topology.alexandroff ⟷ Mathlib.Topology.Compactification.OnePoint

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

@@ -455,7 +455,7 @@ theorem ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
 #print OnePoint.tendsto_nhds_infty' /-
 theorem tendsto_nhds_infty' {α : Type _} {f : OnePoint X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ coe) (coclosedCompact X) l := by
-  simp [nhds_infty_eq, and_comm']
+  simp [nhds_infty_eq, and_comm]
 #align alexandroff.tendsto_nhds_infty' OnePoint.tendsto_nhds_infty'
 -/
 
@@ -465,7 +465,7 @@ theorem tendsto_nhds_infty {α : Type _} {f : OnePoint X → α} {l : Filter α}
       ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ coe) (tᶜ) s :=
   tendsto_nhds_infty'.trans <| by
     simp only [tendsto_pure_left, has_basis_coclosed_compact.tendsto_left_iff, forall_and,
-      and_assoc', exists_prop]
+      and_assoc, exists_prop]
 #align alexandroff.tendsto_nhds_infty OnePoint.tendsto_nhds_infty
 -/
 
@@ -481,7 +481,7 @@ theorem continuousAt_infty {Y : Type _} [TopologicalSpace Y] {f : OnePoint X →
     ContinuousAt f ∞ ↔
       ∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ coe) (tᶜ) s :=
   continuousAt_infty'.trans <| by
-    simp only [has_basis_coclosed_compact.tendsto_left_iff, exists_prop, and_assoc']
+    simp only [has_basis_coclosed_compact.tendsto_left_iff, exists_prop, and_assoc]
 #align alexandroff.continuous_at_infty OnePoint.continuousAt_infty
 -/
 

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

@@ -594,7 +594,7 @@ instance [LocallyCompactSpace X] [T2Space X] : NormalSpace (OnePoint X) :=
     ∀ z : X, ∃ u v : Set (OnePoint X), IsOpen u ∧ IsOpen v ∧ ↑z ∈ u ∧ ∞ ∈ v ∧ Disjoint u v :=
     by
     intro z
-    rcases exists_open_with_compact_closure z with ⟨u, hu, huy', Hu⟩
+    rcases exists_isOpen_mem_isCompact_closure z with ⟨u, hu, huy', Hu⟩
     exact
       ⟨coe '' u, (coe '' closure u)ᶜ, is_open_image_coe.2 hu,
         is_open_compl_image_coe.2 ⟨isClosed_closure, Hu⟩, mem_image_of_mem _ huy',

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

@@ -620,7 +620,7 @@ theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X
     ¬Continuous (@CofiniteTopology.of (OnePoint X)).symm :=
   by
   inhabit X
-  simp only [continuous_iff_continuousAt, ContinuousAt, not_forall]
+  simp only [continuous_iff_continuousAt, ContinuousAt, Classical.not_forall]
   use CofiniteTopology.of ↑(default : X)
   simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using
     (finite_singleton ((default : X) : OnePoint X)).infinite_compl

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

@@ -3,9 +3,9 @@ Copyright (c) 2021 Yourong Zang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yourong Zang, Yury Kudryashov
 -/
-import Mathbin.Data.Fintype.Option
-import Mathbin.Topology.Separation
-import Mathbin.Topology.Sets.Opens
+import Data.Fintype.Option
+import Topology.Separation
+import Topology.Sets.Opens
 
 #align_import topology.alexandroff from "leanprover-community/mathlib"@"ac34df03f74e6f797efd6991df2e3b7f7d8d33e0"
 
@@ -235,7 +235,7 @@ instance : TopologicalSpace (OnePoint X)
     suffices IsOpen (coe ⁻¹' ⋃₀ S : Set X) by
       refine' ⟨_, this⟩
       rintro ⟨s, hsS : s ∈ S, hs : ∞ ∈ s⟩
-      refine' isCompact_of_isClosed_subset ((ho s hsS).1 hs) this.is_closed_compl _
+      refine' IsCompact.of_isClosed_subset ((ho s hsS).1 hs) this.is_closed_compl _
       exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS)
     rw [preimage_sUnion]
     exact isOpen_biUnion fun s hs => (ho s hs).2

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

@@ -599,7 +599,7 @@ instance [LocallyCompactSpace X] [T2Space X] : NormalSpace (OnePoint X) :=
       ⟨coe '' u, (coe '' closure u)ᶜ, is_open_image_coe.2 hu,
         is_open_compl_image_coe.2 ⟨isClosed_closure, Hu⟩, mem_image_of_mem _ huy',
         mem_compl infty_not_mem_image_coe, (image_subset _ subset_closure).disjoint_compl_right⟩
-  refine' @normalOfCompactT2 _ _ _ ⟨fun x y hxy => _⟩
+  refine' @T4Space.of_compactSpace_t2Space _ _ _ ⟨fun x y hxy => _⟩
   induction x using OnePoint.rec <;> induction y using OnePoint.rec
   · exact (hxy rfl).elim
   · rcases key y with ⟨u, v, hu, hv, hxu, hyv, huv⟩

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Yourong Zang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yourong Zang, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.alexandroff
-! leanprover-community/mathlib commit ac34df03f74e6f797efd6991df2e3b7f7d8d33e0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Fintype.Option
 import Mathbin.Topology.Separation
 import Mathbin.Topology.Sets.Opens
 
+#align_import topology.alexandroff from "leanprover-community/mathlib"@"ac34df03f74e6f797efd6991df2e3b7f7d8d33e0"
+
 /-!
 # The Alexandroff Compactification
 

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

@@ -79,7 +79,6 @@ def infty : OnePoint X :=
 #align alexandroff.infty OnePoint.infty
 -/
 
--- mathport name: alexandroff.infty
 scoped notation "∞" => OnePoint.infty
 
 instance : CoeTC X (OnePoint X) :=
@@ -133,32 +132,44 @@ protected def rec (C : OnePoint X → Sort _) (h₁ : C ∞) (h₂ : ∀ x : X,
 #align alexandroff.rec OnePoint.rec
 -/
 
+#print OnePoint.isCompl_range_coe_infty /-
 theorem isCompl_range_coe_infty : IsCompl (range (coe : X → OnePoint X)) {∞} :=
   isCompl_range_some_none X
 #align alexandroff.is_compl_range_coe_infty OnePoint.isCompl_range_coe_infty
+-/
 
+#print OnePoint.range_coe_union_infty /-
 @[simp]
 theorem range_coe_union_infty : range (coe : X → OnePoint X) ∪ {∞} = univ :=
   range_some_union_none X
 #align alexandroff.range_coe_union_infty OnePoint.range_coe_union_infty
+-/
 
