.separation_prod
+#align uniform_space.separation_prod inseparable_prod
 -/
 
-#print UniformSpace.Separated.prod /-
-instance Separated.prod [SeparatedSpace α] [SeparatedSpace β] : SeparatedSpace (α × β) :=
-  separated_def.2 fun x y H =>
+#print Prod.instT0Space /-
+instance Prod.instT0Space [T0Space α] [T0Space β] : T0Space (α × β) :=
+  t0Space_iff_uniformity.2 fun x y H =>
     Prod.ext (eq_of_separated_of_uniformContinuous uniformContinuous_fst H)
       (eq_of_separated_of_uniformContinuous uniformContinuous_snd H)
-#align uniform_space.separated.prod UniformSpace.Separated.prod
+#align uniform_space.separated.prod Prod.instT0Space
 -/
 
 end UniformSpace

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

@@ -92,7 +92,7 @@ variable [UniformSpace α] [UniformSpace β] [UniformSpace γ]
 
 #print UniformSpace.to_regularSpace /-
 instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
-  RegularSpace.ofBasis
+  RegularSpace.of_hasBasis
     (fun a => by rw [nhds_eq_comap_uniformity]; exact uniformity_has_basis_closed.comap _)
     fun a V hV => hV.2.Preimage <| continuous_const.prod_mk continuous_id
 #align uniform_space.to_regular_space UniformSpace.to_regularSpace

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

@@ -249,7 +249,16 @@ theorem isClosed_separationRel : IsClosed (𝓢 α) :=
 -/
 
 #print separated_iff_t2 /-
-theorem separated_iff_t2 : SeparatedSpace α ↔ T2Space α := by classical
+theorem separated_iff_t2 : SeparatedSpace α ↔ T2Space α := by
+  classical
+  constructor <;> intro h
+  · rw [t2_iff_isClosed_diagonal, ← show 𝓢 α = diagonal α from h.1]
+    exact isClosed_separationRel
+  · rw [separated_def']
+    intro x y hxy
+    rcases t2_separation hxy with ⟨u, v, uo, vo, hx, hy, h⟩
+    rcases isOpen_iff_ball_subset.1 uo x hx with ⟨r, hrU, hr⟩
+    exact ⟨r, hrU, fun H => h.le_bot ⟨hr H, hy⟩⟩
 #align separated_iff_t2 separated_iff_t2
 -/
 

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

@@ -249,16 +249,7 @@ theorem isClosed_separationRel : IsClosed (𝓢 α) :=
 -/
 
 #print separated_iff_t2 /-
-theorem separated_iff_t2 : SeparatedSpace α ↔ T2Space α := by
-  classical
-  constructor <;> intro h
-  · rw [t2_iff_isClosed_diagonal, ← show 𝓢 α = diagonal α from h.1]
-    exact isClosed_separationRel
-  · rw [separated_def']
-    intro x y hxy
-    rcases t2_separation hxy with ⟨u, v, uo, vo, hx, hy, h⟩
-    rcases isOpen_iff_ball_subset.1 uo x hx with ⟨r, hrU, hr⟩
-    exact ⟨r, hrU, fun H => h.le_bot ⟨hr H, hy⟩⟩
+theorem separated_iff_t2 : SeparatedSpace α ↔ T2Space α := by classical
 #align separated_iff_t2 separated_iff_t2
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/b1abe23ae96fef89ad30d9f4362c307f72a55010

@@ -165,7 +165,8 @@ theorem separated_def {α : Type u} [UniformSpace α] :
 #print separated_def' /-
 theorem separated_def' {α : Type u} [UniformSpace α] :
     SeparatedSpace α ↔ ∀ x y, x ≠ y → ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
-  separated_def.trans <| forall₂_congr fun x y => by rw [← not_imp_not] <;> simp [not_forall]
+  separated_def.trans <|
+    forall₂_congr fun x y => by rw [← not_imp_not] <;> simp [Classical.not_forall]
 #align separated_def' separated_def'
 -/
 

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

@@ -3,9 +3,9 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Patrick Massot
 -/
-import Mathbin.Tactic.ApplyFun
-import Mathbin.Topology.UniformSpace.Basic
-import Mathbin.Topology.Separation
+import Tactic.ApplyFun
+import Topology.UniformSpace.Basic
+import Topology.Separation
 
 #align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
 
@@ -76,7 +76,7 @@ open scoped Classical Topology uniformity Filter
 
 noncomputable section
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:334:40: warning: unsupported option eqn_compiler.zeta -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:339:40: warning: unsupported option eqn_compiler.zeta -/
 set_option eqn_compiler.zeta true
 
 universe u v w

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

@@ -363,7 +363,7 @@ instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
         have ht' : ∀ {a₁ a₂}, a₁ ≈ a₂ → (a₁, a₂) ∈ t := fun a₁ a₂ h => sInter_subset_of_mem ht h
         u.uniformity.sets_of_superset ht fun ⟨a₁, a₂⟩ h₁ h₂ => hts (ht' <| Setoid.symm h₂) h₁,
         fun h => u.uniformity.sets_of_superset h <| by simp (config := { contextual := true })⟩
-    simp only [isOpen_coinduced, isOpen_uniformity, uniformity, forall_quotient_iff, mem_preimage,
+    simp only [isOpen_coinduced, isOpen_uniformity, uniformity, Quotient.forall, mem_preimage,
       mem_map, preimage_set_of_eq, Quotient.eq']
     exact ⟨fun h a ha => (this a ha).mp <| h a ha, fun h a ha => (this a ha).mpr <| h a ha⟩
 #align uniform_space.separation_setoid.uniform_space UniformSpace.separationSetoid.uniformSpace

mathlib commit https://github.com/leanprover-community/mathlib/commit/001ffdc42920050657fd45bd2b8bfbec8eaaeb29

@@ -218,8 +218,8 @@ theorem separationRel_comap {f : α → β}
   by
   subst h
   dsimp [separationRel]
-  simp_rw [uniformity_comap, (Filter.comap_hasBasis (Prod.map f f) (𝓤 β)).sInter_sets, ←
-    preimage_Inter, sInter_eq_bInter]
+  simp_rw [uniformity_comap, (Filter.comap_hasBasis (Prod.map f f) (𝓤 β)).ker, ← preimage_Inter,
+    sInter_eq_bInter]
   rfl
 #align separation_rel_comap separationRel_comap
 -/

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

@@ -2,16 +2,13 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Patrick Massot
-
-! This file was ported from Lean 3 source module topology.uniform_space.separation
-! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Tactic.ApplyFun
 import Mathbin.Topology.UniformSpace.Basic
 import Mathbin.Topology.Separation
 
+#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
+
 /-!
 # Hausdorff properties of uniform spaces. Separation quotient.
 

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

@@ -110,7 +110,6 @@ protected def separationRel (α : Type u) [u : UniformSpace α] :=
 #align separation_rel separationRel
 -/
 
--- mathport name: separation_rel
 scoped[uniformity] notation "𝓢" => separationRel
 
 #print separated_equiv /-
@@ -124,10 +123,12 @@ theorem separated_equiv : Equivalence fun x y => (x, y) ∈ 𝓢 α :=
 #align separated_equiv separated_equiv
 -/
 
+#print Filter.HasBasis.mem_separationRel /-
 theorem Filter.HasBasis.mem_separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : (𝓤 α).HasBasis p s) {a : α × α} : a ∈ 𝓢 α ↔ ∀ i, p i → a ∈ s i :=
   h.forall_mem_mem
 #align filter.has_basis.mem_separation_rel Filter.HasBasis.mem_separationRel
+-/
 
 #print separationRel_iff_specializes /-
 theorem separationRel_iff_specializes {a b : α} : (a, b) ∈ 𝓢 α ↔ a ⤳ b := by
@@ -164,10 +165,12 @@ theorem separated_def {α : Type u} [UniformSpace α] :
 #align separated_def separated_def
 -/
 
+#print separated_def' /-
 theorem separated_def' {α : Type u} [UniformSpace α] :
     SeparatedSpace α ↔ ∀ x y, x ≠ y → ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
   separated_def.trans <| forall₂_congr fun x y => by rw [← not_imp_not] <;> simp [not_forall]
 #align separated_def' separated_def'
+-/
 
 #print eq_of_uniformity /-
 theorem eq_of_uniformity {α : Type _} [UniformSpace α] [SeparatedSpace α] {x y : α}
@@ -176,6 +179,7 @@ theorem eq_of_uniformity {α : Type _} [UniformSpace α] [SeparatedSpace α] {x
 #align eq_of_uniformity eq_of_uniformity
 -/
 
+#print eq_of_uniformity_basis /-
 theorem eq_of_uniformity_basis {α : Type _} [UniformSpace α] [SeparatedSpace α] {ι : Type _}
     {p : ι → Prop} {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α}
     (h : ∀ {i}, p i → (x, y) ∈ s i) : x = y :=
@@ -183,6 +187,7 @@ theorem eq_of_uniformity_basis {α : Type _} [UniformSpace α] [SeparatedSpace 
     let ⟨i, hi, H⟩ := hs.mem_iff.mp V_in
     H (h hi)
 #align eq_of_uniformity_basis eq_of_uniformity_basis
+-/
 
 #print eq_of_forall_symmetric /-
 theorem eq_of_forall_symmetric {α : Type _} [UniformSpace α] [SeparatedSpace α] {x y : α}
@@ -191,11 +196,13 @@ theorem eq_of_forall_symmetric {α : Type _} [UniformSpace α] [SeparatedSpace 
 #align eq_of_forall_symmetric eq_of_forall_symmetric
 -/
 
+#print eq_of_clusterPt_uniformity /-
 theorem eq_of_clusterPt_uniformity [SeparatedSpace α] {x y : α} (h : ClusterPt (x, y) (𝓤 α)) :
     x = y :=
   eq_of_uniformity_basis uniformity_hasBasis_closed fun V ⟨hV, hVc⟩ =>
     isClosed_iff_clusterPt.1 hVc _ <| h.mono <| le_principal_iff.2 hV
 #align eq_of_cluster_pt_uniformity eq_of_clusterPt_uniformity
+-/
 
 #print idRel_sub_separationRel /-
 theorem idRel_sub_separationRel (α : Type _) [UniformSpace α] : idRel ⊆ 𝓢 α :=
@@ -220,15 +227,20 @@ theorem separationRel_comap {f : α → β}
 #align separation_rel_comap separationRel_comap
 -/
 
+#print Filter.HasBasis.separationRel /-
 protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (hi : p i), s i := by unfold separationRel;
   rw [h.sInter_sets]
 #align filter.has_basis.separation_rel Filter.HasBasis.separationRel
+-/
 
+#print separationRel_eq_inter_closure /-
 theorem separationRel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) := by
   simp [uniformity_has_basis_closure.separation_rel]
 #align separation_rel_eq_inter_closure separationRel_eq_inter_closure
+-/
 
+#print isClosed_separationRel /-
 theorem isClosed_separationRel : IsClosed (𝓢 α) :=
   by
   rw [separationRel_eq_inter_closure]
@@ -236,6 +248,7 @@ theorem isClosed_separationRel : IsClosed (𝓢 α) :=
   rintro _ ⟨t, t_in, rfl⟩
   exact isClosed_closure
 #align is_closed_separation_rel isClosed_separationRel
+-/
 
 #print separated_iff_t2 /-
 theorem separated_iff_t2 : SeparatedSpace α ↔ T2Space α := by
@@ -285,10 +298,12 @@ theorem isClosed_of_spaced_out [SeparatedSpace α] {V₀ : Set (α × α)} (V₀
 #align is_closed_of_spaced_out isClosed_of_spaced_out
 -/
 
+#print isClosed_range_of_spaced_out /-
 theorem isClosed_range_of_spaced_out {ι} [SeparatedSpace α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
     {f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (range f) :=
   isClosed_of_spaced_out V₀_in <| by rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h; exact hf (ne_of_apply_ne f h)
 #align is_closed_range_of_spaced_out isClosed_range_of_spaced_out
+-/
 
 /-!
 ### Separation quotient
@@ -385,6 +400,7 @@ theorem uniformContinuous_quotient_lift {f : α → β} {h : ∀ a b, (a, b) ∈
 #align uniform_space.uniform_continuous_quotient_lift UniformSpace.uniformContinuous_quotient_lift
 -/
 
+#print UniformSpace.uniformContinuous_quotient_lift₂ /-
 theorem uniformContinuous_quotient_lift₂ {f : α → β → γ}
     {h : ∀ a c b d, (a, b) ∈ 𝓢 α → (c, d) ∈ 𝓢 β → f a c = f b d}
     (hf : UniformContinuous fun p : α × β => f p.1 p.2) :
@@ -394,7 +410,9 @@ theorem uniformContinuous_quotient_lift₂ {f : α → β → γ}
     Filter.prod_map_map_eq, Filter.tendsto_map'_iff, Filter.tendsto_map'_iff]
   rwa [UniformContinuous, uniformity_prod_eq_prod, Filter.tendsto_map'_iff] at hf 
 #align uniform_space.uniform_continuous_quotient_lift₂ UniformSpace.uniformContinuous_quotient_lift₂
+-/
 
+#print UniformSpace.comap_quotient_le_uniformity /-
 theorem comap_quotient_le_uniformity :
     ((𝓤 <| Quotient <| separationSetoid α).comap fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) ≤ 𝓤 α :=
   fun t' ht' =>
@@ -409,6 +427,7 @@ theorem comap_quotient_le_uniformity :
     tt_t'
       ⟨b₁, show ((a₁, a₂).1, b₁) ∈ t from ab₁, ss_t ⟨b₂, show ((b₁, a₂).1, b₂) ∈ s from hb, ba₂⟩⟩⟩
 #align uniform_space.comap_quotient_le_uniformity UniformSpace.comap_quotient_le_uniformity
+-/
 
