topology.maps ⟷ Mathlib.Topology.Maps

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

@@ -643,10 +643,10 @@ theorem of_nonempty {f : α → β} (h : ∀ s, IsClosed s → s.Nonempty → Is
 #align is_closed_map.of_nonempty IsClosedMap.of_nonempty
 -/
 
-#print IsClosedMap.closed_range /-
-theorem closed_range {f : α → β} (hf : IsClosedMap f) : IsClosed (range f) :=
+#print IsClosedMap.isClosed_range /-
+theorem isClosed_range {f : α → β} (hf : IsClosedMap f) : IsClosed (range f) :=
   @image_univ _ _ f ▸ hf _ isClosed_univ
-#align is_closed_map.closed_range IsClosedMap.closed_range
+#align is_closed_map.closed_range IsClosedMap.isClosed_range
 -/
 
 #print IsClosedMap.to_quotientMap /-
@@ -815,7 +815,7 @@ variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 /-- A closed embedding is an embedding with closed image. -/
 @[mk_iff]
 structure ClosedEmbedding (f : α → β) extends Embedding f : Prop where
-  closed_range : IsClosed <| range f
+  isClosed_range : IsClosed <| range f
 #align closed_embedding ClosedEmbedding
 -/
 
@@ -836,7 +836,7 @@ theorem ClosedEmbedding.continuous (hf : ClosedEmbedding f) : Continuous f :=
 
 #print ClosedEmbedding.isClosedMap /-
 theorem ClosedEmbedding.isClosedMap (hf : ClosedEmbedding f) : IsClosedMap f :=
-  hf.toEmbedding.to_inducing.IsClosedMap hf.closed_range
+  hf.toEmbedding.to_inducing.IsClosedMap hf.isClosed_range
 #align closed_embedding.is_closed_map ClosedEmbedding.isClosedMap
 -/
 

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

@@ -3,8 +3,8 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 -/
-import Mathbin.Topology.Order
-import Mathbin.Topology.NhdsSet
+import Topology.Order
+import Topology.NhdsSet
 
 #align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
 

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

@@ -336,7 +336,7 @@ def QuotientMap {α : Type _} {β : Type _} [tα : TopologicalSpace α] [tβ : T
 #print quotientMap_iff /-
 theorem quotientMap_iff [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
-  and_congr Iff.rfl topologicalSpace_eq_iff
+  and_congr Iff.rfl TopologicalSpace.ext_iff
 #align quotient_map_iff quotientMap_iff
 -/
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-
-! This file was ported from Lean 3 source module topology.maps
-! leanprover-community/mathlib commit d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.Order
 import Mathbin.Topology.NhdsSet
 
+#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
+
 /-!
 # Specific classes of maps between topological spaces
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/2a0ce625dbb0ffbc7d1316597de0b25c1ec75303

@@ -511,7 +511,7 @@ theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv :
 theorem to_quotientMap {f : α → β} (open_map : IsOpenMap f) (cont : Continuous f)
     (surj : Surjective f) : QuotientMap f :=
   quotientMap_iff.2
-    ⟨surj, fun s => ⟨fun h => h.Preimage cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩
+    ⟨surj, fun s => ⟨fun h => h.Preimage Cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩
 #align is_open_map.to_quotient_map IsOpenMap.to_quotientMap
 -/
 

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

@@ -77,68 +77,95 @@ theorem inducing_id : Inducing (@id α) :=
 #align inducing_id inducing_id
 -/
 
+#print Inducing.comp /-
 protected theorem Inducing.comp {g : β → γ} {f : α → β} (hg : Inducing g) (hf : Inducing f) :
     Inducing (g ∘ f) :=
   ⟨by rw [hf.induced, hg.induced, induced_compose]⟩
 #align inducing.comp Inducing.comp
+-/
 
+#print inducing_of_inducing_compose /-
 theorem inducing_of_inducing_compose {f : α → β} {g : β → γ} (hf : Continuous f) (hg : Continuous g)
     (hgf : Inducing (g ∘ f)) : Inducing f :=
   ⟨le_antisymm (by rwa [← continuous_iff_le_induced])
       (by rw [hgf.induced, ← continuous_iff_le_induced]; apply hg.comp continuous_induced_dom)⟩
 #align inducing_of_inducing_compose inducing_of_inducing_compose
+-/
 
+#print inducing_iff_nhds /-
 theorem inducing_iff_nhds {f : α → β} : Inducing f ↔ ∀ a, 𝓝 a = comap f (𝓝 (f a)) :=
   (inducing_iff _).trans (induced_iff_nhds_eq f)
 #align inducing_iff_nhds inducing_iff_nhds
+-/
 
+#print Inducing.nhds_eq_comap /-
 theorem Inducing.nhds_eq_comap {f : α → β} (hf : Inducing f) : ∀ a : α, 𝓝 a = comap f (𝓝 <| f a) :=
   inducing_iff_nhds.1 hf
 #align inducing.nhds_eq_comap Inducing.nhds_eq_comap
+-/
 
+#print Inducing.nhdsSet_eq_comap /-
 theorem Inducing.nhdsSet_eq_comap {f : α → β} (hf : Inducing f) (s : Set α) :
     𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
   simp only [nhdsSet, sSup_image, comap_supr, hf.nhds_eq_comap, iSup_image]
 #align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comap
+-/
 
+#print Inducing.map_nhds_eq /-
 theorem Inducing.map_nhds_eq {f : α → β} (hf : Inducing f) (a : α) : (𝓝 a).map f = 𝓝[range f] f a :=
   hf.induced.symm ▸ map_nhds_induced_eq a
 #align inducing.map_nhds_eq Inducing.map_nhds_eq
+-/
 
+#print Inducing.map_nhds_of_mem /-
 theorem Inducing.map_nhds_of_mem {f : α → β} (hf : Inducing f) (a : α) (h : range f ∈ 𝓝 (f a)) :
     (𝓝 a).map f = 𝓝 (f a) :=
   hf.induced.symm ▸ map_nhds_induced_of_mem h
 #align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem
+-/
 
+#print Inducing.image_mem_nhdsWithin /-
 theorem Inducing.image_mem_nhdsWithin {f : α → β} (hf : Inducing f) {a : α} {s : Set α}
     (hs : s ∈ 𝓝 a) : f '' s ∈ 𝓝[range f] f a :=
   hf.map_nhds_eq a ▸ image_mem_map hs
 #align inducing.image_mem_nhds_within Inducing.image_mem_nhdsWithin
+-/
 
+#print Inducing.tendsto_nhds_iff /-
 theorem Inducing.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : Inducing g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) := by
   rw [hg.nhds_eq_comap, tendsto_comap_iff]
 #align inducing.tendsto_nhds_iff Inducing.tendsto_nhds_iff
+-/
 
+#print Inducing.continuousAt_iff /-
 theorem Inducing.continuousAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {x : α} :
     ContinuousAt f x ↔ ContinuousAt (g ∘ f) x := by
   simp_rw [ContinuousAt, Inducing.tendsto_nhds_iff hg]
 #align inducing.continuous_at_iff Inducing.continuousAt_iff
+-/
 
+#print Inducing.continuous_iff /-
 theorem Inducing.continuous_iff {f : α → β} {g : β → γ} (hg : Inducing g) :
     Continuous f ↔ Continuous (g ∘ f) := by
   simp_rw [continuous_iff_continuousAt, hg.continuous_at_iff]
 #align inducing.continuous_iff Inducing.continuous_iff
+-/
 
+#print Inducing.continuousAt_iff' /-
 theorem Inducing.continuousAt_iff' {f : α → β} {g : β → γ} (hf : Inducing f) {x : α}
     (h : range f ∈ 𝓝 (f x)) : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
   simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map]
 #align inducing.continuous_at_iff' Inducing.continuousAt_iff'
+-/
 
+#print Inducing.continuous /-
 protected theorem Inducing.continuous {f : α → β} (hf : Inducing f) : Continuous f :=
   hf.continuous_iff.mp continuous_id
 #align inducing.continuous Inducing.continuous
+-/
 
+#print Inducing.inducing_iff /-
 protected theorem Inducing.inducing_iff {f : α → β} {g : β → γ} (hg : Inducing g) :
     Inducing f ↔ Inducing (g ∘ f) :=
   by
@@ -146,33 +173,46 @@ protected theorem Inducing.inducing_iff {f : α → β} {g : β → γ} (hg : In
   rw [hg.continuous_iff]
   exact hgf.continuous
 #align inducing.inducing_iff Inducing.inducing_iff
+-/
 
+#print Inducing.closure_eq_preimage_closure_image /-
 theorem Inducing.closure_eq_preimage_closure_image {f : α → β} (hf : Inducing f) (s : Set α) :
     closure s = f ⁻¹' closure (f '' s) := by ext x;
   rw [Set.mem_preimage, ← closure_induced, hf.induced]
 #align inducing.closure_eq_preimage_closure_image Inducing.closure_eq_preimage_closure_image
+-/
 
+#print Inducing.isClosed_iff /-
 theorem Inducing.isClosed_iff {f : α → β} (hf : Inducing f) {s : Set α} :
     IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by rw [hf.induced, isClosed_induced_iff]
 #align inducing.is_closed_iff Inducing.isClosed_iff
+-/
 
+#print Inducing.isClosed_iff' /-
 theorem Inducing.isClosed_iff' {f : α → β} (hf : Inducing f) {s : Set α} :
     IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by rw [hf.induced, isClosed_induced_iff']
 #align inducing.is_closed_iff' Inducing.isClosed_iff'
+-/
 
+#print Inducing.isClosed_preimage /-
 theorem Inducing.isClosed_preimage {f : α → β} (h : Inducing f) (s : Set β) (hs : IsClosed s) :
     IsClosed (f ⁻¹' s) :=
   (Inducing.isClosed_iff h).mpr ⟨s, hs, rfl⟩
 #align inducing.is_closed_preimage Inducing.isClosed_preimage
+-/
 
+#print Inducing.isOpen_iff /-
 theorem Inducing.isOpen_iff {f : α → β} (hf : Inducing f) {s : Set α} :
     IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := by rw [hf.induced, isOpen_induced_iff]
 #align inducing.is_open_iff Inducing.isOpen_iff
+-/
 
+#print Inducing.dense_iff /-
 theorem Inducing.dense_iff {f : α → β} (hf : Inducing f) {s : Set α} :
     Dense s ↔ ∀ x, f x ∈ closure (f '' s) := by
   simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage]
 #align inducing.dense_iff Inducing.dense_iff
+-/
 
 end Inducing
 
@@ -198,10 +238,12 @@ theorem Function.Injective.embedding_induced [t : TopologicalSpace β] {f : α 
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
+#print Embedding.mk' /-
 theorem Embedding.mk' (f : α → β) (inj : Injective f) (induced : ∀ a, comap f (𝓝 (f a)) = 𝓝 a) :
     Embedding f :=
   ⟨inducing_iff_nhds.2 fun a => (induced a).symm, inj⟩
 #align embedding.mk' Embedding.mk'
+-/
 
 #print embedding_id /-
 theorem embedding_id : Embedding (@id α) :=
@@ -209,16 +251,20 @@ theorem embedding_id : Embedding (@id α) :=
 #align embedding_id embedding_id
 -/
 
+#print Embedding.comp /-
 theorem Embedding.comp {g : β → γ} {f : α → β} (hg : Embedding g) (hf : Embedding f) :
     Embedding (g ∘ f) :=
   { hg.to_inducing.comp hf.to_inducing with inj := fun a₁ a₂ h => hf.inj <| hg.inj h }
 #align embedding.comp Embedding.comp
+-/
 
+#print embedding_of_embedding_compose /-
 theorem embedding_of_embedding_compose {f : α → β} {g : β → γ} (hf : Continuous f)
     (hg : Continuous g) (hgf : Embedding (g ∘ f)) : Embedding f :=
   { induced := (inducing_of_inducing_compose hf hg hgf.to_inducing).induced
     inj := fun a₁ a₂ h => hgf.inj <| by simp [h, (· ∘ ·)] }
 #align embedding_of_embedding_compose embedding_of_embedding_compose
+-/
 
 #print Function.LeftInverse.embedding /-
 protected theorem Function.LeftInverse.embedding {f : α → β} {g : β → α} (h : LeftInverse f g)
@@ -227,35 +273,48 @@ protected theorem Function.LeftInverse.embedding {f : α → β} {g : β → α}
 #align function.left_inverse.embedding Function.LeftInverse.embedding
 -/
 
+#print Embedding.map_nhds_eq /-
 theorem Embedding.map_nhds_eq {f : α → β} (hf : Embedding f) (a : α) :
     (𝓝 a).map f = 𝓝[range f] f a :=
   hf.1.map_nhds_eq a
 #align embedding.map_nhds_eq Embedding.map_nhds_eq
+-/
 
+#print Embedding.map_nhds_of_mem /-
 theorem Embedding.map_nhds_of_mem {f : α → β} (hf : Embedding f) (a : α) (h : range f ∈ 𝓝 (f a)) :
     (𝓝 a).map f = 𝓝 (f a) :=
   hf.1.map_nhds_of_mem a h
 #align embedding.map_nhds_of_mem Embedding.map_nhds_of_mem
+-/
 
+#print Embedding.tendsto_nhds_iff /-
 theorem Embedding.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : Embedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
   hg.to_inducing.tendsto_nhds_iff
 #align embedding.tendsto_nhds_iff Embedding.tendsto_nhds_iff
+-/
 
+#print Embedding.continuous_iff /-
 theorem Embedding.continuous_iff {f : α → β} {g : β → γ} (hg : Embedding g) :
     Continuous f ↔ Continuous (g ∘ f) :=
   Inducing.continuous_iff hg.1
 #align embedding.continuous_iff Embedding.continuous_iff
+-/
 
+#print Embedding.continuous /-
 theorem Embedding.continuous {f : α → β} (hf : Embedding f) : Continuous f :=
   Inducing.continuous hf.1
 #align embedding.continuous Embedding.continuous
+-/
 
+#print Embedding.closure_eq_preimage_closure_image /-
 theorem Embedding.closure_eq_preimage_closure_image {e : α → β} (he : Embedding e) (s : Set α) :
     closure s = e ⁻¹' closure (e '' s) :=
   he.1.closure_eq_preimage_closure_image s
 #align embedding.closure_eq_preimage_closure_image Embedding.closure_eq_preimage_closure_image
+-/
 
+#print Embedding.discreteTopology /-
 /-- The topology induced under an inclusion `f : X → Y` from the discrete topological space `Y`
 is the discrete topology on `X`. -/
 theorem Embedding.discreteTopology {X Y : Type _} [TopologicalSpace X] [tY : TopologicalSpace Y]
@@ -264,6 +323,7 @@ theorem Embedding.discreteTopology {X Y : Type _} [TopologicalSpace X] [tY : Top
     rw [hf.nhds_eq_comap, nhds_discrete, comap_pure, ← image_singleton, hf.inj.preimage_image,
       principal_singleton]
 #align embedding.discrete_topology Embedding.discreteTopology
+-/
 
 end Embedding
 
@@ -276,16 +336,20 @@ def QuotientMap {α : Type _} {β : Type _} [tα : TopologicalSpace α] [tβ : T
 #align quotient_map QuotientMap
 -/
 
+#print quotientMap_iff /-
 theorem quotientMap_iff [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   and_congr Iff.rfl topologicalSpace_eq_iff
 #align quotient_map_iff quotientMap_iff
+-/
 
+#print quotientMap_iff_closed /-
 theorem quotientMap_iff_closed [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsClosed s ↔ IsClosed (f ⁻¹' s) :=
   quotientMap_iff.trans <|
     Iff.rfl.And <| compl_surjective.forall.trans <| by simp only [isOpen_compl_iff, preimage_compl]
 #align quotient_map_iff_closed quotientMap_iff_closed
+-/
 
 namespace QuotientMap
 
@@ -298,10 +362,13 @@ protected theorem id : QuotientMap (@id α) :=
 #align quotient_map.id QuotientMap.id
 -/
 
+#print QuotientMap.comp /-
 protected theorem comp (hg : QuotientMap g) (hf : QuotientMap f) : QuotientMap (g ∘ f) :=
   ⟨hg.left.comp hf.left, by rw [hg.right, hf.right, coinduced_compose]⟩
 #align quotient_map.comp QuotientMap.comp
+-/
 
+#print QuotientMap.of_quotientMap_compose /-
 protected theorem of_quotientMap_compose (hf : Continuous f) (hg : Continuous g)
     (hgf : QuotientMap (g ∘ f)) : QuotientMap g :=
   ⟨hgf.1.of_comp,
@@ -309,32 +376,45 @@ protected theorem of_quotientMap_compose (hf : Continuous f) (hg : Continuous g)
       (by rw [hgf.right, ← continuous_iff_coinduced_le]; apply continuous_coinduced_rng.comp hf)
       (by rwa [← continuous_iff_coinduced_le])⟩
 #align quotient_map.of_quotient_map_compose QuotientMap.of_quotientMap_compose
+-/
 
+#print QuotientMap.of_inverse /-
 theorem of_inverse {g : β → α} (hf : Continuous f) (hg : Continuous g) (h : LeftInverse g f) :
     QuotientMap g :=
   QuotientMap.of_quotientMap_compose hf hg <| h.comp_eq_id.symm ▸ QuotientMap.id
 #align quotient_map.of_inverse QuotientMap.of_inverse
+-/
 
+#print QuotientMap.continuous_iff /-
 protected theorem continuous_iff (hf : QuotientMap f) : Continuous g ↔ Continuous (g ∘ f) := by
   rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.right, coinduced_compose]
 #align quotient_map.continuous_iff QuotientMap.continuous_iff
+-/
 
+#print QuotientMap.continuous /-
 protected theorem continuous (hf : QuotientMap f) : Continuous f :=
   hf.continuous_iff.mp continuous_id
 #align quotient_map.continuous QuotientMap.continuous
+-/
 
+#print QuotientMap.surjective /-
 protected theorem surjective (hf : QuotientMap f) : Surjective f :=
   hf.1
 #align quotient_map.surjective QuotientMap.surjective
+-/
 
+#print QuotientMap.isOpen_preimage /-
 protected theorem isOpen_preimage (hf : QuotientMap f) {s : Set β} : IsOpen (f ⁻¹' s) ↔ IsOpen s :=
   ((quotientMap_iff.1 hf).2 s).symm
 #align quotient_map.is_open_preimage QuotientMap.isOpen_preimage
+-/
 
+#print QuotientMap.isClosed_preimage /-
 protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set β} :
     IsClosed (f ⁻¹' s) ↔ IsClosed s :=
   ((quotientMap_iff_closed.1 hf).2 s).symm
 #align quotient_map.is_closed_preimage QuotientMap.isClosed_preimage
+-/
 
 end QuotientMap
 
@@ -355,42 +435,59 @@ protected theorem id : IsOpenMap (@id α) := fun s hs => by rwa [image_id]
 #align is_open_map.id IsOpenMap.id
 -/
 
+#print IsOpenMap.comp /-
 protected theorem comp {g : β → γ} {f : α → β} (hg : IsOpenMap g) (hf : IsOpenMap f) :
     IsOpenMap (g ∘ f) := by intro s hs <;> rw [image_comp] <;> exact hg _ (hf _ hs)
 #align is_open_map.comp IsOpenMap.comp
+-/
 
+#print IsOpenMap.isOpen_range /-
 theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by rw [← image_univ];
   exact hf _ isOpen_univ
 #align is_open_map.is_open_range IsOpenMap.isOpen_range
+-/
 
+#print IsOpenMap.image_mem_nhds /-
 theorem image_mem_nhds (hf : IsOpenMap f) {x : α} {s : Set α} (hx : s ∈ 𝓝 x) : f '' s ∈ 𝓝 (f x) :=
   let ⟨t, hts, ht, hxt⟩ := mem_nhds_iff.1 hx
   mem_of_superset (IsOpen.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts)
 #align is_open_map.image_mem_nhds IsOpenMap.image_mem_nhds
+-/
 
+#print IsOpenMap.range_mem_nhds /-
 theorem range_mem_nhds (hf : IsOpenMap f) (x : α) : range f ∈ 𝓝 (f x) :=
   hf.isOpen_range.mem_nhds <| mem_range_self _
 #align is_open_map.range_mem_nhds IsOpenMap.range_mem_nhds
+-/
 
+#print IsOpenMap.mapsTo_interior /-
 theorem mapsTo_interior (hf : IsOpenMap f) {s : Set α} {t : Set β} (h : MapsTo f s t) :
     MapsTo f (interior s) (interior t) :=
   mapsTo'.2 <|
     interior_maximal (h.mono interior_subset Subset.rfl).image_subset (hf _ isOpen_interior)
 #align is_open_map.maps_to_interior IsOpenMap.mapsTo_interior
+-/
 
+#print IsOpenMap.image_interior_subset /-
 theorem image_interior_subset (hf : IsOpenMap f) (s : Set α) :
     f '' interior s ⊆ interior (f '' s) :=
   (hf.mapsTo_interior (mapsTo_image f s)).image_subset
 #align is_open_map.image_interior_subset IsOpenMap.image_interior_subset
+-/
 
+#print IsOpenMap.nhds_le /-
 theorem nhds_le (hf : IsOpenMap f) (a : α) : 𝓝 (f a) ≤ (𝓝 a).map f :=
   le_map fun s => hf.image_mem_nhds
 #align is_open_map.nhds_le IsOpenMap.nhds_le
+-/
 
+#print IsOpenMap.of_nhds_le /-
 theorem of_nhds_le (hf : ∀ a, 𝓝 (f a) ≤ map f (𝓝 a)) : IsOpenMap f := fun s hs =>
   isOpen_iff_mem_nhds.2 fun b ⟨a, has, hab⟩ => hab ▸ hf _ (image_mem_map <| IsOpen.mem_nhds hs has)
 #align is_open_map.of_nhds_le IsOpenMap.of_nhds_le
+-/
 
+#print IsOpenMap.of_sections /-
 theorem of_sections {f : α → β}
     (h : ∀ x, ∃ g : β → α, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f :=
   of_nhds_le fun x =>
@@ -400,6 +497,7 @@ theorem of_sections {f : α → β}
       _ ≤ map f (𝓝 (g (f x))) := (map_mono hgc)
       _ = map f (𝓝 x) := by rw [hgx]
 #align is_open_map.of_sections IsOpenMap.of_sections
+-/
 
 #print IsOpenMap.of_inverse /-
 theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
@@ -408,56 +506,73 @@ theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv :
 #align is_open_map.of_inverse IsOpenMap.of_inverse
 -/
 
+#print IsOpenMap.to_quotientMap /-
 /-- A continuous surjective open map is a quotient map. -/
 theorem to_quotientMap {f : α → β} (open_map : IsOpenMap f) (cont : Continuous f)
     (surj : Surjective f) : QuotientMap f :=
   quotientMap_iff.2
     ⟨surj, fun s => ⟨fun h => h.Preimage cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩
 #align is_open_map.to_quotient_map IsOpenMap.to_quotientMap
+-/
 
+#print IsOpenMap.interior_preimage_subset_preimage_interior /-
 theorem interior_preimage_subset_preimage_interior (hf : IsOpenMap f) {s : Set β} :
     interior (f ⁻¹' s) ⊆ f ⁻¹' interior s :=
   hf.mapsTo_interior (mapsTo_preimage _ _)
 #align is_open_map.interior_preimage_subset_preimage_interior IsOpenMap.interior_preimage_subset_preimage_interior
+-/
 
+#print IsOpenMap.preimage_interior_eq_interior_preimage /-
 theorem preimage_interior_eq_interior_preimage (hf₁ : IsOpenMap f) (hf₂ : Continuous f)
     (s : Set β) : f ⁻¹' interior s = interior (f ⁻¹' s) :=
   Subset.antisymm (preimage_interior_subset_interior_preimage hf₂)
     (interior_preimage_subset_preimage_interior hf₁)
 #align is_open_map.preimage_interior_eq_interior_preimage IsOpenMap.preimage_interior_eq_interior_preimage
+-/
 
+#print IsOpenMap.preimage_closure_subset_closure_preimage /-
 theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set β} :
     f ⁻¹' closure s ⊆ closure (f ⁻¹' s) :=
   by
   rw [← compl_subset_compl]
   simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior]
 #align is_open_map.preimage_closure_subset_closure_preimage IsOpenMap.preimage_closure_subset_closure_preimage
+-/
 
+#print IsOpenMap.preimage_closure_eq_closure_preimage /-
 theorem preimage_closure_eq_closure_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set β) :
     f ⁻¹' closure s = closure (f ⁻¹' s) :=
   hf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s)
 #align is_open_map.preimage_closure_eq_closure_preimage IsOpenMap.preimage_closure_eq_closure_preimage
+-/
 
+#print IsOpenMap.preimage_frontier_subset_frontier_preimage /-
 theorem preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set β} :
     f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s) := by
   simpa only [frontier_eq_closure_inter_closure, preimage_inter] using
     inter_subset_inter hf.preimage_closure_subset_closure_preimage
       hf.preimage_closure_subset_closure_preimage
 #align is_open_map.preimage_frontier_subset_frontier_preimage IsOpenMap.preimage_frontier_subset_frontier_preimage
+-/
 
+#print IsOpenMap.preimage_frontier_eq_frontier_preimage /-
 theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set β) :
     f ⁻¹' frontier s = frontier (f ⁻¹' s) := by
   simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl,
     hf.preimage_closure_eq_closure_preimage hfc]
 #align is_open_map.preimage_frontier_eq_frontier_preimage IsOpenMap.preimage_frontier_eq_frontier_preimage
+-/
 
 end IsOpenMap
 
+#print isOpenMap_iff_nhds_le /-
 theorem isOpenMap_iff_nhds_le [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     IsOpenMap f ↔ ∀ a : α, 𝓝 (f a) ≤ (𝓝 a).map f :=
   ⟨fun hf => hf.nhds_le, IsOpenMap.of_nhds_le⟩
 #align is_open_map_iff_nhds_le isOpenMap_iff_nhds_le
+-/
 
+#print isOpenMap_iff_interior /-
 theorem isOpenMap_iff_interior [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     IsOpenMap f ↔ ∀ s, f '' interior s ⊆ interior (f '' s) :=
   ⟨IsOpenMap.image_interior_subset, fun hs u hu =>
@@ -466,12 +581,15 @@ theorem isOpenMap_iff_interior [TopologicalSpace α] [TopologicalSpace β] {f :
         f '' u = f '' interior u := by rw [hu.interior_eq]
         _ ⊆ interior (f '' u) := hs u⟩
 #align is_open_map_iff_interior isOpenMap_iff_interior
+-/
 
+#print Inducing.isOpenMap /-
 /-- An inducing map with an open range is an open map. -/
 protected theorem Inducing.isOpenMap [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
     (hi : Inducing f) (ho : IsOpen (range f)) : IsOpenMap f :=
   IsOpenMap.of_nhds_le fun x => (hi.map_nhds_of_mem _ <| IsOpen.mem_nhds ho <| mem_range_self _).ge
 #align inducing.is_open_map Inducing.isOpenMap
+-/
 
 section IsClosedMap
 
@@ -498,14 +616,18 @@ protected theorem id : IsClosedMap (@id α) := fun s hs => by rwa [image_id]
 #align is_closed_map.id IsClosedMap.id
 -/
 
+#print IsClosedMap.comp /-
 protected theorem comp {g : β → γ} {f : α → β} (hg : IsClosedMap g) (hf : IsClosedMap f) :
     IsClosedMap (g ∘ f) := by intro s hs; rw [image_comp]; exact hg _ (hf _ hs)
 #align is_closed_map.comp IsClosedMap.comp
+-/
 