+#print OnePoint.range_coe_inter_infty /-
 @[simp]
 theorem range_coe_inter_infty : range (coe : X → OnePoint X) ∩ {∞} = ∅ :=
   range_some_inter_none X
 #align alexandroff.range_coe_inter_infty OnePoint.range_coe_inter_infty
+-/
 
+#print OnePoint.compl_range_coe /-
 @[simp]
 theorem compl_range_coe : range (coe : X → OnePoint X)ᶜ = {∞} :=
   compl_range_some X
 #align alexandroff.compl_range_coe OnePoint.compl_range_coe
+-/
 
+#print OnePoint.compl_infty /-
 theorem compl_infty : ({∞}ᶜ : Set (OnePoint X)) = range (coe : X → OnePoint X) :=
   (@isCompl_range_coe_infty X).symm.compl_eq
 #align alexandroff.compl_infty OnePoint.compl_infty
+-/
 
+#print OnePoint.compl_image_coe /-
 theorem compl_image_coe (s : Set X) : (coe '' s : Set (OnePoint X))ᶜ = coe '' sᶜ ∪ {∞} := by
   rw [coe_injective.compl_image_eq, compl_range_coe]
 #align alexandroff.compl_image_coe OnePoint.compl_image_coe
+-/
 
 #print OnePoint.ne_infty_iff_exists /-
 theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x := by
@@ -234,59 +245,79 @@ instance : TopologicalSpace (OnePoint X)
 
 variable {s : Set (OnePoint X)} {t : Set X}
 
+#print OnePoint.isOpen_def /-
 theorem isOpen_def :
     IsOpen s ↔ (∞ ∈ s → IsCompact ((coe ⁻¹' s : Set X)ᶜ)) ∧ IsOpen (coe ⁻¹' s : Set X) :=
   Iff.rfl
 #align alexandroff.is_open_def OnePoint.isOpen_def
+-/
 
+#print OnePoint.isOpen_iff_of_mem' /-
 theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
     IsOpen s ↔ IsCompact ((coe ⁻¹' s : Set X)ᶜ) ∧ IsOpen (coe ⁻¹' s : Set X) := by
   simp [is_open_def, h]
 #align alexandroff.is_open_iff_of_mem' OnePoint.isOpen_iff_of_mem'
+-/
 
+#print OnePoint.isOpen_iff_of_mem /-
 theorem isOpen_iff_of_mem (h : ∞ ∈ s) :
     IsOpen s ↔ IsClosed ((coe ⁻¹' s : Set X)ᶜ) ∧ IsCompact ((coe ⁻¹' s : Set X)ᶜ) := by
   simp only [is_open_iff_of_mem' h, isClosed_compl_iff, and_comm]
 #align alexandroff.is_open_iff_of_mem OnePoint.isOpen_iff_of_mem
+-/
 
+#print OnePoint.isOpen_iff_of_not_mem /-
 theorem isOpen_iff_of_not_mem (h : ∞ ∉ s) : IsOpen s ↔ IsOpen (coe ⁻¹' s : Set X) := by
   simp [is_open_def, h]
 #align alexandroff.is_open_iff_of_not_mem OnePoint.isOpen_iff_of_not_mem
+-/
 
+#print OnePoint.isClosed_iff_of_mem /-
 theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed (coe ⁻¹' s : Set X) :=
   by
   have : ∞ ∉ sᶜ := fun H => H h
   rw [← isOpen_compl_iff, is_open_iff_of_not_mem this, ← isOpen_compl_iff, preimage_compl]
 #align alexandroff.is_closed_iff_of_mem OnePoint.isClosed_iff_of_mem
+-/
 
+#print OnePoint.isClosed_iff_of_not_mem /-
 theorem isClosed_iff_of_not_mem (h : ∞ ∉ s) :
     IsClosed s ↔ IsClosed (coe ⁻¹' s : Set X) ∧ IsCompact (coe ⁻¹' s : Set X) := by
   rw [← isOpen_compl_iff, is_open_iff_of_mem (mem_compl h), ← preimage_compl, compl_compl]
 #align alexandroff.is_closed_iff_of_not_mem OnePoint.isClosed_iff_of_not_mem
+-/
 
+#print OnePoint.isOpen_image_coe /-
 @[simp]
 theorem isOpen_image_coe {s : Set X} : IsOpen (coe '' s : Set (OnePoint X)) ↔ IsOpen s := by
   rw [is_open_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective]
 #align alexandroff.is_open_image_coe OnePoint.isOpen_image_coe
+-/
 
+#print OnePoint.isOpen_compl_image_coe /-
 theorem isOpen_compl_image_coe {s : Set X} :
     IsOpen ((coe '' s : Set (OnePoint X))ᶜ) ↔ IsClosed s ∧ IsCompact s :=
   by
   rw [is_open_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective]
   exact infty_not_mem_image_coe
 #align alexandroff.is_open_compl_image_coe OnePoint.isOpen_compl_image_coe
+-/
 
+#print OnePoint.isClosed_image_coe /-
 @[simp]
 theorem isClosed_image_coe {s : Set X} :
     IsClosed (coe '' s : Set (OnePoint X)) ↔ IsClosed s ∧ IsCompact s := by
   rw [← isOpen_compl_iff, is_open_compl_image_coe]
 #align alexandroff.is_closed_image_coe OnePoint.isClosed_image_coe
+-/
 
+#print OnePoint.opensOfCompl /-
 /-- An open set in `alexandroff X` constructed from a closed compact set in `X` -/
 def opensOfCompl (s : Set X) (h₁ : IsClosed s) (h₂ : IsCompact s) :
     TopologicalSpace.Opens (OnePoint X) :=
   ⟨(coe '' s)ᶜ, isOpen_compl_image_coe.2 ⟨h₁, h₂⟩⟩
 #align alexandroff.opens_of_compl OnePoint.opensOfCompl
+-/
 
 #print OnePoint.infty_mem_opensOfCompl /-
 theorem infty_mem_opensOfCompl {s : Set X} (h₁ : IsClosed s) (h₂ : IsCompact s) :
@@ -295,49 +326,70 @@ theorem infty_mem_opensOfCompl {s : Set X} (h₁ : IsClosed s) (h₂ : IsCompact
 #align alexandroff.infty_mem_opens_of_compl OnePoint.infty_mem_opensOfCompl
 -/
 
+#print OnePoint.continuous_coe /-
 @[continuity]
 theorem continuous_coe : Continuous (coe : X → OnePoint X) :=
   continuous_def.mpr fun s hs => hs.right
 #align alexandroff.continuous_coe OnePoint.continuous_coe
+-/
 