 #print UniformSpace.comap_quotient_eq_uniformity /-
 theorem comap_quotient_eq_uniformity :
@@ -538,6 +557,7 @@ theorem map_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg
 
 end SeparationQuotient
 
+#print UniformSpace.separation_prod /-
 theorem separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a₂, b₂) ↔ a₁ ≈ a₂ ∧ b₁ ≈ b₂ :=
   by
   constructor
@@ -554,12 +574,15 @@ theorem separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a
     have key_β : p_β ((a₁, b₁), (a₂, b₂)) ∈ r_β := by simp [p_β, eqv_β r_β r_β_in]
     exact ⟨h_α key_α, h_β key_β⟩
 #align uniform_space.separation_prod UniformSpace.separation_prod
+-/
 
+#print UniformSpace.Separated.prod /-
 instance Separated.prod [SeparatedSpace α] [SeparatedSpace β] : SeparatedSpace (α × β) :=
   separated_def.2 fun x y H =>
     Prod.ext (eq_of_separated_of_uniformContinuous uniformContinuous_fst H)
       (eq_of_separated_of_uniformContinuous uniformContinuous_snd H)
 #align uniform_space.separated.prod UniformSpace.Separated.prod
+-/
 
 end UniformSpace
 

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

@@ -337,7 +337,6 @@ instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
             (u.uniformity.lift' fun s : Set (α × α) => compRel s (compRel s s)) :=
         by rw [map_lift'_eq] <;> exact monotone_id.comp_rel (monotone_id.comp_rel monotone_id)
       _ ≤ map (fun p : α × α => (⟦p.1⟧, ⟦p.2⟧)) u.uniformity := map_mono comp_le_uniformity3
-      
   isOpen_uniformity s :=
     by
     have :
@@ -521,7 +520,6 @@ theorem map_unique {f : α → β} (hf : UniformContinuous f)
     calc
       map f ⟦a⟧ = ⟦f a⟧ := map_mk hf a
       _ = g ⟦a⟧ := congr_fun comm a
-      
 #align uniform_space.separation_quotient.map_unique UniformSpace.SeparationQuotient.map_unique
 -/
 

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

@@ -240,14 +240,14 @@ theorem isClosed_separationRel : IsClosed (𝓢 α) :=
 #print separated_iff_t2 /-
 theorem separated_iff_t2 : SeparatedSpace α ↔ T2Space α := by
   classical
-    constructor <;> intro h
-    · rw [t2_iff_isClosed_diagonal, ← show 𝓢 α = diagonal α from h.1]
-      exact isClosed_separationRel
-    · rw [separated_def']
-      intro x y hxy
-      rcases t2_separation hxy with ⟨u, v, uo, vo, hx, hy, h⟩
-      rcases isOpen_iff_ball_subset.1 uo x hx with ⟨r, hrU, hr⟩
-      exact ⟨r, hrU, fun H => h.le_bot ⟨hr H, hy⟩⟩
+  constructor <;> intro h
+  · rw [t2_iff_isClosed_diagonal, ← show 𝓢 α = diagonal α from h.1]
+    exact isClosed_separationRel
+  · rw [separated_def']
+    intro x y hxy
+    rcases t2_separation hxy with ⟨u, v, uo, vo, hx, hy, h⟩
+    rcases isOpen_iff_ball_subset.1 uo x hx with ⟨r, hrU, hr⟩
+    exact ⟨r, hrU, fun H => h.le_bot ⟨hr H, hy⟩⟩
 #align separated_iff_t2 separated_iff_t2
 -/
 
@@ -343,7 +343,7 @@ instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
     have :
       ∀ a,
         ⟦a⟧ ∈ s →
-          ({ p : α × α | p.1 = a → ⟦p.2⟧ ∈ s } ∈ 𝓤 α ↔ { p : α × α | p.1 ≈ a → ⟦p.2⟧ ∈ s } ∈ 𝓤 α) :=
+          ({p : α × α | p.1 = a → ⟦p.2⟧ ∈ s} ∈ 𝓤 α ↔ {p : α × α | p.1 ≈ a → ⟦p.2⟧ ∈ s} ∈ 𝓤 α) :=
       fun a ha =>
       ⟨fun h =>
         let ⟨t, ht, hts⟩ := comp_mem_uniformity_sets h

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

@@ -272,7 +272,7 @@ theorem isClosed_of_spaced_out [SeparatedSpace α] {V₀ : Set (α × α)} (V₀
   rcases comp_symm_mem_uniformity_sets V₀_in with ⟨V₁, V₁_in, V₁_symm, h_comp⟩
   apply isClosed_of_closure_subset
   intro x hx
-  rw [mem_closure_iff_ball] at hx
+  rw [mem_closure_iff_ball] at hx 
   rcases hx V₁_in with ⟨y, hy, hy'⟩
   suffices x = y by rwa [this]
   apply eq_of_forall_symmetric
@@ -326,8 +326,8 @@ instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
               compRel s (compRel s s)) :=
         (lift'_mono' fun s hs ⟨a, b⟩ ⟨c, ⟨⟨a₁, a₂⟩, ha, a_eq⟩, ⟨⟨b₁, b₂⟩, hb, b_eq⟩⟩ =>
           by
-          simp at a_eq
-          simp at b_eq
+          simp at a_eq 
+          simp at b_eq 
           have h : ⟦a₂⟧ = ⟦b₁⟧ := by rw [a_eq.right, b_eq.left]
           have h : (a₂, b₁) ∈ 𝓢 α := Quotient.exact h
           simp [Function.comp, Set.image, compRel, and_comm, and_left_comm, and_assoc]
@@ -393,7 +393,7 @@ theorem uniformContinuous_quotient_lift₂ {f : α → β → γ}
   by
   rw [UniformContinuous, uniformity_prod_eq_prod, uniformity_quotient, uniformity_quotient,
     Filter.prod_map_map_eq, Filter.tendsto_map'_iff, Filter.tendsto_map'_iff]
-  rwa [UniformContinuous, uniformity_prod_eq_prod, Filter.tendsto_map'_iff] at hf
+  rwa [UniformContinuous, uniformity_prod_eq_prod, Filter.tendsto_map'_iff] at hf 
 #align uniform_space.uniform_continuous_quotient_lift₂ UniformSpace.uniformContinuous_quotient_lift₂
 
 theorem comap_quotient_le_uniformity :
@@ -548,7 +548,7 @@ theorem separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a
       ⟨separated_of_uniform_continuous uniformContinuous_fst h,
         separated_of_uniform_continuous uniformContinuous_snd h⟩
   · rintro ⟨eqv_α, eqv_β⟩ r r_in
-    rw [uniformity_prod] at r_in
+    rw [uniformity_prod] at r_in 
     rcases r_in with ⟨t_α, ⟨r_α, r_α_in, h_α⟩, t_β, ⟨r_β, r_β_in, h_β⟩, rfl⟩
     let p_α := fun p : (α × β) × α × β => (p.1.1, p.2.1)
     let p_β := fun p : (α × β) × α × β => (p.1.2, p.2.2)

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

@@ -75,7 +75,7 @@ uniformly continuous).
 
 open Filter TopologicalSpace Set Classical Function UniformSpace
 
-open Classical Topology uniformity Filter
+open scoped Classical Topology uniformity Filter
 
 noncomputable section
 

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

@@ -124,12 +124,6 @@ theorem separated_equiv : Equivalence fun x y => (x, y) ∈ 𝓢 α :=
 #align separated_equiv separated_equiv
 -/
 
-/- warning: filter.has_basis.mem_separation_rel -> Filter.HasBasis.mem_separationRel is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α] {ι : Sort.{u2}} {p : ι -> Prop} {s : ι -> (Set.{u1} (Prod.{u1, u1} α α))}, (Filter.HasBasis.{u1, u2} (Prod.{u1, u1} α α) ι (uniformity.{u1} α _inst_1) p s) -> (forall {a : Prod.{u1, u1} α α}, Iff (Membership.Mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.hasMem.{u1} (Prod.{u1, u1} α α)) a (separationRel.{u1} α _inst_1)) (forall (i : ι), (p i) -> (Membership.Mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.hasMem.{u1} (Prod.{u1, u1} α α)) a (s i))))
-but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : UniformSpace.{u2} α] {ι : Sort.{u1}} {p : ι -> Prop} {s : ι -> (Set.{u2} (Prod.{u2, u2} α α))}, (Filter.HasBasis.{u2, u1} (Prod.{u2, u2} α α) ι (uniformity.{u2} α _inst_1) p s) -> (forall {a : Prod.{u2, u2} α α}, Iff (Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) a (separationRel.{u2} α _inst_1)) (forall (i : ι), (p i) -> (Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) a (s i))))
-Case conversion may be inaccurate. Consider using '#align filter.has_basis.mem_separation_rel Filter.HasBasis.mem_separationRelₓ'. -/
 theorem Filter.HasBasis.mem_separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : (𝓤 α).HasBasis p s) {a : α × α} : a ∈ 𝓢 α ↔ ∀ i, p i → a ∈ s i :=
   h.forall_mem_mem
@@ -170,12 +164,6 @@ theorem separated_def {α : Type u} [UniformSpace α] :
 #align separated_def separated_def
 -/
 
-/- warning: separated_def' -> separated_def' is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_4 : UniformSpace.{u1} α], Iff (SeparatedSpace.{u1} α _inst_4) (forall (x : α) (y : α), (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} (Prod.{u1, u1} α α)) (fun (r : Set.{u1} (Prod.{u1, u1} α α)) => Exists.{0} (Membership.Mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.hasMem.{u1} (Prod.{u1, u1} α α)) r (uniformity.{u1} α _inst_4)) (fun (H : Membership.Mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.hasMem.{u1} (Prod.{u1, u1} α α)) r (uniformity.{u1} α _inst_4)) => Not (Membership.Mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.hasMem.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α x y) r)))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_4 : UniformSpace.{u1} α], Iff (SeparatedSpace.{u1} α _inst_4) (forall (x : α) (y : α), (Ne.{succ u1} α x y) -> (Exists.{succ u1} (Set.{u1} (Prod.{u1, u1} α α)) (fun (r : Set.{u1} (Prod.{u1, u1} α α)) => And (Membership.mem.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Filter.{u1} (Prod.{u1, u1} α α)) (instMembershipSetFilter.{u1} (Prod.{u1, u1} α α)) r (uniformity.{u1} α _inst_4)) (Not (Membership.mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.instMembershipSet.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α x y) r)))))
-Case conversion may be inaccurate. Consider using '#align separated_def' separated_def'ₓ'. -/
 theorem separated_def' {α : Type u} [UniformSpace α] :
     SeparatedSpace α ↔ ∀ x y, x ≠ y → ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
   separated_def.trans <| forall₂_congr fun x y => by rw [← not_imp_not] <;> simp [not_forall]
@@ -188,12 +176,6 @@ theorem eq_of_uniformity {α : Type _} [UniformSpace α] [SeparatedSpace α] {x
 #align eq_of_uniformity eq_of_uniformity
 -/
 
-/- warning: eq_of_uniformity_basis -> eq_of_uniformity_basis is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_4 : UniformSpace.{u1} α] [_inst_5 : SeparatedSpace.{u1} α _inst_4] {ι : Type.{u2}} {p : ι -> Prop} {s : ι -> (Set.{u1} (Prod.{u1, u1} α α))}, (Filter.HasBasis.{u1, succ u2} (Prod.{u1, u1} α α) ι (uniformity.{u1} α _inst_4) p s) -> (forall {x : α} {y : α}, (forall {i : ι}, (p i) -> (Membership.Mem.{u1, u1} (Prod.{u1, u1} α α) (Set.{u1} (Prod.{u1, u1} α α)) (Set.hasMem.{u1} (Prod.{u1, u1} α α)) (Prod.mk.{u1, u1} α α x y) (s i))) -> (Eq.{succ u1} α x y))
-but is expected to have type
-  forall {α : Type.{u2}} [_inst_4 : UniformSpace.{u2} α] [_inst_5 : SeparatedSpace.{u2} α _inst_4] {ι : Type.{u1}} {p : ι -> Prop} {s : ι -> (Set.{u2} (Prod.{u2, u2} α α))}, (Filter.HasBasis.{u2, succ u1} (Prod.{u2, u2} α α) ι (uniformity.{u2} α _inst_4) p s) -> (forall {x : α} {y : α}, (forall {i : ι}, (p i) -> (Membership.mem.{u2, u2} (Prod.{u2, u2} α α) (Set.{u2} (Prod.{u2, u2} α α)) (Set.instMembershipSet.{u2} (Prod.{u2, u2} α α)) (Prod.mk.{u2, u2} α α x y) (s i))) -> (Eq.{succ u2} α x y))
-Case conversion may be inaccurate. Consider using '#align eq_of_uniformity_basis eq_of_uniformity_basisₓ'. -/
 theorem eq_of_uniformity_basis {α : Type _} [UniformSpace α] [SeparatedSpace α] {ι : Type _}
     {p : ι → Prop} {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α}
     (h : ∀ {i}, p i → (x, y) ∈ s i) : x = y :=
@@ -209,12 +191,6 @@ theorem eq_of_forall_symmetric {α : Type _} [UniformSpace α] [SeparatedSpace 
 #align eq_of_forall_symmetric eq_of_forall_symmetric
 -/
 
-/- warning: eq_of_cluster_pt_uniformity -> eq_of_clusterPt_uniformity is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α] [_inst_4 : SeparatedSpace.{u1} α _inst_1] {x : α} {y : α}, (ClusterPt.{u1} (Prod.{u1, u1} α α) (Prod.topologicalSpace.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α _inst_1) (UniformSpace.toTopologicalSpace.{u1} α _inst_1)) (Prod.mk.{u1, u1} α α x y) (uniformity.{u1} α _inst_1)) -> (Eq.{succ u1} α x y)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α] [_inst_4 : SeparatedSpace.{u1} α _inst_1] {x : α} {y : α}, (ClusterPt.{u1} (Prod.{u1, u1} α α) (instTopologicalSpaceProd.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α _inst_1) (UniformSpace.toTopologicalSpace.{u1} α _inst_1)) (Prod.mk.{u1, u1} α α x y) (uniformity.{u1} α _inst_1)) -> (Eq.{succ u1} α x y)
-Case conversion may be inaccurate. Consider using '#align eq_of_cluster_pt_uniformity eq_of_clusterPt_uniformityₓ'. -/
 theorem eq_of_clusterPt_uniformity [SeparatedSpace α] {x y : α} (h : ClusterPt (x, y) (𝓤 α)) :
     x = y :=
   eq_of_uniformity_basis uniformity_hasBasis_closed fun V ⟨hV, hVc⟩ =>
@@ -244,33 +220,15 @@ theorem separationRel_comap {f : α → β}
 #align separation_rel_comap separationRel_comap
 -/
 