+#print IsClosedMap.closure_image_subset /-
 theorem closure_image_subset {f : α → β} (hf : IsClosedMap f) (s : Set α) :
     closure (f '' s) ⊆ f '' closure s :=
   closure_minimal (image_subset _ subset_closure) (hf _ isClosed_closure)
 #align is_closed_map.closure_image_subset IsClosedMap.closure_image_subset
+-/
 
 #print IsClosedMap.of_inverse /-
 theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
@@ -515,25 +637,32 @@ theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv :
 #align is_closed_map.of_inverse IsClosedMap.of_inverse
 -/
 
+#print IsClosedMap.of_nonempty /-
 theorem of_nonempty {f : α → β} (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
     IsClosedMap f := by
   intro s hs; cases' eq_empty_or_nonempty s with h2s h2s
   · simp_rw [h2s, image_empty, isClosed_empty]
   · exact h s hs h2s
 #align is_closed_map.of_nonempty IsClosedMap.of_nonempty
+-/
 
+#print IsClosedMap.closed_range /-
 theorem closed_range {f : α → β} (hf : IsClosedMap f) : IsClosed (range f) :=
   @image_univ _ _ f ▸ hf _ isClosed_univ
 #align is_closed_map.closed_range IsClosedMap.closed_range
+-/
 
+#print IsClosedMap.to_quotientMap /-
 theorem to_quotientMap {f : α → β} (hcl : IsClosedMap f) (hcont : Continuous f)
     (hsurj : Surjective f) : QuotientMap f :=
   quotientMap_iff_closed.2
     ⟨hsurj, fun s => ⟨fun hs => hs.Preimage hcont, fun hs => hsurj.image_preimage s ▸ hcl _ hs⟩⟩
 #align is_closed_map.to_quotient_map IsClosedMap.to_quotientMap
+-/
 
 end IsClosedMap
 
+#print Inducing.isClosedMap /-
 theorem Inducing.isClosedMap [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Inducing f)
     (h : IsClosed (range f)) : IsClosedMap f :=
   by
@@ -542,7 +671,9 @@ theorem Inducing.isClosedMap [TopologicalSpace α] [TopologicalSpace β] {f : α
   rw [image_preimage_eq_inter_range]
   exact ht.inter h
 #align inducing.is_closed_map Inducing.isClosedMap
+-/
 
+#print isClosedMap_iff_closure_image /-
 theorem isClosedMap_iff_closure_image [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     IsClosedMap f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s :=
   ⟨IsClosedMap.closure_image_subset, fun hs c hc =>
@@ -551,6 +682,7 @@ theorem isClosedMap_iff_closure_image [TopologicalSpace α] [TopologicalSpace β
         closure (f '' c) ⊆ f '' closure c := hs c
         _ = f '' c := by rw [hc.closure_eq]⟩
 #align is_closed_map_iff_closure_image isClosedMap_iff_closure_image
+-/
 
 section OpenEmbedding
 
@@ -564,15 +696,20 @@ structure OpenEmbedding (f : α → β) extends Embedding f : Prop where
 #align open_embedding OpenEmbedding
 -/
 
+#print OpenEmbedding.isOpenMap /-
 theorem OpenEmbedding.isOpenMap {f : α → β} (hf : OpenEmbedding f) : IsOpenMap f :=
   hf.toEmbedding.to_inducing.IsOpenMap hf.open_range
 #align open_embedding.is_open_map OpenEmbedding.isOpenMap
+-/
 
+#print OpenEmbedding.map_nhds_eq /-
 theorem OpenEmbedding.map_nhds_eq {f : α → β} (hf : OpenEmbedding f) (a : α) :
     map f (𝓝 a) = 𝓝 (f a) :=
   hf.toEmbedding.map_nhds_of_mem _ <| hf.open_range.mem_nhds <| mem_range_self _
 #align open_embedding.map_nhds_eq OpenEmbedding.map_nhds_eq
+-/
 
+#print OpenEmbedding.open_iff_image_open /-
 theorem OpenEmbedding.open_iff_image_open {f : α → β} (hf : OpenEmbedding f) {s : Set α} :
     IsOpen s ↔ IsOpen (f '' s) :=
   ⟨hf.IsOpenMap s, fun h =>
@@ -580,33 +717,45 @@ theorem OpenEmbedding.open_iff_image_open {f : α → β} (hf : OpenEmbedding f)
     convert ← h.preimage hf.to_embedding.continuous
     apply preimage_image_eq _ hf.inj⟩
 #align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_open
+-/
 
+#print OpenEmbedding.tendsto_nhds_iff /-
 theorem OpenEmbedding.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : OpenEmbedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
   hg.toEmbedding.tendsto_nhds_iff
 #align open_embedding.tendsto_nhds_iff OpenEmbedding.tendsto_nhds_iff
+-/
 
+#print OpenEmbedding.continuous /-
 theorem OpenEmbedding.continuous {f : α → β} (hf : OpenEmbedding f) : Continuous f :=
   hf.toEmbedding.Continuous
 #align open_embedding.continuous OpenEmbedding.continuous
+-/
 
+#print OpenEmbedding.open_iff_preimage_open /-
 theorem OpenEmbedding.open_iff_preimage_open {f : α → β} (hf : OpenEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   by
   convert ← hf.open_iff_image_open.symm
   rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
 #align open_embedding.open_iff_preimage_open OpenEmbedding.open_iff_preimage_open
+-/
 
+#print openEmbedding_of_embedding_open /-
 theorem openEmbedding_of_embedding_open {f : α → β} (h₁ : Embedding f) (h₂ : IsOpenMap f) :
     OpenEmbedding f :=
   ⟨h₁, h₂.isOpen_range⟩
 #align open_embedding_of_embedding_open openEmbedding_of_embedding_open
+-/
 
+#print openEmbedding_iff_embedding_open /-
 theorem openEmbedding_iff_embedding_open {f : α → β} :
     OpenEmbedding f ↔ Embedding f ∧ IsOpenMap f :=
   ⟨fun h => ⟨h.1, h.IsOpenMap⟩, fun h => openEmbedding_of_embedding_open h.1 h.2⟩
 #align open_embedding_iff_embedding_open openEmbedding_iff_embedding_open
+-/
 
+#print openEmbedding_of_continuous_injective_open /-
 theorem openEmbedding_of_continuous_injective_open {f : α → β} (h₁ : Continuous f)
     (h₂ : Injective f) (h₃ : IsOpenMap f) : OpenEmbedding f :=
   by
@@ -614,12 +763,15 @@ theorem openEmbedding_of_continuous_injective_open {f : α → β} (h₁ : Conti
   exact fun a =>
     le_antisymm (h₁.tendsto _).le_comap (@comap_map _ _ (𝓝 a) _ h₂ ▸ comap_mono (h₃.nhds_le _))
 #align open_embedding_of_continuous_injective_open openEmbedding_of_continuous_injective_open
+-/
 
+#print openEmbedding_iff_continuous_injective_open /-
 theorem openEmbedding_iff_continuous_injective_open {f : α → β} :
     OpenEmbedding f ↔ Continuous f ∧ Injective f ∧ IsOpenMap f :=
   ⟨fun h => ⟨h.Continuous, h.inj, h.IsOpenMap⟩, fun h =>
     openEmbedding_of_continuous_injective_open h.1 h.2.1 h.2.2⟩
 #align open_embedding_iff_continuous_injective_open openEmbedding_iff_continuous_injective_open
+-/
 
 #print openEmbedding_id /-
 theorem openEmbedding_id : OpenEmbedding (@id α) :=
@@ -627,26 +779,34 @@ theorem openEmbedding_id : OpenEmbedding (@id α) :=
 #align open_embedding_id openEmbedding_id
 -/
 
+#print OpenEmbedding.comp /-
 theorem OpenEmbedding.comp {g : β → γ} {f : α → β} (hg : OpenEmbedding g) (hf : OpenEmbedding f) :
     OpenEmbedding (g ∘ f) :=
   ⟨hg.1.comp hf.1, (hg.IsOpenMap.comp hf.IsOpenMap).isOpen_range⟩
 #align open_embedding.comp OpenEmbedding.comp
+-/
 
+#print OpenEmbedding.isOpenMap_iff /-
 theorem OpenEmbedding.isOpenMap_iff {g : β → γ} {f : α → β} (hg : OpenEmbedding g) :
     IsOpenMap f ↔ IsOpenMap (g ∘ f) := by
   simp only [isOpenMap_iff_nhds_le, ← @map_map _ _ _ _ f g, ← hg.map_nhds_eq, map_le_map_iff hg.inj]
 #align open_embedding.is_open_map_iff OpenEmbedding.isOpenMap_iff
+-/
 
+#print OpenEmbedding.of_comp_iff /-
 theorem OpenEmbedding.of_comp_iff (f : α → β) {g : β → γ} (hg : OpenEmbedding g) :
     OpenEmbedding (g ∘ f) ↔ OpenEmbedding f := by
   simp only [openEmbedding_iff_continuous_injective_open, ← hg.is_open_map_iff, ←
     hg.1.continuous_iff, hg.inj.of_comp_iff]
 #align open_embedding.of_comp_iff OpenEmbedding.of_comp_iff
+-/
 
+#print OpenEmbedding.of_comp /-
 theorem OpenEmbedding.of_comp (f : α → β) {g : β → γ} (hg : OpenEmbedding g)
     (h : OpenEmbedding (g ∘ f)) : OpenEmbedding f :=
   (OpenEmbedding.of_comp_iff f hg).1 h
 #align open_embedding.of_comp OpenEmbedding.of_comp
+-/
 
 end OpenEmbedding
 
@@ -664,19 +824,26 @@ structure ClosedEmbedding (f : α → β) extends Embedding f : Prop where
 
 variable {f : α → β}
 
+#print ClosedEmbedding.tendsto_nhds_iff /-
 theorem ClosedEmbedding.tendsto_nhds_iff {ι : Type _} {g : ι → α} {a : Filter ι} {b : α}
     (hf : ClosedEmbedding f) : Tendsto g a (𝓝 b) ↔ Tendsto (f ∘ g) a (𝓝 (f b)) :=
   hf.toEmbedding.tendsto_nhds_iff
 #align closed_embedding.tendsto_nhds_iff ClosedEmbedding.tendsto_nhds_iff
+-/
 
+#print ClosedEmbedding.continuous /-
 theorem ClosedEmbedding.continuous (hf : ClosedEmbedding f) : Continuous f :=
   hf.toEmbedding.Continuous
 #align closed_embedding.continuous ClosedEmbedding.continuous
+-/
 
+#print ClosedEmbedding.isClosedMap /-
 theorem ClosedEmbedding.isClosedMap (hf : ClosedEmbedding f) : IsClosedMap f :=
   hf.toEmbedding.to_inducing.IsClosedMap hf.closed_range
 #align closed_embedding.is_closed_map ClosedEmbedding.isClosedMap
+-/
 
+#print ClosedEmbedding.closed_iff_image_closed /-
 theorem ClosedEmbedding.closed_iff_image_closed (hf : ClosedEmbedding f) {s : Set α} :
     IsClosed s ↔ IsClosed (f '' s) :=
   ⟨hf.IsClosedMap s, fun h =>
@@ -684,19 +851,25 @@ theorem ClosedEmbedding.closed_iff_image_closed (hf : ClosedEmbedding f) {s : Se
     convert ← continuous_iff_is_closed.mp hf.continuous _ h
     apply preimage_image_eq _ hf.inj⟩
 #align closed_embedding.closed_iff_image_closed ClosedEmbedding.closed_iff_image_closed
+-/
 
+#print ClosedEmbedding.closed_iff_preimage_closed /-
 theorem ClosedEmbedding.closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) :=
   by
   convert ← hf.closed_iff_image_closed.symm
   rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
 #align closed_embedding.closed_iff_preimage_closed ClosedEmbedding.closed_iff_preimage_closed
+-/
 
+#print closedEmbedding_of_embedding_closed /-
 theorem closedEmbedding_of_embedding_closed (h₁ : Embedding f) (h₂ : IsClosedMap f) :
     ClosedEmbedding f :=
   ⟨h₁, by convert h₂ univ isClosed_univ <;> simp⟩
 #align closed_embedding_of_embedding_closed closedEmbedding_of_embedding_closed
+-/
 
+#print closedEmbedding_of_continuous_injective_closed /-
 theorem closedEmbedding_of_continuous_injective_closed (h₁ : Continuous f) (h₂ : Injective f)
     (h₃ : IsClosedMap f) : ClosedEmbedding f :=
   by
@@ -710,6 +883,7 @@ theorem closedEmbedding_of_continuous_injective_closed (h₁ : Continuous f) (h
   refine' fun hs => ⟨f '' s, h₃ s hs, _⟩
   rw [preimage_image_eq _ h₂]
 #align closed_embedding_of_continuous_injective_closed closedEmbedding_of_continuous_injective_closed
+-/
 
 #print closedEmbedding_id /-
 theorem closedEmbedding_id : ClosedEmbedding (@id α) :=
@@ -717,18 +891,22 @@ theorem closedEmbedding_id : ClosedEmbedding (@id α) :=
 #align closed_embedding_id closedEmbedding_id
 -/
 
+#print ClosedEmbedding.comp /-
 theorem ClosedEmbedding.comp {g : β → γ} {f : α → β} (hg : ClosedEmbedding g)
     (hf : ClosedEmbedding f) : ClosedEmbedding (g ∘ f) :=
   ⟨hg.toEmbedding.comp hf.toEmbedding,
     show IsClosed (range (g ∘ f)) by
       rw [range_comp, ← hg.closed_iff_image_closed] <;> exact hf.closed_range⟩
 #align closed_embedding.comp ClosedEmbedding.comp
+-/
 
+#print ClosedEmbedding.closure_image_eq /-
 theorem ClosedEmbedding.closure_image_eq {f : α → β} (hf : ClosedEmbedding f) (s : Set α) :
     closure (f '' s) = f '' closure s :=
   (hf.IsClosedMap.closure_image_subset _).antisymm
     (image_closure_subset_closure_image hf.Continuous)
 #align closed_embedding.closure_image_eq ClosedEmbedding.closure_image_eq
+-/
 
 end ClosedEmbedding
 

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

@@ -399,7 +399,6 @@ theorem of_sections {f : α → β}
       𝓝 (f x) = map f (map g (𝓝 (f x))) := by rw [map_map, hgf.comp_eq_id, map_id]
       _ ≤ map f (𝓝 (g (f x))) := (map_mono hgc)
       _ = map f (𝓝 x) := by rw [hgx]
-      
 #align is_open_map.of_sections IsOpenMap.of_sections
 
 #print IsOpenMap.of_inverse /-
@@ -465,8 +464,7 @@ theorem isOpenMap_iff_interior [TopologicalSpace α] [TopologicalSpace β] {f :
     subset_interior_iff_isOpen.mp <|
       calc
         f '' u = f '' interior u := by rw [hu.interior_eq]
-        _ ⊆ interior (f '' u) := hs u
-        ⟩
+        _ ⊆ interior (f '' u) := hs u⟩
 #align is_open_map_iff_interior isOpenMap_iff_interior
 
 /-- An inducing map with an open range is an open map. -/
@@ -551,8 +549,7 @@ theorem isClosedMap_iff_closure_image [TopologicalSpace α] [TopologicalSpace β
     isClosed_of_closure_subset <|
       calc
         closure (f '' c) ⊆ f '' closure c := hs c
-        _ = f '' c := by rw [hc.closure_eq]
-        ⟩
+        _ = f '' c := by rw [hc.closure_eq]⟩
 #align is_closed_map_iff_closure_image isClosedMap_iff_closure_image
 
 section OpenEmbedding

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

@@ -580,7 +580,7 @@ theorem OpenEmbedding.open_iff_image_open {f : α → β} (hf : OpenEmbedding f)
     IsOpen s ↔ IsOpen (f '' s) :=
   ⟨hf.IsOpenMap s, fun h =>
     by
-    convert← h.preimage hf.to_embedding.continuous
+    convert ← h.preimage hf.to_embedding.continuous
     apply preimage_image_eq _ hf.inj⟩
 #align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_open
 
@@ -596,7 +596,7 @@ theorem OpenEmbedding.continuous {f : α → β} (hf : OpenEmbedding f) : Contin
 theorem OpenEmbedding.open_iff_preimage_open {f : α → β} (hf : OpenEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   by
-  convert← hf.open_iff_image_open.symm
+  convert ← hf.open_iff_image_open.symm
   rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
 #align open_embedding.open_iff_preimage_open OpenEmbedding.open_iff_preimage_open
 
@@ -684,14 +684,14 @@ theorem ClosedEmbedding.closed_iff_image_closed (hf : ClosedEmbedding f) {s : Se
     IsClosed s ↔ IsClosed (f '' s) :=
   ⟨hf.IsClosedMap s, fun h =>
     by
-    convert← continuous_iff_is_closed.mp hf.continuous _ h
+    convert ← continuous_iff_is_closed.mp hf.continuous _ h
     apply preimage_image_eq _ hf.inj⟩
 #align closed_embedding.closed_iff_image_closed ClosedEmbedding.closed_iff_image_closed
 
 theorem ClosedEmbedding.closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) :=
   by
-  convert← hf.closed_iff_image_closed.symm
+  convert ← hf.closed_iff_image_closed.symm
   rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
 #align closed_embedding.closed_iff_preimage_closed ClosedEmbedding.closed_iff_preimage_closed
 

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

@@ -183,7 +183,7 @@ section Embedding
   and for all `s : set α`, `s` is open iff it is the preimage of an open set. -/
 @[mk_iff]
 structure Embedding [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : α → β) extends
-  Inducing f : Prop where
+    Inducing f : Prop where
   inj : Injective f
 #align embedding Embedding
 -/

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

@@ -53,7 +53,7 @@ open map, closed map, embedding, quotient map, identification map
 
 open Set Filter Function
 
-open Topology Filter
+open scoped Topology Filter
 
 variable {α : Type _} {β : Type _} {γ : Type _} {δ : Type _}
 

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

@@ -77,152 +77,68 @@ theorem inducing_id : Inducing (@id α) :=
 #align inducing_id inducing_id
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.comp Inducing.compₓ'. -/
 protected theorem Inducing.comp {g : β → γ} {f : α → β} (hg : Inducing g) (hf : Inducing f) :
     Inducing (g ∘ f) :=
   ⟨by rw [hf.induced, hg.induced, induced_compose]⟩
 #align inducing.comp Inducing.comp
 
-/- warning: inducing_of_inducing_compose -> inducing_of_inducing_compose is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align inducing_of_inducing_compose inducing_of_inducing_composeₓ'. -/
 theorem inducing_of_inducing_compose {f : α → β} {g : β → γ} (hf : Continuous f) (hg : Continuous g)
     (hgf : Inducing (g ∘ f)) : Inducing f :=
   ⟨le_antisymm (by rwa [← continuous_iff_le_induced])
       (by rw [hgf.induced, ← continuous_iff_le_induced]; apply hg.comp continuous_induced_dom)⟩
 #align inducing_of_inducing_compose inducing_of_inducing_compose
 
-/- warning: inducing_iff_nhds -> inducing_iff_nhds is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing_iff_nhds inducing_iff_nhdsₓ'. -/
 theorem inducing_iff_nhds {f : α → β} : Inducing f ↔ ∀ a, 𝓝 a = comap f (𝓝 (f a)) :=
   (inducing_iff _).trans (induced_iff_nhds_eq f)
 #align inducing_iff_nhds inducing_iff_nhds
 
-/- warning: inducing.nhds_eq_comap -> Inducing.nhds_eq_comap is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align inducing.nhds_eq_comap Inducing.nhds_eq_comapₓ'. -/
 theorem Inducing.nhds_eq_comap {f : α → β} (hf : Inducing f) : ∀ a : α, 𝓝 a = comap f (𝓝 <| f a) :=
   inducing_iff_nhds.1 hf
 #align inducing.nhds_eq_comap Inducing.nhds_eq_comap
 
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-Case conversion may be inaccurate. Consider using '#align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comapₓ'. -/
 theorem Inducing.nhdsSet_eq_comap {f : α → β} (hf : Inducing f) (s : Set α) :
     𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
   simp only [nhdsSet, sSup_image, comap_supr, hf.nhds_eq_comap, iSup_image]
 #align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comap
 
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-Case conversion may be inaccurate. Consider using '#align inducing.map_nhds_eq Inducing.map_nhds_eqₓ'. -/
 theorem Inducing.map_nhds_eq {f : α → β} (hf : Inducing f) (a : α) : (𝓝 a).map f = 𝓝[range f] f a :=
   hf.induced.symm ▸ map_nhds_induced_eq a
 #align inducing.map_nhds_eq Inducing.map_nhds_eq
 
-/- warning: inducing.map_nhds_of_mem -> Inducing.map_nhds_of_mem is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align inducing.map_nhds_of_mem Inducing.map_nhds_of_memₓ'. -/
 theorem Inducing.map_nhds_of_mem {f : α → β} (hf : Inducing f) (a : α) (h : range f ∈ 𝓝 (f a)) :
     (𝓝 a).map f = 𝓝 (f a) :=
   hf.induced.symm ▸ map_nhds_induced_of_mem h
 #align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem
 
-/- warning: inducing.image_mem_nhds_within -> Inducing.image_mem_nhdsWithin is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align inducing.image_mem_nhds_within Inducing.image_mem_nhdsWithinₓ'. -/
 theorem Inducing.image_mem_nhdsWithin {f : α → β} (hf : Inducing f) {a : α} {s : Set α}
     (hs : s ∈ 𝓝 a) : f '' s ∈ 𝓝[range f] f a :=
   hf.map_nhds_eq a ▸ image_mem_map hs
 #align inducing.image_mem_nhds_within Inducing.image_mem_nhdsWithin
 
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-Case conversion may be inaccurate. Consider using '#align inducing.tendsto_nhds_iff Inducing.tendsto_nhds_iffₓ'. -/
 theorem Inducing.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : Inducing g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) := by
   rw [hg.nhds_eq_comap, tendsto_comap_iff]
 #align inducing.tendsto_nhds_iff Inducing.tendsto_nhds_iff
 
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-Case conversion may be inaccurate. Consider using '#align inducing.continuous_at_iff Inducing.continuousAt_iffₓ'. -/
 theorem Inducing.continuousAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {x : α} :
     ContinuousAt f x ↔ ContinuousAt (g ∘ f) x := by
   simp_rw [ContinuousAt, Inducing.tendsto_nhds_iff hg]
 #align inducing.continuous_at_iff Inducing.continuousAt_iff
 
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-Case conversion may be inaccurate. Consider using '#align inducing.continuous_iff Inducing.continuous_iffₓ'. -/
 theorem Inducing.continuous_iff {f : α → β} {g : β → γ} (hg : Inducing g) :
     Continuous f ↔ Continuous (g ∘ f) := by
   simp_rw [continuous_iff_continuousAt, hg.continuous_at_iff]
 #align inducing.continuous_iff Inducing.continuous_iff
 
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 theorem Inducing.continuousAt_iff' {f : α → β} {g : β → γ} (hf : Inducing f) {x : α}
     (h : range f ∈ 𝓝 (f x)) : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
   simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map]
 #align inducing.continuous_at_iff' Inducing.continuousAt_iff'
 
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-Case conversion may be inaccurate. Consider using '#align inducing.continuous Inducing.continuousₓ'. -/
 protected theorem Inducing.continuous {f : α → β} (hf : Inducing f) : Continuous f :=
   hf.continuous_iff.mp continuous_id
 #align inducing.continuous Inducing.continuous
 
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-Case conversion may be inaccurate. Consider using '#align inducing.inducing_iff Inducing.inducing_iffₓ'. -/
 protected theorem Inducing.inducing_iff {f : α → β} {g : β → γ} (hg : Inducing g) :
     Inducing f ↔ Inducing (g ∘ f) :=
   by
@@ -231,64 +147,28 @@ protected theorem Inducing.inducing_iff {f : α → β} {g : β → γ} (hg : In
   exact hgf.continuous
 #align inducing.inducing_iff Inducing.inducing_iff
 
-/- warning: inducing.closure_eq_preimage_closure_image -> Inducing.closure_eq_preimage_closure_image is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (Inducing.{u1, u2} α β _inst_1 _inst_2 f) -> (forall (s : Set.{u1} α), Eq.{succ u1} (Set.{u1} α) (closure.{u1} α _inst_1 s) (Set.preimage.{u1, u2} α β f (closure.{u2} β _inst_2 (Set.image.{u1, u2} α β f s))))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.closure_eq_preimage_closure_image Inducing.closure_eq_preimage_closure_imageₓ'. -/
 theorem Inducing.closure_eq_preimage_closure_image {f : α → β} (hf : Inducing f) (s : Set α) :
     closure s = f ⁻¹' closure (f '' s) := by ext x;
   rw [Set.mem_preimage, ← closure_induced, hf.induced]
 #align inducing.closure_eq_preimage_closure_image Inducing.closure_eq_preimage_closure_image
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.is_closed_iff Inducing.isClosed_iffₓ'. -/
 theorem Inducing.isClosed_iff {f : α → β} (hf : Inducing f) {s : Set α} :
     IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by rw [hf.induced, isClosed_induced_iff]
 #align inducing.is_closed_iff Inducing.isClosed_iff
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.is_closed_iff' Inducing.isClosed_iff'ₓ'. -/
 theorem Inducing.isClosed_iff' {f : α → β} (hf : Inducing f) {s : Set α} :
     IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by rw [hf.induced, isClosed_induced_iff']
 #align inducing.is_closed_iff' Inducing.isClosed_iff'
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.is_closed_preimage Inducing.isClosed_preimageₓ'. -/
 theorem Inducing.isClosed_preimage {f : α → β} (h : Inducing f) (s : Set β) (hs : IsClosed s) :
     IsClosed (f ⁻¹' s) :=
   (Inducing.isClosed_iff h).mpr ⟨s, hs, rfl⟩
 #align inducing.is_closed_preimage Inducing.isClosed_preimage
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.is_open_iff Inducing.isOpen_iffₓ'. -/
 theorem Inducing.isOpen_iff {f : α → β} (hf : Inducing f) {s : Set α} :
     IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := by rw [hf.induced, isOpen_induced_iff]
 #align inducing.is_open_iff Inducing.isOpen_iff
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.dense_iff Inducing.dense_iffₓ'. -/
 theorem Inducing.dense_iff {f : α → β} (hf : Inducing f) {s : Set α} :
     Dense s ↔ ∀ x, f x ∈ closure (f '' s) := by
   simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage]
@@ -318,12 +198,6 @@ theorem Function.Injective.embedding_induced [t : TopologicalSpace β] {f : α 
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align embedding.mk' Embedding.mk'ₓ'. -/
 theorem Embedding.mk' (f : α → β) (inj : Injective f) (induced : ∀ a, comap f (𝓝 (f a)) = 𝓝 a) :
     Embedding f :=
   ⟨inducing_iff_nhds.2 fun a => (induced a).symm, inj⟩
@@ -335,23 +209,11 @@ theorem embedding_id : Embedding (@id α) :=
 #align embedding_id embedding_id
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align embedding.comp Embedding.compₓ'. -/
 theorem Embedding.comp {g : β → γ} {f : α → β} (hg : Embedding g) (hf : Embedding f) :
     Embedding (g ∘ f) :=
   { hg.to_inducing.comp hf.to_inducing with inj := fun a₁ a₂ h => hf.inj <| hg.inj h }
 #align embedding.comp Embedding.comp
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align embedding_of_embedding_compose embedding_of_embedding_composeₓ'. -/
 theorem embedding_of_embedding_compose {f : α → β} {g : β → γ} (hf : Continuous f)
     (hg : Continuous g) (hgf : Embedding (g ∘ f)) : Embedding f :=
   { induced := (inducing_of_inducing_compose hf hg hgf.to_inducing).induced
@@ -365,77 +227,35 @@ protected theorem Function.LeftInverse.embedding {f : α → β} {g : β → α}
 #align function.left_inverse.embedding Function.LeftInverse.embedding
 -/
 