+#print OnePoint.isOpenMap_coe /-
 theorem isOpenMap_coe : IsOpenMap (coe : X → OnePoint X) := fun s => isOpen_image_coe.2
 #align alexandroff.is_open_map_coe OnePoint.isOpenMap_coe
+-/
 
+#print OnePoint.openEmbedding_coe /-
 theorem openEmbedding_coe : OpenEmbedding (coe : X → OnePoint X) :=
   openEmbedding_of_continuous_injective_open continuous_coe coe_injective isOpenMap_coe
 #align alexandroff.open_embedding_coe OnePoint.openEmbedding_coe
+-/
 
+#print OnePoint.isOpen_range_coe /-
 theorem isOpen_range_coe : IsOpen (range (coe : X → OnePoint X)) :=
   openEmbedding_coe.open_range
 #align alexandroff.is_open_range_coe OnePoint.isOpen_range_coe
+-/
 
+#print OnePoint.isClosed_infty /-
 theorem isClosed_infty : IsClosed ({∞} : Set (OnePoint X)) := by
   rw [← compl_range_coe, isClosed_compl_iff]; exact is_open_range_coe
 #align alexandroff.is_closed_infty OnePoint.isClosed_infty
+-/
 
+#print OnePoint.nhds_coe_eq /-
 theorem nhds_coe_eq (x : X) : 𝓝 ↑x = map (coe : X → OnePoint X) (𝓝 x) :=
   (openEmbedding_coe.map_nhds_eq x).symm
 #align alexandroff.nhds_coe_eq OnePoint.nhds_coe_eq
+-/
 
+#print OnePoint.nhdsWithin_coe_image /-
 theorem nhdsWithin_coe_image (s : Set X) (x : X) :
     𝓝[coe '' s] (x : OnePoint X) = map coe (𝓝[s] x) :=
   (openEmbedding_coe.toEmbedding.map_nhdsWithin_eq _ _).symm
 #align alexandroff.nhds_within_coe_image OnePoint.nhdsWithin_coe_image
+-/
 
+#print OnePoint.nhdsWithin_coe /-
 theorem nhdsWithin_coe (s : Set (OnePoint X)) (x : X) : 𝓝[s] ↑x = map coe (𝓝[coe ⁻¹' s] x) :=
   (openEmbedding_coe.map_nhdsWithin_preimage_eq _ _).symm
 #align alexandroff.nhds_within_coe OnePoint.nhdsWithin_coe
+-/
 
+#print OnePoint.comap_coe_nhds /-
 theorem comap_coe_nhds (x : X) : comap (coe : X → OnePoint X) (𝓝 x) = 𝓝 x :=
   (openEmbedding_coe.to_inducing.nhds_eq_comap x).symm
 #align alexandroff.comap_coe_nhds OnePoint.comap_coe_nhds
+-/
 
+#print OnePoint.nhdsWithin_compl_coe_neBot /-
 /-- If `x` is not an isolated point of `X`, then `x : alexandroff X` is not an isolated point
 of `alexandroff X`. -/
 instance nhdsWithin_compl_coe_neBot (x : X) [h : NeBot (𝓝[≠] x)] : NeBot (𝓝[≠] (x : OnePoint X)) :=
   by simpa [nhds_within_coe, preimage, coe_eq_coe] using h.map coe
 #align alexandroff.nhds_within_compl_coe_ne_bot OnePoint.nhdsWithin_compl_coe_neBot
+-/
 
+#print OnePoint.nhdsWithin_compl_infty_eq /-
 theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map coe (coclosedCompact X) :=
   by
   refine' (nhdsWithin_basis_open ∞ _).ext (has_basis_coclosed_compact.map _) _ _
@@ -348,22 +400,30 @@ theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map coe (cocl
     refine' ⟨_, ⟨mem_compl infty_not_mem_image_coe, is_open_compl_image_coe.2 ⟨h₁, h₂⟩⟩, _⟩
     simp [compl_image_coe, ← diff_eq, subset_preimage_image]
 #align alexandroff.nhds_within_compl_infty_eq OnePoint.nhdsWithin_compl_infty_eq
+-/
 
+#print OnePoint.nhdsWithin_compl_infty_neBot /-
 /-- If `X` is a non-compact space, then `∞` is not an isolated point of `alexandroff X`. -/
 instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : OnePoint X)) := by
   rw [nhds_within_compl_infty_eq]; infer_instance
 #align alexandroff.nhds_within_compl_infty_ne_bot OnePoint.nhdsWithin_compl_infty_neBot
+-/
 
+#print OnePoint.nhdsWithin_compl_neBot /-
 instance (priority := 900) nhdsWithin_compl_neBot [∀ x : X, NeBot (𝓝[≠] x)] [NoncompactSpace X]
     (x : OnePoint X) : NeBot (𝓝[≠] x) :=
   OnePoint.rec _ OnePoint.nhdsWithin_compl_infty_neBot
     (fun y => OnePoint.nhdsWithin_compl_coe_neBot y) x
 #align alexandroff.nhds_within_compl_ne_bot OnePoint.nhdsWithin_compl_neBot
+-/
 
+#print OnePoint.nhds_infty_eq /-
 theorem nhds_infty_eq : 𝓝 (∞ : OnePoint X) = map coe (coclosedCompact X) ⊔ pure ∞ := by
   rw [← nhds_within_compl_infty_eq, nhdsWithin_compl_singleton_sup_pure]
 #align alexandroff.nhds_infty_eq OnePoint.nhds_infty_eq
+-/
 
+#print OnePoint.hasBasis_nhds_infty /-
 theorem hasBasis_nhds_infty :
     (𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
       coe '' sᶜ ∪ {∞} :=
@@ -371,28 +431,38 @@ theorem hasBasis_nhds_infty :
   rw [nhds_infty_eq]
   exact (has_basis_coclosed_compact.map _).sup_pure _
 #align alexandroff.has_basis_nhds_infty OnePoint.hasBasis_nhds_infty
+-/
 
+#print OnePoint.comap_coe_nhds_infty /-
 @[simp]
 theorem comap_coe_nhds_infty : comap (coe : X → OnePoint X) (𝓝 ∞) = coclosedCompact X := by
   simp [nhds_infty_eq, comap_sup, comap_map coe_injective]
 #align alexandroff.comap_coe_nhds_infty OnePoint.comap_coe_nhds_infty
+-/
 
+#print OnePoint.le_nhds_infty /-
 theorem le_nhds_infty {f : Filter (OnePoint X)} :
     f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → coe '' sᶜ ∪ {∞} ∈ f := by
   simp only [has_basis_nhds_infty.ge_iff, and_imp]
 #align alexandroff.le_nhds_infty OnePoint.le_nhds_infty
+-/
 