-/- warning: filter.has_basis.separation_rel -> Filter.HasBasis.separationRel is a dubious translation:
-lean 3 declaration is
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 protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (hi : p i), s i := by unfold separationRel;
   rw [h.sInter_sets]
 #align filter.has_basis.separation_rel Filter.HasBasis.separationRel
 
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 theorem separationRel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) := by
   simp [uniformity_has_basis_closure.separation_rel]
 #align separation_rel_eq_inter_closure separationRel_eq_inter_closure
 
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 theorem isClosed_separationRel : IsClosed (𝓢 α) :=
   by
   rw [separationRel_eq_inter_closure]
@@ -327,12 +285,6 @@ theorem isClosed_of_spaced_out [SeparatedSpace α] {V₀ : Set (α × α)} (V₀
 #align is_closed_of_spaced_out isClosed_of_spaced_out
 -/
 
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 theorem isClosed_range_of_spaced_out {ι} [SeparatedSpace α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
     {f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (range f) :=
   isClosed_of_spaced_out V₀_in <| by rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h; exact hf (ne_of_apply_ne f h)
@@ -434,12 +386,6 @@ theorem uniformContinuous_quotient_lift {f : α → β} {h : ∀ a b, (a, b) ∈
 #align uniform_space.uniform_continuous_quotient_lift UniformSpace.uniformContinuous_quotient_lift
 -/
 
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-Case conversion may be inaccurate. Consider using '#align uniform_space.uniform_continuous_quotient_lift₂ UniformSpace.uniformContinuous_quotient_lift₂ₓ'. -/
 theorem uniformContinuous_quotient_lift₂ {f : α → β → γ}
     {h : ∀ a c b d, (a, b) ∈ 𝓢 α → (c, d) ∈ 𝓢 β → f a c = f b d}
     (hf : UniformContinuous fun p : α × β => f p.1 p.2) :
@@ -450,12 +396,6 @@ theorem uniformContinuous_quotient_lift₂ {f : α → β → γ}
   rwa [UniformContinuous, uniformity_prod_eq_prod, Filter.tendsto_map'_iff] at hf
 #align uniform_space.uniform_continuous_quotient_lift₂ UniformSpace.uniformContinuous_quotient_lift₂
 
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 theorem comap_quotient_le_uniformity :
     ((𝓤 <| Quotient <| separationSetoid α).comap fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) ≤ 𝓤 α :=
   fun t' ht' =>
@@ -600,12 +540,6 @@ theorem map_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg
 
 end SeparationQuotient
 
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 theorem separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a₂, b₂) ↔ a₁ ≈ a₂ ∧ b₁ ≈ b₂ :=
   by
   constructor
@@ -623,12 +557,6 @@ theorem separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a
     exact ⟨h_α key_α, h_β key_β⟩
 #align uniform_space.separation_prod UniformSpace.separation_prod
 
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 instance Separated.prod [SeparatedSpace α] [SeparatedSpace β] : SeparatedSpace (α × β) :=
   separated_def.2 fun x y H =>
     Prod.ext (eq_of_separated_of_uniformContinuous uniformContinuous_fst H)

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

@@ -96,9 +96,7 @@ variable [UniformSpace α] [UniformSpace β] [UniformSpace γ]
 #print UniformSpace.to_regularSpace /-
 instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
   RegularSpace.ofBasis
-    (fun a => by
-      rw [nhds_eq_comap_uniformity]
-      exact uniformity_has_basis_closed.comap _)
+    (fun a => by rw [nhds_eq_comap_uniformity]; exact uniformity_has_basis_closed.comap _)
     fun a V hV => hV.2.Preimage <| continuous_const.prod_mk continuous_id
 #align uniform_space.to_regular_space UniformSpace.to_regularSpace
 -/
@@ -253,9 +251,7 @@ but is expected to have type
   forall {α : Type.{u2}} [_inst_1 : UniformSpace.{u2} α] {ι : Sort.{u1}} {p : ι -> Prop} {s : ι -> (Set.{u2} (Prod.{u2, u2} α α))}, (Filter.HasBasis.{u2, u1} (Prod.{u2, u2} α α) ι (uniformity.{u2} α _inst_1) p s) -> (Eq.{succ u2} (Set.{u2} (Prod.{u2, u2} α α)) (separationRel.{u2} α _inst_1) (Set.iInter.{u2, u1} (Prod.{u2, u2} α α) ι (fun (i : ι) => Set.iInter.{u2, 0} (Prod.{u2, u2} α α) (p i) (fun (hi : p i) => s i))))
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.separation_rel Filter.HasBasis.separationRelₓ'. -/
 protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
-    (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (hi : p i), s i :=
-  by
-  unfold separationRel
+    (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (hi : p i), s i := by unfold separationRel;
   rw [h.sInter_sets]
 #align filter.has_basis.separation_rel Filter.HasBasis.separationRel
 
@@ -339,9 +335,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align is_closed_range_of_spaced_out isClosed_range_of_spaced_outₓ'. -/
 theorem isClosed_range_of_spaced_out {ι} [SeparatedSpace α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
     {f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (range f) :=
-  isClosed_of_spaced_out V₀_in <| by
-    rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h
-    exact hf (ne_of_apply_ne f h)
+  isClosed_of_spaced_out V₀_in <| by rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h; exact hf (ne_of_apply_ne f h)
 #align is_closed_range_of_spaced_out isClosed_range_of_spaced_out
 
 /-!
@@ -555,10 +549,8 @@ theorem lift_mk [SeparatedSpace β] {f : α → β} (h : UniformContinuous f) (a
 theorem uniformContinuous_lift [SeparatedSpace β] (f : α → β) : UniformContinuous (lift f) :=
   by
   by_cases hf : UniformContinuous f
-  · rw [lift, dif_pos hf]
-    exact uniform_continuous_quotient_lift hf
-  · rw [lift, dif_neg hf]
-    exact uniformContinuous_of_const fun a b => rfl
+  · rw [lift, dif_pos hf]; exact uniform_continuous_quotient_lift hf
+  · rw [lift, dif_neg hf]; exact uniformContinuous_of_const fun a b => rfl
 #align uniform_space.separation_quotient.uniform_continuous_lift UniformSpace.SeparationQuotient.uniformContinuous_lift
 -/
 

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

@@ -458,7 +458,7 @@ theorem uniformContinuous_quotient_lift₂ {f : α → β → γ}
 
 /- warning: uniform_space.comap_quotient_le_uniformity -> UniformSpace.comap_quotient_le_uniformity is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α], LE.le.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Preorder.toLE.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.partialOrder.{u1} (Prod.{u1, u1} α α)))) (Filter.comap.{u1, u1} (Prod.{u1, u1} α α) (Prod.{u1, u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1))) (fun (p : Prod.{u1, u1} α α) => Prod.mk.{u1, u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.mk'.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p)) (Quotient.mk'.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1) (Prod.snd.{u1, u1} α α p))) (uniformity.{u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (UniformSpace.separationSetoid.uniformSpace.{u1} α _inst_1))) (uniformity.{u1} α _inst_1)
+  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α], LE.le.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Preorder.toHasLe.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.partialOrder.{u1} (Prod.{u1, u1} α α)))) (Filter.comap.{u1, u1} (Prod.{u1, u1} α α) (Prod.{u1, u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1))) (fun (p : Prod.{u1, u1} α α) => Prod.mk.{u1, u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.mk'.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p)) (Quotient.mk'.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1) (Prod.snd.{u1, u1} α α p))) (uniformity.{u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (UniformSpace.separationSetoid.uniformSpace.{u1} α _inst_1))) (uniformity.{u1} α _inst_1)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α], LE.le.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Preorder.toLE.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Prod.{u1, u1} α α)) (Filter.instPartialOrderFilter.{u1} (Prod.{u1, u1} α α)))) (Filter.comap.{u1, u1} (Prod.{u1, u1} α α) (Prod.{u1, u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1))) (fun (p : Prod.{u1, u1} α α) => Prod.mk.{u1, u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (Quotient.mk.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1) (Prod.fst.{u1, u1} α α p)) (Quotient.mk.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1) (Prod.snd.{u1, u1} α α p))) (uniformity.{u1} (Quotient.{succ u1} α (UniformSpace.separationSetoid.{u1} α _inst_1)) (UniformSpace.separationSetoid.uniformSpace.{u1} α _inst_1))) (uniformity.{u1} α _inst_1)
 Case conversion may be inaccurate. Consider using '#align uniform_space.comap_quotient_le_uniformity UniformSpace.comap_quotient_le_uniformityₓ'. -/

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

@@ -240,7 +240,7 @@ theorem separationRel_comap {f : α → β}
   by
   subst h
   dsimp [separationRel]
-  simp_rw [uniformity_comap, (Filter.comap_hasBasis (Prod.map f f) (𝓤 β)).interₛ_sets, ←
+  simp_rw [uniformity_comap, (Filter.comap_hasBasis (Prod.map f f) (𝓤 β)).sInter_sets, ←
     preimage_Inter, sInter_eq_bInter]
   rfl
 #align separation_rel_comap separationRel_comap
@@ -248,9 +248,9 @@ theorem separationRel_comap {f : α → β}
 
 /- warning: filter.has_basis.separation_rel -> Filter.HasBasis.separationRel is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α] {ι : Sort.{u2}} {p : ι -> Prop} {s : ι -> (Set.{u1} (Prod.{u1, u1} α α))}, (Filter.HasBasis.{u1, u2} (Prod.{u1, u1} α α) ι (uniformity.{u1} α _inst_1) p s) -> (Eq.{succ u1} (Set.{u1} (Prod.{u1, u1} α α)) (separationRel.{u1} α _inst_1) (Set.interᵢ.{u1, u2} (Prod.{u1, u1} α α) ι (fun (i : ι) => Set.interᵢ.{u1, 0} (Prod.{u1, u1} α α) (p i) (fun (hi : p i) => s i))))
+  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α] {ι : Sort.{u2}} {p : ι -> Prop} {s : ι -> (Set.{u1} (Prod.{u1, u1} α α))}, (Filter.HasBasis.{u1, u2} (Prod.{u1, u1} α α) ι (uniformity.{u1} α _inst_1) p s) -> (Eq.{succ u1} (Set.{u1} (Prod.{u1, u1} α α)) (separationRel.{u1} α _inst_1) (Set.iInter.{u1, u2} (Prod.{u1, u1} α α) ι (fun (i : ι) => Set.iInter.{u1, 0} (Prod.{u1, u1} α α) (p i) (fun (hi : p i) => s i))))
 but is expected to have type
-  forall {α : Type.{u2}} [_inst_1 : UniformSpace.{u2} α] {ι : Sort.{u1}} {p : ι -> Prop} {s : ι -> (Set.{u2} (Prod.{u2, u2} α α))}, (Filter.HasBasis.{u2, u1} (Prod.{u2, u2} α α) ι (uniformity.{u2} α _inst_1) p s) -> (Eq.{succ u2} (Set.{u2} (Prod.{u2, u2} α α)) (separationRel.{u2} α _inst_1) (Set.interᵢ.{u2, u1} (Prod.{u2, u2} α α) ι (fun (i : ι) => Set.interᵢ.{u2, 0} (Prod.{u2, u2} α α) (p i) (fun (hi : p i) => s i))))
+  forall {α : Type.{u2}} [_inst_1 : UniformSpace.{u2} α] {ι : Sort.{u1}} {p : ι -> Prop} {s : ι -> (Set.{u2} (Prod.{u2, u2} α α))}, (Filter.HasBasis.{u2, u1} (Prod.{u2, u2} α α) ι (uniformity.{u2} α _inst_1) p s) -> (Eq.{succ u2} (Set.{u2} (Prod.{u2, u2} α α)) (separationRel.{u2} α _inst_1) (Set.iInter.{u2, u1} (Prod.{u2, u2} α α) ι (fun (i : ι) => Set.iInter.{u2, 0} (Prod.{u2, u2} α α) (p i) (fun (hi : p i) => s i))))
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.separation_rel Filter.HasBasis.separationRelₓ'. -/
 protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (hi : p i), s i :=
@@ -261,9 +261,9 @@ protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop}
 