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-Case conversion may be inaccurate. Consider using '#align embedding.map_nhds_eq Embedding.map_nhds_eqₓ'. -/
 theorem Embedding.map_nhds_eq {f : α → β} (hf : Embedding f) (a : α) :
     (𝓝 a).map f = 𝓝[range f] f a :=
   hf.1.map_nhds_eq a
 #align embedding.map_nhds_eq Embedding.map_nhds_eq
 
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-Case conversion may be inaccurate. Consider using '#align embedding.map_nhds_of_mem Embedding.map_nhds_of_memₓ'. -/
 theorem Embedding.map_nhds_of_mem {f : α → β} (hf : Embedding f) (a : α) (h : range f ∈ 𝓝 (f a)) :
     (𝓝 a).map f = 𝓝 (f a) :=
   hf.1.map_nhds_of_mem a h
 #align embedding.map_nhds_of_mem Embedding.map_nhds_of_mem
 
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-Case conversion may be inaccurate. Consider using '#align embedding.tendsto_nhds_iff Embedding.tendsto_nhds_iffₓ'. -/
 theorem Embedding.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : Embedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
   hg.to_inducing.tendsto_nhds_iff
 #align embedding.tendsto_nhds_iff Embedding.tendsto_nhds_iff
 
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-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align embedding.continuous_iff Embedding.continuous_iffₓ'. -/
 theorem Embedding.continuous_iff {f : α → β} {g : β → γ} (hg : Embedding g) :
     Continuous f ↔ Continuous (g ∘ f) :=
   Inducing.continuous_iff hg.1
 #align embedding.continuous_iff Embedding.continuous_iff
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align embedding.continuous Embedding.continuousₓ'. -/
 theorem Embedding.continuous {f : α → β} (hf : Embedding f) : Continuous f :=
   Inducing.continuous hf.1
 #align embedding.continuous Embedding.continuous
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align embedding.closure_eq_preimage_closure_image Embedding.closure_eq_preimage_closure_imageₓ'. -/
 theorem Embedding.closure_eq_preimage_closure_image {e : α → β} (he : Embedding e) (s : Set α) :
     closure s = e ⁻¹' closure (e '' s) :=
   he.1.closure_eq_preimage_closure_image s
 #align embedding.closure_eq_preimage_closure_image Embedding.closure_eq_preimage_closure_image
 
-/- warning: embedding.discrete_topology -> Embedding.discreteTopology is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} {Y : Type.{u2}} [_inst_4 : TopologicalSpace.{u1} X] [tY : TopologicalSpace.{u2} Y] [_inst_5 : DiscreteTopology.{u2} Y tY] {f : X -> Y}, (Embedding.{u1, u2} X Y _inst_4 tY f) -> (DiscreteTopology.{u1} X _inst_4)
-but is expected to have type
-  forall {X : Type.{u2}} {Y : Type.{u1}} [_inst_4 : TopologicalSpace.{u2} X] [tY : TopologicalSpace.{u1} Y] [_inst_5 : DiscreteTopology.{u1} Y tY] {f : X -> Y}, (Embedding.{u2, u1} X Y _inst_4 tY f) -> (DiscreteTopology.{u2} X _inst_4)
-Case conversion may be inaccurate. Consider using '#align embedding.discrete_topology Embedding.discreteTopologyₓ'. -/
 /-- The topology induced under an inclusion `f : X → Y` from the discrete topological space `Y`
 is the discrete topology on `X`. -/
 theorem Embedding.discreteTopology {X Y : Type _} [TopologicalSpace X] [tY : TopologicalSpace Y]
@@ -456,12 +276,6 @@ def QuotientMap {α : Type _} {β : Type _} [tα : TopologicalSpace α] [tβ : T
 #align quotient_map QuotientMap
 -/
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align quotient_map_iff quotientMap_iffₓ'. -/
 theorem quotientMap_iff [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   and_congr Iff.rfl topologicalSpace_eq_iff
@@ -484,22 +298,10 @@ protected theorem id : QuotientMap (@id α) :=
 #align quotient_map.id QuotientMap.id
 -/
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align quotient_map.comp QuotientMap.compₓ'. -/
 protected theorem comp (hg : QuotientMap g) (hf : QuotientMap f) : QuotientMap (g ∘ f) :=
   ⟨hg.left.comp hf.left, by rw [hg.right, hf.right, coinduced_compose]⟩
 #align quotient_map.comp QuotientMap.comp
 
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-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Continuous.{u2, u3} β γ _inst_2 _inst_3 g) -> (QuotientMap.{u1, u3} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u2, succ u3} α β γ g f)) -> (QuotientMap.{u2, u3} β γ _inst_2 _inst_3 g)
-but is expected to have type
-  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u1} γ] {g : β -> γ} {f : α -> β}, (Continuous.{u3, u2} α β _inst_1 _inst_2 f) -> (Continuous.{u2, u1} β γ _inst_2 _inst_3 g) -> (QuotientMap.{u3, u1} α γ _inst_1 _inst_3 (Function.comp.{succ u3, succ u2, succ u1} α β γ g f)) -> (QuotientMap.{u2, u1} β γ _inst_2 _inst_3 g)
-Case conversion may be inaccurate. Consider using '#align quotient_map.of_quotient_map_compose QuotientMap.of_quotientMap_composeₓ'. -/
 protected theorem of_quotientMap_compose (hf : Continuous f) (hg : Continuous g)
     (hgf : QuotientMap (g ∘ f)) : QuotientMap g :=
   ⟨hgf.1.of_comp,
@@ -508,63 +310,27 @@ protected theorem of_quotientMap_compose (hf : Continuous f) (hg : Continuous g)
       (by rwa [← continuous_iff_coinduced_le])⟩
 #align quotient_map.of_quotient_map_compose QuotientMap.of_quotientMap_compose
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align quotient_map.of_inverse QuotientMap.of_inverseₓ'. -/
 theorem of_inverse {g : β → α} (hf : Continuous f) (hg : Continuous g) (h : LeftInverse g f) :
     QuotientMap g :=
   QuotientMap.of_quotientMap_compose hf hg <| h.comp_eq_id.symm ▸ QuotientMap.id
 #align quotient_map.of_inverse QuotientMap.of_inverse
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align quotient_map.continuous_iff QuotientMap.continuous_iffₓ'. -/
 protected theorem continuous_iff (hf : QuotientMap f) : Continuous g ↔ Continuous (g ∘ f) := by
   rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.right, coinduced_compose]
 #align quotient_map.continuous_iff QuotientMap.continuous_iff
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align quotient_map.continuous QuotientMap.continuousₓ'. -/
 protected theorem continuous (hf : QuotientMap f) : Continuous f :=
   hf.continuous_iff.mp continuous_id
 #align quotient_map.continuous QuotientMap.continuous
 
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-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (QuotientMap.{u1, u2} α β _inst_1 _inst_2 f) -> (Function.Surjective.{succ u1, succ u2} α β f)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align quotient_map.surjective QuotientMap.surjectiveₓ'. -/
 protected theorem surjective (hf : QuotientMap f) : Surjective f :=
   hf.1
 #align quotient_map.surjective QuotientMap.surjective
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align quotient_map.is_open_preimage QuotientMap.isOpen_preimageₓ'. -/
 protected theorem isOpen_preimage (hf : QuotientMap f) {s : Set β} : IsOpen (f ⁻¹' s) ↔ IsOpen s :=
   ((quotientMap_iff.1 hf).2 s).symm
 #align quotient_map.is_open_preimage QuotientMap.isOpen_preimage
 
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-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (QuotientMap.{u1, u2} α β _inst_1 _inst_2 f) -> (forall {s : Set.{u2} β}, Iff (IsClosed.{u1} α _inst_1 (Set.preimage.{u1, u2} α β f s)) (IsClosed.{u2} β _inst_2 s))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align quotient_map.is_closed_preimage QuotientMap.isClosed_preimageₓ'. -/
 protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set β} :
     IsClosed (f ⁻¹' s) ↔ IsClosed s :=
   ((quotientMap_iff_closed.1 hf).2 s).symm
@@ -589,96 +355,42 @@ protected theorem id : IsOpenMap (@id α) := fun s hs => by rwa [image_id]
 #align is_open_map.id IsOpenMap.id
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open_map.comp IsOpenMap.compₓ'. -/
 protected theorem comp {g : β → γ} {f : α → β} (hg : IsOpenMap g) (hf : IsOpenMap f) :
     IsOpenMap (g ∘ f) := by intro s hs <;> rw [image_comp] <;> exact hg _ (hf _ hs)
 #align is_open_map.comp IsOpenMap.comp
 
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-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f) -> (IsOpen.{u2} β _inst_2 (Set.range.{u2, succ u1} β α f))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open_map.is_open_range IsOpenMap.isOpen_rangeₓ'. -/
 theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by rw [← image_univ];
   exact hf _ isOpen_univ
 #align is_open_map.is_open_range IsOpenMap.isOpen_range
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open_map.image_mem_nhds IsOpenMap.image_mem_nhdsₓ'. -/
 theorem image_mem_nhds (hf : IsOpenMap f) {x : α} {s : Set α} (hx : s ∈ 𝓝 x) : f '' s ∈ 𝓝 (f x) :=
   let ⟨t, hts, ht, hxt⟩ := mem_nhds_iff.1 hx
   mem_of_superset (IsOpen.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts)
 #align is_open_map.image_mem_nhds IsOpenMap.image_mem_nhds
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open_map.range_mem_nhds IsOpenMap.range_mem_nhdsₓ'. -/
 theorem range_mem_nhds (hf : IsOpenMap f) (x : α) : range f ∈ 𝓝 (f x) :=
   hf.isOpen_range.mem_nhds <| mem_range_self _
 #align is_open_map.range_mem_nhds IsOpenMap.range_mem_nhds
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open_map.maps_to_interior IsOpenMap.mapsTo_interiorₓ'. -/
 theorem mapsTo_interior (hf : IsOpenMap f) {s : Set α} {t : Set β} (h : MapsTo f s t) :
     MapsTo f (interior s) (interior t) :=
   mapsTo'.2 <|
     interior_maximal (h.mono interior_subset Subset.rfl).image_subset (hf _ isOpen_interior)
 #align is_open_map.maps_to_interior IsOpenMap.mapsTo_interior
 
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-Case conversion may be inaccurate. Consider using '#align is_open_map.image_interior_subset IsOpenMap.image_interior_subsetₓ'. -/
 theorem image_interior_subset (hf : IsOpenMap f) (s : Set α) :
     f '' interior s ⊆ interior (f '' s) :=
   (hf.mapsTo_interior (mapsTo_image f s)).image_subset
 #align is_open_map.image_interior_subset IsOpenMap.image_interior_subset
 
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-Case conversion may be inaccurate. Consider using '#align is_open_map.nhds_le IsOpenMap.nhds_leₓ'. -/
 theorem nhds_le (hf : IsOpenMap f) (a : α) : 𝓝 (f a) ≤ (𝓝 a).map f :=
   le_map fun s => hf.image_mem_nhds
 #align is_open_map.nhds_le IsOpenMap.nhds_le
 
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 theorem of_nhds_le (hf : ∀ a, 𝓝 (f a) ≤ map f (𝓝 a)) : IsOpenMap f := fun s hs =>
   isOpen_iff_mem_nhds.2 fun b ⟨a, has, hab⟩ => hab ▸ hf _ (image_mem_map <| IsOpen.mem_nhds hs has)
 #align is_open_map.of_nhds_le IsOpenMap.of_nhds_le
 
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 theorem of_sections {f : α → β}
     (h : ∀ x, ∃ g : β → α, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f :=
   of_nhds_le fun x =>
@@ -697,12 +409,6 @@ theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv :
 #align is_open_map.of_inverse IsOpenMap.of_inverse
 -/
 
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 /-- A continuous surjective open map is a quotient map. -/
 theorem to_quotientMap {f : α → β} (open_map : IsOpenMap f) (cont : Continuous f)
     (surj : Surjective f) : QuotientMap f :=
@@ -710,35 +416,17 @@ theorem to_quotientMap {f : α → β} (open_map : IsOpenMap f) (cont : Continuo
     ⟨surj, fun s => ⟨fun h => h.Preimage cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩
 #align is_open_map.to_quotient_map IsOpenMap.to_quotientMap
 
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 theorem interior_preimage_subset_preimage_interior (hf : IsOpenMap f) {s : Set β} :
     interior (f ⁻¹' s) ⊆ f ⁻¹' interior s :=
   hf.mapsTo_interior (mapsTo_preimage _ _)
 #align is_open_map.interior_preimage_subset_preimage_interior IsOpenMap.interior_preimage_subset_preimage_interior
 
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 theorem preimage_interior_eq_interior_preimage (hf₁ : IsOpenMap f) (hf₂ : Continuous f)
     (s : Set β) : f ⁻¹' interior s = interior (f ⁻¹' s) :=
   Subset.antisymm (preimage_interior_subset_interior_preimage hf₂)
     (interior_preimage_subset_preimage_interior hf₁)
 #align is_open_map.preimage_interior_eq_interior_preimage IsOpenMap.preimage_interior_eq_interior_preimage
 
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 theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set β} :
     f ⁻¹' closure s ⊆ closure (f ⁻¹' s) :=
   by
@@ -746,23 +434,11 @@ theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set β}
   simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior]
 #align is_open_map.preimage_closure_subset_closure_preimage IsOpenMap.preimage_closure_subset_closure_preimage
 
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-Case conversion may be inaccurate. Consider using '#align is_open_map.preimage_closure_eq_closure_preimage IsOpenMap.preimage_closure_eq_closure_preimageₓ'. -/
 theorem preimage_closure_eq_closure_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set β) :
     f ⁻¹' closure s = closure (f ⁻¹' s) :=
   hf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s)
 #align is_open_map.preimage_closure_eq_closure_preimage IsOpenMap.preimage_closure_eq_closure_preimage
 
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 theorem preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set β} :
     f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s) := by
   simpa only [frontier_eq_closure_inter_closure, preimage_inter] using
@@ -770,12 +446,6 @@ theorem preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set 
       hf.preimage_closure_subset_closure_preimage
 #align is_open_map.preimage_frontier_subset_frontier_preimage IsOpenMap.preimage_frontier_subset_frontier_preimage
 
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-Case conversion may be inaccurate. Consider using '#align is_open_map.preimage_frontier_eq_frontier_preimage IsOpenMap.preimage_frontier_eq_frontier_preimageₓ'. -/
 theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set β) :
     f ⁻¹' frontier s = frontier (f ⁻¹' s) := by
   simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl,
@@ -784,23 +454,11 @@ theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continu
 
 end IsOpenMap
 
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-Case conversion may be inaccurate. Consider using '#align is_open_map_iff_nhds_le isOpenMap_iff_nhds_leₓ'. -/
 theorem isOpenMap_iff_nhds_le [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     IsOpenMap f ↔ ∀ a : α, 𝓝 (f a) ≤ (𝓝 a).map f :=
   ⟨fun hf => hf.nhds_le, IsOpenMap.of_nhds_le⟩
 #align is_open_map_iff_nhds_le isOpenMap_iff_nhds_le
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_open_map_iff_interior isOpenMap_iff_interiorₓ'. -/
 theorem isOpenMap_iff_interior [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     IsOpenMap f ↔ ∀ s, f '' interior s ⊆ interior (f '' s) :=
   ⟨IsOpenMap.image_interior_subset, fun hs u hu =>
@@ -811,12 +469,6 @@ theorem isOpenMap_iff_interior [TopologicalSpace α] [TopologicalSpace β] {f :
         ⟩
 #align is_open_map_iff_interior isOpenMap_iff_interior
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.is_open_map Inducing.isOpenMapₓ'. -/
 /-- An inducing map with an open range is an open map. -/
 protected theorem Inducing.isOpenMap [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
     (hi : Inducing f) (ho : IsOpen (range f)) : IsOpenMap f :=
@@ -848,22 +500,10 @@ protected theorem id : IsClosedMap (@id α) := fun s hs => by rwa [image_id]
 #align is_closed_map.id IsClosedMap.id
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_closed_map.comp IsClosedMap.compₓ'. -/
 protected theorem comp {g : β → γ} {f : α → β} (hg : IsClosedMap g) (hf : IsClosedMap f) :
     IsClosedMap (g ∘ f) := by intro s hs; rw [image_comp]; exact hg _ (hf _ hs)
 #align is_closed_map.comp IsClosedMap.comp
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_closed_map.closure_image_subset IsClosedMap.closure_image_subsetₓ'. -/
 theorem closure_image_subset {f : α → β} (hf : IsClosedMap f) (s : Set α) :
     closure (f '' s) ⊆ f '' closure s :=
   closure_minimal (image_subset _ subset_closure) (hf _ isClosed_closure)
@@ -877,12 +517,6 @@ theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv :
 #align is_closed_map.of_inverse IsClosedMap.of_inverse
 -/
 
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_closed_map.of_nonempty IsClosedMap.of_nonemptyₓ'. -/
 theorem of_nonempty {f : α → β} (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
     IsClosedMap f := by
   intro s hs; cases' eq_empty_or_nonempty s with h2s h2s
@@ -890,12 +524,6 @@ theorem of_nonempty {f : α → β} (h : ∀ s, IsClosed s → s.Nonempty → Is
   · exact h s hs h2s
 #align is_closed_map.of_nonempty IsClosedMap.of_nonempty
 
-/- warning: is_closed_map.closed_range -> IsClosedMap.closed_range is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_closed_map.closed_range IsClosedMap.closed_rangeₓ'. -/
 theorem closed_range {f : α → β} (hf : IsClosedMap f) : IsClosed (range f) :=
   @image_univ _ _ f ▸ hf _ isClosed_univ
 #align is_closed_map.closed_range IsClosedMap.closed_range
@@ -908,12 +536,6 @@ theorem to_quotientMap {f : α → β} (hcl : IsClosedMap f) (hcont : Continuous
 
 end IsClosedMap
 
-/- warning: inducing.is_closed_map -> Inducing.isClosedMap is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align inducing.is_closed_map Inducing.isClosedMapₓ'. -/
 theorem Inducing.isClosedMap [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Inducing f)
     (h : IsClosed (range f)) : IsClosedMap f :=
   by
@@ -923,12 +545,6 @@ theorem Inducing.isClosedMap [TopologicalSpace α] [TopologicalSpace β] {f : α
   exact ht.inter h
 #align inducing.is_closed_map Inducing.isClosedMap
 
-/- warning: is_closed_map_iff_closure_image -> isClosedMap_iff_closure_image is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_closed_map_iff_closure_image isClosedMap_iff_closure_imageₓ'. -/
 theorem isClosedMap_iff_closure_image [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     IsClosedMap f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s :=
   ⟨IsClosedMap.closure_image_subset, fun hs c hc =>
@@ -951,33 +567,15 @@ structure OpenEmbedding (f : α → β) extends Embedding f : Prop where
 #align open_embedding OpenEmbedding
 -/
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding.is_open_map OpenEmbedding.isOpenMapₓ'. -/
 theorem OpenEmbedding.isOpenMap {f : α → β} (hf : OpenEmbedding f) : IsOpenMap f :=
   hf.toEmbedding.to_inducing.IsOpenMap hf.open_range
 #align open_embedding.is_open_map OpenEmbedding.isOpenMap
 
-/- warning: open_embedding.map_nhds_eq -> OpenEmbedding.map_nhds_eq is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding.map_nhds_eq OpenEmbedding.map_nhds_eqₓ'. -/
 theorem OpenEmbedding.map_nhds_eq {f : α → β} (hf : OpenEmbedding f) (a : α) :
     map f (𝓝 a) = 𝓝 (f a) :=
   hf.toEmbedding.map_nhds_of_mem _ <| hf.open_range.mem_nhds <| mem_range_self _
 #align open_embedding.map_nhds_eq OpenEmbedding.map_nhds_eq
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_openₓ'. -/
 theorem OpenEmbedding.open_iff_image_open {f : α → β} (hf : OpenEmbedding f) {s : Set α} :
     IsOpen s ↔ IsOpen (f '' s) :=
   ⟨hf.IsOpenMap s, fun h =>
@@ -986,33 +584,15 @@ theorem OpenEmbedding.open_iff_image_open {f : α → β} (hf : OpenEmbedding f)
     apply preimage_image_eq _ hf.inj⟩
 #align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_open
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding.tendsto_nhds_iff OpenEmbedding.tendsto_nhds_iffₓ'. -/
 theorem OpenEmbedding.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : OpenEmbedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
   hg.toEmbedding.tendsto_nhds_iff
 #align open_embedding.tendsto_nhds_iff OpenEmbedding.tendsto_nhds_iff
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding.continuous OpenEmbedding.continuousₓ'. -/
 theorem OpenEmbedding.continuous {f : α → β} (hf : OpenEmbedding f) : Continuous f :=
   hf.toEmbedding.Continuous
 #align open_embedding.continuous OpenEmbedding.continuous
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding.open_iff_preimage_open OpenEmbedding.open_iff_preimage_openₓ'. -/
 theorem OpenEmbedding.open_iff_preimage_open {f : α → β} (hf : OpenEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   by
@@ -1020,34 +600,16 @@ theorem OpenEmbedding.open_iff_preimage_open {f : α → β} (hf : OpenEmbedding
   rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
 #align open_embedding.open_iff_preimage_open OpenEmbedding.open_iff_preimage_open
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding_of_embedding_open openEmbedding_of_embedding_openₓ'. -/
 theorem openEmbedding_of_embedding_open {f : α → β} (h₁ : Embedding f) (h₂ : IsOpenMap f) :
     OpenEmbedding f :=
   ⟨h₁, h₂.isOpen_range⟩
 #align open_embedding_of_embedding_open openEmbedding_of_embedding_open
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding_iff_embedding_open openEmbedding_iff_embedding_openₓ'. -/
 theorem openEmbedding_iff_embedding_open {f : α → β} :
     OpenEmbedding f ↔ Embedding f ∧ IsOpenMap f :=
   ⟨fun h => ⟨h.1, h.IsOpenMap⟩, fun h => openEmbedding_of_embedding_open h.1 h.2⟩
 #align open_embedding_iff_embedding_open openEmbedding_iff_embedding_open
 
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-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding_of_continuous_injective_open openEmbedding_of_continuous_injective_openₓ'. -/
 theorem openEmbedding_of_continuous_injective_open {f : α → β} (h₁ : Continuous f)
     (h₂ : Injective f) (h₃ : IsOpenMap f) : OpenEmbedding f :=
   by
@@ -1056,12 +618,6 @@ theorem openEmbedding_of_continuous_injective_open {f : α → β} (h₁ : Conti
     le_antisymm (h₁.tendsto _).le_comap (@comap_map _ _ (𝓝 a) _ h₂ ▸ comap_mono (h₃.nhds_le _))
 #align open_embedding_of_continuous_injective_open openEmbedding_of_continuous_injective_open
 
-/- warning: open_embedding_iff_continuous_injective_open -> openEmbedding_iff_continuous_injective_open is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, Iff (OpenEmbedding.{u1, u2} α β _inst_1 _inst_2 f) (And (Continuous.{u1, u2} α β _inst_1 _inst_2 f) (And (Function.Injective.{succ u1, succ u2} α β f) (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f)))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, Iff (OpenEmbedding.{u2, u1} α β _inst_1 _inst_2 f) (And (Continuous.{u2, u1} α β _inst_1 _inst_2 f) (And (Function.Injective.{succ u2, succ u1} α β f) (IsOpenMap.{u2, u1} α β _inst_1 _inst_2 f)))
-Case conversion may be inaccurate. Consider using '#align open_embedding_iff_continuous_injective_open openEmbedding_iff_continuous_injective_openₓ'. -/
 theorem openEmbedding_iff_continuous_injective_open {f : α → β} :
     OpenEmbedding f ↔ Continuous f ∧ Injective f ∧ IsOpenMap f :=
   ⟨fun h => ⟨h.Continuous, h.inj, h.IsOpenMap⟩, fun h =>
@@ -1074,46 +630,22 @@ theorem openEmbedding_id : OpenEmbedding (@id α) :=
 #align open_embedding_id openEmbedding_id
 -/
 
-/- warning: open_embedding.comp -> OpenEmbedding.comp is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding.comp OpenEmbedding.compₓ'. -/
 theorem OpenEmbedding.comp {g : β → γ} {f : α → β} (hg : OpenEmbedding g) (hf : OpenEmbedding f) :
     OpenEmbedding (g ∘ f) :=
   ⟨hg.1.comp hf.1, (hg.IsOpenMap.comp hf.IsOpenMap).isOpen_range⟩
 #align open_embedding.comp OpenEmbedding.comp
 
-/- warning: open_embedding.is_open_map_iff -> OpenEmbedding.isOpenMap_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {g : β -> γ} {f : α -> β}, (OpenEmbedding.{u2, u3} β γ _inst_2 _inst_3 g) -> (Iff (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f) (IsOpenMap.{u1, u3} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u2, succ u3} α β γ g f)))
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align open_embedding.is_open_map_iff OpenEmbedding.isOpenMap_iffₓ'. -/
 theorem OpenEmbedding.isOpenMap_iff {g : β → γ} {f : α → β} (hg : OpenEmbedding g) :
     IsOpenMap f ↔ IsOpenMap (g ∘ f) := by
   simp only [isOpenMap_iff_nhds_le, ← @map_map _ _ _ _ f g, ← hg.map_nhds_eq, map_le_map_iff hg.inj]
 #align open_embedding.is_open_map_iff OpenEmbedding.isOpenMap_iff
 
-/- warning: open_embedding.of_comp_iff -> OpenEmbedding.of_comp_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] (f : α -> β) {g : β -> γ}, (OpenEmbedding.{u2, u3} β γ _inst_2 _inst_3 g) -> (Iff (OpenEmbedding.{u1, u3} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u2, succ u3} α β γ g f)) (OpenEmbedding.{u1, u2} α β _inst_1 _inst_2 f))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] (f : α -> β) {g : β -> γ}, (OpenEmbedding.{u3, u2} β γ _inst_2 _inst_3 g) -> (Iff (OpenEmbedding.{u1, u2} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u3, succ u2} α β γ g f)) (OpenEmbedding.{u1, u3} α β _inst_1 _inst_2 f))
-Case conversion may be inaccurate. Consider using '#align open_embedding.of_comp_iff OpenEmbedding.of_comp_iffₓ'. -/
 theorem OpenEmbedding.of_comp_iff (f : α → β) {g : β → γ} (hg : OpenEmbedding g) :
     OpenEmbedding (g ∘ f) ↔ OpenEmbedding f := by
   simp only [openEmbedding_iff_continuous_injective_open, ← hg.is_open_map_iff, ←
     hg.1.continuous_iff, hg.inj.of_comp_iff]
 #align open_embedding.of_comp_iff OpenEmbedding.of_comp_iff
 