+#print OnePoint.ultrafilter_le_nhds_infty /-
 theorem ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
     (f : Filter (OnePoint X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → coe '' s ∉ f := by
   simp only [le_nhds_infty, ← compl_image_coe, Ultrafilter.mem_coe,
     Ultrafilter.compl_mem_iff_not_mem]
 #align alexandroff.ultrafilter_le_nhds_infty OnePoint.ultrafilter_le_nhds_infty
+-/
 
+#print OnePoint.tendsto_nhds_infty' /-
 theorem tendsto_nhds_infty' {α : Type _} {f : OnePoint X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ coe) (coclosedCompact X) l := by
   simp [nhds_infty_eq, and_comm']
 #align alexandroff.tendsto_nhds_infty' OnePoint.tendsto_nhds_infty'
+-/
 
+#print OnePoint.tendsto_nhds_infty /-
 theorem tendsto_nhds_infty {α : Type _} {f : OnePoint X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔
       ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ coe) (tᶜ) s :=
@@ -400,24 +470,32 @@ theorem tendsto_nhds_infty {α : Type _} {f : OnePoint X → α} {l : Filter α}
     simp only [tendsto_pure_left, has_basis_coclosed_compact.tendsto_left_iff, forall_and,
       and_assoc', exists_prop]
 #align alexandroff.tendsto_nhds_infty OnePoint.tendsto_nhds_infty
+-/
 
+#print OnePoint.continuousAt_infty' /-
 theorem continuousAt_infty' {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} :
     ContinuousAt f ∞ ↔ Tendsto (f ∘ coe) (coclosedCompact X) (𝓝 (f ∞)) :=
   tendsto_nhds_infty'.trans <| and_iff_right (tendsto_pure_nhds _ _)
 #align alexandroff.continuous_at_infty' OnePoint.continuousAt_infty'
+-/
 
+#print OnePoint.continuousAt_infty /-
 theorem continuousAt_infty {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} :
     ContinuousAt f ∞ ↔
       ∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ coe) (tᶜ) s :=
   continuousAt_infty'.trans <| by
     simp only [has_basis_coclosed_compact.tendsto_left_iff, exists_prop, and_assoc']
 #align alexandroff.continuous_at_infty OnePoint.continuousAt_infty
+-/
 
+#print OnePoint.continuousAt_coe /-
 theorem continuousAt_coe {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
     ContinuousAt f x ↔ ContinuousAt (f ∘ coe) x := by
   rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]
 #align alexandroff.continuous_at_coe OnePoint.continuousAt_coe
+-/
 
+#print OnePoint.denseRange_coe /-
 /-- If `X` is not a compact space, then the natural embedding `X → alexandroff X` has dense range.
 -/
 theorem denseRange_coe [NoncompactSpace X] : DenseRange (coe : X → OnePoint X) :=
@@ -425,38 +503,53 @@ theorem denseRange_coe [NoncompactSpace X] : DenseRange (coe : X → OnePoint X)
   rw [DenseRange, ← compl_infty]
   exact dense_compl_singleton _
 #align alexandroff.dense_range_coe OnePoint.denseRange_coe
+-/
 
+#print OnePoint.denseEmbedding_coe /-
 theorem denseEmbedding_coe [NoncompactSpace X] : DenseEmbedding (coe : X → OnePoint X) :=
   { openEmbedding_coe with dense := denseRange_coe }
 #align alexandroff.dense_embedding_coe OnePoint.denseEmbedding_coe
+-/
 
+#print OnePoint.specializes_coe /-
 @[simp]
 theorem specializes_coe {x y : X} : (x : OnePoint X) ⤳ y ↔ x ⤳ y :=
   openEmbedding_coe.to_inducing.specializes_iff
 #align alexandroff.specializes_coe OnePoint.specializes_coe
+-/
 
+#print OnePoint.inseparable_coe /-
 @[simp]
 theorem inseparable_coe {x y : X} : Inseparable (x : OnePoint X) y ↔ Inseparable x y :=
   openEmbedding_coe.to_inducing.inseparable_iff
 #align alexandroff.inseparable_coe OnePoint.inseparable_coe
+-/
 
+#print OnePoint.not_specializes_infty_coe /-
 theorem not_specializes_infty_coe {x : X} : ¬Specializes ∞ (x : OnePoint X) :=
   isClosed_infty.not_specializes rfl (coe_ne_infty x)
 #align alexandroff.not_specializes_infty_coe OnePoint.not_specializes_infty_coe
+-/
 
+#print OnePoint.not_inseparable_infty_coe /-
 theorem not_inseparable_infty_coe {x : X} : ¬Inseparable ∞ (x : OnePoint X) := fun h =>
   not_specializes_infty_coe h.Specializes
 #align alexandroff.not_inseparable_infty_coe OnePoint.not_inseparable_infty_coe
+-/
 
+#print OnePoint.not_inseparable_coe_infty /-
 theorem not_inseparable_coe_infty {x : X} : ¬Inseparable (x : OnePoint X) ∞ := fun h =>
   not_specializes_infty_coe h.specializes'
 #align alexandroff.not_inseparable_coe_infty OnePoint.not_inseparable_coe_infty
+-/
 
+#print OnePoint.inseparable_iff /-
 theorem inseparable_iff {x y : OnePoint X} :
     Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
   induction x using OnePoint.rec <;> induction y using OnePoint.rec <;>
     simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe]
 #align alexandroff.inseparable_iff OnePoint.inseparable_iff
+-/
 
 /-!
 ### Compactness and separation properties
@@ -523,6 +616,7 @@ instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (OnePoint X)
   to_preconnectedSpace := denseEmbedding_coe.to_denseInducing.PreconnectedSpace
   to_nonempty := inferInstance
 
+#print OnePoint.not_continuous_cofiniteTopology_of_symm /-
 /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
 `cofinite_topology (alexandroff X)` to `alexandroff X` is not continuous. -/
 theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
@@ -534,6 +628,7 @@ theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X
   simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using
     (finite_singleton ((default : X) : OnePoint X)).infinite_compl
 #align alexandroff.not_continuous_cofinite_topology_of_symm OnePoint.not_continuous_cofiniteTopology_of_symm
+-/
 
 end OnePoint
 

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

@@ -478,7 +478,7 @@ instance : CompactSpace (OnePoint X)
       by
       rw [nhds_infty_eq]
       exact (tendsto_map.mono_left cocompact_le_coclosed_compact).mono_right le_sup_left
-    convert← this.is_compact_insert_range_of_cocompact continuous_coe
+    convert ← this.is_compact_insert_range_of_cocompact continuous_coe
     exact insert_none_range_some X
 