 /- warning: separation_rel_eq_inter_closure -> separationRel_eq_inter_closure is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α], Eq.{succ u1} (Set.{u1} (Prod.{u1, u1} α α)) (separationRel.{u1} α _inst_1) (Set.interₛ.{u1} (Prod.{u1, u1} α α) (Set.image.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Set.{u1} (Prod.{u1, u1} α α)) (closure.{u1} (Prod.{u1, u1} α α) (Prod.topologicalSpace.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α _inst_1) (UniformSpace.toTopologicalSpace.{u1} α _inst_1))) (Filter.sets.{u1} (Prod.{u1, u1} α α) (uniformity.{u1} α _inst_1))))
+  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α], Eq.{succ u1} (Set.{u1} (Prod.{u1, u1} α α)) (separationRel.{u1} α _inst_1) (Set.sInter.{u1} (Prod.{u1, u1} α α) (Set.image.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Set.{u1} (Prod.{u1, u1} α α)) (closure.{u1} (Prod.{u1, u1} α α) (Prod.topologicalSpace.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α _inst_1) (UniformSpace.toTopologicalSpace.{u1} α _inst_1))) (Filter.sets.{u1} (Prod.{u1, u1} α α) (uniformity.{u1} α _inst_1))))
 but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α], Eq.{succ u1} (Set.{u1} (Prod.{u1, u1} α α)) (separationRel.{u1} α _inst_1) (Set.interₛ.{u1} (Prod.{u1, u1} α α) (Set.image.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Set.{u1} (Prod.{u1, u1} α α)) (closure.{u1} (Prod.{u1, u1} α α) (instTopologicalSpaceProd.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α _inst_1) (UniformSpace.toTopologicalSpace.{u1} α _inst_1))) (Filter.sets.{u1} (Prod.{u1, u1} α α) (uniformity.{u1} α _inst_1))))
+  forall {α : Type.{u1}} [_inst_1 : UniformSpace.{u1} α], Eq.{succ u1} (Set.{u1} (Prod.{u1, u1} α α)) (separationRel.{u1} α _inst_1) (Set.sInter.{u1} (Prod.{u1, u1} α α) (Set.image.{u1, u1} (Set.{u1} (Prod.{u1, u1} α α)) (Set.{u1} (Prod.{u1, u1} α α)) (closure.{u1} (Prod.{u1, u1} α α) (instTopologicalSpaceProd.{u1, u1} α α (UniformSpace.toTopologicalSpace.{u1} α _inst_1) (UniformSpace.toTopologicalSpace.{u1} α _inst_1))) (Filter.sets.{u1} (Prod.{u1, u1} α α) (uniformity.{u1} α _inst_1))))
 Case conversion may be inaccurate. Consider using '#align separation_rel_eq_inter_closure separationRel_eq_inter_closureₓ'. -/
 theorem separationRel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) := by
   simp [uniformity_has_basis_closure.separation_rel]
@@ -278,7 +278,7 @@ Case conversion may be inaccurate. Consider using '#align is_closed_separation_r
 theorem isClosed_separationRel : IsClosed (𝓢 α) :=
   by
   rw [separationRel_eq_inter_closure]
-  apply isClosed_interₛ
+  apply isClosed_sInter
   rintro _ ⟨t, t_in, rfl⟩
   exact isClosed_closure
 #align is_closed_separation_rel isClosed_separationRel
@@ -403,7 +403,7 @@ instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
         let ⟨t, ht, hts⟩ := comp_mem_uniformity_sets h
         have hts : ∀ {a₁ a₂}, (a, a₁) ∈ t → (a₁, a₂) ∈ t → ⟦a₂⟧ ∈ s := fun a₁ a₂ ha₁ ha₂ =>
           @hts (a, a₂) ⟨a₁, ha₁, ha₂⟩ rfl
-        have ht' : ∀ {a₁ a₂}, a₁ ≈ a₂ → (a₁, a₂) ∈ t := fun a₁ a₂ h => interₛ_subset_of_mem ht h
+        have ht' : ∀ {a₁ a₂}, a₁ ≈ a₂ → (a₁, a₂) ∈ t := fun a₁ a₂ h => sInter_subset_of_mem ht h
         u.uniformity.sets_of_superset ht fun ⟨a₁, a₂⟩ h₁ h₂ => hts (ht' <| Setoid.symm h₂) h₁,
         fun h => u.uniformity.sets_of_superset h <| by simp (config := { contextual := true })⟩
     simp only [isOpen_coinduced, isOpen_uniformity, uniformity, forall_quotient_iff, mem_preimage,

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

@@ -378,14 +378,14 @@ instance separationSetoid.uniformSpace {α : Type u} [u : UniformSpace α] :
           u.uniformity.lift'
             ((image fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) ∘ fun s : Set (α × α) =>
               compRel s (compRel s s)) :=
-        lift'_mono' fun s hs ⟨a, b⟩ ⟨c, ⟨⟨a₁, a₂⟩, ha, a_eq⟩, ⟨⟨b₁, b₂⟩, hb, b_eq⟩⟩ =>
+        (lift'_mono' fun s hs ⟨a, b⟩ ⟨c, ⟨⟨a₁, a₂⟩, ha, a_eq⟩, ⟨⟨b₁, b₂⟩, hb, b_eq⟩⟩ =>
           by
           simp at a_eq
           simp at b_eq
           have h : ⟦a₂⟧ = ⟦b₁⟧ := by rw [a_eq.right, b_eq.left]
           have h : (a₂, b₁) ∈ 𝓢 α := Quotient.exact h
           simp [Function.comp, Set.image, compRel, and_comm, and_left_comm, and_assoc]
-          exact ⟨a₁, a_eq.left, b₂, b_eq.right, a₂, ha, b₁, h s hs, hb⟩
+          exact ⟨a₁, a_eq.left, b₂, b_eq.right, a₂, ha, b₁, h s hs, hb⟩)
       _ =
           map (fun p : α × α => (⟦p.1⟧, ⟦p.2⟧))
             (u.uniformity.lift' fun s : Set (α × α) => compRel s (compRel s s)) :=

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

@@ -531,9 +531,11 @@ instance : SeparatedSpace (SeparationQuotient α) :=
 instance [Inhabited α] : Inhabited (SeparationQuotient α) :=
   Quotient.inhabited (separationSetoid α)
 
-theorem mk'_eq_mk' {x y : α} : (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=
+#print UniformSpace.SeparationQuotient.mk_eq_mk /-
+theorem mk_eq_mk {x y : α} : (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=
   Quotient.eq''.trans separationRel_iff_inseparable
-#align uniform_space.separation_quotient.mk_eq_mk UniformSpace.SeparationQuotient.mk'_eq_mk'
+#align uniform_space.separation_quotient.mk_eq_mk UniformSpace.SeparationQuotient.mk_eq_mk
+-/
 
 #print UniformSpace.SeparationQuotient.lift /-
 /-- Factoring functions to a separated space through the separation quotient. -/

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.uniform_space.separation
-! leanprover-community/mathlib commit 4c19a16e4b705bf135cf9a80ac18fcc99c438514
+! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -137,6 +137,19 @@ theorem Filter.HasBasis.mem_separationRel {ι : Sort _} {p : ι → Prop} {s : 
   h.forall_mem_mem
 #align filter.has_basis.mem_separation_rel Filter.HasBasis.mem_separationRel
 
+#print separationRel_iff_specializes /-
+theorem separationRel_iff_specializes {a b : α} : (a, b) ∈ 𝓢 α ↔ a ⤳ b := by
+  simp only [(𝓤 α).basis_sets.mem_separationRel, id, mem_set_of_eq,
+    (nhds_basis_uniformity (𝓤 α).basis_sets).specializes_iff]
+#align separation_rel_iff_specializes separationRel_iff_specializes
+-/
+
+#print separationRel_iff_inseparable /-
+theorem separationRel_iff_inseparable {a b : α} : (a, b) ∈ 𝓢 α ↔ Inseparable a b :=
+  separationRel_iff_specializes.trans specializes_iff_inseparable
+#align separation_rel_iff_inseparable separationRel_iff_inseparable
+-/
+
 #print SeparatedSpace /-
 /-- A uniform space is separated if its separation relation is trivial (each point
 is related only to itself). -/
@@ -518,6 +531,10 @@ instance : SeparatedSpace (SeparationQuotient α) :=
 instance [Inhabited α] : Inhabited (SeparationQuotient α) :=
   Quotient.inhabited (separationSetoid α)
 
+theorem mk'_eq_mk' {x y : α} : (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=
+  Quotient.eq''.trans separationRel_iff_inseparable
+#align uniform_space.separation_quotient.mk_eq_mk UniformSpace.SeparationQuotient.mk'_eq_mk'
+
 #print UniformSpace.SeparationQuotient.lift /-
 /-- Factoring functions to a separated space through the separation quotient. -/
 def lift [SeparatedSpace β] (f : α → β) : SeparationQuotient α → β :=

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

@@ -250,7 +250,7 @@ instance instUniformSpace : UniformSpace (SeparationQuotient α) where
     exact @hUt (x, z) ⟨y', this.mem_open (UniformSpace.isOpen_ball _ hUo) hxyU, hyzU⟩
   nhds_eq_comap_uniformity := surjective_mk.forall.2 fun x ↦ comap_injective surjective_mk <| by
     conv_lhs => rw [comap_mk_nhds_mk, nhds_eq_comap_uniformity, ← comap_map_mk_uniformity]
-    simp only [Filter.comap_comap]; rfl
+    simp only [Filter.comap_comap, Function.comp, Prod_map]
 
 theorem uniformity_eq : 𝓤 (SeparationQuotient α) = (𝓤 α).map (Prod.map mk mk) := rfl
 #align uniform_space.uniformity_quotient SeparationQuotient.uniformity_eq

• replace isOpen_uniformity with nhds_eq_comap_uniformity as I suggested in #2028
• don't extend UniformSpace.Core so that we can drop refl, as it follows from nhds_eq_comap_uniformity;
• drop UniformSpace.mk' - can't be a match_pattern anymore;
• deprecate UniformSpace.ofNhdsEqComap.

@@ -238,21 +238,19 @@ theorem comap_map_mk_uniformity : comap (Prod.map mk mk) (map (Prod.map mk mk) (
   simp only [Prod.map, Prod.ext_iff, mk_eq_mk] at hxy
   exact ((hxy.1.prod hxy.2).mem_open_iff hU.2).1 hyU
 
-instance instUniformSpace : UniformSpace (SeparationQuotient α) :=
-  .ofNhdsEqComap
-    { uniformity := map (Prod.map mk mk) (𝓤 α)
-      refl := le_trans (by simpa using surjective_mk) (Filter.map_mono refl_le_uniformity)
-      symm := tendsto_map' <| tendsto_map.comp tendsto_swap_uniformity
-      comp := fun t ht ↦ by
-        rcases comp_open_symm_mem_uniformity_sets ht with ⟨U, hU, hUo, -, hUt⟩
-        refine mem_of_superset (mem_lift' <| image_mem_map hU) ?_
-        simp only [subset_def, Prod.forall, mem_compRel, mem_image, Prod.ext_iff]
-        rintro _ _ ⟨_, ⟨⟨x, y⟩, hxyU, rfl, rfl⟩, ⟨⟨y', z⟩, hyzU, hy, rfl⟩⟩
-        have : y' ⤳ y := (mk_eq_mk.1 hy).specializes
-        exact @hUt (x, z) ⟨y', this.mem_open (UniformSpace.isOpen_ball _ hUo) hxyU, hyzU⟩ }
-    inferInstance <| surjective_mk.forall.2 fun x ↦ comap_injective surjective_mk <| by
-      conv_lhs => rw [comap_mk_nhds_mk, nhds_eq_comap_uniformity, ← comap_map_mk_uniformity]
-      simp only [Filter.comap_comap]; rfl
+instance instUniformSpace : UniformSpace (SeparationQuotient α) where
+  uniformity := map (Prod.map mk mk) (𝓤 α)
+  symm := tendsto_map' <| tendsto_map.comp tendsto_swap_uniformity
+  comp := fun t ht ↦ by
+    rcases comp_open_symm_mem_uniformity_sets ht with ⟨U, hU, hUo, -, hUt⟩
+    refine mem_of_superset (mem_lift' <| image_mem_map hU) ?_
+    simp only [subset_def, Prod.forall, mem_compRel, mem_image, Prod.ext_iff]
+    rintro _ _ ⟨_, ⟨⟨x, y⟩, hxyU, rfl, rfl⟩, ⟨⟨y', z⟩, hyzU, hy, rfl⟩⟩
+    have : y' ⤳ y := (mk_eq_mk.1 hy).specializes
+    exact @hUt (x, z) ⟨y', this.mem_open (UniformSpace.isOpen_ball _ hUo) hxyU, hyzU⟩
+  nhds_eq_comap_uniformity := surjective_mk.forall.2 fun x ↦ comap_injective surjective_mk <| by
+    conv_lhs => rw [comap_mk_nhds_mk, nhds_eq_comap_uniformity, ← comap_map_mk_uniformity]
+    simp only [Filter.comap_comap]; rfl
 
 theorem uniformity_eq : 𝓤 (SeparationQuotient α) = (𝓤 α).map (Prod.map mk mk) := rfl
 #align uniform_space.uniformity_quotient SeparationQuotient.uniformity_eq

We had duplicated API between topological spaces and uniform spaces. In this PR I mostly deduplicate it with some exceptions:

• SeparationQuotient.lift' and SeparationQuotient.map are leftovers from the old version that are designed to work with uniform spaces;
• probably, some theorems/instances still assume UniformSpace when TopologicalSpace would work.

Outside of UniformSpace/Separation, I mostly changed SeparatedSpace to T0Space and separationRel to Inseparable. I also rewrote a few proofs that were broken by the API change.

Fixes #2031

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johannes Hölzl, Patrick Massot
+Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
 -/
 import Mathlib.Tactic.ApplyFun
 import Mathlib.Topology.UniformSpace.Basic
@@ -12,58 +12,89 @@ import Mathlib.Topology.Separation
 /-!
 # Hausdorff properties of uniform spaces. Separation quotient.
 
-This file studies uniform spaces whose underlying topological spaces are separated
-(also known as Hausdorff or T₂).
-This turns out to be equivalent to asking that the intersection of all entourages
-is the diagonal only. This condition actually implies the stronger separation property
-that the space is T₃, hence those conditions are equivalent for topologies coming from
-a uniform structure.
+Two points of a topological space are called `Inseparable`,
+if their neighborhoods filter are equal.
+Equivalently, `Inseparable x y` means that any open set that contains `x` must contain `y`
+and vice versa.
+
+In a uniform space, points `x` and `y` are inseparable
+if and only if `(x, y)` belongs to all entourages,
+see `inseparable_iff_ker_uniformity`.
+
+A uniform space is a regular topological space,
+hence separation axioms `T0Space`, `T1Space`, `T2Space`, and `T3Space`
+are equivalent for uniform spaces,
+and Lean typeclass search can automatically convert from one assumption to another.
+We say that a uniform space is *separated*, if it satisfies these axioms.
+If you need an `Iff` statement (e.g., to rewrite),
+then see `R1Space.t0Space_iff_t2Space` and `RegularSpace.t0Space_iff_t3Space`.
+
+In this file we prove several facts
+that relate `Inseparable` and `Specializes` to the uniformity filter.
+Most of them are simple corollaries of `Filter.HasBasis.inseparable_iff_uniformity`
+for different filter bases of `𝓤 α`.
+
+Then we study the Kolmogorov quotient `SeparationQuotient X` of a uniform space.
+For a general topological space,
+this quotient is defined as the quotient by `Inseparable` equivalence relation.
+It is the maximal T₀ quotient of a topological space.
+
+In case of a uniform space, we equip this quotient with a `UniformSpace` structure
+that agrees with the quotient topology.
+We also prove that the quotient map induces uniformity on the original space.
+
+Finally, we turn `SeparationQuotient` into a functor
+(not in terms of `CategoryTheory.Functor` to avoid extra imports)
+by defining `SeparationQuotient.lift'` and `SeparationQuotient.map` operations.
 