-/- warning: open_embedding.of_comp -> OpenEmbedding.of_comp is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] (f : α -> β) {g : β -> γ}, (OpenEmbedding.{u2, u3} β γ _inst_2 _inst_3 g) -> (OpenEmbedding.{u1, u3} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u2, succ u3} α β γ g f)) -> (OpenEmbedding.{u1, u2} α β _inst_1 _inst_2 f)
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] (f : α -> β) {g : β -> γ}, (OpenEmbedding.{u3, u2} β γ _inst_2 _inst_3 g) -> (OpenEmbedding.{u1, u2} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u3, succ u2} α β γ g f)) -> (OpenEmbedding.{u1, u3} α β _inst_1 _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align open_embedding.of_comp OpenEmbedding.of_compₓ'. -/
 theorem OpenEmbedding.of_comp (f : α → β) {g : β → γ} (hg : OpenEmbedding g)
     (h : OpenEmbedding (g ∘ f)) : OpenEmbedding f :=
   (OpenEmbedding.of_comp_iff f hg).1 h
@@ -1135,43 +667,19 @@ structure ClosedEmbedding (f : α → β) extends Embedding f : Prop where
 
 variable {f : α → β}
 
-/- warning: closed_embedding.tendsto_nhds_iff -> ClosedEmbedding.tendsto_nhds_iff is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β} {ι : Type.{u3}} {g : ι -> α} {a : Filter.{u3} ι} {b : α}, (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (Iff (Filter.Tendsto.{u3, u1} ι α g a (nhds.{u1} α _inst_1 b)) (Filter.Tendsto.{u3, u2} ι β (Function.comp.{succ u3, succ u1, succ u2} ι α β f g) a (nhds.{u2} β _inst_2 (f b))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β} {ι : Type.{u3}} {g : ι -> α} {a : Filter.{u3} ι} {b : α}, (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) -> (Iff (Filter.Tendsto.{u3, u2} ι α g a (nhds.{u2} α _inst_1 b)) (Filter.Tendsto.{u3, u1} ι β (Function.comp.{succ u3, succ u2, succ u1} ι α β f g) a (nhds.{u1} β _inst_2 (f b))))
-Case conversion may be inaccurate. Consider using '#align closed_embedding.tendsto_nhds_iff ClosedEmbedding.tendsto_nhds_iffₓ'. -/
 theorem ClosedEmbedding.tendsto_nhds_iff {ι : Type _} {g : ι → α} {a : Filter ι} {b : α}
     (hf : ClosedEmbedding f) : Tendsto g a (𝓝 b) ↔ Tendsto (f ∘ g) a (𝓝 (f b)) :=
   hf.toEmbedding.tendsto_nhds_iff
 #align closed_embedding.tendsto_nhds_iff ClosedEmbedding.tendsto_nhds_iff
 
-/- warning: closed_embedding.continuous -> ClosedEmbedding.continuous is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (Continuous.{u1, u2} α β _inst_1 _inst_2 f)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) -> (Continuous.{u2, u1} α β _inst_1 _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align closed_embedding.continuous ClosedEmbedding.continuousₓ'. -/
 theorem ClosedEmbedding.continuous (hf : ClosedEmbedding f) : Continuous f :=
   hf.toEmbedding.Continuous
 #align closed_embedding.continuous ClosedEmbedding.continuous
 
-/- warning: closed_embedding.is_closed_map -> ClosedEmbedding.isClosedMap is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (IsClosedMap.{u1, u2} α β _inst_1 _inst_2 f)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) -> (IsClosedMap.{u2, u1} α β _inst_1 _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align closed_embedding.is_closed_map ClosedEmbedding.isClosedMapₓ'. -/
 theorem ClosedEmbedding.isClosedMap (hf : ClosedEmbedding f) : IsClosedMap f :=
   hf.toEmbedding.to_inducing.IsClosedMap hf.closed_range
 #align closed_embedding.is_closed_map ClosedEmbedding.isClosedMap
 
-/- warning: closed_embedding.closed_iff_image_closed -> ClosedEmbedding.closed_iff_image_closed is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (forall {s : Set.{u1} α}, Iff (IsClosed.{u1} α _inst_1 s) (IsClosed.{u2} β _inst_2 (Set.image.{u1, u2} α β f s)))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) -> (forall {s : Set.{u2} α}, Iff (IsClosed.{u2} α _inst_1 s) (IsClosed.{u1} β _inst_2 (Set.image.{u2, u1} α β f s)))
-Case conversion may be inaccurate. Consider using '#align closed_embedding.closed_iff_image_closed ClosedEmbedding.closed_iff_image_closedₓ'. -/
 theorem ClosedEmbedding.closed_iff_image_closed (hf : ClosedEmbedding f) {s : Set α} :
     IsClosed s ↔ IsClosed (f '' s) :=
   ⟨hf.IsClosedMap s, fun h =>
@@ -1180,12 +688,6 @@ theorem ClosedEmbedding.closed_iff_image_closed (hf : ClosedEmbedding f) {s : Se
     apply preimage_image_eq _ hf.inj⟩
 #align closed_embedding.closed_iff_image_closed ClosedEmbedding.closed_iff_image_closed
 
-/- warning: closed_embedding.closed_iff_preimage_closed -> ClosedEmbedding.closed_iff_preimage_closed is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (forall {s : Set.{u2} β}, (HasSubset.Subset.{u2} (Set.{u2} β) (Set.hasSubset.{u2} β) s (Set.range.{u2, succ u1} β α f)) -> (Iff (IsClosed.{u2} β _inst_2 s) (IsClosed.{u1} α _inst_1 (Set.preimage.{u1, u2} α β f s))))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) -> (forall {s : Set.{u1} β}, (HasSubset.Subset.{u1} (Set.{u1} β) (Set.instHasSubsetSet.{u1} β) s (Set.range.{u1, succ u2} β α f)) -> (Iff (IsClosed.{u1} β _inst_2 s) (IsClosed.{u2} α _inst_1 (Set.preimage.{u2, u1} α β f s))))
-Case conversion may be inaccurate. Consider using '#align closed_embedding.closed_iff_preimage_closed ClosedEmbedding.closed_iff_preimage_closedₓ'. -/
 theorem ClosedEmbedding.closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) :=
   by
@@ -1193,23 +695,11 @@ theorem ClosedEmbedding.closed_iff_preimage_closed (hf : ClosedEmbedding f) {s :
   rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
 #align closed_embedding.closed_iff_preimage_closed ClosedEmbedding.closed_iff_preimage_closed
 
-/- warning: closed_embedding_of_embedding_closed -> closedEmbedding_of_embedding_closed is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (Embedding.{u1, u2} α β _inst_1 _inst_2 f) -> (IsClosedMap.{u1, u2} α β _inst_1 _inst_2 f) -> (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (Embedding.{u2, u1} α β _inst_1 _inst_2 f) -> (IsClosedMap.{u2, u1} α β _inst_1 _inst_2 f) -> (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align closed_embedding_of_embedding_closed closedEmbedding_of_embedding_closedₓ'. -/
 theorem closedEmbedding_of_embedding_closed (h₁ : Embedding f) (h₂ : IsClosedMap f) :
     ClosedEmbedding f :=
   ⟨h₁, by convert h₂ univ isClosed_univ <;> simp⟩
 #align closed_embedding_of_embedding_closed closedEmbedding_of_embedding_closed
 
-/- warning: closed_embedding_of_continuous_injective_closed -> closedEmbedding_of_continuous_injective_closed is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (Continuous.{u1, u2} α β _inst_1 _inst_2 f) -> (Function.Injective.{succ u1, succ u2} α β f) -> (IsClosedMap.{u1, u2} α β _inst_1 _inst_2 f) -> (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f)
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (Continuous.{u2, u1} α β _inst_1 _inst_2 f) -> (Function.Injective.{succ u2, succ u1} α β f) -> (IsClosedMap.{u2, u1} α β _inst_1 _inst_2 f) -> (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align closed_embedding_of_continuous_injective_closed closedEmbedding_of_continuous_injective_closedₓ'. -/
 theorem closedEmbedding_of_continuous_injective_closed (h₁ : Continuous f) (h₂ : Injective f)
     (h₃ : IsClosedMap f) : ClosedEmbedding f :=
   by
@@ -1230,12 +720,6 @@ theorem closedEmbedding_id : ClosedEmbedding (@id α) :=
 #align closed_embedding_id closedEmbedding_id
 -/
 
-/- warning: closed_embedding.comp -> ClosedEmbedding.comp is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] [_inst_3 : TopologicalSpace.{u3} γ] {g : β -> γ} {f : α -> β}, (ClosedEmbedding.{u2, u3} β γ _inst_2 _inst_3 g) -> (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (ClosedEmbedding.{u1, u3} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u2, succ u3} α β γ g f))
-but is expected to have type
-  forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {g : β -> γ} {f : α -> β}, (ClosedEmbedding.{u3, u2} β γ _inst_2 _inst_3 g) -> (ClosedEmbedding.{u1, u3} α β _inst_1 _inst_2 f) -> (ClosedEmbedding.{u1, u2} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u3, succ u2} α β γ g f))
-Case conversion may be inaccurate. Consider using '#align closed_embedding.comp ClosedEmbedding.compₓ'. -/
 theorem ClosedEmbedding.comp {g : β → γ} {f : α → β} (hg : ClosedEmbedding g)
     (hf : ClosedEmbedding f) : ClosedEmbedding (g ∘ f) :=
   ⟨hg.toEmbedding.comp hf.toEmbedding,
@@ -1243,12 +727,6 @@ theorem ClosedEmbedding.comp {g : β → γ} {f : α → β} (hg : ClosedEmbeddi
       rw [range_comp, ← hg.closed_iff_image_closed] <;> exact hf.closed_range⟩
 #align closed_embedding.comp ClosedEmbedding.comp
 
-/- warning: closed_embedding.closure_image_eq -> ClosedEmbedding.closure_image_eq is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (ClosedEmbedding.{u1, u2} α β _inst_1 _inst_2 f) -> (forall (s : Set.{u1} α), Eq.{succ u2} (Set.{u2} β) (closure.{u2} β _inst_2 (Set.image.{u1, u2} α β f s)) (Set.image.{u1, u2} α β f (closure.{u1} α _inst_1 s)))
-but is expected to have type
-  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (ClosedEmbedding.{u2, u1} α β _inst_1 _inst_2 f) -> (forall (s : Set.{u2} α), Eq.{succ u1} (Set.{u1} β) (closure.{u1} β _inst_2 (Set.image.{u2, u1} α β f s)) (Set.image.{u2, u1} α β f (closure.{u2} α _inst_1 s)))
-Case conversion may be inaccurate. Consider using '#align closed_embedding.closure_image_eq ClosedEmbedding.closure_image_eqₓ'. -/
 theorem ClosedEmbedding.closure_image_eq {f : α → β} (hf : ClosedEmbedding f) (s : Set α) :
     closure (f '' s) = f '' closure s :=
   (hf.IsClosedMap.closure_image_subset _).antisymm

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

@@ -97,9 +97,7 @@ Case conversion may be inaccurate. Consider using '#align inducing_of_inducing_c
 theorem inducing_of_inducing_compose {f : α → β} {g : β → γ} (hf : Continuous f) (hg : Continuous g)
     (hgf : Inducing (g ∘ f)) : Inducing f :=
   ⟨le_antisymm (by rwa [← continuous_iff_le_induced])
-      (by
-        rw [hgf.induced, ← continuous_iff_le_induced]
-        apply hg.comp continuous_induced_dom)⟩
+      (by rw [hgf.induced, ← continuous_iff_le_induced]; apply hg.comp continuous_induced_dom)⟩
 #align inducing_of_inducing_compose inducing_of_inducing_compose
 
 /- warning: inducing_iff_nhds -> inducing_iff_nhds is a dubious translation:
@@ -240,8 +238,7 @@ but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (Inducing.{u2, u1} α β _inst_1 _inst_2 f) -> (forall (s : Set.{u2} α), Eq.{succ u2} (Set.{u2} α) (closure.{u2} α _inst_1 s) (Set.preimage.{u2, u1} α β f (closure.{u1} β _inst_2 (Set.image.{u2, u1} α β f s))))
 Case conversion may be inaccurate. Consider using '#align inducing.closure_eq_preimage_closure_image Inducing.closure_eq_preimage_closure_imageₓ'. -/
 theorem Inducing.closure_eq_preimage_closure_image {f : α → β} (hf : Inducing f) (s : Set α) :
-    closure s = f ⁻¹' closure (f '' s) := by
-  ext x
+    closure s = f ⁻¹' closure (f '' s) := by ext x;
   rw [Set.mem_preimage, ← closure_induced, hf.induced]
 #align inducing.closure_eq_preimage_closure_image Inducing.closure_eq_preimage_closure_image
 
@@ -507,9 +504,7 @@ protected theorem of_quotientMap_compose (hf : Continuous f) (hg : Continuous g)
     (hgf : QuotientMap (g ∘ f)) : QuotientMap g :=
   ⟨hgf.1.of_comp,
     le_antisymm
-      (by
-        rw [hgf.right, ← continuous_iff_coinduced_le]
-        apply continuous_coinduced_rng.comp hf)
+      (by rw [hgf.right, ← continuous_iff_coinduced_le]; apply continuous_coinduced_rng.comp hf)
       (by rwa [← continuous_iff_coinduced_le])⟩
 #align quotient_map.of_quotient_map_compose QuotientMap.of_quotientMap_compose
 
@@ -610,9 +605,7 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (IsOpenMap.{u2, u1} α β _inst_1 _inst_2 f) -> (IsOpen.{u1} β _inst_2 (Set.range.{u1, succ u2} β α f))
 Case conversion may be inaccurate. Consider using '#align is_open_map.is_open_range IsOpenMap.isOpen_rangeₓ'. -/
-theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) :=
-  by
-  rw [← image_univ]
+theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by rw [← image_univ];
   exact hf _ isOpen_univ
 #align is_open_map.is_open_range IsOpenMap.isOpen_range
 
@@ -862,10 +855,7 @@ but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u3}} {γ : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u3} β] [_inst_3 : TopologicalSpace.{u2} γ] {g : β -> γ} {f : α -> β}, (IsClosedMap.{u3, u2} β γ _inst_2 _inst_3 g) -> (IsClosedMap.{u1, u3} α β _inst_1 _inst_2 f) -> (IsClosedMap.{u1, u2} α γ _inst_1 _inst_3 (Function.comp.{succ u1, succ u3, succ u2} α β γ g f))
 Case conversion may be inaccurate. Consider using '#align is_closed_map.comp IsClosedMap.compₓ'. -/
 protected theorem comp {g : β → γ} {f : α → β} (hg : IsClosedMap g) (hf : IsClosedMap f) :
-    IsClosedMap (g ∘ f) := by
-  intro s hs
-  rw [image_comp]
-  exact hg _ (hf _ hs)
+    IsClosedMap (g ∘ f) := by intro s hs; rw [image_comp]; exact hg _ (hf _ hs)
 #align is_closed_map.comp IsClosedMap.comp
 
 /- warning: is_closed_map.closure_image_subset -> IsClosedMap.closure_image_subset is a dubious translation:

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.maps
-! leanprover-community/mathlib commit e46da4e335b8671848ac711ccb34b42538c0d800
+! leanprover-community/mathlib commit d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -465,11 +465,17 @@ lean 3 declaration is
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, Iff (QuotientMap.{u2, u1} α β _inst_1 _inst_2 f) (And (Function.Surjective.{succ u2, succ u1} α β f) (forall (s : Set.{u1} β), Iff (IsOpen.{u1} β _inst_2 s) (IsOpen.{u2} α _inst_1 (Set.preimage.{u2, u1} α β f s))))
 Case conversion may be inaccurate. Consider using '#align quotient_map_iff quotientMap_iffₓ'. -/
-theorem quotientMap_iff {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
+theorem quotientMap_iff [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   and_congr Iff.rfl topologicalSpace_eq_iff
 #align quotient_map_iff quotientMap_iff
 
+theorem quotientMap_iff_closed [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
+    QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsClosed s ↔ IsClosed (f ⁻¹' s) :=
+  quotientMap_iff.trans <|
+    Iff.rfl.And <| compl_surjective.forall.trans <| by simp only [isOpen_compl_iff, preimage_compl]
+#align quotient_map_iff_closed quotientMap_iff_closed
+
 namespace QuotientMap
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
@@ -565,8 +571,8 @@ but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (QuotientMap.{u2, u1} α β _inst_1 _inst_2 f) -> (forall {s : Set.{u1} β}, Iff (IsClosed.{u2} α _inst_1 (Set.preimage.{u2, u1} α β f s)) (IsClosed.{u1} β _inst_2 s))
 Case conversion may be inaccurate. Consider using '#align quotient_map.is_closed_preimage QuotientMap.isClosed_preimageₓ'. -/
 protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set β} :
-    IsClosed (f ⁻¹' s) ↔ IsClosed s := by
-  simp only [← isOpen_compl_iff, ← preimage_compl, hf.is_open_preimage]
+    IsClosed (f ⁻¹' s) ↔ IsClosed s :=
+  ((quotientMap_iff_closed.1 hf).2 s).symm
 #align quotient_map.is_closed_preimage QuotientMap.isClosed_preimage
 
 end QuotientMap
@@ -904,6 +910,12 @@ theorem closed_range {f : α → β} (hf : IsClosedMap f) : IsClosed (range f) :
   @image_univ _ _ f ▸ hf _ isClosed_univ
 #align is_closed_map.closed_range IsClosedMap.closed_range
 
+theorem to_quotientMap {f : α → β} (hcl : IsClosedMap f) (hcont : Continuous f)
+    (hsurj : Surjective f) : QuotientMap f :=
+  quotientMap_iff_closed.2
+    ⟨hsurj, fun s => ⟨fun hs => hs.Preimage hcont, fun hs => hsurj.image_preimage s ▸ hcl _ hs⟩⟩
+#align is_closed_map.to_quotient_map IsClosedMap.to_quotientMap
+
 end IsClosedMap
 
 /- warning: inducing.is_closed_map -> Inducing.isClosedMap is a dubious translation:

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

@@ -656,7 +656,7 @@ theorem image_interior_subset (hf : IsOpenMap f) (s : Set α) :
 
 /- warning: is_open_map.nhds_le -> IsOpenMap.nhds_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f) -> (forall (a : α), LE.le.{u2} (Filter.{u2} β) (Preorder.toLE.{u2} (Filter.{u2} β) (PartialOrder.toPreorder.{u2} (Filter.{u2} β) (Filter.partialOrder.{u2} β))) (nhds.{u2} β _inst_2 (f a)) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f) -> (forall (a : α), LE.le.{u2} (Filter.{u2} β) (Preorder.toHasLe.{u2} (Filter.{u2} β) (PartialOrder.toPreorder.{u2} (Filter.{u2} β) (Filter.partialOrder.{u2} β))) (nhds.{u2} β _inst_2 (f a)) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a)))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, (IsOpenMap.{u2, u1} α β _inst_1 _inst_2 f) -> (forall (a : α), LE.le.{u1} (Filter.{u1} β) (Preorder.toLE.{u1} (Filter.{u1} β) (PartialOrder.toPreorder.{u1} (Filter.{u1} β) (Filter.instPartialOrderFilter.{u1} β))) (nhds.{u1} β _inst_2 (f a)) (Filter.map.{u2, u1} α β f (nhds.{u2} α _inst_1 a)))
 Case conversion may be inaccurate. Consider using '#align is_open_map.nhds_le IsOpenMap.nhds_leₓ'. -/
@@ -666,7 +666,7 @@ theorem nhds_le (hf : IsOpenMap f) (a : α) : 𝓝 (f a) ≤ (𝓝 a).map f :=
 
 /- warning: is_open_map.of_nhds_le -> IsOpenMap.of_nhds_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (forall (a : α), LE.le.{u2} (Filter.{u2} β) (Preorder.toLE.{u2} (Filter.{u2} β) (PartialOrder.toPreorder.{u2} (Filter.{u2} β) (Filter.partialOrder.{u2} β))) (nhds.{u2} β _inst_2 (f a)) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a))) -> (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f)
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (forall (a : α), LE.le.{u2} (Filter.{u2} β) (Preorder.toHasLe.{u2} (Filter.{u2} β) (PartialOrder.toPreorder.{u2} (Filter.{u2} β) (Filter.partialOrder.{u2} β))) (nhds.{u2} β _inst_2 (f a)) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a))) -> (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f)
 but is expected to have type
   forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, (forall (a : α), LE.le.{u2} (Filter.{u2} β) (Preorder.toLE.{u2} (Filter.{u2} β) (PartialOrder.toPreorder.{u2} (Filter.{u2} β) (Filter.instPartialOrderFilter.{u2} β))) (nhds.{u2} β _inst_2 (f a)) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a))) -> (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f)
 Case conversion may be inaccurate. Consider using '#align is_open_map.of_nhds_le IsOpenMap.of_nhds_leₓ'. -/
@@ -787,7 +787,7 @@ end IsOpenMap
 
 /- warning: is_open_map_iff_nhds_le -> isOpenMap_iff_nhds_le is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, Iff (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f) (forall (a : α), LE.le.{u2} (Filter.{u2} β) (Preorder.toLE.{u2} (Filter.{u2} β) (PartialOrder.toPreorder.{u2} (Filter.{u2} β) (Filter.partialOrder.{u2} β))) (nhds.{u2} β _inst_2 (f a)) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a)))
+  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} α] [_inst_2 : TopologicalSpace.{u2} β] {f : α -> β}, Iff (IsOpenMap.{u1, u2} α β _inst_1 _inst_2 f) (forall (a : α), LE.le.{u2} (Filter.{u2} β) (Preorder.toHasLe.{u2} (Filter.{u2} β) (PartialOrder.toPreorder.{u2} (Filter.{u2} β) (Filter.partialOrder.{u2} β))) (nhds.{u2} β _inst_2 (f a)) (Filter.map.{u1, u2} α β f (nhds.{u1} α _inst_1 a)))
 but is expected to have type
   forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} α] [_inst_2 : TopologicalSpace.{u1} β] {f : α -> β}, Iff (IsOpenMap.{u2, u1} α β _inst_1 _inst_2 f) (forall (a : α), LE.le.{u1} (Filter.{u1} β) (Preorder.toLE.{u1} (Filter.{u1} β) (PartialOrder.toPreorder.{u1} (Filter.{u1} β) (Filter.instPartialOrderFilter.{u1} β))) (nhds.{u1} β _inst_2 (f a)) (Filter.map.{u2, u1} α β f (nhds.{u2} α _inst_1 a)))
 Case conversion may be inaccurate. Consider using '#align is_open_map_iff_nhds_le isOpenMap_iff_nhds_leₓ'. -/

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

@@ -130,7 +130,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comapₓ'. -/
 theorem Inducing.nhdsSet_eq_comap {f : α → β} (hf : Inducing f) (s : Set α) :
     𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
-  simp only [nhdsSet, supₛ_image, comap_supr, hf.nhds_eq_comap, supᵢ_image]
+  simp only [nhdsSet, sSup_image, comap_supr, hf.nhds_eq_comap, iSup_image]
 #align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comap
 
 /- warning: inducing.map_nhds_eq -> Inducing.map_nhds_eq is a dubious translation:

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

@@ -980,7 +980,7 @@ theorem OpenEmbedding.open_iff_image_open {f : α → β} (hf : OpenEmbedding f)
     IsOpen s ↔ IsOpen (f '' s) :=
   ⟨hf.IsOpenMap s, fun h =>
     by
-    convert ← h.preimage hf.to_embedding.continuous
+    convert← h.preimage hf.to_embedding.continuous
     apply preimage_image_eq _ hf.inj⟩
 #align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_open
 
@@ -1014,7 +1014,7 @@ Case conversion may be inaccurate. Consider using '#align open_embedding.open_if
 theorem OpenEmbedding.open_iff_preimage_open {f : α → β} (hf : OpenEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   by
-  convert ← hf.open_iff_image_open.symm
+  convert← hf.open_iff_image_open.symm
   rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
 #align open_embedding.open_iff_preimage_open OpenEmbedding.open_iff_preimage_open
 
@@ -1174,7 +1174,7 @@ theorem ClosedEmbedding.closed_iff_image_closed (hf : ClosedEmbedding f) {s : Se
     IsClosed s ↔ IsClosed (f '' s) :=
   ⟨hf.IsClosedMap s, fun h =>
     by
-    convert ← continuous_iff_is_closed.mp hf.continuous _ h
+    convert← continuous_iff_is_closed.mp hf.continuous _ h
     apply preimage_image_eq _ hf.inj⟩
 #align closed_embedding.closed_iff_image_closed ClosedEmbedding.closed_iff_image_closed
 
@@ -1187,7 +1187,7 @@ Case conversion may be inaccurate. Consider using '#align closed_embedding.close
 theorem ClosedEmbedding.closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) :=
   by
-  convert ← hf.closed_iff_image_closed.symm
+  convert← hf.closed_iff_image_closed.symm
   rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
 #align closed_embedding.closed_iff_preimage_closed ClosedEmbedding.closed_iff_preimage_closed
 

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

@@ -686,7 +686,7 @@ theorem of_sections {f : α → β}
     let ⟨g, hgc, hgx, hgf⟩ := h x
     calc
       𝓝 (f x) = map f (map g (𝓝 (f x))) := by rw [map_map, hgf.comp_eq_id, map_id]
-      _ ≤ map f (𝓝 (g (f x))) := map_mono hgc
+      _ ≤ map f (𝓝 (g (f x))) := (map_mono hgc)
       _ = map f (𝓝 x) := by rw [hgx]
       
 #align is_open_map.of_sections IsOpenMap.of_sections

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

Also use OpenEmbedding.of_subsingleton to golf AlgebraicGeometry.isOpenImmersion_of_isEmpty.

@@ -176,6 +176,9 @@ theorem dense_iff (hf : Inducing f) {s : Set X} :
   simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage]
 #align inducing.dense_iff Inducing.dense_iff
 
+theorem of_subsingleton [Subsingleton X] (f : X → Y) : Inducing f :=
+  ⟨Subsingleton.elim _ _⟩
+
 end Inducing
 
 end Inducing
@@ -251,6 +254,9 @@ theorem Embedding.discreteTopology [DiscreteTopology Y] (hf : Embedding f) : Dis
   .of_continuous_injective hf.continuous hf.inj
 #align embedding.discrete_topology Embedding.discreteTopology
 
+theorem Embedding.of_subsingleton [Subsingleton X] (f : X → Y) : Embedding f :=
+  ⟨.of_subsingleton f, f.injective_of_subsingleton⟩
+
 end Embedding
 
 section QuotientMap
@@ -417,6 +423,8 @@ theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continu
     hf.preimage_closure_eq_closure_preimage hfc]
 #align is_open_map.preimage_frontier_eq_frontier_preimage IsOpenMap.preimage_frontier_eq_frontier_preimage
 
+theorem of_isEmpty [h : IsEmpty X] (f : X → Y) : IsOpenMap f := of_nhds_le h.elim
+
 end IsOpenMap
 
 theorem isOpenMap_iff_nhds_le : IsOpenMap f ↔ ∀ x : X, 𝓝 (f x) ≤ (𝓝 x).map f :=
@@ -630,6 +638,9 @@ theorem of_comp (f : X → Y) (hg : OpenEmbedding g)
   (OpenEmbedding.of_comp_iff f hg).1 h
 #align open_embedding.of_comp OpenEmbedding.of_comp
 
+theorem of_isEmpty [IsEmpty X] (f : X → Y) : OpenEmbedding f :=
+  openEmbedding_of_embedding_open (.of_subsingleton f) (IsOpenMap.of_isEmpty f)
+
 end OpenEmbedding
 
 end OpenEmbedding

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

@@ -472,9 +472,11 @@ theorem of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s))
   · exact h s hs h2s
 #align is_closed_map.of_nonempty IsClosedMap.of_nonempty
 
-theorem closed_range (hf : IsClosedMap f) : IsClosed (range f) :=
+theorem isClosed_range (hf : IsClosedMap f) : IsClosed (range f) :=
   @image_univ _ _ f ▸ hf _ isClosed_univ
-#align is_closed_map.closed_range IsClosedMap.closed_range
+#align is_closed_map.closed_range IsClosedMap.isClosed_range
+
+@[deprecated] alias closed_range := isClosed_range -- 2024-03-17
 
 theorem to_quotientMap (hcl : IsClosedMap f) (hcont : Continuous f)
     (hsurj : Surjective f) : QuotientMap f :=
@@ -535,12 +537,12 @@ section OpenEmbedding
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
 theorem OpenEmbedding.isOpenMap (hf : OpenEmbedding f) : IsOpenMap f :=
-  hf.toEmbedding.toInducing.isOpenMap hf.open_range
+  hf.toEmbedding.toInducing.isOpenMap hf.isOpen_range
 #align open_embedding.is_open_map OpenEmbedding.isOpenMap
 
 theorem OpenEmbedding.map_nhds_eq (hf : OpenEmbedding f) (x : X) :
     map f (𝓝 x) = 𝓝 (f x) :=
-  hf.toEmbedding.map_nhds_of_mem _ <| hf.open_range.mem_nhds <| mem_range_self _
+  hf.toEmbedding.map_nhds_of_mem _ <| hf.isOpen_range.mem_nhds <| mem_range_self _
 #align open_embedding.map_nhds_eq OpenEmbedding.map_nhds_eq
 
 theorem OpenEmbedding.open_iff_image_open (hf : OpenEmbedding f) {s : Set X} :
@@ -648,7 +650,7 @@ theorem continuous (hf : ClosedEmbedding f) : Continuous f :=
 #align closed_embedding.continuous ClosedEmbedding.continuous
 
 theorem isClosedMap (hf : ClosedEmbedding f) : IsClosedMap f :=
-  hf.toEmbedding.toInducing.isClosedMap hf.closed_range
+  hf.toEmbedding.toInducing.isClosedMap hf.isClosed_range
 #align closed_embedding.is_closed_map ClosedEmbedding.isClosedMap
 
 theorem closed_iff_image_closed (hf : ClosedEmbedding f) {s : Set X} :
@@ -677,12 +679,12 @@ theorem _root_.closedEmbedding_of_continuous_injective_closed (h₁ : Continuous
 #align closed_embedding_of_continuous_injective_closed closedEmbedding_of_continuous_injective_closed
 
 theorem _root_.closedEmbedding_id : ClosedEmbedding (@id X) :=
-  ⟨embedding_id, IsClosedMap.id.closed_range⟩
+  ⟨embedding_id, IsClosedMap.id.isClosed_range⟩
 #align closed_embedding_id closedEmbedding_id
 
 theorem comp (hg : ClosedEmbedding g) (hf : ClosedEmbedding f) :
     ClosedEmbedding (g ∘ f) :=
-  ⟨hg.toEmbedding.comp hf.toEmbedding, (hg.isClosedMap.comp hf.isClosedMap).closed_range⟩
+  ⟨hg.toEmbedding.comp hf.toEmbedding, (hg.isClosedMap.comp hf.isClosedMap).isClosed_range⟩
 #align closed_embedding.comp ClosedEmbedding.comp
 
 theorem closure_image_eq (hf : ClosedEmbedding f) (s : Set X) :

The main addition is IsTopologicalBasis.inf (see https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Inf.20of.20a.20pair.20of.20topologies/near/419989448), and I also reordered things to be in the more typical order (deducing the Pi version from the iInf version rather than the converse).