 /-- The one point compactification of a `t0_space` space is a `t0_space`. -/

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

@@ -486,7 +486,7 @@ instance [T0Space X] : T0Space (OnePoint X) :=
   by
   refine' ⟨fun x y hxy => _⟩
   rcases inseparable_iff.1 hxy with (⟨rfl, rfl⟩ | ⟨x, rfl, y, rfl, h⟩)
-  exacts[rfl, congr_arg coe h.eq]
+  exacts [rfl, congr_arg coe h.eq]
 
 /-- The one point compactification of a `t1_space` space is a `t1_space`. -/
 instance [T1Space X] : T1Space (OnePoint X)
@@ -546,7 +546,7 @@ Let `α = alexandroff ℕ` be the one-point compactification of `ℕ`, and let `
 `id : α → β` is a continuous equivalence that is not a homeomorphism.
 -/
 theorem Continuous.homeoOfEquivCompactToT2.t1_counterexample :
-    ∃ (α β : Type)(Iα : TopologicalSpace α)(Iβ : TopologicalSpace β),
+    ∃ (α β : Type) (Iα : TopologicalSpace α) (Iβ : TopologicalSpace β),
       CompactSpace α ∧ T1Space β ∧ ∃ f : α ≃ β, Continuous f ∧ ¬Continuous f.symm :=
   ⟨OnePoint ℕ, CofiniteTopology (OnePoint ℕ), inferInstance, inferInstance, inferInstance,
     inferInstance, CofiniteTopology.of, CofiniteTopology.continuous_of,

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

@@ -56,144 +56,144 @@ In this section we define `alexandroff X` to be the disjoint union of `X` and `
 
 variable {X : Type _}
 
-#print Alexandroff /-
+#print OnePoint /-
 /-- The Alexandroff extension of an arbitrary topological space `X` -/
-def Alexandroff (X : Type _) :=
+def OnePoint (X : Type _) :=
   Option X
-#align alexandroff Alexandroff
+#align alexandroff OnePoint
 -/
 
 /-- The repr uses the notation from the `alexandroff` locale. -/
-instance [Repr X] : Repr (Alexandroff X) :=
+instance [Repr X] : Repr (OnePoint X) :=
   ⟨fun o =>
     match o with
     | none => "∞"
     | some a => "↑" ++ repr a⟩
 
-namespace Alexandroff
+namespace OnePoint
 
-#print Alexandroff.infty /-
+#print OnePoint.infty /-
 /-- The point at infinity -/
-def infty : Alexandroff X :=
+def infty : OnePoint X :=
   none
-#align alexandroff.infty Alexandroff.infty
+#align alexandroff.infty OnePoint.infty
 -/
 
 -- mathport name: alexandroff.infty
-scoped notation "∞" => Alexandroff.infty
+scoped notation "∞" => OnePoint.infty
 
-instance : CoeTC X (Alexandroff X) :=
+instance : CoeTC X (OnePoint X) :=
   ⟨Option.some⟩
 
-instance : Inhabited (Alexandroff X) :=
+instance : Inhabited (OnePoint X) :=
   ⟨∞⟩
 
-instance [Fintype X] : Fintype (Alexandroff X) :=
+instance [Fintype X] : Fintype (OnePoint X) :=
   Option.fintype
 
-#print Alexandroff.infinite /-
-instance infinite [Infinite X] : Infinite (Alexandroff X) :=
+#print OnePoint.infinite /-
+instance infinite [Infinite X] : Infinite (OnePoint X) :=
   Option.infinite
-#align alexandroff.infinite Alexandroff.infinite
+#align alexandroff.infinite OnePoint.infinite
 -/
 
-#print Alexandroff.coe_injective /-
-theorem coe_injective : Function.Injective (coe : X → Alexandroff X) :=
+#print OnePoint.coe_injective /-
+theorem coe_injective : Function.Injective (coe : X → OnePoint X) :=
   Option.some_injective X
-#align alexandroff.coe_injective Alexandroff.coe_injective
+#align alexandroff.coe_injective OnePoint.coe_injective
 -/
 
-#print Alexandroff.coe_eq_coe /-
+#print OnePoint.coe_eq_coe /-
 @[norm_cast]
-theorem coe_eq_coe {x y : X} : (x : Alexandroff X) = y ↔ x = y :=
+theorem coe_eq_coe {x y : X} : (x : OnePoint X) = y ↔ x = y :=
   coe_injective.eq_iff
-#align alexandroff.coe_eq_coe Alexandroff.coe_eq_coe
+#align alexandroff.coe_eq_coe OnePoint.coe_eq_coe
 -/
 
-#print Alexandroff.coe_ne_infty /-
+#print OnePoint.coe_ne_infty /-
 @[simp]
-theorem coe_ne_infty (x : X) : (x : Alexandroff X) ≠ ∞ :=
+theorem coe_ne_infty (x : X) : (x : OnePoint X) ≠ ∞ :=
   fun.
-#align alexandroff.coe_ne_infty Alexandroff.coe_ne_infty
+#align alexandroff.coe_ne_infty OnePoint.coe_ne_infty
 -/
 
-#print Alexandroff.infty_ne_coe /-
+#print OnePoint.infty_ne_coe /-
 @[simp]
-theorem infty_ne_coe (x : X) : ∞ ≠ (x : Alexandroff X) :=
+theorem infty_ne_coe (x : X) : ∞ ≠ (x : OnePoint X) :=
   fun.
-#align alexandroff.infty_ne_coe Alexandroff.infty_ne_coe
+#align alexandroff.infty_ne_coe OnePoint.infty_ne_coe
 -/
 
-#print Alexandroff.rec /-
+#print OnePoint.rec /-
 /-- Recursor for `alexandroff` using the preferred forms `∞` and `↑x`. -/
 @[elab_as_elim]
-protected def rec (C : Alexandroff X → Sort _) (h₁ : C ∞) (h₂ : ∀ x : X, C x) :
-    ∀ z : Alexandroff X, C z :=
+protected def rec (C : OnePoint X → Sort _) (h₁ : C ∞) (h₂ : ∀ x : X, C x) :
+    ∀ z : OnePoint X, C z :=
   Option.rec h₁ h₂
-#align alexandroff.rec Alexandroff.rec
+#align alexandroff.rec OnePoint.rec
 -/
 
-theorem isCompl_range_coe_infty : IsCompl (range (coe : X → Alexandroff X)) {∞} :=
+theorem isCompl_range_coe_infty : IsCompl (range (coe : X → OnePoint X)) {∞} :=
   isCompl_range_some_none X
-#align alexandroff.is_compl_range_coe_infty Alexandroff.isCompl_range_coe_infty
+#align alexandroff.is_compl_range_coe_infty OnePoint.isCompl_range_coe_infty
 