-More generally, the intersection `𝓢 X` of all entourages of `X`, which has type `Set (X × X)` is an
-equivalence relation on `X`. Points which are equivalent under the relation are basically
-undistinguishable from the point of view of the uniform structure. For instance any uniformly
-continuous function will send equivalent points to the same value.
-
-The quotient `SeparationQuotient X` of `X` by `𝓢 X` has a natural uniform structure which is
-separated, and satisfies a universal property: every uniformly continuous function
-from `X` to a separated uniform space uniquely factors through `SeparationQuotient X`.
-As usual, this allows to turn `SeparationQuotient` into a functor (but we don't use the
-category theory library in this file).
+## Main definitions
 
-These notions admit relative versions, one can ask that `s : Set X` is separated, this
-is equivalent to asking that the uniform structure induced on `s` is separated.
+* `SeparationQuotient.instUniformSpace`: uniform space structure on `SeparationQuotient α`,
+  where `α` is a uniform space;
 
-## Main definitions
+* `SeparationQuotient.lift'`: given a map `f : α → β`
+  from a uniform space to a separated uniform space,
+  lift it to a map `SeparationQuotient α → β`;
+  if the original map is not uniformly continuous, then returns a constant map.
 
-* `separationRel X : Set (X × X)`: the separation relation
-* `SeparatedSpace X`: a predicate class asserting that `X` is separated
-* `SeparationQuotient X`: the maximal separated quotient of `X`.
-* `SeparationQuotient.lift f`: factors a map `f : X → Y` through the separation quotient of `X`.
-* `SeparationQuotient.map f`: turns a map `f : X → Y` into a map between the separation quotients
-  of `X` and `Y`.
+* `SeparationQuotient.map`: given a map `f : α → β` between uniform spaces,
+  returns a map `SeparationQuotient α → SeparationQuotient β`.
+  If the original map is not uniformly continuous, then returns a constant map.
+  Otherwise, `SeparationQuotient.map f (SeparationQuotient.mk x) = SeparationQuotient.mk (f x)`.
 
 ## Main results
 
-* `separated_iff_t2`: the equivalence between being separated and being Hausdorff for uniform
-  spaces.
-* `SeparationQuotient.uniformContinuous_lift`: factoring a uniformly continuous map through the
+* `SeparationQuotient.uniformity_eq`: the uniformity filter on `SeparationQuotient α`
+  is the push forward of the uniformity filter on `α`.
+* `SeparationQuotient.comap_mk_uniformity`: the quotient map `α → SeparationQuotient α`
+  induces uniform space structure on the original space.
+* `SeparationQuotient.uniformContinuous_lift'`: factoring a uniformly continuous map through the
   separation quotient gives a uniformly continuous map.
 * `SeparationQuotient.uniformContinuous_map`: maps induced between separation quotients are
   uniformly continuous.
 
-## Notations
+## Implementation notes
+
+This files used to contain definitions of `separationRel α` and `UniformSpace.SeparationQuotient α`.
+These definitions were equal (but not definitionally equal)
+to `{x : α × α | Inseparable x.1 x.2}` and `SeparationQuotient α`, respectively,
+and were added to the library before their geneeralizations to topological spaces.
 
-Localized in `uniformity`, we have the notation `𝓢 X` for the separation relation
-on a uniform space `X`,
+In #10644, we migrated from these definitions
+to more general `Inseparable` and `SeparationQuotient`.
 
-## Implementation notes
+## TODO
 
-The separation setoid `separationSetoid` is not declared as a global instance.
-It is made a local instance while building the theory of `SeparationQuotient`.
-The factored map `SeparationQuotient.lift f` is defined without imposing any condition on
-`f`, but returns junk if `f` is not uniformly continuous (constant junk hence it is always
-uniformly continuous).
+Definitions `SeparationQuotient.lift'` and `SeparationQuotient.map`
+rely on `UniformSpace` structures in the domain and in the codomain.
+We should generalize them to topological spaces.
+This generalization will drop `UniformContinuous` assumptions in some lemmas,
+and add these assumptions in other lemmas,
+so it was not done in #10644 to keep it reasonably sized.
 
+## Keywords
+
+uniform space, separated space, Hausdorff space, separation quotient
 -/
 
 open Filter Set Function Topology Uniformity UniformSpace
@@ -86,136 +117,89 @@ instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
     fun a _V hV ↦ isClosed_ball a hV.2
 #align uniform_space.to_regular_space UniformSpace.to_regularSpace
 
--- Porting note (#11215): TODO: use `Inseparable`
-/-- The separation relation is the intersection of all entourages.
-  Two points which are related by the separation relation are "indistinguishable"
-  according to the uniform structure. -/
-def separationRel (α : Type u) [UniformSpace α] := (𝓤 α).ker
-#align separation_rel separationRel
-
-@[inherit_doc]
-scoped[Uniformity] notation "𝓢" => separationRel
-
-theorem separated_equiv : Equivalence fun x y => (x, y) ∈ 𝓢 α :=
-  ⟨fun _ _ => refl_mem_uniformity, fun h _s hs => h _ (symm_le_uniformity hs),
-    fun {x y z} (hxy : (x, y) ∈ 𝓢 α) (hyz : (y, z) ∈ 𝓢 α) s (hs : s ∈ 𝓤 α) =>
-    let ⟨t, ht, (h_ts : compRel t t ⊆ s)⟩ := comp_mem_uniformity_sets hs
-    h_ts <| show (x, z) ∈ compRel t t from ⟨y, hxy t ht, hyz t ht⟩⟩
-#align separated_equiv separated_equiv
-
-theorem Filter.HasBasis.mem_separationRel {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
-    (h : (𝓤 α).HasBasis p s) {a : α × α} : a ∈ 𝓢 α ↔ ∀ i, p i → a ∈ s i :=
-  h.forall_mem_mem
-#align filter.has_basis.mem_separation_rel Filter.HasBasis.mem_separationRel
-
-theorem separationRel_iff_specializes {a b : α} : (a, b) ∈ 𝓢 α ↔ a ⤳ b := by
-  simp only [(𝓤 α).basis_sets.mem_separationRel, id, mem_setOf_eq,
-    (nhds_basis_uniformity (𝓤 α).basis_sets).specializes_iff]
-#align separation_rel_iff_specializes separationRel_iff_specializes
-
-theorem separationRel_iff_inseparable {a b : α} : (a, b) ∈ 𝓢 α ↔ Inseparable a b :=
-  separationRel_iff_specializes.trans specializes_iff_inseparable
-#align separation_rel_iff_inseparable separationRel_iff_inseparable
-
-/-- A uniform space is separated if its separation relation is trivial (each point
-is related only to itself). -/
-class SeparatedSpace (α : Type u) [UniformSpace α] : Prop where
-  /-- The separation relation is equal to the diagonal `idRel`. -/
-  out : 𝓢 α = idRel
-#align separated_space SeparatedSpace
-
-theorem separatedSpace_iff {α : Type u} [UniformSpace α] : SeparatedSpace α ↔ 𝓢 α = idRel :=
-  ⟨fun h => h.1, fun h => ⟨h⟩⟩
-#align separated_space_iff separatedSpace_iff
-
-theorem separated_def {α : Type u} [UniformSpace α] :
-    SeparatedSpace α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by
-  simp only [separatedSpace_iff, Set.ext_iff, Prod.forall, mem_idRel, separationRel, mem_sInter]
-  exact forall₂_congr fun _ _ => ⟨Iff.mp, fun h => ⟨h, fun H U hU => H ▸ refl_mem_uniformity hU⟩⟩
-#align separated_def separated_def
-
-theorem separated_def' {α : Type u} [UniformSpace α] :
-    SeparatedSpace α ↔ Pairwise fun x y => ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
-  separated_def.trans <| forall₂_congr fun x y => by rw [← not_imp_not]; simp [not_forall]
-#align separated_def' separated_def'
-
-theorem eq_of_uniformity {α : Type*} [UniformSpace α] [SeparatedSpace α] {x y : α}
+#align separation_rel Inseparable
+#noalign separated_equiv
+#align separation_rel_iff_specializes specializes_iff_inseparable
+#noalign separation_rel_iff_inseparable
+
+theorem Filter.HasBasis.specializes_iff_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
+    (h : (𝓤 α).HasBasis p s) {x y : α} : x ⤳ y ↔ ∀ i, p i → (x, y) ∈ s i :=
+  (nhds_basis_uniformity h).specializes_iff
+
+theorem Filter.HasBasis.inseparable_iff_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
+    (h : (𝓤 α).HasBasis p s) {x y : α} : Inseparable x y ↔ ∀ i, p i → (x, y) ∈ s i :=
+  specializes_iff_inseparable.symm.trans h.specializes_iff_uniformity
+#align filter.has_basis.mem_separation_rel Filter.HasBasis.inseparable_iff_uniformity
+
+theorem inseparable_iff_ker_uniformity {x y : α} : Inseparable x y ↔ (x, y) ∈ (𝓤 α).ker :=
+  (𝓤 α).basis_sets.inseparable_iff_uniformity
+
+protected theorem Inseparable.nhds_le_uniformity {x y : α} (h : Inseparable x y) :
+    𝓝 (x, y) ≤ 𝓤 α := by
+  rw [h.prod rfl]
+  apply nhds_le_uniformity
+
+theorem inseparable_iff_clusterPt_uniformity {x y : α} :
+    Inseparable x y ↔ ClusterPt (x, y) (𝓤 α) := by
+  refine ⟨fun h ↦ .of_nhds_le h.nhds_le_uniformity, fun h ↦ ?_⟩
+  simp_rw [uniformity_hasBasis_closed.inseparable_iff_uniformity, isClosed_iff_clusterPt]
+  exact fun U ⟨hU, hUc⟩ ↦ hUc _ <| h.mono <| le_principal_iff.2 hU
+
+#align separated_space T0Space
+
+theorem t0Space_iff_uniformity :
+    T0Space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by
+  simp only [t0Space_iff_inseparable, inseparable_iff_ker_uniformity, mem_ker, id]
+#align separated_def t0Space_iff_uniformity
+
+theorem t0Space_iff_uniformity' :
+    T0Space α ↔ Pairwise fun x y ↦ ∃ r ∈ 𝓤 α, (x, y) ∉ r := by
+  simp [t0Space_iff_not_inseparable, inseparable_iff_ker_uniformity]
+#align separated_def' t0Space_iff_uniformity'
+
+theorem t0Space_iff_ker_uniformity : T0Space α ↔ (𝓤 α).ker = diagonal α := by
+  simp_rw [t0Space_iff_uniformity, subset_antisymm_iff, diagonal_subset_iff, subset_def,
+    Prod.forall, Filter.mem_ker, mem_diagonal_iff, iff_self_and]
+  exact fun _ x s hs ↦ refl_mem_uniformity hs
+#align separated_space_iff t0Space_iff_ker_uniformity
+
+theorem eq_of_uniformity {α : Type*} [UniformSpace α] [T0Space α] {x y : α}
     (h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y :=
-  separated_def.mp ‹SeparatedSpace α› x y fun _ => h
+  t0Space_iff_uniformity.mp ‹T0Space α› x y @h
 #align eq_of_uniformity eq_of_uniformity
 
-theorem eq_of_uniformity_basis {α : Type*} [UniformSpace α] [SeparatedSpace α] {ι : Type*}
+theorem eq_of_uniformity_basis {α : Type*} [UniformSpace α] [T0Space α] {ι : Sort*}
     {p : ι → Prop} {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α}
     (h : ∀ {i}, p i → (x, y) ∈ s i) : x = y :=
-  eq_of_uniformity fun V_in => let ⟨_, hi, H⟩ := hs.mem_iff.mp V_in; H (h hi)
+  (hs.inseparable_iff_uniformity.2 @h).eq
 #align eq_of_uniformity_basis eq_of_uniformity_basis
 
-theorem eq_of_forall_symmetric {α : Type*} [UniformSpace α] [SeparatedSpace α] {x y : α}
+theorem eq_of_forall_symmetric {α : Type*} [UniformSpace α] [T0Space α] {x y : α}
     (h : ∀ {V}, V ∈ 𝓤 α → SymmetricRel V → (x, y) ∈ V) : x = y :=
-  eq_of_uniformity_basis hasBasis_symmetric (by simpa [and_imp])
+  eq_of_uniformity_basis hasBasis_symmetric (by simpa)
 #align eq_of_forall_symmetric eq_of_forall_symmetric
 
-theorem eq_of_clusterPt_uniformity [SeparatedSpace α] {x y : α} (h : ClusterPt (x, y) (𝓤 α)) :
-    x = y :=
-  eq_of_uniformity_basis uniformity_hasBasis_closed fun ⟨hV, hVc⟩ =>
-    isClosed_iff_clusterPt.1 hVc _ <| h.mono <| le_principal_iff.2 hV
+theorem eq_of_clusterPt_uniformity [T0Space α] {x y : α} (h : ClusterPt (x, y) (𝓤 α)) : x = y :=
+  (inseparable_iff_clusterPt_uniformity.2 h).eq
 #align eq_of_cluster_pt_uniformity eq_of_clusterPt_uniformity
 