Also a few extra golfs and variations.

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Christopher Hoskin <christopher.hoskin@gmail.com> Co-authored-by: Christopher Hoskin <christopher.hoskin@overleaf.com>

@@ -84,6 +84,10 @@ theorem nhds_eq_comap (hf : Inducing f) : ∀ x : X, 𝓝 x = comap f (𝓝 <| f
   inducing_iff_nhds.1 hf
 #align inducing.nhds_eq_comap Inducing.nhds_eq_comap
 
+theorem basis_nhds {p : ι → Prop} {s : ι → Set Y} (hf : Inducing f) {x : X}
+    (h_basis : (𝓝 (f x)).HasBasis p s) : (𝓝 x).HasBasis p (preimage f ∘ s) :=
+  hf.nhds_eq_comap x ▸ h_basis.comap f
+
 theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set X) :
     𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
   simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image]

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

• "new lemma"
• "added lemma"

@@ -98,7 +98,7 @@ theorem map_nhds_of_mem (hf : Inducing f) (x : X) (h : range f ∈ 𝓝 (f x)) :
   hf.induced.symm ▸ map_nhds_induced_of_mem h
 #align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem
 
--- Porting note: new lemma
+-- Porting note (#10756): new lemma
 theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} :
     MapClusterPt (f x) l f ↔ ClusterPt x l := by
   delta MapClusterPt ClusterPt

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".

@@ -98,7 +98,7 @@ theorem map_nhds_of_mem (hf : Inducing f) (x : X) (h : range f ∈ 𝓝 (f x)) :
   hf.induced.symm ▸ map_nhds_induced_of_mem h
 #align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem
 
--- porting note: new lemma
+-- Porting note: new lemma
 theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} :
     MapClusterPt (f x) l f ↔ ClusterPt x l := by
   delta MapClusterPt ClusterPt

In some cases, the order of implicit arguments changed because now they appear in a different order in variables.

Also, some definitions used greek letters for topological spaces, changed to X/Y.

@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 -/
 import Mathlib.Topology.Order
-import Mathlib.Topology.NhdsSet
 
 #align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
 
@@ -53,16 +52,6 @@ variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y 
 
 section Inducing
 
-/-- A function `f : X → Y` between topological spaces is inducing if the topology on `X` is induced
-by the topology on `Y` through `f`, meaning that a set `s : Set X` is open iff it is the preimage
-under `f` of some open set `t : Set Y`. -/
-@[mk_iff]
-structure Inducing [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : X → Y) : Prop where
-  /-- The topology on the domain is equal to the induced topology. -/
-  induced : tX = tY.induced f
-#align inducing Inducing
-#align inducing_iff inducing_iff
-
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
 theorem inducing_induced (f : X → Y) : @Inducing X Y (TopologicalSpace.induced f ‹_›) _ f :=
@@ -189,16 +178,6 @@ end Inducing
 
 section Embedding
 
-/-- A function between topological spaces is an embedding if it is injective,
-  and for all `s : Set X`, `s` is open iff it is the preimage of an open set. -/
-@[mk_iff]
-structure Embedding [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) extends
-  Inducing f : Prop where
-  /-- A topological embedding is injective. -/
-  inj : Injective f
-#align embedding Embedding
-#align embedding_iff embedding_iff
-
 theorem Function.Injective.embedding_induced [t : TopologicalSpace Y] (hf : Injective f) :
     @_root_.Embedding X Y (t.induced f) t f :=
   @_root_.Embedding.mk X Y (t.induced f) t _ (inducing_induced f) hf
@@ -271,12 +250,6 @@ theorem Embedding.discreteTopology [DiscreteTopology Y] (hf : Embedding f) : Dis
 end Embedding
 
 section QuotientMap
-/-- A function between topological spaces is a quotient map if it is surjective,
-  and for all `s : Set Y`, `s` is open iff its preimage is an open set. -/
-def QuotientMap {X : Type*} {Y : Type*} [tX : TopologicalSpace X] [tY : TopologicalSpace Y]
-    (f : X → Y) : Prop :=
-  Surjective f ∧ tY = tX.coinduced f
-#align quotient_map QuotientMap
 
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
@@ -339,12 +312,6 @@ end QuotientMap
 end QuotientMap
 
 section OpenMap
-/-- A map `f : X → Y` is said to be an *open map*, if the image of any open `U : Set X`
-is open in `Y`. -/
-def IsOpenMap [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) :=
-  ∀ U : Set X, IsOpen U → IsOpen (f '' U)
-#align is_open_map IsOpenMap
-
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
 namespace IsOpenMap
@@ -471,12 +438,6 @@ section IsClosedMap
 
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
-/-- A map `f : X → Y` is said to be a *closed map*, if the image of any closed `U : Set X`
-is closed in `Y`. -/
-def IsClosedMap (f : X → Y) :=
-  ∀ U : Set X, IsClosed U → IsClosed (f '' U)
-#align is_closed_map IsClosedMap
-
 namespace IsClosedMap
 open Function
 
@@ -569,14 +530,6 @@ section OpenEmbedding
 
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
-/-- An open embedding is an embedding with open image. -/
-@[mk_iff]
-structure OpenEmbedding (f : X → Y) extends Embedding f : Prop where
-  /-- The range of an open embedding is an open set. -/
-  open_range : IsOpen <| range f
-#align open_embedding OpenEmbedding
-#align open_embedding_iff openEmbedding_iff
-
 theorem OpenEmbedding.isOpenMap (hf : OpenEmbedding f) : IsOpenMap f :=
   hf.toEmbedding.toInducing.isOpenMap hf.open_range
 #align open_embedding.is_open_map OpenEmbedding.isOpenMap
@@ -679,14 +632,6 @@ section ClosedEmbedding
 
 variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
-/-- A closed embedding is an embedding with closed image. -/
-@[mk_iff]
-structure ClosedEmbedding (f : X → Y) extends Embedding f : Prop where
-  /-- The range of a closed embedding is a closed set. -/
-  closed_range : IsClosed <| range f
-#align closed_embedding ClosedEmbedding
-#align closed_embedding_iff closedEmbedding_iff
-
 namespace ClosedEmbedding
 
 theorem tendsto_nhds_iff {g : ι → X} {l : Filter ι} {x : X} (hf : ClosedEmbedding f) :

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

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

@@ -49,26 +49,26 @@ open Set Filter Function
 
 open TopologicalSpace Topology Filter
 
-variable {α : Type*} {β : Type*} {γ : Type*} {ι : Type*} {f : α → β} {g : β → γ}
+variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
 
 section Inducing
 
-/-- A function `f : α → β` between topological spaces is inducing if the topology on `α` is induced
-by the topology on `β` through `f`, meaning that a set `s : Set α` is open iff it is the preimage
-under `f` of some open set `t : Set β`. -/
+/-- A function `f : X → Y` between topological spaces is inducing if the topology on `X` is induced
+by the topology on `Y` through `f`, meaning that a set `s : Set X` is open iff it is the preimage
+under `f` of some open set `t : Set Y`. -/
 @[mk_iff]
-structure Inducing [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : α → β) : Prop where
+structure Inducing [tX : TopologicalSpace X] [tY : TopologicalSpace Y] (f : X → Y) : Prop where
   /-- The topology on the domain is equal to the induced topology. -/
-  induced : tα = tβ.induced f
+  induced : tX = tY.induced f
 #align inducing Inducing
 #align inducing_iff inducing_iff
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
-theorem inducing_induced (f : α → β) : @Inducing α β (TopologicalSpace.induced f ‹_›) _ f :=
+theorem inducing_induced (f : X → Y) : @Inducing X Y (TopologicalSpace.induced f ‹_›) _ f :=
   @Inducing.mk _ _ (TopologicalSpace.induced f ‹_›) _ _ rfl
 
-theorem inducing_id : Inducing (@id α) :=
+theorem inducing_id : Inducing (@id X) :=
   ⟨induced_id.symm⟩
 #align inducing_id inducing_id
 
@@ -85,47 +85,47 @@ theorem inducing_of_inducing_compose
         exact induced_mono hg.le_induced)⟩
 #align inducing_of_inducing_compose inducing_of_inducing_compose
 
-theorem inducing_iff_nhds : Inducing f ↔ ∀ a, 𝓝 a = comap f (𝓝 (f a)) :=
+theorem inducing_iff_nhds : Inducing f ↔ ∀ x, 𝓝 x = comap f (𝓝 (f x)) :=
   (inducing_iff _).trans (induced_iff_nhds_eq f)
 #align inducing_iff_nhds inducing_iff_nhds
 
 namespace Inducing
 
-theorem nhds_eq_comap (hf : Inducing f) : ∀ a : α, 𝓝 a = comap f (𝓝 <| f a) :=
+theorem nhds_eq_comap (hf : Inducing f) : ∀ x : X, 𝓝 x = comap f (𝓝 <| f x) :=
   inducing_iff_nhds.1 hf
 #align inducing.nhds_eq_comap Inducing.nhds_eq_comap
 
-theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set α) :
+theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set X) :
     𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
   simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image]
 #align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comap
 
-theorem map_nhds_eq (hf : Inducing f) (a : α) : (𝓝 a).map f = 𝓝[range f] f a :=
-  hf.induced.symm ▸ map_nhds_induced_eq a
+theorem map_nhds_eq (hf : Inducing f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x :=
+  hf.induced.symm ▸ map_nhds_induced_eq x
 #align inducing.map_nhds_eq Inducing.map_nhds_eq
 
-theorem map_nhds_of_mem (hf : Inducing f) (a : α) (h : range f ∈ 𝓝 (f a)) :
-    (𝓝 a).map f = 𝓝 (f a) :=
+theorem map_nhds_of_mem (hf : Inducing f) (x : X) (h : range f ∈ 𝓝 (f x)) :
+    (𝓝 x).map f = 𝓝 (f x) :=
   hf.induced.symm ▸ map_nhds_induced_of_mem h
 #align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem
 
 -- porting note: new lemma
-theorem mapClusterPt_iff (hf : Inducing f) {a : α} {l : Filter α} :
-    MapClusterPt (f a) l f ↔ ClusterPt a l := by
+theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} :
+    MapClusterPt (f x) l f ↔ ClusterPt x l := by
   delta MapClusterPt ClusterPt
   rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff]
 
-theorem image_mem_nhdsWithin (hf : Inducing f) {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :
-    f '' s ∈ 𝓝[range f] f a :=
-  hf.map_nhds_eq a ▸ image_mem_map hs
+theorem image_mem_nhdsWithin (hf : Inducing f) {x : X} {s : Set X} (hs : s ∈ 𝓝 x) :
+    f '' s ∈ 𝓝[range f] f x :=
+  hf.map_nhds_eq x ▸ image_mem_map hs
 #align inducing.image_mem_nhds_within Inducing.image_mem_nhdsWithin
 
-theorem tendsto_nhds_iff {f : ι → β} {a : Filter ι} {b : β} (hg : Inducing g) :
-    Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) := by
+theorem tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : Inducing g) :
+    Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := by
   rw [hg.nhds_eq_comap, tendsto_comap_iff]
 #align inducing.tendsto_nhds_iff Inducing.tendsto_nhds_iff
 
-theorem continuousAt_iff (hg : Inducing g) {x : α} :
+theorem continuousAt_iff (hg : Inducing g) {x : X} :
     ContinuousAt f x ↔ ContinuousAt (g ∘ f) x :=
   hg.tendsto_nhds_iff
 #align inducing.continuous_at_iff Inducing.continuousAt_iff
@@ -135,7 +135,7 @@ theorem continuous_iff (hg : Inducing g) :
   simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff]
 #align inducing.continuous_iff Inducing.continuous_iff
 
-theorem continuousAt_iff' (hf : Inducing f) {x : α} (h : range f ∈ 𝓝 (f x)) :
+theorem continuousAt_iff' (hf : Inducing f) {x : X} (h : range f ∈ 𝓝 (f x)) :
     ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
   simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map, comp]
 #align inducing.continuous_at_iff' Inducing.continuousAt_iff'
@@ -151,34 +151,34 @@ protected theorem inducing_iff (hg : Inducing g) :
   exact hgf.continuous
 #align inducing.inducing_iff Inducing.inducing_iff
 
-theorem closure_eq_preimage_closure_image (hf : Inducing f) (s : Set α) :
+theorem closure_eq_preimage_closure_image (hf : Inducing f) (s : Set X) :
     closure s = f ⁻¹' closure (f '' s) := by
   ext x
   rw [Set.mem_preimage, ← closure_induced, hf.induced]
 #align inducing.closure_eq_preimage_closure_image Inducing.closure_eq_preimage_closure_image
 
-theorem isClosed_iff (hf : Inducing f) {s : Set α} :
+theorem isClosed_iff (hf : Inducing f) {s : Set X} :
     IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by rw [hf.induced, isClosed_induced_iff]
 #align inducing.is_closed_iff Inducing.isClosed_iff
 
-theorem isClosed_iff' (hf : Inducing f) {s : Set α} :
+theorem isClosed_iff' (hf : Inducing f) {s : Set X} :
     IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by rw [hf.induced, isClosed_induced_iff']
 #align inducing.is_closed_iff' Inducing.isClosed_iff'
 
-theorem isClosed_preimage (h : Inducing f) (s : Set β) (hs : IsClosed s) :
+theorem isClosed_preimage (h : Inducing f) (s : Set Y) (hs : IsClosed s) :
     IsClosed (f ⁻¹' s) :=
   (isClosed_iff h).mpr ⟨s, hs, rfl⟩
 #align inducing.is_closed_preimage Inducing.isClosed_preimage
 
-theorem isOpen_iff (hf : Inducing f) {s : Set α} :
+theorem isOpen_iff (hf : Inducing f) {s : Set X} :
     IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := by rw [hf.induced, isOpen_induced_iff]
 #align inducing.is_open_iff Inducing.isOpen_iff
 
 theorem setOf_isOpen (hf : Inducing f) :
-    {s : Set α | IsOpen s} = preimage f '' {t | IsOpen t} :=
+    {s : Set X | IsOpen s} = preimage f '' {t | IsOpen t} :=
   Set.ext fun _ ↦ hf.isOpen_iff
 
-theorem dense_iff (hf : Inducing f) {s : Set α} :
+theorem dense_iff (hf : Inducing f) {s : Set X} :
     Dense s ↔ ∀ x, f x ∈ closure (f '' s) := by
   simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage]
 #align inducing.dense_iff Inducing.dense_iff
@@ -190,28 +190,28 @@ end Inducing
 section Embedding
 
 /-- A function between topological spaces is an embedding if it is injective,
-  and for all `s : Set α`, `s` is open iff it is the preimage of an open set. -/
+  and for all `s : Set X`, `s` is open iff it is the preimage of an open set. -/
 @[mk_iff]
-structure Embedding [TopologicalSpace α] [TopologicalSpace β] (f : α → β) extends
+structure Embedding [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) extends
   Inducing f : Prop where
   /-- A topological embedding is injective. -/
   inj : Injective f
 #align embedding Embedding
 #align embedding_iff embedding_iff
 
-theorem Function.Injective.embedding_induced [t : TopologicalSpace β] {f : α → β}
-    (hf : Injective f) : @_root_.Embedding α β (t.induced f) t f :=
-  @_root_.Embedding.mk α β (t.induced f) t _ (inducing_induced f) hf
+theorem Function.Injective.embedding_induced [t : TopologicalSpace Y] (hf : Injective f) :
+    @_root_.Embedding X Y (t.induced f) t f :=
+  @_root_.Embedding.mk X Y (t.induced f) t _ (inducing_induced f) hf
 #align function.injective.embedding_induced Function.Injective.embedding_induced
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
-theorem Embedding.mk' (f : α → β) (inj : Injective f) (induced : ∀ a, comap f (𝓝 (f a)) = 𝓝 a) :
+theorem Embedding.mk' (f : X → Y) (inj : Injective f) (induced : ∀ x, comap f (𝓝 (f x)) = 𝓝 x) :
     Embedding f :=
-  ⟨inducing_iff_nhds.2 fun a => (induced a).symm, inj⟩
+  ⟨inducing_iff_nhds.2 fun x => (induced x).symm, inj⟩
 #align embedding.mk' Embedding.mk'
 
-theorem embedding_id : Embedding (@id α) :=
+theorem embedding_id : Embedding (@id X) :=
   ⟨inducing_id, fun _ _ h => h⟩
 #align embedding_id embedding_id
 
@@ -223,26 +223,26 @@ protected theorem Embedding.comp (hg : Embedding g) (hf : Embedding f) :
 theorem embedding_of_embedding_compose
     (hf : Continuous f) (hg : Continuous g) (hgf : Embedding (g ∘ f)) : Embedding f :=
   { induced := (inducing_of_inducing_compose hf hg hgf.toInducing).induced
-    inj := fun a₁ a₂ h => hgf.inj <| by simp [h, (· ∘ ·)] }
+    inj := fun x₁ x₂ h => hgf.inj <| by simp [h, (· ∘ ·)] }
 #align embedding_of_embedding_compose embedding_of_embedding_compose
 
-protected theorem Function.LeftInverse.embedding {f : α → β} {g : β → α} (h : LeftInverse f g)
+protected theorem Function.LeftInverse.embedding {f : X → Y} {g : Y → X} (h : LeftInverse f g)
     (hf : Continuous f) (hg : Continuous g) : _root_.Embedding g :=
   embedding_of_embedding_compose hg hf <| h.comp_eq_id.symm ▸ embedding_id
 #align function.left_inverse.embedding Function.LeftInverse.embedding
 
-theorem Embedding.map_nhds_eq (hf : Embedding f) (a : α) :
-    (𝓝 a).map f = 𝓝[range f] f a :=
-  hf.1.map_nhds_eq a
+theorem Embedding.map_nhds_eq (hf : Embedding f) (x : X) :
+    (𝓝 x).map f = 𝓝[range f] f x :=
+  hf.1.map_nhds_eq x
 #align embedding.map_nhds_eq Embedding.map_nhds_eq
 
-theorem Embedding.map_nhds_of_mem (hf : Embedding f) (a : α) (h : range f ∈ 𝓝 (f a)) :
-    (𝓝 a).map f = 𝓝 (f a) :=
-  hf.1.map_nhds_of_mem a h
+theorem Embedding.map_nhds_of_mem (hf : Embedding f) (x : X) (h : range f ∈ 𝓝 (f x)) :
+    (𝓝 x).map f = 𝓝 (f x) :=
+  hf.1.map_nhds_of_mem x h
 #align embedding.map_nhds_of_mem Embedding.map_nhds_of_mem
 
-theorem Embedding.tendsto_nhds_iff {f : ι → β} {a : Filter ι} {b : β}
-    (hg : Embedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
+theorem Embedding.tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y}
+    (hg : Embedding g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) :=
   hg.toInducing.tendsto_nhds_iff
 #align embedding.tendsto_nhds_iff Embedding.tendsto_nhds_iff
 
@@ -255,17 +255,16 @@ theorem Embedding.continuous (hf : Embedding f) : Continuous f :=
   Inducing.continuous hf.1
 #align embedding.continuous Embedding.continuous
 
-theorem Embedding.closure_eq_preimage_closure_image {e : α → β} (he : Embedding e) (s : Set α) :
-    closure s = e ⁻¹' closure (e '' s) :=
-  he.1.closure_eq_preimage_closure_image s
+theorem Embedding.closure_eq_preimage_closure_image (hf : Embedding f) (s : Set X) :
+    closure s = f ⁻¹' closure (f '' s) :=
+  hf.1.closure_eq_preimage_closure_image s
 #align embedding.closure_eq_preimage_closure_image Embedding.closure_eq_preimage_closure_image
 
 /-- The topology induced under an inclusion `f : X → Y` from a discrete topological space `Y`
 is the discrete topology on `X`.
 
 See also `DiscreteTopology.of_continuous_injective`. -/
-theorem Embedding.discreteTopology {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-    [DiscreteTopology Y] {f : X → Y} (hf : Embedding f) : DiscreteTopology X :=
+theorem Embedding.discreteTopology [DiscreteTopology Y] (hf : Embedding f) : DiscreteTopology X :=
   .of_continuous_injective hf.continuous hf.inj
 #align embedding.discrete_topology Embedding.discreteTopology
 
@@ -273,28 +272,28 @@ end Embedding
 
 section QuotientMap
 /-- A function between topological spaces is a quotient map if it is surjective,
-  and for all `s : Set β`, `s` is open iff its preimage is an open set. -/
-def QuotientMap {α : Type*} {β : Type*} [tα : TopologicalSpace α] [tβ : TopologicalSpace β]
-    (f : α → β) : Prop :=
-  Surjective f ∧ tβ = tα.coinduced f
+  and for all `s : Set Y`, `s` is open iff its preimage is an open set. -/
+def QuotientMap {X : Type*} {Y : Type*} [tX : TopologicalSpace X] [tY : TopologicalSpace Y]
+    (f : X → Y) : Prop :=
+  Surjective f ∧ tY = tX.coinduced f
 #align quotient_map QuotientMap
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
-theorem quotientMap_iff : QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
+theorem quotientMap_iff : QuotientMap f ↔ Surjective f ∧ ∀ s : Set Y, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   and_congr Iff.rfl TopologicalSpace.ext_iff
 #align quotient_map_iff quotientMap_iff
 
 theorem quotientMap_iff_closed :
-    QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsClosed s ↔ IsClosed (f ⁻¹' s) :=
+    QuotientMap f ↔ Surjective f ∧ ∀ s : Set Y, IsClosed s ↔ IsClosed (f ⁻¹' s) :=
   quotientMap_iff.trans <| Iff.rfl.and <| compl_surjective.forall.trans <| by
     simp only [isOpen_compl_iff, preimage_compl]
 #align quotient_map_iff_closed quotientMap_iff_closed
 
 namespace QuotientMap
 
-protected theorem id : QuotientMap (@id α) :=
-  ⟨fun a => ⟨a, rfl⟩, coinduced_id.symm⟩
+protected theorem id : QuotientMap (@id X) :=
+  ⟨fun x => ⟨x, rfl⟩, coinduced_id.symm⟩
 #align quotient_map.id QuotientMap.id
 
 protected theorem comp (hg : QuotientMap g) (hf : QuotientMap f) : QuotientMap (g ∘ f) :=
@@ -309,7 +308,7 @@ protected theorem of_quotientMap_compose (hf : Continuous f) (hg : Continuous g)
       hg.coinduced_le⟩
 #align quotient_map.of_quotient_map_compose QuotientMap.of_quotientMap_compose
 
-theorem of_inverse {g : β → α} (hf : Continuous f) (hg : Continuous g) (h : LeftInverse g f) :
+theorem of_inverse {g : Y → X} (hf : Continuous f) (hg : Continuous g) (h : LeftInverse g f) :
     QuotientMap g :=
   QuotientMap.of_quotientMap_compose hf hg <| h.comp_eq_id.symm ▸ QuotientMap.id
 #align quotient_map.of_inverse QuotientMap.of_inverse
@@ -326,11 +325,11 @@ protected theorem surjective (hf : QuotientMap f) : Surjective f :=
   hf.1
 #align quotient_map.surjective QuotientMap.surjective
 
-protected theorem isOpen_preimage (hf : QuotientMap f) {s : Set β} : IsOpen (f ⁻¹' s) ↔ IsOpen s :=
+protected theorem isOpen_preimage (hf : QuotientMap f) {s : Set Y} : IsOpen (f ⁻¹' s) ↔ IsOpen s :=
   ((quotientMap_iff.1 hf).2 s).symm
 #align quotient_map.is_open_preimage QuotientMap.isOpen_preimage
 
-protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set β} :
+protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set Y} :
     IsClosed (f ⁻¹' s) ↔ IsClosed s :=
   ((quotientMap_iff_closed.1 hf).2 s).symm
 #align quotient_map.is_closed_preimage QuotientMap.isClosed_preimage
@@ -340,17 +339,17 @@ end QuotientMap
 end QuotientMap
 
 section OpenMap
-/-- A map `f : α → β` is said to be an *open map*, if the image of any open `U : Set α`
-is open in `β`. -/
-def IsOpenMap [TopologicalSpace α] [TopologicalSpace β] (f : α → β) :=
-  ∀ U : Set α, IsOpen U → IsOpen (f '' U)
+/-- A map `f : X → Y` is said to be an *open map*, if the image of any open `U : Set X`
+is open in `Y`. -/
+def IsOpenMap [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) :=
+  ∀ U : Set X, IsOpen U → IsOpen (f '' U)
 #align is_open_map IsOpenMap
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
 namespace IsOpenMap
 
-protected theorem id : IsOpenMap (@id α) := fun s hs => by rwa [image_id]
+protected theorem id : IsOpenMap (@id X) := fun s hs => by rwa [image_id]
 #align is_open_map.id IsOpenMap.id
 
 protected theorem comp (hg : IsOpenMap g) (hf : IsOpenMap f) :
@@ -362,36 +361,36 @@ theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by
   exact hf _ isOpen_univ
 #align is_open_map.is_open_range IsOpenMap.isOpen_range
 