 @[simp]
-theorem range_coe_union_infty : range (coe : X → Alexandroff X) ∪ {∞} = univ :=
+theorem range_coe_union_infty : range (coe : X → OnePoint X) ∪ {∞} = univ :=
   range_some_union_none X
-#align alexandroff.range_coe_union_infty Alexandroff.range_coe_union_infty
+#align alexandroff.range_coe_union_infty OnePoint.range_coe_union_infty
 
 @[simp]
-theorem range_coe_inter_infty : range (coe : X → Alexandroff X) ∩ {∞} = ∅ :=
+theorem range_coe_inter_infty : range (coe : X → OnePoint X) ∩ {∞} = ∅ :=
   range_some_inter_none X
-#align alexandroff.range_coe_inter_infty Alexandroff.range_coe_inter_infty
+#align alexandroff.range_coe_inter_infty OnePoint.range_coe_inter_infty
 
 @[simp]
-theorem compl_range_coe : range (coe : X → Alexandroff X)ᶜ = {∞} :=
+theorem compl_range_coe : range (coe : X → OnePoint X)ᶜ = {∞} :=
   compl_range_some X
-#align alexandroff.compl_range_coe Alexandroff.compl_range_coe
+#align alexandroff.compl_range_coe OnePoint.compl_range_coe
 
-theorem compl_infty : ({∞}ᶜ : Set (Alexandroff X)) = range (coe : X → Alexandroff X) :=
+theorem compl_infty : ({∞}ᶜ : Set (OnePoint X)) = range (coe : X → OnePoint X) :=
   (@isCompl_range_coe_infty X).symm.compl_eq
-#align alexandroff.compl_infty Alexandroff.compl_infty
+#align alexandroff.compl_infty OnePoint.compl_infty
 
-theorem compl_image_coe (s : Set X) : (coe '' s : Set (Alexandroff X))ᶜ = coe '' sᶜ ∪ {∞} := by
+theorem compl_image_coe (s : Set X) : (coe '' s : Set (OnePoint X))ᶜ = coe '' sᶜ ∪ {∞} := by
   rw [coe_injective.compl_image_eq, compl_range_coe]
-#align alexandroff.compl_image_coe Alexandroff.compl_image_coe
+#align alexandroff.compl_image_coe OnePoint.compl_image_coe
 
-#print Alexandroff.ne_infty_iff_exists /-
-theorem ne_infty_iff_exists {x : Alexandroff X} : x ≠ ∞ ↔ ∃ y : X, (y : Alexandroff X) = x := by
-  induction x using Alexandroff.rec <;> simp
-#align alexandroff.ne_infty_iff_exists Alexandroff.ne_infty_iff_exists
+#print OnePoint.ne_infty_iff_exists /-
+theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x := by
+  induction x using OnePoint.rec <;> simp
+#align alexandroff.ne_infty_iff_exists OnePoint.ne_infty_iff_exists
 -/
 
-#print Alexandroff.canLift /-
-instance canLift : CanLift (Alexandroff X) X coe fun x => x ≠ ∞ :=
+#print OnePoint.canLift /-
+instance canLift : CanLift (OnePoint X) X coe fun x => x ≠ ∞ :=
   WithTop.canLift
-#align alexandroff.can_lift Alexandroff.canLift
+#align alexandroff.can_lift OnePoint.canLift
 -/
 
-#print Alexandroff.not_mem_range_coe_iff /-
-theorem not_mem_range_coe_iff {x : Alexandroff X} : x ∉ range (coe : X → Alexandroff X) ↔ x = ∞ :=
-  by rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff]
-#align alexandroff.not_mem_range_coe_iff Alexandroff.not_mem_range_coe_iff
+#print OnePoint.not_mem_range_coe_iff /-
+theorem not_mem_range_coe_iff {x : OnePoint X} : x ∉ range (coe : X → OnePoint X) ↔ x = ∞ := by
+  rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff]
+#align alexandroff.not_mem_range_coe_iff OnePoint.not_mem_range_coe_iff
 -/
 
-#print Alexandroff.infty_not_mem_range_coe /-
-theorem infty_not_mem_range_coe : ∞ ∉ range (coe : X → Alexandroff X) :=
+#print OnePoint.infty_not_mem_range_coe /-
+theorem infty_not_mem_range_coe : ∞ ∉ range (coe : X → OnePoint X) :=
   not_mem_range_coe_iff.2 rfl
-#align alexandroff.infty_not_mem_range_coe Alexandroff.infty_not_mem_range_coe
+#align alexandroff.infty_not_mem_range_coe OnePoint.infty_not_mem_range_coe
 -/
 
-#print Alexandroff.infty_not_mem_image_coe /-
-theorem infty_not_mem_image_coe {s : Set X} : ∞ ∉ (coe : X → Alexandroff X) '' s :=
+#print OnePoint.infty_not_mem_image_coe /-
+theorem infty_not_mem_image_coe {s : Set X} : ∞ ∉ (coe : X → OnePoint X) '' s :=
   not_mem_subset (image_subset_range _ _) infty_not_mem_range_coe
-#align alexandroff.infty_not_mem_image_coe Alexandroff.infty_not_mem_image_coe
+#align alexandroff.infty_not_mem_image_coe OnePoint.infty_not_mem_image_coe
 -/
 
-#print Alexandroff.coe_preimage_infty /-
+#print OnePoint.coe_preimage_infty /-
 @[simp]
-theorem coe_preimage_infty : (coe : X → Alexandroff X) ⁻¹' {∞} = ∅ := by ext; simp
-#align alexandroff.coe_preimage_infty Alexandroff.coe_preimage_infty
+theorem coe_preimage_infty : (coe : X → OnePoint X) ⁻¹' {∞} = ∅ := by ext; simp
+#align alexandroff.coe_preimage_infty OnePoint.coe_preimage_infty
 -/
 
 /-!
@@ -212,11 +212,10 @@ that `coe` has dense range, so it is a dense embedding.
 
 variable [TopologicalSpace X]
 
-instance : TopologicalSpace (Alexandroff X)
+instance : TopologicalSpace (OnePoint X)
     where
   IsOpen s :=
-    (∞ ∈ s → IsCompact (((coe : X → Alexandroff X) ⁻¹' s)ᶜ)) ∧
-      IsOpen ((coe : X → Alexandroff X) ⁻¹' s)
+    (∞ ∈ s → IsCompact (((coe : X → OnePoint X) ⁻¹' s)ᶜ)) ∧ IsOpen ((coe : X → OnePoint X) ⁻¹' s)
   isOpen_univ := by simp
   isOpen_inter s t := by
     rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
@@ -233,114 +232,113 @@ instance : TopologicalSpace (Alexandroff X)
     rw [preimage_sUnion]
     exact isOpen_biUnion fun s hs => (ho s hs).2
 