-theorem idRel_sub_separationRel (α : Type*) [UniformSpace α] : idRel ⊆ 𝓢 α := by
-  unfold separationRel
-  rw [idRel_subset]
-  intro x
-  suffices ∀ t ∈ 𝓤 α, (x, x) ∈ t by simpa only [refl_mem_uniformity]
-  exact fun t => refl_mem_uniformity
-#align id_rel_sub_separation_relation idRel_sub_separationRel
-
-theorem separationRel_comap {f : α → β}
-    (h : ‹UniformSpace α› = UniformSpace.comap f ‹UniformSpace β›) :
-    𝓢 α = Prod.map f f ⁻¹' 𝓢 β := by
-  subst h
-  dsimp [separationRel]
-  simp_rw [uniformity_comap, ((𝓤 β).comap_hasBasis <| Prod.map f f).ker, ker_def, preimage_iInter]
-#align separation_rel_comap separationRel_comap
-
-protected theorem Filter.HasBasis.separationRel {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
-    (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (_ : p i), s i := by
-  unfold separationRel
-  rw [h.ker]
-#align filter.has_basis.separation_rel Filter.HasBasis.separationRel
-
-theorem separationRel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) := by
-  simp [uniformity_hasBasis_closure.separationRel]
-#align separation_rel_eq_inter_closure separationRel_eq_inter_closure
-
-theorem isClosed_separationRel : IsClosed (𝓢 α) := by
-  rw [separationRel_eq_inter_closure]
-  apply isClosed_sInter
-  rintro _ ⟨t, -, rfl⟩
-  exact isClosed_closure
-#align is_closed_separation_rel isClosed_separationRel
-
-theorem separated_iff_t2 : SeparatedSpace α ↔ T2Space α := by
-  constructor <;> intro h
-  · rw [t2_iff_isClosed_diagonal, ← show 𝓢 α = diagonal α from h.1]
-    exact isClosed_separationRel
-  · rw [separated_def']
-    intro x y hxy
-    rcases t2_separation hxy with ⟨u, v, uo, -, hx, hy, h⟩
-    rcases isOpen_iff_ball_subset.1 uo x hx with ⟨r, hrU, hr⟩
-    exact ⟨r, hrU, fun H => h.le_bot ⟨hr H, hy⟩⟩
-#align separated_iff_t2 separated_iff_t2
-
--- see Note [lower instance priority]
-instance (priority := 100) separated_t3 [SeparatedSpace α] : T3Space α :=
-  haveI := separated_iff_t2.mp ‹_›
-  ⟨⟩
-#align separated_t3 separated_t3
-
-instance Subtype.separatedSpace [SeparatedSpace α] (s : Set α) : SeparatedSpace s :=
-  separated_iff_t2.mpr inferInstance
-#align subtype.separated_space Subtype.separatedSpace
-
-theorem isClosed_of_spaced_out [SeparatedSpace α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α) {s : Set α}
+theorem Filter.Tendsto.inseparable_iff_uniformity {l : Filter β} [NeBot l] {f g : β → α} {a b : α}
+    (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) :
+    Inseparable a b ↔ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α) := by
+  refine ⟨fun h ↦ (ha.prod_mk_nhds hb).mono_right h.nhds_le_uniformity, fun h ↦ ?_⟩
+  rw [inseparable_iff_clusterPt_uniformity]
+  exact (ClusterPt.of_le_nhds (ha.prod_mk_nhds hb)).mono h
+
+#align id_rel_sub_separation_relation Inseparable.rfl
+#align separation_rel_comap Inducing.inseparable_iff
+#align filter.has_basis.separation_rel Filter.HasBasis.ker
+#noalign separation_rel_eq_inter_closure
+#align is_closed_separation_rel isClosed_setOf_inseparable
+#align separated_iff_t2 R1Space.t2Space_iff_t0Space
+#align separated_t3 RegularSpace.t3Space_iff_t0Space
+#align subtype.separated_space Subtype.t0Space
+
+theorem isClosed_of_spaced_out [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α) {s : Set α}
     (hs : s.Pairwise fun x y => (x, y) ∉ V₀) : IsClosed s := by
   rcases comp_symm_mem_uniformity_sets V₀_in with ⟨V₁, V₁_in, V₁_symm, h_comp⟩
   apply isClosed_of_closure_subset
@@ -232,7 +216,7 @@ theorem isClosed_of_spaced_out [SeparatedSpace α] {V₀ : Set (α × α)} (V₀
   exact ball_inter_right x _ _ hz
 #align is_closed_of_spaced_out isClosed_of_spaced_out
 
-theorem isClosed_range_of_spaced_out {ι} [SeparatedSpace α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
+theorem isClosed_range_of_spaced_out {ι} [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
     {f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (range f) :=
   isClosed_of_spaced_out V₀_in <| by
     rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h
@@ -243,197 +227,125 @@ theorem isClosed_range_of_spaced_out {ι} [SeparatedSpace α] {V₀ : Set (α ×
 ### Separation quotient
 -/
 
-namespace UniformSpace
-
-/-- The separation relation of a uniform space seen as a setoid. -/
-def separationSetoid (α : Type u) [UniformSpace α] : Setoid α :=
-  ⟨fun x y => (x, y) ∈ 𝓢 α, separated_equiv⟩
-#align uniform_space.separation_setoid UniformSpace.separationSetoid
-
-attribute [local instance] separationSetoid
-
-instance separationSetoid.uniformSpace {α : Type u} [UniformSpace α] :
-    UniformSpace (Quotient (separationSetoid α)) where
-  toTopologicalSpace := instTopologicalSpaceQuotient
-  uniformity := map (fun p : α × α => (⟦p.1⟧, ⟦p.2⟧)) (𝓤 α)
-  refl := le_trans (by simp [Quotient.exists_rep]) (Filter.map_mono refl_le_uniformity)
-  symm := tendsto_map' <| tendsto_map.comp tendsto_swap_uniformity
-  comp s hs := by
-    rcases comp_open_symm_mem_uniformity_sets hs with ⟨U, hU, hUo, -, hUs⟩
-    refine' mem_of_superset (mem_lift' <| image_mem_map hU) ?_
-    simp only [subset_def, Prod.forall, mem_compRel, mem_image, Prod.ext_iff]
-    rintro _ _ ⟨_, ⟨⟨x, y⟩, hxyU, rfl, rfl⟩, ⟨⟨y', z⟩, hyzU, hy, rfl⟩⟩
-    have : y' ⤳ y := separationRel_iff_specializes.1 (Quotient.exact hy)
-    exact @hUs (x, z) ⟨y', this.mem_open (UniformSpace.isOpen_ball _ hUo) hxyU, hyzU⟩
-  isOpen_uniformity s := isOpen_coinduced.trans <| by
-    simp only [_root_.isOpen_uniformity, Quotient.forall, mem_map', mem_setOf_eq]
-    refine forall₂_congr fun x _ => ⟨fun h => ?_, fun h => mem_of_superset h ?_⟩
-    · rcases comp_mem_uniformity_sets h with ⟨t, ht, hts⟩
-      refine mem_of_superset ht fun (y, z) hyz hyx => @hts (x, z) ⟨y, ?_, hyz⟩ rfl
-      exact Quotient.exact hyx.symm _ ht
-    · exact fun y hy hyx => hy <| congr_arg _ hyx
-#align uniform_space.separation_setoid.uniform_space UniformSpace.separationSetoid.uniformSpace
-
-theorem uniformity_quotient :
-    𝓤 (Quotient (separationSetoid α)) = (𝓤 α).map fun p : α × α => (⟦p.1⟧, ⟦p.2⟧) :=
-  rfl
-#align uniform_space.uniformity_quotient UniformSpace.uniformity_quotient
+#align uniform_space.separation_setoid inseparableSetoid
 
-theorem uniformContinuous_quotient_mk' :
-    UniformContinuous (Quotient.mk' : α → Quotient (separationSetoid α)) :=
-  le_rfl
-#align uniform_space.uniform_continuous_quotient_mk UniformSpace.uniformContinuous_quotient_mk'
+namespace SeparationQuotient
 
-theorem uniformContinuous_quotient_mk : UniformContinuous (Quotient.mk (separationSetoid α)) :=
+theorem comap_map_mk_uniformity : comap (Prod.map mk mk) (map (Prod.map mk mk) (𝓤 α)) = 𝓤 α := by
+  refine le_antisymm ?_ le_comap_map
+  refine ((((𝓤 α).basis_sets.map _).comap _).le_basis_iff uniformity_hasBasis_open).2 fun U hU ↦ ?_
+  refine ⟨U, hU.1, fun (x₁, x₂) ⟨(y₁, y₂), hyU, hxy⟩ ↦ ?_⟩
+  simp only [Prod.map, Prod.ext_iff, mk_eq_mk] at hxy
+  exact ((hxy.1.prod hxy.2).mem_open_iff hU.2).1 hyU
+
+instance instUniformSpace : UniformSpace (SeparationQuotient α) :=
+  .ofNhdsEqComap
+    { uniformity := map (Prod.map mk mk) (𝓤 α)
+      refl := le_trans (by simpa using surjective_mk) (Filter.map_mono refl_le_uniformity)
+      symm := tendsto_map' <| tendsto_map.comp tendsto_swap_uniformity
+      comp := fun t ht ↦ by
+        rcases comp_open_symm_mem_uniformity_sets ht with ⟨U, hU, hUo, -, hUt⟩
+        refine mem_of_superset (mem_lift' <| image_mem_map hU) ?_
+        simp only [subset_def, Prod.forall, mem_compRel, mem_image, Prod.ext_iff]
+        rintro _ _ ⟨_, ⟨⟨x, y⟩, hxyU, rfl, rfl⟩, ⟨⟨y', z⟩, hyzU, hy, rfl⟩⟩
+        have : y' ⤳ y := (mk_eq_mk.1 hy).specializes
+        exact @hUt (x, z) ⟨y', this.mem_open (UniformSpace.isOpen_ball _ hUo) hxyU, hyzU⟩ }
+    inferInstance <| surjective_mk.forall.2 fun x ↦ comap_injective surjective_mk <| by
+      conv_lhs => rw [comap_mk_nhds_mk, nhds_eq_comap_uniformity, ← comap_map_mk_uniformity]
+      simp only [Filter.comap_comap]; rfl
+
+theorem uniformity_eq : 𝓤 (SeparationQuotient α) = (𝓤 α).map (Prod.map mk mk) := rfl
+#align uniform_space.uniformity_quotient SeparationQuotient.uniformity_eq
+
+theorem uniformContinuous_mk : UniformContinuous (mk : α → SeparationQuotient α) :=
   le_rfl
+#align uniform_space.uniform_continuous_quotient_mk SeparationQuotient.uniformContinuous_mk
 
-theorem uniformContinuous_quotient {f : Quotient (separationSetoid α) → β}
-    (hf : UniformContinuous fun x => f ⟦x⟧) : UniformContinuous f :=
-  hf
-#align uniform_space.uniform_continuous_quotient UniformSpace.uniformContinuous_quotient
-
-theorem uniformContinuous_quotient_lift {f : α → β} {h : ∀ a b, (a, b) ∈ 𝓢 α → f a = f b}
-    (hf : UniformContinuous f) : UniformContinuous fun a => Quotient.lift f h a :=
-  uniformContinuous_quotient hf
-#align uniform_space.uniform_continuous_quotient_lift UniformSpace.uniformContinuous_quotient_lift
-
-theorem uniformContinuous_quotient_lift₂ {f : α → β → γ}
-    {h : ∀ a c b d, (a, b) ∈ 𝓢 α → (c, d) ∈ 𝓢 β → f a c = f b d}
-    (hf : UniformContinuous fun p : α × β => f p.1 p.2) :
-    UniformContinuous fun p : _ × _ => Quotient.lift₂ f h p.1 p.2 := by
-  rw [UniformContinuous, uniformity_prod_eq_prod, uniformity_quotient, uniformity_quotient,
-    Filter.prod_map_map_eq, Filter.tendsto_map'_iff, Filter.tendsto_map'_iff]
-  rwa [UniformContinuous, uniformity_prod_eq_prod, Filter.tendsto_map'_iff] at hf
-#align uniform_space.uniform_continuous_quotient_lift₂ UniformSpace.uniformContinuous_quotient_lift₂
-
-theorem comap_quotient_le_uniformity :
-    ((𝓤 <| Quotient <| separationSetoid α).comap fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) ≤ 𝓤 α :=
-  ((((𝓤 α).basis_sets.map _).comap _).le_basis_iff uniformity_hasBasis_open).2 fun U hU =>
-    ⟨U, hU.1, fun ⟨x, y⟩ ⟨⟨x', y'⟩, hx', h⟩ => by
-      simp only [Prod.ext_iff, Quotient.eq] at h
-      exact (((separationRel_iff_inseparable.1 h.1).prod
-        (separationRel_iff_inseparable.1 h.2)).mem_open_iff hU.2).1 hx'⟩
-#align uniform_space.comap_quotient_le_uniformity UniformSpace.comap_quotient_le_uniformity
-
-theorem comap_quotient_eq_uniformity :
-    ((𝓤 <| Quotient <| separationSetoid α).comap fun p : α × α => (⟦p.fst⟧, ⟦p.snd⟧)) = 𝓤 α :=
-  le_antisymm comap_quotient_le_uniformity le_comap_map
-#align uniform_space.comap_quotient_eq_uniformity UniformSpace.comap_quotient_eq_uniformity
-
-instance separated_separation : SeparatedSpace (Quotient (separationSetoid α)) :=
-  ⟨Set.ext fun ⟨a, b⟩ =>
-      Quotient.inductionOn₂ a b fun a b =>
-        ⟨fun h =>
-          have : a ≈ b := fun s hs =>
-            have :
-              s ∈ (𝓤 <| Quotient <| separationSetoid α).comap fun p : α × α => (⟦p.1⟧, ⟦p.2⟧) :=
-              comap_quotient_le_uniformity hs
-            let ⟨t, ht, hts⟩ := this
-            hts (by dsimp [preimage]; exact h t ht)
-          show ⟦a⟧ = ⟦b⟧ from Quotient.sound this,
-          fun heq : ⟦a⟧ = ⟦b⟧ => fun h hs => heq ▸ refl_mem_uniformity hs⟩⟩
-#align uniform_space.separated_separation UniformSpace.separated_separation
-
-theorem separated_of_uniformContinuous {f : α → β} {x y : α} (H : UniformContinuous f) (h : x ≈ y) :
-    f x ≈ f y := fun _ h' => h _ (H h')
-#align uniform_space.separated_of_uniform_continuous UniformSpace.separated_of_uniformContinuous
-
-theorem eq_of_separated_of_uniformContinuous [SeparatedSpace β] {f : α → β} {x y : α}
-    (H : UniformContinuous f) (h : x ≈ y) : f x = f y :=
-  separated_def.1 (by infer_instance) _ _ <| separated_of_uniformContinuous H h
-#align uniform_space.eq_of_separated_of_uniform_continuous UniformSpace.eq_of_separated_of_uniformContinuous
-
-/-- The maximal separated quotient of a uniform space `α`. -/
-def SeparationQuotient (α : Type*) [UniformSpace α] :=
-  Quotient (separationSetoid α)
-#align uniform_space.separation_quotient UniformSpace.SeparationQuotient
+theorem uniformContinuous_dom {f : SeparationQuotient α → β} :
+    UniformContinuous f ↔ UniformContinuous (f ∘ mk) :=
+  .rfl
+#align uniform_space.uniform_continuous_quotient SeparationQuotient.uniformContinuous_dom
 