-theorem image_mem_nhds (hf : IsOpenMap f) {x : α} {s : Set α} (hx : s ∈ 𝓝 x) : f '' s ∈ 𝓝 (f x) :=
+theorem image_mem_nhds (hf : IsOpenMap f) {x : X} {s : Set X} (hx : s ∈ 𝓝 x) : f '' s ∈ 𝓝 (f x) :=
   let ⟨t, hts, ht, hxt⟩ := mem_nhds_iff.1 hx
   mem_of_superset (IsOpen.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts)
 #align is_open_map.image_mem_nhds IsOpenMap.image_mem_nhds
 
-theorem range_mem_nhds (hf : IsOpenMap f) (x : α) : range f ∈ 𝓝 (f x) :=
+theorem range_mem_nhds (hf : IsOpenMap f) (x : X) : range f ∈ 𝓝 (f x) :=
   hf.isOpen_range.mem_nhds <| mem_range_self _
 #align is_open_map.range_mem_nhds IsOpenMap.range_mem_nhds
 
-theorem mapsTo_interior (hf : IsOpenMap f) {s : Set α} {t : Set β} (h : MapsTo f s t) :
+theorem mapsTo_interior (hf : IsOpenMap f) {s : Set X} {t : Set Y} (h : MapsTo f s t) :
     MapsTo f (interior s) (interior t) :=
   mapsTo'.2 <|
     interior_maximal (h.mono interior_subset Subset.rfl).image_subset (hf _ isOpen_interior)
 #align is_open_map.maps_to_interior IsOpenMap.mapsTo_interior
 
-theorem image_interior_subset (hf : IsOpenMap f) (s : Set α) :
+theorem image_interior_subset (hf : IsOpenMap f) (s : Set X) :
     f '' interior s ⊆ interior (f '' s) :=
   (hf.mapsTo_interior (mapsTo_image f s)).image_subset
 #align is_open_map.image_interior_subset IsOpenMap.image_interior_subset
 
-theorem nhds_le (hf : IsOpenMap f) (a : α) : 𝓝 (f a) ≤ (𝓝 a).map f :=
+theorem nhds_le (hf : IsOpenMap f) (x : X) : 𝓝 (f x) ≤ (𝓝 x).map f :=
   le_map fun _ => hf.image_mem_nhds
 #align is_open_map.nhds_le IsOpenMap.nhds_le
 
-theorem of_nhds_le (hf : ∀ a, 𝓝 (f a) ≤ map f (𝓝 a)) : IsOpenMap f := fun _s hs =>
-  isOpen_iff_mem_nhds.2 fun _b ⟨_a, has, hab⟩ => hab ▸ hf _ (image_mem_map <| hs.mem_nhds has)
+theorem of_nhds_le (hf : ∀ x, 𝓝 (f x) ≤ map f (𝓝 x)) : IsOpenMap f := fun _s hs =>
+  isOpen_iff_mem_nhds.2 fun _y ⟨_x, hxs, hxy⟩ => hxy ▸ hf _ (image_mem_map <| hs.mem_nhds hxs)
 #align is_open_map.of_nhds_le IsOpenMap.of_nhds_le
 
 theorem of_sections
-    (h : ∀ x, ∃ g : β → α, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f :=
+    (h : ∀ x, ∃ g : Y → X, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f :=
   of_nhds_le fun x =>
     let ⟨g, hgc, hgx, hgf⟩ := h x
     calc
@@ -400,7 +399,7 @@ theorem of_sections
       _ = map f (𝓝 x) := by rw [hgx]
 #align is_open_map.of_sections IsOpenMap.of_sections
 
-theorem of_inverse {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
+theorem of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f')
     (r_inv : RightInverse f f') : IsOpenMap f :=
   of_sections fun _ => ⟨f', h.continuousAt, r_inv _, l_inv⟩
 #align is_open_map.of_inverse IsOpenMap.of_inverse
@@ -412,36 +411,36 @@ theorem to_quotientMap (open_map : IsOpenMap f) (cont : Continuous f) (surj : Su
     ⟨surj, fun s => ⟨fun h => h.preimage cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩
 #align is_open_map.to_quotient_map IsOpenMap.to_quotientMap
 
-theorem interior_preimage_subset_preimage_interior (hf : IsOpenMap f) {s : Set β} :
+theorem interior_preimage_subset_preimage_interior (hf : IsOpenMap f) {s : Set Y} :
     interior (f ⁻¹' s) ⊆ f ⁻¹' interior s :=
   hf.mapsTo_interior (mapsTo_preimage _ _)
 #align is_open_map.interior_preimage_subset_preimage_interior IsOpenMap.interior_preimage_subset_preimage_interior
 
 theorem preimage_interior_eq_interior_preimage (hf₁ : IsOpenMap f) (hf₂ : Continuous f)
-    (s : Set β) : f ⁻¹' interior s = interior (f ⁻¹' s) :=
+    (s : Set Y) : f ⁻¹' interior s = interior (f ⁻¹' s) :=
   Subset.antisymm (preimage_interior_subset_interior_preimage hf₂)
     (interior_preimage_subset_preimage_interior hf₁)
 #align is_open_map.preimage_interior_eq_interior_preimage IsOpenMap.preimage_interior_eq_interior_preimage
 
-theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set β} :
+theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set Y} :
     f ⁻¹' closure s ⊆ closure (f ⁻¹' s) := by
   rw [← compl_subset_compl]
   simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior]
 #align is_open_map.preimage_closure_subset_closure_preimage IsOpenMap.preimage_closure_subset_closure_preimage
 
-theorem preimage_closure_eq_closure_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set β) :
+theorem preimage_closure_eq_closure_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) :
     f ⁻¹' closure s = closure (f ⁻¹' s) :=
   hf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s)
 #align is_open_map.preimage_closure_eq_closure_preimage IsOpenMap.preimage_closure_eq_closure_preimage
 
-theorem preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set β} :
+theorem preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set Y} :
     f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s) := by
   simpa only [frontier_eq_closure_inter_closure, preimage_inter] using
     inter_subset_inter hf.preimage_closure_subset_closure_preimage
       hf.preimage_closure_subset_closure_preimage
 #align is_open_map.preimage_frontier_subset_frontier_preimage IsOpenMap.preimage_frontier_subset_frontier_preimage
 
-theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set β) :
+theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) :
     f ⁻¹' frontier s = frontier (f ⁻¹' s) := by
   simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl,
     hf.preimage_closure_eq_closure_preimage hfc]
@@ -449,7 +448,7 @@ theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continu
 
 end IsOpenMap
 
-theorem isOpenMap_iff_nhds_le : IsOpenMap f ↔ ∀ a : α, 𝓝 (f a) ≤ (𝓝 a).map f :=
+theorem isOpenMap_iff_nhds_le : IsOpenMap f ↔ ∀ x : X, 𝓝 (f x) ≤ (𝓝 x).map f :=
   ⟨fun hf => hf.nhds_le, IsOpenMap.of_nhds_le⟩
 #align is_open_map_iff_nhds_le isOpenMap_iff_nhds_le
 
@@ -470,18 +469,18 @@ end OpenMap
 
 section IsClosedMap
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
-/-- A map `f : α → β` is said to be a *closed map*, if the image of any closed `U : Set α`
-is closed in `β`. -/
-def IsClosedMap (f : α → β) :=
-  ∀ U : Set α, IsClosed U → IsClosed (f '' U)
+/-- A map `f : X → Y` is said to be a *closed map*, if the image of any closed `U : Set X`
+is closed in `Y`. -/
+def IsClosedMap (f : X → Y) :=
+  ∀ U : Set X, IsClosed U → IsClosed (f '' U)
 #align is_closed_map IsClosedMap
 
 namespace IsClosedMap
 open Function
 
-protected theorem id : IsClosedMap (@id α) := fun s hs => by rwa [image_id]
+protected theorem id : IsClosedMap (@id X) := fun s hs => by rwa [image_id]
 #align is_closed_map.id IsClosedMap.id
 
 protected theorem comp (hg : IsClosedMap g) (hf : IsClosedMap f) : IsClosedMap (g ∘ f) := by
@@ -490,12 +489,12 @@ protected theorem comp (hg : IsClosedMap g) (hf : IsClosedMap f) : IsClosedMap (
   exact hg _ (hf _ hs)
 #align is_closed_map.comp IsClosedMap.comp
 
-theorem closure_image_subset (hf : IsClosedMap f) (s : Set α) :
+theorem closure_image_subset (hf : IsClosedMap f) (s : Set X) :
     closure (f '' s) ⊆ f '' closure s :=
   closure_minimal (image_subset _ subset_closure) (hf _ isClosed_closure)
 #align is_closed_map.closure_image_subset IsClosedMap.closure_image_subset
 
-theorem of_inverse {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
+theorem of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f')
     (r_inv : RightInverse f f') : IsClosedMap f := fun s hs => by
   rw [image_eq_preimage_of_inverse r_inv l_inv]
   exact hs.preimage h
@@ -546,20 +545,20 @@ theorem isClosedMap_iff_clusterPt :
     and_comm]
 
 theorem IsClosedMap.closure_image_eq_of_continuous
-    (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set α) :
+    (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set X) :
     closure (f '' s) = f '' closure s :=
   subset_antisymm (f_closed.closure_image_subset s) (image_closure_subset_closure_image f_cont)
 
 theorem IsClosedMap.lift'_closure_map_eq
-    (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter α) :
+    (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter X) :
     (map f F).lift' closure = map f (F.lift' closure) := by
-  rw [map_lift'_eq2 (monotone_closure β), map_lift'_eq (monotone_closure α)]
+  rw [map_lift'_eq2 (monotone_closure Y), map_lift'_eq (monotone_closure X)]
   congr
   ext s : 1
   exact f_closed.closure_image_eq_of_continuous f_cont s
 
 theorem IsClosedMap.mapClusterPt_iff_lift'_closure
-    {F : Filter α} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : β} :
+    {F : Filter X} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : Y} :
     MapClusterPt y F f ↔ ((F.lift' closure) ⊓ 𝓟 (f ⁻¹' {y})).NeBot := by
   rw [MapClusterPt, clusterPt_iff_lift'_closure', f_closed.lift'_closure_map_eq f_cont,
       ← comap_principal, ← map_neBot_iff f, Filter.push_pull, principal_singleton]
@@ -568,11 +567,11 @@ end IsClosedMap
 
 section OpenEmbedding
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
 /-- An open embedding is an embedding with open image. -/
 @[mk_iff]
-structure OpenEmbedding (f : α → β) extends Embedding f : Prop where
+structure OpenEmbedding (f : X → Y) extends Embedding f : Prop where
   /-- The range of an open embedding is an open set. -/
   open_range : IsOpen <| range f
 #align open_embedding OpenEmbedding
@@ -582,28 +581,28 @@ theorem OpenEmbedding.isOpenMap (hf : OpenEmbedding f) : IsOpenMap f :=
   hf.toEmbedding.toInducing.isOpenMap hf.open_range
 #align open_embedding.is_open_map OpenEmbedding.isOpenMap
 
-theorem OpenEmbedding.map_nhds_eq (hf : OpenEmbedding f) (a : α) :
-    map f (𝓝 a) = 𝓝 (f a) :=
+theorem OpenEmbedding.map_nhds_eq (hf : OpenEmbedding f) (x : X) :
+    map f (𝓝 x) = 𝓝 (f x) :=
   hf.toEmbedding.map_nhds_of_mem _ <| hf.open_range.mem_nhds <| mem_range_self _
 #align open_embedding.map_nhds_eq OpenEmbedding.map_nhds_eq
 
-theorem OpenEmbedding.open_iff_image_open (hf : OpenEmbedding f) {s : Set α} :
+theorem OpenEmbedding.open_iff_image_open (hf : OpenEmbedding f) {s : Set X} :
     IsOpen s ↔ IsOpen (f '' s) :=
   ⟨hf.isOpenMap s, fun h => by
     convert ← h.preimage hf.toEmbedding.continuous
     apply preimage_image_eq _ hf.inj⟩
 #align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_open
 
-theorem OpenEmbedding.tendsto_nhds_iff {f : ι → β} {a : Filter ι} {b : β} (hg : OpenEmbedding g) :
-    Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
+theorem OpenEmbedding.tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : OpenEmbedding g) :
+    Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) :=
   hg.toEmbedding.tendsto_nhds_iff
 #align open_embedding.tendsto_nhds_iff OpenEmbedding.tendsto_nhds_iff
 
-theorem OpenEmbedding.tendsto_nhds_iff' (hf : OpenEmbedding f) {l : Filter γ} {a : α} :
-    Tendsto (g ∘ f) (𝓝 a) l ↔ Tendsto g (𝓝 (f a)) l := by
+theorem OpenEmbedding.tendsto_nhds_iff' (hf : OpenEmbedding f) {l : Filter Z} {x : X} :
+    Tendsto (g ∘ f) (𝓝 x) l ↔ Tendsto g (𝓝 (f x)) l := by
   rw [Tendsto, ← map_map, hf.map_nhds_eq]; rfl
 
-theorem OpenEmbedding.continuousAt_iff (hf : OpenEmbedding f) {x : α} :
+theorem OpenEmbedding.continuousAt_iff (hf : OpenEmbedding f) {x : X} :
     ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) :=
   hf.tendsto_nhds_iff'
 #align open_embedding.continuous_at_iff OpenEmbedding.continuousAt_iff
@@ -612,7 +611,7 @@ theorem OpenEmbedding.continuous (hf : OpenEmbedding f) : Continuous f :=
   hf.toEmbedding.continuous
 #align open_embedding.continuous OpenEmbedding.continuous
 
-theorem OpenEmbedding.open_iff_preimage_open (hf : OpenEmbedding f) {s : Set β}
+theorem OpenEmbedding.open_iff_preimage_open (hf : OpenEmbedding f) {s : Set Y}
     (hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) := by
   rw [hf.open_iff_image_open, image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
 #align open_embedding.open_iff_preimage_open OpenEmbedding.open_iff_preimage_open
@@ -635,8 +634,8 @@ theorem openEmbedding_iff_embedding_open :
 theorem openEmbedding_of_continuous_injective_open
     (h₁ : Continuous f) (h₂ : Injective f) (h₃ : IsOpenMap f) : OpenEmbedding f := by
   simp only [openEmbedding_iff_embedding_open, embedding_iff, inducing_iff_nhds, *, and_true_iff]
-  exact fun a =>
-    le_antisymm (h₁.tendsto _).le_comap (@comap_map _ _ (𝓝 a) _ h₂ ▸ comap_mono (h₃.nhds_le _))
+  exact fun x =>
+    le_antisymm (h₁.tendsto _).le_comap (@comap_map _ _ (𝓝 x) _ h₂ ▸ comap_mono (h₃.nhds_le _))
 #align open_embedding_of_continuous_injective_open openEmbedding_of_continuous_injective_open
 
 theorem openEmbedding_iff_continuous_injective_open :
@@ -645,7 +644,7 @@ theorem openEmbedding_iff_continuous_injective_open :
     openEmbedding_of_continuous_injective_open h.1 h.2.1 h.2.2⟩
 #align open_embedding_iff_continuous_injective_open openEmbedding_iff_continuous_injective_open
 
-theorem openEmbedding_id : OpenEmbedding (@id α) :=
+theorem openEmbedding_id : OpenEmbedding (@id X) :=
   ⟨embedding_id, IsOpenMap.id.isOpen_range⟩
 #align open_embedding_id openEmbedding_id
 
@@ -661,13 +660,13 @@ theorem isOpenMap_iff (hg : OpenEmbedding g) :
   simp_rw [isOpenMap_iff_nhds_le, ← map_map, comp, ← hg.map_nhds_eq, Filter.map_le_map_iff hg.inj]
 #align open_embedding.is_open_map_iff OpenEmbedding.isOpenMap_iff
 
-theorem of_comp_iff (f : α → β) (hg : OpenEmbedding g) :
+theorem of_comp_iff (f : X → Y) (hg : OpenEmbedding g) :
     OpenEmbedding (g ∘ f) ↔ OpenEmbedding f := by
   simp only [openEmbedding_iff_continuous_injective_open, ← hg.isOpenMap_iff, ←
     hg.1.continuous_iff, hg.inj.of_comp_iff]
 #align open_embedding.of_comp_iff OpenEmbedding.of_comp_iff
 
-theorem of_comp (f : α → β) (hg : OpenEmbedding g)
+theorem of_comp (f : X → Y) (hg : OpenEmbedding g)
     (h : OpenEmbedding (g ∘ f)) : OpenEmbedding f :=
   (OpenEmbedding.of_comp_iff f hg).1 h
 #align open_embedding.of_comp OpenEmbedding.of_comp
@@ -678,11 +677,11 @@ end OpenEmbedding
 
 section ClosedEmbedding
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
 
 /-- A closed embedding is an embedding with closed image. -/
 @[mk_iff]
-structure ClosedEmbedding (f : α → β) extends Embedding f : Prop where
+structure ClosedEmbedding (f : X → Y) extends Embedding f : Prop where
   /-- The range of a closed embedding is a closed set. -/
   closed_range : IsClosed <| range f
 #align closed_embedding ClosedEmbedding
@@ -690,8 +689,8 @@ structure ClosedEmbedding (f : α → β) extends Embedding f : Prop where
 
 namespace ClosedEmbedding
 
-theorem tendsto_nhds_iff {g : ι → α} {a : Filter ι} {b : α} (hf : ClosedEmbedding f) :
-    Tendsto g a (𝓝 b) ↔ Tendsto (f ∘ g) a (𝓝 (f b)) :=
+theorem tendsto_nhds_iff {g : ι → X} {l : Filter ι} {x : X} (hf : ClosedEmbedding f) :
+    Tendsto g l (𝓝 x) ↔ Tendsto (f ∘ g) l (𝓝 (f x)) :=
   hf.toEmbedding.tendsto_nhds_iff
 #align closed_embedding.tendsto_nhds_iff ClosedEmbedding.tendsto_nhds_iff
 
@@ -703,14 +702,14 @@ theorem isClosedMap (hf : ClosedEmbedding f) : IsClosedMap f :=
   hf.toEmbedding.toInducing.isClosedMap hf.closed_range
 #align closed_embedding.is_closed_map ClosedEmbedding.isClosedMap
 
-theorem closed_iff_image_closed (hf : ClosedEmbedding f) {s : Set α} :
+theorem closed_iff_image_closed (hf : ClosedEmbedding f) {s : Set X} :
     IsClosed s ↔ IsClosed (f '' s) :=
   ⟨hf.isClosedMap s, fun h => by
     rw [← preimage_image_eq s hf.inj]
     exact h.preimage hf.continuous⟩
 #align closed_embedding.closed_iff_image_closed ClosedEmbedding.closed_iff_image_closed
 
-theorem closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set β}
+theorem closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set Y}
     (hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) := by
   rw [hf.closed_iff_image_closed, image_preimage_eq_of_subset hs]
 #align closed_embedding.closed_iff_preimage_closed ClosedEmbedding.closed_iff_preimage_closed
@@ -728,7 +727,7 @@ theorem _root_.closedEmbedding_of_continuous_injective_closed (h₁ : Continuous
   rw [preimage_compl, preimage_image_eq _ h₂, compl_compl]
 #align closed_embedding_of_continuous_injective_closed closedEmbedding_of_continuous_injective_closed
 
-theorem _root_.closedEmbedding_id : ClosedEmbedding (@id α) :=
+theorem _root_.closedEmbedding_id : ClosedEmbedding (@id X) :=
   ⟨embedding_id, IsClosedMap.id.closed_range⟩
 #align closed_embedding_id closedEmbedding_id
 
@@ -737,7 +736,7 @@ theorem comp (hg : ClosedEmbedding g) (hf : ClosedEmbedding f) :
   ⟨hg.toEmbedding.comp hf.toEmbedding, (hg.isClosedMap.comp hf.isClosedMap).closed_range⟩
 #align closed_embedding.comp ClosedEmbedding.comp
 
-theorem closure_image_eq (hf : ClosedEmbedding f) (s : Set α) :
+theorem closure_image_eq (hf : ClosedEmbedding f) (s : Set X) :
     closure (f '' s) = f '' closure s :=
   hf.isClosedMap.closure_image_eq_of_continuous hf.continuous s
 #align closed_embedding.closure_image_eq ClosedEmbedding.closure_image_eq

• Make sure each new definition is in a separate section.
• Add corresponding namespaces where missing.
• Collect TopologicalSpace assumptions.
• Collect variables ${f : \alpha \to \beta}$ and ${g : \beta \to \gamma}$ in theorems; we leave definitions alone.
• In a later PR, we will rename the type variables in this file: this reduces the diff in doing so.

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

@@ -49,7 +49,7 @@ open Set Filter Function
 
 open TopologicalSpace Topology Filter
 
-variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
+variable {α : Type*} {β : Type*} {γ : Type*} {ι : Type*} {f : α → β} {g : β → γ}
 
 section Inducing
 
@@ -63,7 +63,7 @@ structure Inducing [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f :
 #align inducing Inducing
 #align inducing_iff inducing_iff
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
+variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
 theorem inducing_induced (f : α → β) : @Inducing α β (TopologicalSpace.induced f ‹_›) _ f :=
   @Inducing.mk _ _ (TopologicalSpace.induced f ‹_›) _ _ rfl
@@ -72,117 +72,121 @@ theorem inducing_id : Inducing (@id α) :=
   ⟨induced_id.symm⟩
 #align inducing_id inducing_id
 
-protected theorem Inducing.comp {g : β → γ} {f : α → β} (hg : Inducing g) (hf : Inducing f) :
+protected theorem Inducing.comp (hg : Inducing g) (hf : Inducing f) :
     Inducing (g ∘ f) :=
   ⟨by rw [hf.induced, hg.induced, induced_compose]⟩
 #align inducing.comp Inducing.comp
 
-theorem inducing_of_inducing_compose {f : α → β} {g : β → γ} (hf : Continuous f) (hg : Continuous g)
-    (hgf : Inducing (g ∘ f)) : Inducing f :=
+theorem inducing_of_inducing_compose
+    (hf : Continuous f) (hg : Continuous g) (hgf : Inducing (g ∘ f)) : Inducing f :=
   ⟨le_antisymm (by rwa [← continuous_iff_le_induced])
       (by
         rw [hgf.induced, ← induced_compose]
         exact induced_mono hg.le_induced)⟩
 #align inducing_of_inducing_compose inducing_of_inducing_compose
 
-theorem inducing_iff_nhds {f : α → β} : Inducing f ↔ ∀ a, 𝓝 a = comap f (𝓝 (f a)) :=
+theorem inducing_iff_nhds : Inducing f ↔ ∀ a, 𝓝 a = comap f (𝓝 (f a)) :=
   (inducing_iff _).trans (induced_iff_nhds_eq f)
 #align inducing_iff_nhds inducing_iff_nhds
 
-theorem Inducing.nhds_eq_comap {f : α → β} (hf : Inducing f) : ∀ a : α, 𝓝 a = comap f (𝓝 <| f a) :=
+namespace Inducing
+
+theorem nhds_eq_comap (hf : Inducing f) : ∀ a : α, 𝓝 a = comap f (𝓝 <| f a) :=
   inducing_iff_nhds.1 hf
 #align inducing.nhds_eq_comap Inducing.nhds_eq_comap
 
-theorem Inducing.nhdsSet_eq_comap {f : α → β} (hf : Inducing f) (s : Set α) :
+theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set α) :
     𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
   simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image]
 #align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comap
 
-theorem Inducing.map_nhds_eq {f : α → β} (hf : Inducing f) (a : α) : (𝓝 a).map f = 𝓝[range f] f a :=
+theorem map_nhds_eq (hf : Inducing f) (a : α) : (𝓝 a).map f = 𝓝[range f] f a :=
   hf.induced.symm ▸ map_nhds_induced_eq a
 #align inducing.map_nhds_eq Inducing.map_nhds_eq
 
-theorem Inducing.map_nhds_of_mem {f : α → β} (hf : Inducing f) (a : α) (h : range f ∈ 𝓝 (f a)) :
+theorem map_nhds_of_mem (hf : Inducing f) (a : α) (h : range f ∈ 𝓝 (f a)) :
     (𝓝 a).map f = 𝓝 (f a) :=
   hf.induced.symm ▸ map_nhds_induced_of_mem h
 #align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem
 
 -- porting note: new lemma
-theorem Inducing.mapClusterPt_iff {f : α → β} (hf : Inducing f) {a : α} {l : Filter α} :
+theorem mapClusterPt_iff (hf : Inducing f) {a : α} {l : Filter α} :
     MapClusterPt (f a) l f ↔ ClusterPt a l := by
   delta MapClusterPt ClusterPt
   rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff]
 
-theorem Inducing.image_mem_nhdsWithin {f : α → β} (hf : Inducing f) {a : α} {s : Set α}
-    (hs : s ∈ 𝓝 a) : f '' s ∈ 𝓝[range f] f a :=
+theorem image_mem_nhdsWithin (hf : Inducing f) {a : α} {s : Set α} (hs : s ∈ 𝓝 a) :
+    f '' s ∈ 𝓝[range f] f a :=
   hf.map_nhds_eq a ▸ image_mem_map hs
 #align inducing.image_mem_nhds_within Inducing.image_mem_nhdsWithin
 
-theorem Inducing.tendsto_nhds_iff {ι : Type*} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
-    (hg : Inducing g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) := by
+theorem tendsto_nhds_iff {f : ι → β} {a : Filter ι} {b : β} (hg : Inducing g) :
+    Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) := by
   rw [hg.nhds_eq_comap, tendsto_comap_iff]
 #align inducing.tendsto_nhds_iff Inducing.tendsto_nhds_iff
 
-theorem Inducing.continuousAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {x : α} :
+theorem continuousAt_iff (hg : Inducing g) {x : α} :
     ContinuousAt f x ↔ ContinuousAt (g ∘ f) x :=
   hg.tendsto_nhds_iff
 #align inducing.continuous_at_iff Inducing.continuousAt_iff
 
-theorem Inducing.continuous_iff {f : α → β} {g : β → γ} (hg : Inducing g) :
+theorem continuous_iff (hg : Inducing g) :
     Continuous f ↔ Continuous (g ∘ f) := by
   simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff]
 #align inducing.continuous_iff Inducing.continuous_iff
 
-theorem Inducing.continuousAt_iff' {f : α → β} {g : β → γ} (hf : Inducing f) {x : α}
-    (h : range f ∈ 𝓝 (f x)) : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
+theorem continuousAt_iff' (hf : Inducing f) {x : α} (h : range f ∈ 𝓝 (f x)) :
+    ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
   simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map, comp]
 #align inducing.continuous_at_iff' Inducing.continuousAt_iff'
 
-protected theorem Inducing.continuous {f : α → β} (hf : Inducing f) : Continuous f :=
+protected theorem continuous (hf : Inducing f) : Continuous f :=
   hf.continuous_iff.mp continuous_id
 #align inducing.continuous Inducing.continuous
 
-protected theorem Inducing.inducing_iff {f : α → β} {g : β → γ} (hg : Inducing g) :
+protected theorem inducing_iff (hg : Inducing g) :
     Inducing f ↔ Inducing (g ∘ f) := by
   refine' ⟨fun h => hg.comp h, fun hgf => inducing_of_inducing_compose _ hg.continuous hgf⟩
   rw [hg.continuous_iff]
   exact hgf.continuous
 #align inducing.inducing_iff Inducing.inducing_iff
 
-theorem Inducing.closure_eq_preimage_closure_image {f : α → β} (hf : Inducing f) (s : Set α) :
+theorem closure_eq_preimage_closure_image (hf : Inducing f) (s : Set α) :
     closure s = f ⁻¹' closure (f '' s) := by
   ext x
   rw [Set.mem_preimage, ← closure_induced, hf.induced]
 #align inducing.closure_eq_preimage_closure_image Inducing.closure_eq_preimage_closure_image
 
-theorem Inducing.isClosed_iff {f : α → β} (hf : Inducing f) {s : Set α} :
+theorem isClosed_iff (hf : Inducing f) {s : Set α} :
     IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by rw [hf.induced, isClosed_induced_iff]
 #align inducing.is_closed_iff Inducing.isClosed_iff
 