-variable {s : Set (Alexandroff X)} {t : Set X}
+variable {s : Set (OnePoint X)} {t : Set X}
 
 theorem isOpen_def :
     IsOpen s ↔ (∞ ∈ s → IsCompact ((coe ⁻¹' s : Set X)ᶜ)) ∧ IsOpen (coe ⁻¹' s : Set X) :=
   Iff.rfl
-#align alexandroff.is_open_def Alexandroff.isOpen_def
+#align alexandroff.is_open_def OnePoint.isOpen_def
 
 theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
     IsOpen s ↔ IsCompact ((coe ⁻¹' s : Set X)ᶜ) ∧ IsOpen (coe ⁻¹' s : Set X) := by
   simp [is_open_def, h]
-#align alexandroff.is_open_iff_of_mem' Alexandroff.isOpen_iff_of_mem'
+#align alexandroff.is_open_iff_of_mem' OnePoint.isOpen_iff_of_mem'
 
 theorem isOpen_iff_of_mem (h : ∞ ∈ s) :
     IsOpen s ↔ IsClosed ((coe ⁻¹' s : Set X)ᶜ) ∧ IsCompact ((coe ⁻¹' s : Set X)ᶜ) := by
   simp only [is_open_iff_of_mem' h, isClosed_compl_iff, and_comm]
-#align alexandroff.is_open_iff_of_mem Alexandroff.isOpen_iff_of_mem
+#align alexandroff.is_open_iff_of_mem OnePoint.isOpen_iff_of_mem
 
 theorem isOpen_iff_of_not_mem (h : ∞ ∉ s) : IsOpen s ↔ IsOpen (coe ⁻¹' s : Set X) := by
   simp [is_open_def, h]
-#align alexandroff.is_open_iff_of_not_mem Alexandroff.isOpen_iff_of_not_mem
+#align alexandroff.is_open_iff_of_not_mem OnePoint.isOpen_iff_of_not_mem
 
 theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed (coe ⁻¹' s : Set X) :=
   by
   have : ∞ ∉ sᶜ := fun H => H h
   rw [← isOpen_compl_iff, is_open_iff_of_not_mem this, ← isOpen_compl_iff, preimage_compl]
-#align alexandroff.is_closed_iff_of_mem Alexandroff.isClosed_iff_of_mem
+#align alexandroff.is_closed_iff_of_mem OnePoint.isClosed_iff_of_mem
 
 theorem isClosed_iff_of_not_mem (h : ∞ ∉ s) :
     IsClosed s ↔ IsClosed (coe ⁻¹' s : Set X) ∧ IsCompact (coe ⁻¹' s : Set X) := by
   rw [← isOpen_compl_iff, is_open_iff_of_mem (mem_compl h), ← preimage_compl, compl_compl]
-#align alexandroff.is_closed_iff_of_not_mem Alexandroff.isClosed_iff_of_not_mem
+#align alexandroff.is_closed_iff_of_not_mem OnePoint.isClosed_iff_of_not_mem
 
 @[simp]
-theorem isOpen_image_coe {s : Set X} : IsOpen (coe '' s : Set (Alexandroff X)) ↔ IsOpen s := by
+theorem isOpen_image_coe {s : Set X} : IsOpen (coe '' s : Set (OnePoint X)) ↔ IsOpen s := by
   rw [is_open_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective]
-#align alexandroff.is_open_image_coe Alexandroff.isOpen_image_coe
+#align alexandroff.is_open_image_coe OnePoint.isOpen_image_coe
 
 theorem isOpen_compl_image_coe {s : Set X} :
-    IsOpen ((coe '' s : Set (Alexandroff X))ᶜ) ↔ IsClosed s ∧ IsCompact s :=
+    IsOpen ((coe '' s : Set (OnePoint X))ᶜ) ↔ IsClosed s ∧ IsCompact s :=
   by
   rw [is_open_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective]
   exact infty_not_mem_image_coe
-#align alexandroff.is_open_compl_image_coe Alexandroff.isOpen_compl_image_coe
+#align alexandroff.is_open_compl_image_coe OnePoint.isOpen_compl_image_coe
 
 @[simp]
 theorem isClosed_image_coe {s : Set X} :
-    IsClosed (coe '' s : Set (Alexandroff X)) ↔ IsClosed s ∧ IsCompact s := by
+    IsClosed (coe '' s : Set (OnePoint X)) ↔ IsClosed s ∧ IsCompact s := by
   rw [← isOpen_compl_iff, is_open_compl_image_coe]
-#align alexandroff.is_closed_image_coe Alexandroff.isClosed_image_coe
+#align alexandroff.is_closed_image_coe OnePoint.isClosed_image_coe
 
 /-- An open set in `alexandroff X` constructed from a closed compact set in `X` -/
 def opensOfCompl (s : Set X) (h₁ : IsClosed s) (h₂ : IsCompact s) :
-    TopologicalSpace.Opens (Alexandroff X) :=
+    TopologicalSpace.Opens (OnePoint X) :=
   ⟨(coe '' s)ᶜ, isOpen_compl_image_coe.2 ⟨h₁, h₂⟩⟩
-#align alexandroff.opens_of_compl Alexandroff.opensOfCompl
+#align alexandroff.opens_of_compl OnePoint.opensOfCompl
 
-#print Alexandroff.infty_mem_opensOfCompl /-
+#print OnePoint.infty_mem_opensOfCompl /-
 theorem infty_mem_opensOfCompl {s : Set X} (h₁ : IsClosed s) (h₂ : IsCompact s) :
     ∞ ∈ opensOfCompl s h₁ h₂ :=
   mem_compl infty_not_mem_image_coe
-#align alexandroff.infty_mem_opens_of_compl Alexandroff.infty_mem_opensOfCompl
+#align alexandroff.infty_mem_opens_of_compl OnePoint.infty_mem_opensOfCompl
 -/
 
 @[continuity]
-theorem continuous_coe : Continuous (coe : X → Alexandroff X) :=
+theorem continuous_coe : Continuous (coe : X → OnePoint X) :=
   continuous_def.mpr fun s hs => hs.right
-#align alexandroff.continuous_coe Alexandroff.continuous_coe
+#align alexandroff.continuous_coe OnePoint.continuous_coe
 