-namespace SeparationQuotient
+theorem uniformContinuous_dom₂ {f : SeparationQuotient α × SeparationQuotient β → γ} :
+    UniformContinuous f ↔ UniformContinuous fun p : α × β ↦ f (mk p.1, mk p.2) := by
+  simp only [UniformContinuous, uniformity_prod_eq_prod, uniformity_eq, prod_map_map_eq,
+    tendsto_map'_iff]
+  rfl
+
+theorem uniformContinuous_lift {f : α → β} (h : ∀ a b, Inseparable a b → f a = f b) :
+    UniformContinuous (lift f h) ↔ UniformContinuous f :=
+  .rfl
+#align uniform_space.uniform_continuous_quotient_lift SeparationQuotient.uniformContinuous_lift
+
+theorem uniformContinuous_uncurry_lift₂ {f : α → β → γ}
+    (h : ∀ a c b d, Inseparable a b → Inseparable c d → f a c = f b d) :
+    UniformContinuous (uncurry <| lift₂ f h) ↔ UniformContinuous (uncurry f) :=
+  uniformContinuous_dom₂
+#align uniform_space.uniform_continuous_quotient_lift₂ SeparationQuotient.uniformContinuous_uncurry_lift₂
+
+#noalign uniform_space.comap_quotient_le_uniformity
+
+theorem comap_mk_uniformity : (𝓤 (SeparationQuotient α)).comap (Prod.map mk mk) = 𝓤 α :=
+  comap_map_mk_uniformity
+#align uniform_space.comap_quotient_eq_uniformity SeparationQuotient.comap_mk_uniformity
 
-instance : UniformSpace (SeparationQuotient α) :=
-  separationSetoid.uniformSpace
+#align uniform_space.separated_separation SeparationQuotient.instT0Space
 
-instance : SeparatedSpace (SeparationQuotient α) :=
-  UniformSpace.separated_separation
+#align uniform_space.separated_of_uniform_continuous Inseparable.map
+#noalign uniform_space.eq_of_separated_of_uniform_continuous
 
-instance [Inhabited α] : Inhabited (SeparationQuotient α) :=
-  inferInstanceAs (Inhabited (Quotient (separationSetoid α)))
+#align uniform_space.separation_quotient SeparationQuotient
+#align uniform_space.separation_quotient.mk_eq_mk SeparationQuotient.mk_eq_mk
 
-lemma mk_eq_mk {x y : α} : (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=
-  Quotient.eq'.trans separationRel_iff_inseparable
-#align uniform_space.separation_quotient.mk_eq_mk UniformSpace.SeparationQuotient.mk_eq_mk
+/-- Factoring functions to a separated space through the separation quotient.
 
-/-- Factoring functions to a separated space through the separation quotient. -/
-def lift [SeparatedSpace β] (f : α → β) : SeparationQuotient α → β :=
-  if h : UniformContinuous f then Quotient.lift f fun _ _ => eq_of_separated_of_uniformContinuous h
-  else fun x => f (Nonempty.some ⟨x.out⟩)
-#align uniform_space.separation_quotient.lift UniformSpace.SeparationQuotient.lift
+TODO: unify with `SeparationQuotient.lift`. -/
+def lift' [T0Space β] (f : α → β) : SeparationQuotient α → β :=
+  if hc : UniformContinuous f then lift f fun _ _ h => (h.map hc.continuous).eq
+  else fun x => f (Nonempty.some ⟨x.out'⟩)
+#align uniform_space.separation_quotient.lift SeparationQuotient.lift'
 
-theorem lift_mk [SeparatedSpace β] {f : α → β} (h : UniformContinuous f) (a : α) :
-    lift f ⟦a⟧ = f a := by rw [lift, dif_pos h]; rfl
-#align uniform_space.separation_quotient.lift_mk UniformSpace.SeparationQuotient.lift_mk
+theorem lift'_mk [T0Space β] {f : α → β} (h : UniformContinuous f) (a : α) :
+    lift' f (mk a) = f a := by rw [lift', dif_pos h]; rfl
+#align uniform_space.separation_quotient.lift_mk SeparationQuotient.lift'_mk
 
-theorem uniformContinuous_lift [SeparatedSpace β] (f : α → β) : UniformContinuous (lift f) := by
+theorem uniformContinuous_lift' [T0Space β] (f : α → β) : UniformContinuous (lift' f) := by
   by_cases hf : UniformContinuous f
-  · rw [lift, dif_pos hf]
-    exact uniformContinuous_quotient_lift hf
-  · rw [lift, dif_neg hf]
+  · rwa [lift', dif_pos hf, uniformContinuous_lift]
+  · rw [lift', dif_neg hf]
     exact uniformContinuous_of_const fun a _ => rfl
-#align uniform_space.separation_quotient.uniform_continuous_lift UniformSpace.SeparationQuotient.uniformContinuous_lift
+#align uniform_space.separation_quotient.uniform_continuous_lift SeparationQuotient.uniformContinuous_lift'
 
 /-- The separation quotient functor acting on functions. -/
-def map (f : α → β) : SeparationQuotient α → SeparationQuotient β :=
-  lift (Quotient.mk' ∘ f)
-#align uniform_space.separation_quotient.map UniformSpace.SeparationQuotient.map
+def map (f : α → β) : SeparationQuotient α → SeparationQuotient β := lift' (mk ∘ f)
+#align uniform_space.separation_quotient.map SeparationQuotient.map
 
-theorem map_mk {f : α → β} (h : UniformContinuous f) (a : α) : map f ⟦a⟧ = ⟦f a⟧ := by
-  rw [map, lift_mk (uniformContinuous_quotient_mk'.comp h)]; rfl
-#align uniform_space.separation_quotient.map_mk UniformSpace.SeparationQuotient.map_mk
+theorem map_mk {f : α → β} (h : UniformContinuous f) (a : α) : map f (mk a) = mk (f a) := by
+  rw [map, lift'_mk (uniformContinuous_mk.comp h)]; rfl
+#align uniform_space.separation_quotient.map_mk SeparationQuotient.map_mk
 
 theorem uniformContinuous_map (f : α → β) : UniformContinuous (map f) :=
-  uniformContinuous_lift (Quotient.mk' ∘ f)
-#align uniform_space.separation_quotient.uniform_continuous_map UniformSpace.SeparationQuotient.uniformContinuous_map
+  uniformContinuous_lift' _
+#align uniform_space.separation_quotient.uniform_continuous_map SeparationQuotient.uniformContinuous_map
 
 theorem map_unique {f : α → β} (hf : UniformContinuous f)
-    {g : SeparationQuotient α → SeparationQuotient β}
-    (comm : Quotient.mk _ ∘ f = g ∘ Quotient.mk _) : map f = g := by
+    {g : SeparationQuotient α → SeparationQuotient β} (comm : mk ∘ f = g ∘ mk) : map f = g := by
   ext ⟨a⟩
   calc
     map f ⟦a⟧ = ⟦f a⟧ := map_mk hf a
     _ = g ⟦a⟧ := congr_fun comm a
-#align uniform_space.separation_quotient.map_unique UniformSpace.SeparationQuotient.map_unique
+#align uniform_space.separation_quotient.map_unique SeparationQuotient.map_unique
 
-theorem map_id : map (@id α) = id :=
-  map_unique uniformContinuous_id rfl
-#align uniform_space.separation_quotient.map_id UniformSpace.SeparationQuotient.map_id
+@[simp]
+theorem map_id : map (@id α) = id := map_unique uniformContinuous_id rfl
+#align uniform_space.separation_quotient.map_id SeparationQuotient.map_id
 
 theorem map_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) :
     map g ∘ map f = map (g ∘ f) :=
   (map_unique (hg.comp hf) <| by simp only [Function.comp, map_mk, hf, hg]).symm
-#align uniform_space.separation_quotient.map_comp UniformSpace.SeparationQuotient.map_comp
+#align uniform_space.separation_quotient.map_comp SeparationQuotient.map_comp
 
 end SeparationQuotient
 
-theorem separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a₂, b₂) ↔ a₁ ≈ a₂ ∧ b₁ ≈ b₂ := by
-  constructor
-  · intro h
-    exact
-      ⟨separated_of_uniformContinuous uniformContinuous_fst h,
-        separated_of_uniformContinuous uniformContinuous_snd h⟩
-  · rintro ⟨eqv_α, eqv_β⟩ r r_in
-    rw [uniformity_prod] at r_in
-    rcases r_in with ⟨t_α, ⟨r_α, r_α_in, h_α⟩, t_β, ⟨r_β, r_β_in, h_β⟩, rfl⟩
-    let p_α := fun p : (α × β) × α × β => (p.1.1, p.2.1)
-    let p_β := fun p : (α × β) × α × β => (p.1.2, p.2.2)
-    have key_α : p_α ((a₁, b₁), (a₂, b₂)) ∈ r_α := by simp [eqv_α r_α r_α_in]
-    have key_β : p_β ((a₁, b₁), (a₂, b₂)) ∈ r_β := by simp [eqv_β r_β r_β_in]
-    exact ⟨h_α key_α, h_β key_β⟩
-#align uniform_space.separation_prod UniformSpace.separation_prod
-
-instance Separated.prod [SeparatedSpace α] [SeparatedSpace β] : SeparatedSpace (α × β) :=
-  separated_def.2 fun _ _ H =>
-    Prod.ext (eq_of_separated_of_uniformContinuous uniformContinuous_fst H)
-      (eq_of_separated_of_uniformContinuous uniformContinuous_snd H)
-#align uniform_space.separated.prod UniformSpace.Separated.prod
-
-end UniformSpace
+#align uniform_space.separation_prod inseparable_prod
+#align uniform_space.separated.prod Prod.instT0Space

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

@@ -86,7 +86,7 @@ instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
     fun a _V hV ↦ isClosed_ball a hV.2
 #align uniform_space.to_regular_space UniformSpace.to_regularSpace
 
--- Porting note: todo: use `Inseparable`
+-- Porting note (#11215): TODO: use `Inseparable`
 /-- The separation relation is the intersection of all entourages.
   Two points which are related by the separation relation are "indistinguishable"
   according to the uniform structure. -/

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".

@@ -86,7 +86,7 @@ instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
     fun a _V hV ↦ isClosed_ball a hV.2
 #align uniform_space.to_regular_space UniformSpace.to_regularSpace
 
--- porting note: todo: use `Inseparable`
+-- Porting note: todo: use `Inseparable`
 /-- The separation relation is the intersection of all entourages.
   Two points which are related by the separation relation are "indistinguishable"
   according to the uniform structure. -/

Mathport chose wrong names for these constructors.

@@ -81,11 +81,9 @@ variable [UniformSpace α] [UniformSpace β] [UniformSpace γ]
 -/
 
 instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
-  RegularSpace.ofBasis
-    (fun a => by
-      rw [nhds_eq_comap_uniformity]
-      exact uniformity_hasBasis_closed.comap _)
-    fun a V hV => by exact hV.2.preimage <| continuous_const.prod_mk continuous_id
+  .of_hasBasis
+    (fun _ ↦ nhds_basis_uniformity' uniformity_hasBasis_closed)
+    fun a _V hV ↦ isClosed_ball a hV.2
 #align uniform_space.to_regular_space UniformSpace.to_regularSpace
 
 -- porting note: todo: use `Inseparable`

See Zulip thread for the discussion.

@@ -176,7 +176,7 @@ theorem separationRel_comap {f : α → β}
     𝓢 α = Prod.map f f ⁻¹' 𝓢 β := by
   subst h
   dsimp [separationRel]
-  simp_rw [uniformity_comap, ((𝓤 β).comap_hasBasis $ Prod.map f f).ker, ker_def, preimage_iInter]
+  simp_rw [uniformity_comap, ((𝓤 β).comap_hasBasis <| Prod.map f f).ker, ker_def, preimage_iInter]
 #align separation_rel_comap separationRel_comap
 
 protected theorem Filter.HasBasis.separationRel {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}

Performed with a regex search for ∀ (.) (.), \1 ≠ \2 →, and a few variants to catch implicit binders and explicit types.

I have deliberately avoided trying to make the analogous Set.Pairwise transformation (or any Pairwise (foo on bar) transformations) in this PR, to keep the diff small.

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

@@ -137,7 +137,7 @@ theorem separated_def {α : Type u} [UniformSpace α] :
 #align separated_def separated_def
 
 theorem separated_def' {α : Type u} [UniformSpace α] :
-    SeparatedSpace α ↔ ∀ x y, x ≠ y → ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
+    SeparatedSpace α ↔ Pairwise fun x y => ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
   separated_def.trans <| forall₂_congr fun x y => by rw [← not_imp_not]; simp [not_forall]
 #align separated_def' separated_def'
 

Also forall_quotient_iff {α : Type*} [r : Setoid α] ... -> Quotient.forall {α : Sort*} {s : Setoid α} ...