-theorem Inducing.isClosed_iff' {f : α → β} (hf : Inducing f) {s : Set α} :
+theorem isClosed_iff' (hf : Inducing f) {s : Set α} :
     IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by rw [hf.induced, isClosed_induced_iff']
 #align inducing.is_closed_iff' Inducing.isClosed_iff'
 
-theorem Inducing.isClosed_preimage {f : α → β} (h : Inducing f) (s : Set β) (hs : IsClosed s) :
+theorem isClosed_preimage (h : Inducing f) (s : Set β) (hs : IsClosed s) :
     IsClosed (f ⁻¹' s) :=
-  (Inducing.isClosed_iff h).mpr ⟨s, hs, rfl⟩
+  (isClosed_iff h).mpr ⟨s, hs, rfl⟩
 #align inducing.is_closed_preimage Inducing.isClosed_preimage
 
-theorem Inducing.isOpen_iff {f : α → β} (hf : Inducing f) {s : Set α} :
+theorem isOpen_iff (hf : Inducing f) {s : Set α} :
     IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := by rw [hf.induced, isOpen_induced_iff]
 #align inducing.is_open_iff Inducing.isOpen_iff
 
-theorem Inducing.setOf_isOpen {f : α → β} (hf : Inducing f) :
+theorem setOf_isOpen (hf : Inducing f) :
     {s : Set α | IsOpen s} = preimage f '' {t | IsOpen t} :=
   Set.ext fun _ ↦ hf.isOpen_iff
 
-theorem Inducing.dense_iff {f : α → β} (hf : Inducing f) {s : Set α} :
+theorem dense_iff (hf : Inducing f) {s : Set α} :
     Dense s ↔ ∀ x, f x ∈ closure (f '' s) := by
   simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage]
 #align inducing.dense_iff Inducing.dense_iff
 
 end Inducing
 
+end Inducing
+
 section Embedding
 
 /-- A function between topological spaces is an embedding if it is injective,
@@ -211,13 +215,13 @@ theorem embedding_id : Embedding (@id α) :=
   ⟨inducing_id, fun _ _ h => h⟩
 #align embedding_id embedding_id
 
-protected theorem Embedding.comp {g : β → γ} {f : α → β} (hg : Embedding g) (hf : Embedding f) :
+protected theorem Embedding.comp (hg : Embedding g) (hf : Embedding f) :
     Embedding (g ∘ f) :=
   { hg.toInducing.comp hf.toInducing with inj := fun _ _ h => hf.inj <| hg.inj h }
 #align embedding.comp Embedding.comp
 
-theorem embedding_of_embedding_compose {f : α → β} {g : β → γ} (hf : Continuous f)
-    (hg : Continuous g) (hgf : Embedding (g ∘ f)) : Embedding f :=
+theorem embedding_of_embedding_compose
+    (hf : Continuous f) (hg : Continuous g) (hgf : Embedding (g ∘ f)) : Embedding f :=
   { induced := (inducing_of_inducing_compose hf hg hgf.toInducing).induced
     inj := fun a₁ a₂ h => hgf.inj <| by simp [h, (· ∘ ·)] }
 #align embedding_of_embedding_compose embedding_of_embedding_compose
@@ -227,27 +231,27 @@ protected theorem Function.LeftInverse.embedding {f : α → β} {g : β → α}
   embedding_of_embedding_compose hg hf <| h.comp_eq_id.symm ▸ embedding_id
 #align function.left_inverse.embedding Function.LeftInverse.embedding
 
-theorem Embedding.map_nhds_eq {f : α → β} (hf : Embedding f) (a : α) :
+theorem Embedding.map_nhds_eq (hf : Embedding f) (a : α) :
     (𝓝 a).map f = 𝓝[range f] f a :=
   hf.1.map_nhds_eq a
 #align embedding.map_nhds_eq Embedding.map_nhds_eq
 
-theorem Embedding.map_nhds_of_mem {f : α → β} (hf : Embedding f) (a : α) (h : range f ∈ 𝓝 (f a)) :
+theorem Embedding.map_nhds_of_mem (hf : Embedding f) (a : α) (h : range f ∈ 𝓝 (f a)) :
     (𝓝 a).map f = 𝓝 (f a) :=
   hf.1.map_nhds_of_mem a h
 #align embedding.map_nhds_of_mem Embedding.map_nhds_of_mem
 
-theorem Embedding.tendsto_nhds_iff {ι : Type*} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
+theorem Embedding.tendsto_nhds_iff {f : ι → β} {a : Filter ι} {b : β}
     (hg : Embedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
   hg.toInducing.tendsto_nhds_iff
 #align embedding.tendsto_nhds_iff Embedding.tendsto_nhds_iff
 
-theorem Embedding.continuous_iff {f : α → β} {g : β → γ} (hg : Embedding g) :
+theorem Embedding.continuous_iff (hg : Embedding g) :
     Continuous f ↔ Continuous (g ∘ f) :=
   Inducing.continuous_iff hg.1
 #align embedding.continuous_iff Embedding.continuous_iff
 
-theorem Embedding.continuous {f : α → β} (hf : Embedding f) : Continuous f :=
+theorem Embedding.continuous (hf : Embedding f) : Continuous f :=
   Inducing.continuous hf.1
 #align embedding.continuous Embedding.continuous
 
@@ -267,6 +271,7 @@ theorem Embedding.discreteTopology {X Y : Type*} [TopologicalSpace X] [Topologic
 
 end Embedding
 
+section QuotientMap
 /-- A function between topological spaces is a quotient map if it is surjective,
   and for all `s : Set β`, `s` is open iff its preimage is an open set. -/
 def QuotientMap {α : Type*} {β : Type*} [tα : TopologicalSpace α] [tβ : TopologicalSpace β]
@@ -274,12 +279,13 @@ def QuotientMap {α : Type*} {β : Type*} [tα : TopologicalSpace α] [tβ : Top
   Surjective f ∧ tβ = tα.coinduced f
 #align quotient_map QuotientMap
 
-theorem quotientMap_iff [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
-    QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
+variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
+
+theorem quotientMap_iff : QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   and_congr Iff.rfl TopologicalSpace.ext_iff
 #align quotient_map_iff quotientMap_iff
 
-theorem quotientMap_iff_closed [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
+theorem quotientMap_iff_closed :
     QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsClosed s ↔ IsClosed (f ⁻¹' s) :=
   quotientMap_iff.trans <| Iff.rfl.and <| compl_surjective.forall.trans <| by
     simp only [isOpen_compl_iff, preimage_compl]
@@ -287,9 +293,6 @@ theorem quotientMap_iff_closed [TopologicalSpace α] [TopologicalSpace β] {f :
 
 namespace QuotientMap
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
-  {g : β → γ} {f : α → β}
-
 protected theorem id : QuotientMap (@id α) :=
   ⟨fun a => ⟨a, rfl⟩, coinduced_id.symm⟩
 #align quotient_map.id QuotientMap.id
@@ -334,20 +337,23 @@ protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set β} :
 
 end QuotientMap
 
+end QuotientMap
+
+section OpenMap
 /-- A map `f : α → β` is said to be an *open map*, if the image of any open `U : Set α`
 is open in `β`. -/
 def IsOpenMap [TopologicalSpace α] [TopologicalSpace β] (f : α → β) :=
   ∀ U : Set α, IsOpen U → IsOpen (f '' U)
 #align is_open_map IsOpenMap
 
-namespace IsOpenMap
+variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {f : α → β}
+namespace IsOpenMap
 
 protected theorem id : IsOpenMap (@id α) := fun s hs => by rwa [image_id]
 #align is_open_map.id IsOpenMap.id
 
-protected theorem comp {g : β → γ} {f : α → β} (hg : IsOpenMap g) (hf : IsOpenMap f) :
+protected theorem comp (hg : IsOpenMap g) (hf : IsOpenMap f) :
     IsOpenMap (g ∘ f) := fun s hs => by rw [image_comp]; exact hg _ (hf _ hs)
 #align is_open_map.comp IsOpenMap.comp
 
@@ -384,7 +390,7 @@ theorem of_nhds_le (hf : ∀ a, 𝓝 (f a) ≤ map f (𝓝 a)) : IsOpenMap f :=
   isOpen_iff_mem_nhds.2 fun _b ⟨_a, has, hab⟩ => hab ▸ hf _ (image_mem_map <| hs.mem_nhds has)
 #align is_open_map.of_nhds_le IsOpenMap.of_nhds_le
 
-theorem of_sections {f : α → β}
+theorem of_sections
     (h : ∀ x, ∃ g : β → α, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f :=
   of_nhds_le fun x =>
     let ⟨g, hgc, hgx, hgf⟩ := h x
@@ -394,14 +400,14 @@ theorem of_sections {f : α → β}
       _ = map f (𝓝 x) := by rw [hgx]
 #align is_open_map.of_sections IsOpenMap.of_sections
 
-theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
+theorem of_inverse {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
     (r_inv : RightInverse f f') : IsOpenMap f :=
   of_sections fun _ => ⟨f', h.continuousAt, r_inv _, l_inv⟩
 #align is_open_map.of_inverse IsOpenMap.of_inverse
 
 /-- A continuous surjective open map is a quotient map. -/
-theorem to_quotientMap {f : α → β} (open_map : IsOpenMap f) (cont : Continuous f)
-    (surj : Surjective f) : QuotientMap f :=
+theorem to_quotientMap (open_map : IsOpenMap f) (cont : Continuous f) (surj : Surjective f) :
+    QuotientMap f :=
   quotientMap_iff.2
     ⟨surj, fun s => ⟨fun h => h.preimage cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩
 #align is_open_map.to_quotient_map IsOpenMap.to_quotientMap
@@ -443,13 +449,11 @@ theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continu
 
 end IsOpenMap
 
-theorem isOpenMap_iff_nhds_le [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
-    IsOpenMap f ↔ ∀ a : α, 𝓝 (f a) ≤ (𝓝 a).map f :=
+theorem isOpenMap_iff_nhds_le : IsOpenMap f ↔ ∀ a : α, 𝓝 (f a) ≤ (𝓝 a).map f :=
   ⟨fun hf => hf.nhds_le, IsOpenMap.of_nhds_le⟩
 #align is_open_map_iff_nhds_le isOpenMap_iff_nhds_le
 
-theorem isOpenMap_iff_interior [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
-    IsOpenMap f ↔ ∀ s, f '' interior s ⊆ interior (f '' s) :=
+theorem isOpenMap_iff_interior : IsOpenMap f ↔ ∀ s, f '' interior s ⊆ interior (f '' s) :=
   ⟨IsOpenMap.image_interior_subset, fun hs u hu =>
     subset_interior_iff_isOpen.mp <|
       calc
@@ -458,14 +462,15 @@ theorem isOpenMap_iff_interior [TopologicalSpace α] [TopologicalSpace β] {f :
 #align is_open_map_iff_interior isOpenMap_iff_interior
 
 /-- An inducing map with an open range is an open map. -/
-protected theorem Inducing.isOpenMap [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
-    (hi : Inducing f) (ho : IsOpen (range f)) : IsOpenMap f :=
+protected theorem Inducing.isOpenMap (hi : Inducing f) (ho : IsOpen (range f)) : IsOpenMap f :=
   IsOpenMap.of_nhds_le fun _ => (hi.map_nhds_of_mem _ <| IsOpen.mem_nhds ho <| mem_range_self _).ge
 #align inducing.is_open_map Inducing.isOpenMap
 
+end OpenMap
+
 section IsClosedMap
 
-variable [TopologicalSpace α] [TopologicalSpace β]
+variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
 /-- A map `f : α → β` is said to be a *closed map*, if the image of any closed `U : Set α`
 is closed in `β`. -/
@@ -473,47 +478,41 @@ def IsClosedMap (f : α → β) :=
   ∀ U : Set α, IsClosed U → IsClosed (f '' U)
 #align is_closed_map IsClosedMap
 
-end IsClosedMap
-
 namespace IsClosedMap
-
-variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
-
 open Function
 
 protected theorem id : IsClosedMap (@id α) := fun s hs => by rwa [image_id]
 #align is_closed_map.id IsClosedMap.id
 
-protected theorem comp {g : β → γ} {f : α → β} (hg : IsClosedMap g) (hf : IsClosedMap f) :
-    IsClosedMap (g ∘ f) := by
+protected theorem comp (hg : IsClosedMap g) (hf : IsClosedMap f) : IsClosedMap (g ∘ f) := by
   intro s hs
   rw [image_comp]
   exact hg _ (hf _ hs)
 #align is_closed_map.comp IsClosedMap.comp
 
-theorem closure_image_subset {f : α → β} (hf : IsClosedMap f) (s : Set α) :
+theorem closure_image_subset (hf : IsClosedMap f) (s : Set α) :
     closure (f '' s) ⊆ f '' closure s :=
   closure_minimal (image_subset _ subset_closure) (hf _ isClosed_closure)
 #align is_closed_map.closure_image_subset IsClosedMap.closure_image_subset
 
-theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
+theorem of_inverse {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
     (r_inv : RightInverse f f') : IsClosedMap f := fun s hs => by
   rw [image_eq_preimage_of_inverse r_inv l_inv]
   exact hs.preimage h
 #align is_closed_map.of_inverse IsClosedMap.of_inverse
 
-theorem of_nonempty {f : α → β} (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
+theorem of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
     IsClosedMap f := by
   intro s hs; rcases eq_empty_or_nonempty s with h2s | h2s
   · simp_rw [h2s, image_empty, isClosed_empty]
   · exact h s hs h2s
 #align is_closed_map.of_nonempty IsClosedMap.of_nonempty
 
-theorem closed_range {f : α → β} (hf : IsClosedMap f) : IsClosed (range f) :=
+theorem closed_range (hf : IsClosedMap f) : IsClosed (range f) :=
   @image_univ _ _ f ▸ hf _ isClosed_univ
 #align is_closed_map.closed_range IsClosedMap.closed_range
 
-theorem to_quotientMap {f : α → β} (hcl : IsClosedMap f) (hcont : Continuous f)
+theorem to_quotientMap (hcl : IsClosedMap f) (hcont : Continuous f)
     (hsurj : Surjective f) : QuotientMap f :=
   quotientMap_iff_closed.2 ⟨hsurj, fun s =>
     ⟨fun hs => hs.preimage hcont, fun hs => hsurj.image_preimage s ▸ hcl _ hs⟩⟩
@@ -521,15 +520,14 @@ theorem to_quotientMap {f : α → β} (hcl : IsClosedMap f) (hcont : Continuous
 
 end IsClosedMap
 
-theorem Inducing.isClosedMap [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Inducing f)
-    (h : IsClosed (range f)) : IsClosedMap f := by
+theorem Inducing.isClosedMap (hf : Inducing f) (h : IsClosed (range f)) : IsClosedMap f := by
   intro s hs
   rcases hf.isClosed_iff.1 hs with ⟨t, ht, rfl⟩
   rw [image_preimage_eq_inter_range]
   exact ht.inter h
 #align inducing.is_closed_map Inducing.isClosedMap
 
-theorem isClosedMap_iff_closure_image [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
+theorem isClosedMap_iff_closure_image :
     IsClosedMap f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s :=
   ⟨IsClosedMap.closure_image_subset, fun hs c hc =>
     isClosed_of_closure_subset <|
@@ -542,30 +540,32 @@ theorem isClosedMap_iff_closure_image [TopologicalSpace α] [TopologicalSpace β
 the image by `f` of some cluster point of `s`.
 If you require this for all filters instead of just principal filters, and also that `f` is
 continuous, you get the notion of **proper map**. See `isProperMap_iff_clusterPt`. -/
-theorem isClosedMap_iff_clusterPt [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
+theorem isClosedMap_iff_clusterPt :
     IsClosedMap f ↔ ∀ s y, MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s) := by
   simp [MapClusterPt, isClosedMap_iff_closure_image, subset_def, mem_closure_iff_clusterPt,
     and_comm]
 
-theorem IsClosedMap.closure_image_eq_of_continuous [TopologicalSpace α] [TopologicalSpace β]
-    {f : α → β} (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set α) :
+theorem IsClosedMap.closure_image_eq_of_continuous
+    (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set α) :
     closure (f '' s) = f '' closure s :=
   subset_antisymm (f_closed.closure_image_subset s) (image_closure_subset_closure_image f_cont)
 
-theorem IsClosedMap.lift'_closure_map_eq [TopologicalSpace α] [TopologicalSpace β]
-    {f : α → β} (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter α) :
+theorem IsClosedMap.lift'_closure_map_eq
+    (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter α) :
     (map f F).lift' closure = map f (F.lift' closure) := by
   rw [map_lift'_eq2 (monotone_closure β), map_lift'_eq (monotone_closure α)]
   congr
   ext s : 1
   exact f_closed.closure_image_eq_of_continuous f_cont s
 
-theorem IsClosedMap.mapClusterPt_iff_lift'_closure [TopologicalSpace α] [TopologicalSpace β]
-    {F : Filter α} {f : α → β} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : β} :
+theorem IsClosedMap.mapClusterPt_iff_lift'_closure
+    {F : Filter α} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : β} :
     MapClusterPt y F f ↔ ((F.lift' closure) ⊓ 𝓟 (f ⁻¹' {y})).NeBot := by
   rw [MapClusterPt, clusterPt_iff_lift'_closure', f_closed.lift'_closure_map_eq f_cont,
       ← comap_principal, ← map_neBot_iff f, Filter.push_pull, principal_singleton]
 
+end IsClosedMap
+
 section OpenEmbedding
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
@@ -578,68 +578,68 @@ structure OpenEmbedding (f : α → β) extends Embedding f : Prop where
 #align open_embedding OpenEmbedding
 #align open_embedding_iff openEmbedding_iff
 
-theorem OpenEmbedding.isOpenMap {f : α → β} (hf : OpenEmbedding f) : IsOpenMap f :=
+theorem OpenEmbedding.isOpenMap (hf : OpenEmbedding f) : IsOpenMap f :=
   hf.toEmbedding.toInducing.isOpenMap hf.open_range
 #align open_embedding.is_open_map OpenEmbedding.isOpenMap
 
-theorem OpenEmbedding.map_nhds_eq {f : α → β} (hf : OpenEmbedding f) (a : α) :
+theorem OpenEmbedding.map_nhds_eq (hf : OpenEmbedding f) (a : α) :
     map f (𝓝 a) = 𝓝 (f a) :=
   hf.toEmbedding.map_nhds_of_mem _ <| hf.open_range.mem_nhds <| mem_range_self _
 #align open_embedding.map_nhds_eq OpenEmbedding.map_nhds_eq
 
-theorem OpenEmbedding.open_iff_image_open {f : α → β} (hf : OpenEmbedding f) {s : Set α} :
+theorem OpenEmbedding.open_iff_image_open (hf : OpenEmbedding f) {s : Set α} :
     IsOpen s ↔ IsOpen (f '' s) :=
   ⟨hf.isOpenMap s, fun h => by
     convert ← h.preimage hf.toEmbedding.continuous
     apply preimage_image_eq _ hf.inj⟩
 #align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_open
 
-theorem OpenEmbedding.tendsto_nhds_iff {ι : Type*} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
-    (hg : OpenEmbedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
+theorem OpenEmbedding.tendsto_nhds_iff {f : ι → β} {a : Filter ι} {b : β} (hg : OpenEmbedding g) :
+    Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
   hg.toEmbedding.tendsto_nhds_iff
 #align open_embedding.tendsto_nhds_iff OpenEmbedding.tendsto_nhds_iff
 
-theorem OpenEmbedding.tendsto_nhds_iff' {f : α → β} (hf : OpenEmbedding f) {g : β → γ}
-    {l : Filter γ} {a : α} : Tendsto (g ∘ f) (𝓝 a) l ↔ Tendsto g (𝓝 (f a)) l := by
+theorem OpenEmbedding.tendsto_nhds_iff' (hf : OpenEmbedding f) {l : Filter γ} {a : α} :
+    Tendsto (g ∘ f) (𝓝 a) l ↔ Tendsto g (𝓝 (f a)) l := by
   rw [Tendsto, ← map_map, hf.map_nhds_eq]; rfl
 
-theorem OpenEmbedding.continuousAt_iff {f : α → β} (hf : OpenEmbedding f) {g : β → γ} {x : α} :
+theorem OpenEmbedding.continuousAt_iff (hf : OpenEmbedding f) {x : α} :
     ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) :=
   hf.tendsto_nhds_iff'
 #align open_embedding.continuous_at_iff OpenEmbedding.continuousAt_iff
 
-theorem OpenEmbedding.continuous {f : α → β} (hf : OpenEmbedding f) : Continuous f :=
+theorem OpenEmbedding.continuous (hf : OpenEmbedding f) : Continuous f :=
   hf.toEmbedding.continuous
 #align open_embedding.continuous OpenEmbedding.continuous
 
-theorem OpenEmbedding.open_iff_preimage_open {f : α → β} (hf : OpenEmbedding f) {s : Set β}
+theorem OpenEmbedding.open_iff_preimage_open (hf : OpenEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) := by
   rw [hf.open_iff_image_open, image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs]
 #align open_embedding.open_iff_preimage_open OpenEmbedding.open_iff_preimage_open
 
-theorem openEmbedding_of_embedding_open {f : α → β} (h₁ : Embedding f) (h₂ : IsOpenMap f) :
+theorem openEmbedding_of_embedding_open (h₁ : Embedding f) (h₂ : IsOpenMap f) :
     OpenEmbedding f :=
   ⟨h₁, h₂.isOpen_range⟩
 #align open_embedding_of_embedding_open openEmbedding_of_embedding_open
 
 /-- A surjective embedding is an `OpenEmbedding`. -/
-theorem _root_.Embedding.toOpenEmbedding_of_surjective {f : α → β}
-    (hf : Embedding f) (hsurj: f.Surjective) : OpenEmbedding f :=
+theorem _root_.Embedding.toOpenEmbedding_of_surjective (hf : Embedding f) (hsurj: f.Surjective) :
+    OpenEmbedding f :=
   ⟨hf, hsurj.range_eq ▸ isOpen_univ⟩
 
-theorem openEmbedding_iff_embedding_open {f : α → β} :
+theorem openEmbedding_iff_embedding_open :
     OpenEmbedding f ↔ Embedding f ∧ IsOpenMap f :=
   ⟨fun h => ⟨h.1, h.isOpenMap⟩, fun h => openEmbedding_of_embedding_open h.1 h.2⟩
 #align open_embedding_iff_embedding_open openEmbedding_iff_embedding_open
 
-theorem openEmbedding_of_continuous_injective_open {f : α → β} (h₁ : Continuous f)
-    (h₂ : Injective f) (h₃ : IsOpenMap f) : OpenEmbedding f := by
+theorem openEmbedding_of_continuous_injective_open
+    (h₁ : Continuous f) (h₂ : Injective f) (h₃ : IsOpenMap f) : OpenEmbedding f := by
   simp only [openEmbedding_iff_embedding_open, embedding_iff, inducing_iff_nhds, *, and_true_iff]
   exact fun a =>
     le_antisymm (h₁.tendsto _).le_comap (@comap_map _ _ (𝓝 a) _ h₂ ▸ comap_mono (h₃.nhds_le _))
 #align open_embedding_of_continuous_injective_open openEmbedding_of_continuous_injective_open
 
-theorem openEmbedding_iff_continuous_injective_open {f : α → β} :
+theorem openEmbedding_iff_continuous_injective_open :
     OpenEmbedding f ↔ Continuous f ∧ Injective f ∧ IsOpenMap f :=
   ⟨fun h => ⟨h.continuous, h.inj, h.isOpenMap⟩, fun h =>
     openEmbedding_of_continuous_injective_open h.1 h.2.1 h.2.2⟩
@@ -649,29 +649,33 @@ theorem openEmbedding_id : OpenEmbedding (@id α) :=
   ⟨embedding_id, IsOpenMap.id.isOpen_range⟩
 #align open_embedding_id openEmbedding_id
 
-protected theorem OpenEmbedding.comp {g : β → γ} {f : α → β} (hg : OpenEmbedding g)
+namespace OpenEmbedding
+
+protected theorem comp (hg : OpenEmbedding g)
     (hf : OpenEmbedding f) : OpenEmbedding (g ∘ f) :=
   ⟨hg.1.comp hf.1, (hg.isOpenMap.comp hf.isOpenMap).isOpen_range⟩
 #align open_embedding.comp OpenEmbedding.comp
 
-theorem OpenEmbedding.isOpenMap_iff {g : β → γ} {f : α → β} (hg : OpenEmbedding g) :
+theorem isOpenMap_iff (hg : OpenEmbedding g) :
     IsOpenMap f ↔ IsOpenMap (g ∘ f) := by
   simp_rw [isOpenMap_iff_nhds_le, ← map_map, comp, ← hg.map_nhds_eq, Filter.map_le_map_iff hg.inj]
 #align open_embedding.is_open_map_iff OpenEmbedding.isOpenMap_iff
 
-theorem OpenEmbedding.of_comp_iff (f : α → β) {g : β → γ} (hg : OpenEmbedding g) :
+theorem of_comp_iff (f : α → β) (hg : OpenEmbedding g) :
     OpenEmbedding (g ∘ f) ↔ OpenEmbedding f := by
   simp only [openEmbedding_iff_continuous_injective_open, ← hg.isOpenMap_iff, ←
     hg.1.continuous_iff, hg.inj.of_comp_iff]
 #align open_embedding.of_comp_iff OpenEmbedding.of_comp_iff
 
-theorem OpenEmbedding.of_comp (f : α → β) {g : β → γ} (hg : OpenEmbedding g)
+theorem of_comp (f : α → β) (hg : OpenEmbedding g)
     (h : OpenEmbedding (g ∘ f)) : OpenEmbedding f :=
   (OpenEmbedding.of_comp_iff f hg).1 h
 #align open_embedding.of_comp OpenEmbedding.of_comp
 
 end OpenEmbedding
 
+end OpenEmbedding
+
 section ClosedEmbedding
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
@@ -684,39 +688,39 @@ structure ClosedEmbedding (f : α → β) extends Embedding f : Prop where
 #align closed_embedding ClosedEmbedding
 #align closed_embedding_iff closedEmbedding_iff
 
-variable {f : α → β}
+namespace ClosedEmbedding
 
-theorem ClosedEmbedding.tendsto_nhds_iff {ι : Type*} {g : ι → α} {a : Filter ι} {b : α}
-    (hf : ClosedEmbedding f) : Tendsto g a (𝓝 b) ↔ Tendsto (f ∘ g) a (𝓝 (f b)) :=
+theorem tendsto_nhds_iff {g : ι → α} {a : Filter ι} {b : α} (hf : ClosedEmbedding f) :
+    Tendsto g a (𝓝 b) ↔ Tendsto (f ∘ g) a (𝓝 (f b)) :=
   hf.toEmbedding.tendsto_nhds_iff
 #align closed_embedding.tendsto_nhds_iff ClosedEmbedding.tendsto_nhds_iff
 
-theorem ClosedEmbedding.continuous (hf : ClosedEmbedding f) : Continuous f :=
+theorem continuous (hf : ClosedEmbedding f) : Continuous f :=
   hf.toEmbedding.continuous
 #align closed_embedding.continuous ClosedEmbedding.continuous
 
-theorem ClosedEmbedding.isClosedMap (hf : ClosedEmbedding f) : IsClosedMap f :=
+theorem isClosedMap (hf : ClosedEmbedding f) : IsClosedMap f :=
   hf.toEmbedding.toInducing.isClosedMap hf.closed_range
 #align closed_embedding.is_closed_map ClosedEmbedding.isClosedMap
 
-theorem ClosedEmbedding.closed_iff_image_closed (hf : ClosedEmbedding f) {s : Set α} :
+theorem closed_iff_image_closed (hf : ClosedEmbedding f) {s : Set α} :
     IsClosed s ↔ IsClosed (f '' s) :=
   ⟨hf.isClosedMap s, fun h => by
     rw [← preimage_image_eq s hf.inj]
     exact h.preimage hf.continuous⟩
 #align closed_embedding.closed_iff_image_closed ClosedEmbedding.closed_iff_image_closed
 
-theorem ClosedEmbedding.closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set β}
+theorem closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set β}
     (hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) := by
   rw [hf.closed_iff_image_closed, image_preimage_eq_of_subset hs]
 #align closed_embedding.closed_iff_preimage_closed ClosedEmbedding.closed_iff_preimage_closed
 
-theorem closedEmbedding_of_embedding_closed (h₁ : Embedding f) (h₂ : IsClosedMap f) :
+theorem _root_.closedEmbedding_of_embedding_closed (h₁ : Embedding f) (h₂ : IsClosedMap f) :
     ClosedEmbedding f :=
   ⟨h₁, image_univ (f := f) ▸ h₂ univ isClosed_univ⟩
 #align closed_embedding_of_embedding_closed closedEmbedding_of_embedding_closed
 
-theorem closedEmbedding_of_continuous_injective_closed (h₁ : Continuous f) (h₂ : Injective f)
+theorem _root_.closedEmbedding_of_continuous_injective_closed (h₁ : Continuous f) (h₂ : Injective f)
     (h₃ : IsClosedMap f) : ClosedEmbedding f := by
   refine closedEmbedding_of_embedding_closed ⟨⟨?_⟩, h₂⟩ h₃
   refine h₁.le_induced.antisymm fun s hs => ?_
@@ -724,18 +728,20 @@ theorem closedEmbedding_of_continuous_injective_closed (h₁ : Continuous f) (h
   rw [preimage_compl, preimage_image_eq _ h₂, compl_compl]
 #align closed_embedding_of_continuous_injective_closed closedEmbedding_of_continuous_injective_closed
 
-theorem closedEmbedding_id : ClosedEmbedding (@id α) :=
+theorem _root_.closedEmbedding_id : ClosedEmbedding (@id α) :=
   ⟨embedding_id, IsClosedMap.id.closed_range⟩
 #align closed_embedding_id closedEmbedding_id
 
-theorem ClosedEmbedding.comp {g : β → γ} {f : α → β} (hg : ClosedEmbedding g)
-    (hf : ClosedEmbedding f) : ClosedEmbedding (g ∘ f) :=
+theorem comp (hg : ClosedEmbedding g) (hf : ClosedEmbedding f) :
+    ClosedEmbedding (g ∘ f) :=
   ⟨hg.toEmbedding.comp hf.toEmbedding, (hg.isClosedMap.comp hf.isClosedMap).closed_range⟩
 #align closed_embedding.comp ClosedEmbedding.comp
 
-theorem ClosedEmbedding.closure_image_eq {f : α → β} (hf : ClosedEmbedding f) (s : Set α) :
+theorem closure_image_eq (hf : ClosedEmbedding f) (s : Set α) :
     closure (f '' s) = f '' closure s :=
   hf.isClosedMap.closure_image_eq_of_continuous hf.continuous s
 #align closed_embedding.closure_image_eq ClosedEmbedding.closure_image_eq
 
 end ClosedEmbedding
+
+end ClosedEmbedding

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

Partially addresses issue #9129

@@ -56,7 +56,7 @@ section Inducing
 /-- A function `f : α → β` between topological spaces is inducing if the topology on `α` is induced
 by the topology on `β` through `f`, meaning that a set `s : Set α` is open iff it is the preimage
 under `f` of some open set `t : Set β`. -/
-@[mk_iff inducing_iff]
+@[mk_iff]
 structure Inducing [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : α → β) : Prop where
   /-- The topology on the domain is equal to the induced topology. -/
   induced : tα = tβ.induced f
@@ -187,7 +187,7 @@ section Embedding
 
 /-- A function between topological spaces is an embedding if it is injective,
   and for all `s : Set α`, `s` is open iff it is the preimage of an open set. -/
-@[mk_iff embedding_iff]
+@[mk_iff]
 structure Embedding [TopologicalSpace α] [TopologicalSpace β] (f : α → β) extends
   Inducing f : Prop where
   /-- A topological embedding is injective. -/
@@ -571,7 +571,7 @@ section OpenEmbedding
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
 /-- An open embedding is an embedding with open image. -/
-@[mk_iff openEmbedding_iff]
+@[mk_iff]
 structure OpenEmbedding (f : α → β) extends Embedding f : Prop where
   /-- The range of an open embedding is an open set. -/
   open_range : IsOpen <| range f
@@ -677,7 +677,7 @@ section ClosedEmbedding
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
 
 /-- A closed embedding is an embedding with closed image. -/
-@[mk_iff closedEmbedding_iff]
+@[mk_iff]
 structure ClosedEmbedding (f : α → β) extends Embedding f : Prop where
   /-- The range of a closed embedding is a closed set. -/
   closed_range : IsClosed <| range f

Basis for generalising the result for local diffeomorphism to local structomorphisms.

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

@@ -622,6 +622,11 @@ theorem openEmbedding_of_embedding_open {f : α → β} (h₁ : Embedding f) (h
   ⟨h₁, h₂.isOpen_range⟩
 #align open_embedding_of_embedding_open openEmbedding_of_embedding_open
 
+/-- A surjective embedding is an `OpenEmbedding`. -/
+theorem _root_.Embedding.toOpenEmbedding_of_surjective {f : α → β}
+    (hf : Embedding f) (hsurj: f.Surjective) : OpenEmbedding f :=
+  ⟨hf, hsurj.range_eq ▸ isOpen_univ⟩
+
 theorem openEmbedding_iff_embedding_open {f : α → β} :
     OpenEmbedding f ↔ Embedding f ∧ IsOpenMap f :=
   ⟨fun h => ⟨h.1, h.isOpenMap⟩, fun h => openEmbedding_of_embedding_open h.1 h.2⟩

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -504,7 +504,7 @@ theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv :
 
 theorem of_nonempty {f : α → β} (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
     IsClosedMap f := by
-  intro s hs; cases' eq_empty_or_nonempty s with h2s h2s
+  intro s hs; rcases eq_empty_or_nonempty s with h2s | h2s
   · simp_rw [h2s, image_empty, isClosed_empty]
   · exact h s hs h2s
 #align is_closed_map.of_nonempty IsClosedMap.of_nonempty

A stronger version of #8800, the differences are:

• assume either IsSigmaCompact U or SigmaCompactSpace M;

• only need test functions satisfying tsupport g ⊆ U rather than support g ⊆ U;

• requires LocallyIntegrableOn U rather than LocallyIntegrable on the whole space.

Also fills in some missing APIs around the manifold and measure theory libraries.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -603,6 +603,11 @@ theorem OpenEmbedding.tendsto_nhds_iff' {f : α → β} (hf : OpenEmbedding f) {
     {l : Filter γ} {a : α} : Tendsto (g ∘ f) (𝓝 a) l ↔ Tendsto g (𝓝 (f a)) l := by
   rw [Tendsto, ← map_map, hf.map_nhds_eq]; rfl
 
+theorem OpenEmbedding.continuousAt_iff {f : α → β} (hf : OpenEmbedding f) {g : β → γ} {x : α} :
+    ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) :=
+  hf.tendsto_nhds_iff'
+#align open_embedding.continuous_at_iff OpenEmbedding.continuousAt_iff
+
 theorem OpenEmbedding.continuous {f : α → β} (hf : OpenEmbedding f) : Continuous f :=
   hf.toEmbedding.continuous
 #align open_embedding.continuous OpenEmbedding.continuous

@@ -256,13 +256,13 @@ theorem Embedding.closure_eq_preimage_closure_image {e : α → β} (he : Embedd
   he.1.closure_eq_preimage_closure_image s
 #align embedding.closure_eq_preimage_closure_image Embedding.closure_eq_preimage_closure_image
 
-/-- The topology induced under an inclusion `f : X → Y` from the discrete topological space `Y`
-is the discrete topology on `X`. -/
-theorem Embedding.discreteTopology {X Y : Type*} [TopologicalSpace X] [tY : TopologicalSpace Y]
+/-- The topology induced under an inclusion `f : X → Y` from a discrete topological space `Y`
+is the discrete topology on `X`.
+
+See also `DiscreteTopology.of_continuous_injective`. -/
+theorem Embedding.discreteTopology {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
     [DiscreteTopology Y] {f : X → Y} (hf : Embedding f) : DiscreteTopology X :=
-  discreteTopology_iff_nhds.2 fun x => by
-    rw [hf.nhds_eq_comap, nhds_discrete, comap_pure, ← image_singleton, hf.inj.preimage_image,
-      principal_singleton]
+  .of_continuous_injective hf.continuous hf.inj
 #align embedding.discrete_topology Embedding.discreteTopology
 
 end Embedding

@@ -276,7 +276,7 @@ def QuotientMap {α : Type*} {β : Type*} [tα : TopologicalSpace α] [tβ : Top
 
 theorem quotientMap_iff [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
-  and_congr Iff.rfl topologicalSpace_eq_iff
+  and_congr Iff.rfl TopologicalSpace.ext_iff
 #align quotient_map_iff quotientMap_iff
 
 theorem quotientMap_iff_closed [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :

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

This has nice performance benefits.

@@ -49,7 +49,7 @@ open Set Filter Function
 
 open TopologicalSpace Topology Filter
 
-variable {α : Type _} {β : Type _} {γ : Type _} {δ : Type _}
+variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
 
 section Inducing
 
@@ -118,7 +118,7 @@ theorem Inducing.image_mem_nhdsWithin {f : α → β} (hf : Inducing f) {a : α}
   hf.map_nhds_eq a ▸ image_mem_map hs
 #align inducing.image_mem_nhds_within Inducing.image_mem_nhdsWithin
 
-theorem Inducing.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
+theorem Inducing.tendsto_nhds_iff {ι : Type*} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : Inducing g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) := by
   rw [hg.nhds_eq_comap, tendsto_comap_iff]
 #align inducing.tendsto_nhds_iff Inducing.tendsto_nhds_iff
@@ -237,7 +237,7 @@ theorem Embedding.map_nhds_of_mem {f : α → β} (hf : Embedding f) (a : α) (h
   hf.1.map_nhds_of_mem a h
 #align embedding.map_nhds_of_mem Embedding.map_nhds_of_mem
 
-theorem Embedding.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
+theorem Embedding.tendsto_nhds_iff {ι : Type*} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : Embedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
   hg.toInducing.tendsto_nhds_iff
 #align embedding.tendsto_nhds_iff Embedding.tendsto_nhds_iff
@@ -258,7 +258,7 @@ theorem Embedding.closure_eq_preimage_closure_image {e : α → β} (he : Embedd
 
 /-- The topology induced under an inclusion `f : X → Y` from the discrete topological space `Y`
 is the discrete topology on `X`. -/
-theorem Embedding.discreteTopology {X Y : Type _} [TopologicalSpace X] [tY : TopologicalSpace Y]
+theorem Embedding.discreteTopology {X Y : Type*} [TopologicalSpace X] [tY : TopologicalSpace Y]
     [DiscreteTopology Y] {f : X → Y} (hf : Embedding f) : DiscreteTopology X :=
   discreteTopology_iff_nhds.2 fun x => by
     rw [hf.nhds_eq_comap, nhds_discrete, comap_pure, ← image_singleton, hf.inj.preimage_image,
@@ -269,7 +269,7 @@ end Embedding
 
 /-- A function between topological spaces is a quotient map if it is surjective,
   and for all `s : Set β`, `s` is open iff its preimage is an open set. -/
-def QuotientMap {α : Type _} {β : Type _} [tα : TopologicalSpace α] [tβ : TopologicalSpace β]
+def QuotientMap {α : Type*} {β : Type*} [tα : TopologicalSpace α] [tβ : TopologicalSpace β]
     (f : α → β) : Prop :=
   Surjective f ∧ tβ = tα.coinduced f
 #align quotient_map QuotientMap
@@ -594,7 +594,7 @@ theorem OpenEmbedding.open_iff_image_open {f : α → β} (hf : OpenEmbedding f)
     apply preimage_image_eq _ hf.inj⟩
 #align open_embedding.open_iff_image_open OpenEmbedding.open_iff_image_open
 
-theorem OpenEmbedding.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
+theorem OpenEmbedding.tendsto_nhds_iff {ι : Type*} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β}
     (hg : OpenEmbedding g) : Tendsto f a (𝓝 b) ↔ Tendsto (g ∘ f) a (𝓝 (g b)) :=
   hg.toEmbedding.tendsto_nhds_iff
 #align open_embedding.tendsto_nhds_iff OpenEmbedding.tendsto_nhds_iff
@@ -676,7 +676,7 @@ structure ClosedEmbedding (f : α → β) extends Embedding f : Prop where
 
 variable {f : α → β}
 
-theorem ClosedEmbedding.tendsto_nhds_iff {ι : Type _} {g : ι → α} {a : Filter ι} {b : α}
+theorem ClosedEmbedding.tendsto_nhds_iff {ι : Type*} {g : ι → α} {a : Filter ι} {b : α}
     (hf : ClosedEmbedding f) : Tendsto g a (𝓝 b) ↔ Tendsto (f ∘ g) a (𝓝 (f b)) :=
   hf.toEmbedding.tendsto_nhds_iff
 #align closed_embedding.tendsto_nhds_iff ClosedEmbedding.tendsto_nhds_iff

@@ -538,11 +538,34 @@ theorem isClosedMap_iff_closure_image [TopologicalSpace α] [TopologicalSpace β
         _ = f '' c := by rw [hc.closure_eq]⟩
 #align is_closed_map_iff_closure_image isClosedMap_iff_closure_image
 
+/-- A map `f : X → Y` is closed if and only if for all sets `s`, any cluster point of `f '' s` is
+the image by `f` of some cluster point of `s`.
+If you require this for all filters instead of just principal filters, and also that `f` is
+continuous, you get the notion of **proper map**. See `isProperMap_iff_clusterPt`. -/
 theorem isClosedMap_iff_clusterPt [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     IsClosedMap f ↔ ∀ s y, MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s) := by
   simp [MapClusterPt, isClosedMap_iff_closure_image, subset_def, mem_closure_iff_clusterPt,
     and_comm]
 
+theorem IsClosedMap.closure_image_eq_of_continuous [TopologicalSpace α] [TopologicalSpace β]
+    {f : α → β} (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set α) :
+    closure (f '' s) = f '' closure s :=
+  subset_antisymm (f_closed.closure_image_subset s) (image_closure_subset_closure_image f_cont)
+
+theorem IsClosedMap.lift'_closure_map_eq [TopologicalSpace α] [TopologicalSpace β]
+    {f : α → β} (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter α) :
+    (map f F).lift' closure = map f (F.lift' closure) := by
+  rw [map_lift'_eq2 (monotone_closure β), map_lift'_eq (monotone_closure α)]
+  congr
+  ext s : 1
+  exact f_closed.closure_image_eq_of_continuous f_cont s
+
+theorem IsClosedMap.mapClusterPt_iff_lift'_closure [TopologicalSpace α] [TopologicalSpace β]
+    {F : Filter α} {f : α → β} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : β} :
+    MapClusterPt y F f ↔ ((F.lift' closure) ⊓ 𝓟 (f ⁻¹' {y})).NeBot := by
+  rw [MapClusterPt, clusterPt_iff_lift'_closure', f_closed.lift'_closure_map_eq f_cont,
+      ← comap_principal, ← map_neBot_iff f, Filter.push_pull, principal_singleton]
+
 section OpenEmbedding
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
@@ -702,8 +725,7 @@ theorem ClosedEmbedding.comp {g : β → γ} {f : α → β} (hg : ClosedEmbeddi
 
 theorem ClosedEmbedding.closure_image_eq {f : α → β} (hf : ClosedEmbedding f) (s : Set α) :
     closure (f '' s) = f '' closure s :=
-  (hf.isClosedMap.closure_image_subset _).antisymm
-    (image_closure_subset_closure_image hf.continuous)
+  hf.isClosedMap.closure_image_eq_of_continuous hf.continuous s
 #align closed_embedding.closure_image_eq ClosedEmbedding.closure_image_eq
 
 end ClosedEmbedding

@@ -538,6 +538,11 @@ theorem isClosedMap_iff_closure_image [TopologicalSpace α] [TopologicalSpace β
         _ = f '' c := by rw [hc.closure_eq]⟩
 #align is_closed_map_iff_closure_image isClosedMap_iff_closure_image
 
+theorem isClosedMap_iff_clusterPt [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
+    IsClosedMap f ↔ ∀ s y, MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s) := by
+  simp [MapClusterPt, isClosedMap_iff_closure_image, subset_def, mem_closure_iff_clusterPt,
+    and_comm]
+
 section OpenEmbedding
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-
-! This file was ported from Lean 3 source module topology.maps
-! leanprover-community/mathlib commit d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Order
 import Mathlib.Topology.NhdsSet
 
+#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
+
 /-!
 # Specific classes of maps between topological spaces
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -69,7 +69,7 @@ structure Inducing [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f :
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 
 theorem inducing_induced (f : α → β) : @Inducing α β (TopologicalSpace.induced f ‹_›) _ f :=
-  @Inducing.mk  _ _ (TopologicalSpace.induced f ‹_›) _ _ rfl
+  @Inducing.mk _ _ (TopologicalSpace.induced f ‹_›) _ _ rfl
 
 theorem inducing_id : Inducing (@id α) :=
   ⟨induced_id.symm⟩
@@ -551,7 +551,7 @@ structure OpenEmbedding (f : α → β) extends Embedding f : Prop where
   /-- The range of an open embedding is an open set. -/
   open_range : IsOpen <| range f
 #align open_embedding OpenEmbedding
-#align open_embedding_iff  openEmbedding_iff
+#align open_embedding_iff openEmbedding_iff
 
 theorem OpenEmbedding.isOpenMap {f : α → β} (hf : OpenEmbedding f) : IsOpenMap f :=
   hf.toEmbedding.toInducing.isOpenMap hf.open_range

If g : C(β, γ) is inducing, then fun f : C(α, β) ↦ g.comp f is inducing.

@@ -175,6 +175,10 @@ theorem Inducing.isOpen_iff {f : α → β} (hf : Inducing f) {s : Set α} :
     IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := by rw [hf.induced, isOpen_induced_iff]
 #align inducing.is_open_iff Inducing.isOpen_iff
 
+theorem Inducing.setOf_isOpen {f : α → β} (hf : Inducing f) :
+    {s : Set α | IsOpen s} = preimage f '' {t | IsOpen t} :=
+  Set.ext fun _ ↦ hf.isOpen_iff
+
 theorem Inducing.dense_iff {f : α → β} (hf : Inducing f) {s : Set α} :
     Dense s ↔ ∀ x, f x ∈ closure (f '' s) := by
   simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage]

Forward-port of leanprover-community/mathlib#19071

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 
 ! This file was ported from Lean 3 source module topology.maps
-! leanprover-community/mathlib commit bcfa726826abd57587355b4b5b7e78ad6527b7e4
+! leanprover-community/mathlib commit d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -273,11 +273,17 @@ def QuotientMap {α : Type _} {β : Type _} [tα : TopologicalSpace α] [tβ : T
   Surjective f ∧ tβ = tα.coinduced f
 #align quotient_map QuotientMap
 
-theorem quotientMap_iff {α β : Type _} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
+theorem quotientMap_iff [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
     QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
   and_congr Iff.rfl topologicalSpace_eq_iff
 #align quotient_map_iff quotientMap_iff
 
+theorem quotientMap_iff_closed [TopologicalSpace α] [TopologicalSpace β] {f : α → β} :
+    QuotientMap f ↔ Surjective f ∧ ∀ s : Set β, IsClosed s ↔ IsClosed (f ⁻¹' s) :=
+  quotientMap_iff.trans <| Iff.rfl.and <| compl_surjective.forall.trans <| by
+    simp only [isOpen_compl_iff, preimage_compl]
+#align quotient_map_iff_closed quotientMap_iff_closed
+
 namespace QuotientMap
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
@@ -321,8 +327,8 @@ protected theorem isOpen_preimage (hf : QuotientMap f) {s : Set β} : IsOpen (f
 #align quotient_map.is_open_preimage QuotientMap.isOpen_preimage
 
 protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set β} :
-    IsClosed (f ⁻¹' s) ↔ IsClosed s := by
-  simp only [← isOpen_compl_iff, ← preimage_compl, hf.isOpen_preimage]
+    IsClosed (f ⁻¹' s) ↔ IsClosed s :=
+  ((quotientMap_iff_closed.1 hf).2 s).symm
 #align quotient_map.is_closed_preimage QuotientMap.isClosed_preimage
 
 end QuotientMap
@@ -506,6 +512,12 @@ theorem closed_range {f : α → β} (hf : IsClosedMap f) : IsClosed (range f) :
   @image_univ _ _ f ▸ hf _ isClosed_univ
 #align is_closed_map.closed_range IsClosedMap.closed_range
 
+theorem to_quotientMap {f : α → β} (hcl : IsClosedMap f) (hcont : Continuous f)
+    (hsurj : Surjective f) : QuotientMap f :=
+  quotientMap_iff_closed.2 ⟨hsurj, fun s =>
+    ⟨fun hs => hs.preimage hcont, fun hs => hsurj.image_preimage s ▸ hcl _ hs⟩⟩
+#align is_closed_map.to_quotient_map IsClosedMap.to_quotientMap
+
 end IsClosedMap
 
 theorem Inducing.isClosedMap [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Inducing f)

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

@@ -57,8 +57,8 @@ variable {α : Type _} {β : Type _} {γ : Type _} {δ : Type _}
 section Inducing
 
 /-- A function `f : α → β` between topological spaces is inducing if the topology on `α` is induced
-by the topology on `β` through `f`, meaning that a set `s : set α` is open iff it is the preimage
-under `f` of some open set `t : set β`. -/
+by the topology on `β` through `f`, meaning that a set `s : Set α` is open iff it is the preimage
+under `f` of some open set `t : Set β`. -/
 @[mk_iff inducing_iff]
 structure Inducing [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : α → β) : Prop where
   /-- The topology on the domain is equal to the induced topology. -/
@@ -185,7 +185,7 @@ end Inducing
 section Embedding
 
 /-- A function between topological spaces is an embedding if it is injective,
-  and for all `s : set α`, `s` is open iff it is the preimage of an open set. -/
+  and for all `s : Set α`, `s` is open iff it is the preimage of an open set. -/
 @[mk_iff embedding_iff]
 structure Embedding [TopologicalSpace α] [TopologicalSpace β] (f : α → β) extends
   Inducing f : Prop where
@@ -267,7 +267,7 @@ theorem Embedding.discreteTopology {X Y : Type _} [TopologicalSpace X] [tY : Top
 end Embedding
 
 /-- A function between topological spaces is a quotient map if it is surjective,
-  and for all `s : set β`, `s` is open iff its preimage is an open set. -/
+  and for all `s : Set β`, `s` is open iff its preimage is an open set. -/
 def QuotientMap {α : Type _} {β : Type _} [tα : TopologicalSpace α] [tβ : TopologicalSpace β]
     (f : α → β) : Prop :=
   Surjective f ∧ tβ = tα.coinduced f
@@ -327,7 +327,7 @@ protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set β} :
 
 end QuotientMap
 
-/-- A map `f : α → β` is said to be an *open map*, if the image of any open `U : set α`
+/-- A map `f : α → β` is said to be an *open map*, if the image of any open `U : Set α`
 is open in `β`. -/
 def IsOpenMap [TopologicalSpace α] [TopologicalSpace β] (f : α → β) :=
   ∀ U : Set α, IsOpen U → IsOpen (f '' U)
@@ -460,7 +460,7 @@ section IsClosedMap
 
 variable [TopologicalSpace α] [TopologicalSpace β]
 
-/-- A map `f : α → β` is said to be a *closed map*, if the image of any closed `U : set α`
+/-- A map `f : α → β` is said to be a *closed map*, if the image of any closed `U : Set α`
 is closed in `β`. -/
 def IsClosedMap (f : α → β) :=
   ∀ U : Set α, IsClosed U → IsClosed (f '' U)

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>

@@ -98,7 +98,7 @@ theorem Inducing.nhds_eq_comap {f : α → β} (hf : Inducing f) : ∀ a : α, 
 
 theorem Inducing.nhdsSet_eq_comap {f : α → β} (hf : Inducing f) (s : Set α) :
     𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
-  simp only [nhdsSet, supₛ_image, comap_supᵢ, hf.nhds_eq_comap, supᵢ_image]
+  simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image]
 #align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comap
 
 theorem Inducing.map_nhds_eq {f : α → β} (hf : Inducing f) (a : α) : (𝓝 a).map f = 𝓝[range f] f a :=

This PR fixes two things:

• Most align statements for definitions and theorems and instances that are separated by two newlines from the relevant declaration (s/\n\n#align/\n#align). This is often seen in the mathport output after ending calc blocks.
• All remaining more-than-one-line #align statements. (This was needed for a script I wrote for #3630.)

@@ -385,7 +385,6 @@ theorem of_sections {f : α → β}
       𝓝 (f x) = map f (map g (𝓝 (f x))) := by rw [map_map, hgf.comp_eq_id, map_id]
       _ ≤ map f (𝓝 (g (f x))) := map_mono hgc
       _ = map f (𝓝 x) := by rw [hgx]
-
 #align is_open_map.of_sections IsOpenMap.of_sections
 
 theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')

@@ -385,7 +385,7 @@ theorem of_sections {f : α → β}
       𝓝 (f x) = map f (map g (𝓝 (f x))) := by rw [map_map, hgf.comp_eq_id, map_id]
       _ ≤ map f (𝓝 (g (f x))) := map_mono hgc
       _ = map f (𝓝 x) := by rw [hgx]
-      
+
 #align is_open_map.of_sections IsOpenMap.of_sections
 
 theorem of_inverse {f : α → β} {f' : β → α} (h : Continuous f') (l_inv : LeftInverse f f')
@@ -559,6 +559,10 @@ theorem OpenEmbedding.tendsto_nhds_iff {ι : Type _} {f : ι → β} {g : β →
   hg.toEmbedding.tendsto_nhds_iff
 #align open_embedding.tendsto_nhds_iff OpenEmbedding.tendsto_nhds_iff
 
+theorem OpenEmbedding.tendsto_nhds_iff' {f : α → β} (hf : OpenEmbedding f) {g : β → γ}
+    {l : Filter γ} {a : α} : Tendsto (g ∘ f) (𝓝 a) l ↔ Tendsto g (𝓝 (f a)) l := by
+  rw [Tendsto, ← map_map, hf.map_nhds_eq]; rfl
+
 theorem OpenEmbedding.continuous {f : α → β} (hf : OpenEmbedding f) : Continuous f :=
   hf.toEmbedding.continuous
 #align open_embedding.continuous OpenEmbedding.continuous

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

@@ -110,6 +110,12 @@ theorem Inducing.map_nhds_of_mem {f : α → β} (hf : Inducing f) (a : α) (h :
   hf.induced.symm ▸ map_nhds_induced_of_mem h
 #align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem
 
+-- porting note: new lemma
+theorem Inducing.mapClusterPt_iff {f : α → β} (hf : Inducing f) {a : α} {l : Filter α} :
+    MapClusterPt (f a) l f ↔ ClusterPt a l := by
+  delta MapClusterPt ClusterPt
+  rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff]
+
 theorem Inducing.image_mem_nhdsWithin {f : α → β} (hf : Inducing f) {a : α} {s : Set α}
     (hs : s ∈ 𝓝 a) : f '' s ∈ 𝓝[range f] f a :=
   hf.map_nhds_eq a ▸ image_mem_map hs