@@ -268,7 +268,7 @@ instance separationSetoid.uniformSpace {α : Type u} [UniformSpace α] :
     have : y' ⤳ y := separationRel_iff_specializes.1 (Quotient.exact hy)
     exact @hUs (x, z) ⟨y', this.mem_open (UniformSpace.isOpen_ball _ hUo) hxyU, hyzU⟩
   isOpen_uniformity s := isOpen_coinduced.trans <| by
-    simp only [_root_.isOpen_uniformity, forall_quotient_iff, mem_map', mem_setOf_eq]
+    simp only [_root_.isOpen_uniformity, Quotient.forall, mem_map', mem_setOf_eq]
     refine forall₂_congr fun x _ => ⟨fun h => ?_, fun h => mem_of_superset h ?_⟩
     · rcases comp_mem_uniformity_sets h with ⟨t, ht, hts⟩
       refine mem_of_superset ht fun (y, z) hyz hyx => @hts (x, z) ⟨y, ?_, hyz⟩ rfl

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.

@@ -92,7 +92,7 @@ instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
 /-- The separation relation is the intersection of all entourages.
   Two points which are related by the separation relation are "indistinguishable"
   according to the uniform structure. -/
-def separationRel (α : Type u) [UniformSpace α] := ⋂₀ (𝓤 α).sets
+def separationRel (α : Type u) [UniformSpace α] := (𝓤 α).ker
 #align separation_rel separationRel
 
 @[inherit_doc]
@@ -176,15 +176,13 @@ theorem separationRel_comap {f : α → β}
     𝓢 α = Prod.map f f ⁻¹' 𝓢 β := by
   subst h
   dsimp [separationRel]
-  simp_rw [uniformity_comap, (Filter.comap_hasBasis (Prod.map f f) (𝓤 β)).sInter_sets, ←
-    preimage_iInter, sInter_eq_biInter]
-  rfl
+  simp_rw [uniformity_comap, ((𝓤 β).comap_hasBasis $ Prod.map f f).ker, ker_def, preimage_iInter]
 #align separation_rel_comap separationRel_comap
 
 protected theorem Filter.HasBasis.separationRel {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (_ : p i), s i := by
   unfold separationRel
-  rw [h.sInter_sets]
+  rw [h.ker]
 #align filter.has_basis.separation_rel Filter.HasBasis.separationRel
 
 theorem separationRel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) := by

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

This has nice performance benefits.

@@ -105,7 +105,7 @@ theorem separated_equiv : Equivalence fun x y => (x, y) ∈ 𝓢 α :=
     h_ts <| show (x, z) ∈ compRel t t from ⟨y, hxy t ht, hyz t ht⟩⟩
 #align separated_equiv separated_equiv
 
-theorem Filter.HasBasis.mem_separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
+theorem Filter.HasBasis.mem_separationRel {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : (𝓤 α).HasBasis p s) {a : α × α} : a ∈ 𝓢 α ↔ ∀ i, p i → a ∈ s i :=
   h.forall_mem_mem
 #align filter.has_basis.mem_separation_rel Filter.HasBasis.mem_separationRel
@@ -141,18 +141,18 @@ theorem separated_def' {α : Type u} [UniformSpace α] :
   separated_def.trans <| forall₂_congr fun x y => by rw [← not_imp_not]; simp [not_forall]
 #align separated_def' separated_def'
 
-theorem eq_of_uniformity {α : Type _} [UniformSpace α] [SeparatedSpace α] {x y : α}
+theorem eq_of_uniformity {α : Type*} [UniformSpace α] [SeparatedSpace α] {x y : α}
     (h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y :=
   separated_def.mp ‹SeparatedSpace α› x y fun _ => h
 #align eq_of_uniformity eq_of_uniformity
 
-theorem eq_of_uniformity_basis {α : Type _} [UniformSpace α] [SeparatedSpace α] {ι : Type _}
+theorem eq_of_uniformity_basis {α : Type*} [UniformSpace α] [SeparatedSpace α] {ι : Type*}
     {p : ι → Prop} {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α}
     (h : ∀ {i}, p i → (x, y) ∈ s i) : x = y :=
   eq_of_uniformity fun V_in => let ⟨_, hi, H⟩ := hs.mem_iff.mp V_in; H (h hi)
 #align eq_of_uniformity_basis eq_of_uniformity_basis
 
-theorem eq_of_forall_symmetric {α : Type _} [UniformSpace α] [SeparatedSpace α] {x y : α}
+theorem eq_of_forall_symmetric {α : Type*} [UniformSpace α] [SeparatedSpace α] {x y : α}
     (h : ∀ {V}, V ∈ 𝓤 α → SymmetricRel V → (x, y) ∈ V) : x = y :=
   eq_of_uniformity_basis hasBasis_symmetric (by simpa [and_imp])
 #align eq_of_forall_symmetric eq_of_forall_symmetric
@@ -163,7 +163,7 @@ theorem eq_of_clusterPt_uniformity [SeparatedSpace α] {x y : α} (h : ClusterPt
     isClosed_iff_clusterPt.1 hVc _ <| h.mono <| le_principal_iff.2 hV
 #align eq_of_cluster_pt_uniformity eq_of_clusterPt_uniformity
 
-theorem idRel_sub_separationRel (α : Type _) [UniformSpace α] : idRel ⊆ 𝓢 α := by
+theorem idRel_sub_separationRel (α : Type*) [UniformSpace α] : idRel ⊆ 𝓢 α := by
   unfold separationRel
   rw [idRel_subset]
   intro x
@@ -181,7 +181,7 @@ theorem separationRel_comap {f : α → β}
   rfl
 #align separation_rel_comap separationRel_comap
 
-protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
+protected theorem Filter.HasBasis.separationRel {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (_ : p i), s i := by
   unfold separationRel
   rw [h.sInter_sets]
@@ -348,7 +348,7 @@ theorem eq_of_separated_of_uniformContinuous [SeparatedSpace β] {f : α → β}
 #align uniform_space.eq_of_separated_of_uniform_continuous UniformSpace.eq_of_separated_of_uniformContinuous
 
 /-- The maximal separated quotient of a uniform space `α`. -/
-def SeparationQuotient (α : Type _) [UniformSpace α] :=
+def SeparationQuotient (α : Type*) [UniformSpace α] :=
   Quotient (separationSetoid α)
 #align uniform_space.separation_quotient UniformSpace.SeparationQuotient
 

• 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) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Patrick Massot
-
-! This file was ported from Lean 3 source module topology.uniform_space.separation
-! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Tactic.ApplyFun
 import Mathlib.Topology.UniformSpace.Basic
 import Mathlib.Topology.Separation
 
+#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
+
 /-!
 # Hausdorff properties of uniform spaces. Separation quotient.
 

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -367,7 +367,7 @@ instance [Inhabited α] : Inhabited (SeparationQuotient α) :=
   inferInstanceAs (Inhabited (Quotient (separationSetoid α)))
 
 lemma mk_eq_mk {x y : α} : (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=
-Quotient.eq'.trans separationRel_iff_inseparable
+  Quotient.eq'.trans separationRel_iff_inseparable
 #align uniform_space.separation_quotient.mk_eq_mk UniformSpace.SeparationQuotient.mk_eq_mk
 
 /-- Factoring functions to a separated space through the separation quotient. -/

@@ -185,7 +185,7 @@ theorem separationRel_comap {f : α → β}
 #align separation_rel_comap separationRel_comap
 
 protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
-    (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (_hi : p i), s i := by
+    (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (_ : p i), s i := by
   unfold separationRel
   rw [h.sInter_sets]
 #align filter.has_basis.separation_rel Filter.HasBasis.separationRel

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>

@@ -135,7 +135,7 @@ theorem separatedSpace_iff {α : Type u} [UniformSpace α] : SeparatedSpace α 
 
 theorem separated_def {α : Type u} [UniformSpace α] :
     SeparatedSpace α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by
-  simp only [separatedSpace_iff, Set.ext_iff, Prod.forall, mem_idRel, separationRel, mem_interₛ]
+  simp only [separatedSpace_iff, Set.ext_iff, Prod.forall, mem_idRel, separationRel, mem_sInter]
   exact forall₂_congr fun _ _ => ⟨Iff.mp, fun h => ⟨h, fun H U hU => H ▸ refl_mem_uniformity hU⟩⟩
 #align separated_def separated_def
 
@@ -179,15 +179,15 @@ theorem separationRel_comap {f : α → β}
     𝓢 α = Prod.map f f ⁻¹' 𝓢 β := by
   subst h
   dsimp [separationRel]
-  simp_rw [uniformity_comap, (Filter.comap_hasBasis (Prod.map f f) (𝓤 β)).interₛ_sets, ←
-    preimage_interᵢ, interₛ_eq_binterᵢ]
+  simp_rw [uniformity_comap, (Filter.comap_hasBasis (Prod.map f f) (𝓤 β)).sInter_sets, ←
+    preimage_iInter, sInter_eq_biInter]
   rfl
 #align separation_rel_comap separationRel_comap
 
 protected theorem Filter.HasBasis.separationRel {ι : Sort _} {p : ι → Prop} {s : ι → Set (α × α)}
     (h : HasBasis (𝓤 α) p s) : 𝓢 α = ⋂ (i) (_hi : p i), s i := by
   unfold separationRel
-  rw [h.interₛ_sets]
+  rw [h.sInter_sets]
 #align filter.has_basis.separation_rel Filter.HasBasis.separationRel
 
 theorem separationRel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) := by
@@ -196,7 +196,7 @@ theorem separationRel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).
 
 theorem isClosed_separationRel : IsClosed (𝓢 α) := by
   rw [separationRel_eq_inter_closure]
-  apply isClosed_interₛ
+  apply isClosed_sInter
   rintro _ ⟨t, -, rfl⟩
   exact isClosed_closure
 #align is_closed_separation_rel isClosed_separationRel

@@ -42,16 +42,16 @@ is equivalent to asking that the uniform structure induced on `s` is separated.
 * `SeparatedSpace X`: a predicate class asserting that `X` is separated
 * `SeparationQuotient X`: the maximal separated quotient of `X`.
 * `SeparationQuotient.lift f`: factors a map `f : X → Y` through the separation quotient of `X`.
-* `separation_quotient.map f`: turns a map `f : X → Y` into a map between the separation quotients
+* `SeparationQuotient.map f`: turns a map `f : X → Y` into a map between the separation quotients
   of `X` and `Y`.
 
 ## Main results
 
 * `separated_iff_t2`: the equivalence between being separated and being Hausdorff for uniform
   spaces.
-* `separation_quotient.uniform_continuous_lift`: factoring a uniformly continuous map through the
+* `SeparationQuotient.uniformContinuous_lift`: factoring a uniformly continuous map through the
   separation quotient gives a uniformly continuous map.
-* `separation_quotient.uniform_continuous_map`: maps induced between separation quotients are
+* `SeparationQuotient.uniformContinuous_map`: maps induced between separation quotients are
   uniformly continuous.
 
 ## Notations
@@ -61,7 +61,7 @@ on a uniform space `X`,
 
 ## Implementation notes
 
-The separation setoid `separation_setoid` is not declared as a global instance.
+The separation setoid `separationSetoid` is not declared as a global instance.
 It is made a local instance while building the theory of `SeparationQuotient`.
 The factored map `SeparationQuotient.lift f` is defined without imposing any condition on
 `f`, but returns junk if `f` is not uniformly continuous (constant junk hence it is always
@@ -265,7 +265,7 @@ instance separationSetoid.uniformSpace {α : Type u} [UniformSpace α] :
   uniformity := map (fun p : α × α => (⟦p.1⟧, ⟦p.2⟧)) (𝓤 α)
   refl := le_trans (by simp [Quotient.exists_rep]) (Filter.map_mono refl_le_uniformity)
   symm := tendsto_map' <| tendsto_map.comp tendsto_swap_uniformity
-  comp := fun s hs => by
+  comp s hs := by
     rcases comp_open_symm_mem_uniformity_sets hs with ⟨U, hU, hUo, -, hUs⟩
     refine' mem_of_superset (mem_lift' <| image_mem_map hU) ?_
     simp only [subset_def, Prod.forall, mem_compRel, mem_image, Prod.ext_iff]

Some Mathlib 4 lemmas were backported to Mathlib 3 in leanprover-community/mathlib#18502, sync SHAs

• topology.uniform_space.separation@d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46..0c1f285a9f6e608ae2bdffa3f993eafb01eba829
• topology.constructions@dc6c365e751e34d100e80fe6e314c3c3e0fd2988..0c1f285a9f6e608ae2bdffa3f993eafb01eba829

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.uniform_space.separation
-! leanprover-community/mathlib commit d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46
+! leanprover-community/mathlib commit 0c1f285a9f6e608ae2bdffa3f993eafb01eba829
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -113,14 +113,14 @@ theorem Filter.HasBasis.mem_separationRel {ι : Sort _} {p : ι → Prop} {s : 
   h.forall_mem_mem
 #align filter.has_basis.mem_separation_rel Filter.HasBasis.mem_separationRel
 
--- porting note: new lemma
 theorem separationRel_iff_specializes {a b : α} : (a, b) ∈ 𝓢 α ↔ a ⤳ b := by
   simp only [(𝓤 α).basis_sets.mem_separationRel, id, mem_setOf_eq,
     (nhds_basis_uniformity (𝓤 α).basis_sets).specializes_iff]
+#align separation_rel_iff_specializes separationRel_iff_specializes
 
--- porting note: new lemma
 theorem separationRel_iff_inseparable {a b : α} : (a, b) ∈ 𝓢 α ↔ Inseparable a b :=
   separationRel_iff_specializes.trans specializes_iff_inseparable
+#align separation_rel_iff_inseparable separationRel_iff_inseparable
 
 /-- A uniform space is separated if its separation relation is trivial (each point
 is related only to itself). -/
@@ -366,6 +366,10 @@ instance : SeparatedSpace (SeparationQuotient α) :=
 instance [Inhabited α] : Inhabited (SeparationQuotient α) :=
   inferInstanceAs (Inhabited (Quotient (separationSetoid α)))
 
+lemma mk_eq_mk {x y : α} : (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=
+Quotient.eq'.trans separationRel_iff_inseparable
+#align uniform_space.separation_quotient.mk_eq_mk UniformSpace.SeparationQuotient.mk_eq_mk
+
 /-- Factoring functions to a separated space through the separation quotient. -/
 def lift [SeparatedSpace β] (f : α → β) : SeparationQuotient α → β :=
   if h : UniformContinuous f then Quotient.lift f fun _ _ => eq_of_separated_of_uniformContinuous h