-theorem isOpenMap_coe : IsOpenMap (coe : X → Alexandroff X) := fun s => isOpen_image_coe.2
-#align alexandroff.is_open_map_coe Alexandroff.isOpenMap_coe
+theorem isOpenMap_coe : IsOpenMap (coe : X → OnePoint X) := fun s => isOpen_image_coe.2
+#align alexandroff.is_open_map_coe OnePoint.isOpenMap_coe
 
-theorem openEmbedding_coe : OpenEmbedding (coe : X → Alexandroff X) :=
+theorem openEmbedding_coe : OpenEmbedding (coe : X → OnePoint X) :=
   openEmbedding_of_continuous_injective_open continuous_coe coe_injective isOpenMap_coe
-#align alexandroff.open_embedding_coe Alexandroff.openEmbedding_coe
+#align alexandroff.open_embedding_coe OnePoint.openEmbedding_coe
 
-theorem isOpen_range_coe : IsOpen (range (coe : X → Alexandroff X)) :=
+theorem isOpen_range_coe : IsOpen (range (coe : X → OnePoint X)) :=
   openEmbedding_coe.open_range
-#align alexandroff.is_open_range_coe Alexandroff.isOpen_range_coe
+#align alexandroff.is_open_range_coe OnePoint.isOpen_range_coe
 
-theorem isClosed_infty : IsClosed ({∞} : Set (Alexandroff X)) := by
+theorem isClosed_infty : IsClosed ({∞} : Set (OnePoint X)) := by
   rw [← compl_range_coe, isClosed_compl_iff]; exact is_open_range_coe
-#align alexandroff.is_closed_infty Alexandroff.isClosed_infty
+#align alexandroff.is_closed_infty OnePoint.isClosed_infty
 
-theorem nhds_coe_eq (x : X) : 𝓝 ↑x = map (coe : X → Alexandroff X) (𝓝 x) :=
+theorem nhds_coe_eq (x : X) : 𝓝 ↑x = map (coe : X → OnePoint X) (𝓝 x) :=
   (openEmbedding_coe.map_nhds_eq x).symm
-#align alexandroff.nhds_coe_eq Alexandroff.nhds_coe_eq
+#align alexandroff.nhds_coe_eq OnePoint.nhds_coe_eq
 
 theorem nhdsWithin_coe_image (s : Set X) (x : X) :
-    𝓝[coe '' s] (x : Alexandroff X) = map coe (𝓝[s] x) :=
+    𝓝[coe '' s] (x : OnePoint X) = map coe (𝓝[s] x) :=
   (openEmbedding_coe.toEmbedding.map_nhdsWithin_eq _ _).symm
-#align alexandroff.nhds_within_coe_image Alexandroff.nhdsWithin_coe_image
+#align alexandroff.nhds_within_coe_image OnePoint.nhdsWithin_coe_image
 
-theorem nhdsWithin_coe (s : Set (Alexandroff X)) (x : X) : 𝓝[s] ↑x = map coe (𝓝[coe ⁻¹' s] x) :=
+theorem nhdsWithin_coe (s : Set (OnePoint X)) (x : X) : 𝓝[s] ↑x = map coe (𝓝[coe ⁻¹' s] x) :=
   (openEmbedding_coe.map_nhdsWithin_preimage_eq _ _).symm
-#align alexandroff.nhds_within_coe Alexandroff.nhdsWithin_coe
+#align alexandroff.nhds_within_coe OnePoint.nhdsWithin_coe
 
-theorem comap_coe_nhds (x : X) : comap (coe : X → Alexandroff X) (𝓝 x) = 𝓝 x :=
+theorem comap_coe_nhds (x : X) : comap (coe : X → OnePoint X) (𝓝 x) = 𝓝 x :=
   (openEmbedding_coe.to_inducing.nhds_eq_comap x).symm
-#align alexandroff.comap_coe_nhds Alexandroff.comap_coe_nhds
+#align alexandroff.comap_coe_nhds OnePoint.comap_coe_nhds
 
 /-- If `x` is not an isolated point of `X`, then `x : alexandroff X` is not an isolated point
 of `alexandroff X`. -/
-instance nhdsWithin_compl_coe_neBot (x : X) [h : NeBot (𝓝[≠] x)] :
-    NeBot (𝓝[≠] (x : Alexandroff X)) := by
-  simpa [nhds_within_coe, preimage, coe_eq_coe] using h.map coe
-#align alexandroff.nhds_within_compl_coe_ne_bot Alexandroff.nhdsWithin_compl_coe_neBot
+instance nhdsWithin_compl_coe_neBot (x : X) [h : NeBot (𝓝[≠] x)] : NeBot (𝓝[≠] (x : OnePoint X)) :=
+  by simpa [nhds_within_coe, preimage, coe_eq_coe] using h.map coe
+#align alexandroff.nhds_within_compl_coe_ne_bot OnePoint.nhdsWithin_compl_coe_neBot
 
-theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : Alexandroff X) = map coe (coclosedCompact X) :=
+theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map coe (coclosedCompact X) :=
   by
   refine' (nhdsWithin_basis_open ∞ _).ext (has_basis_coclosed_compact.map _) _ _
   · rintro s ⟨hs, hso⟩
@@ -349,116 +347,116 @@ theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : Alexandroff X) = map coe (c
   · rintro s ⟨h₁, h₂⟩
     refine' ⟨_, ⟨mem_compl infty_not_mem_image_coe, is_open_compl_image_coe.2 ⟨h₁, h₂⟩⟩, _⟩
     simp [compl_image_coe, ← diff_eq, subset_preimage_image]
-#align alexandroff.nhds_within_compl_infty_eq Alexandroff.nhdsWithin_compl_infty_eq
+#align alexandroff.nhds_within_compl_infty_eq OnePoint.nhdsWithin_compl_infty_eq
 
 /-- If `X` is a non-compact space, then `∞` is not an isolated point of `alexandroff X`. -/
-instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : Alexandroff X)) := by
+instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : OnePoint X)) := by
   rw [nhds_within_compl_infty_eq]; infer_instance
-#align alexandroff.nhds_within_compl_infty_ne_bot Alexandroff.nhdsWithin_compl_infty_neBot
+#align alexandroff.nhds_within_compl_infty_ne_bot OnePoint.nhdsWithin_compl_infty_neBot
 
 instance (priority := 900) nhdsWithin_compl_neBot [∀ x : X, NeBot (𝓝[≠] x)] [NoncompactSpace X]
-    (x : Alexandroff X) : NeBot (𝓝[≠] x) :=
-  Alexandroff.rec _ Alexandroff.nhdsWithin_compl_infty_neBot
-    (fun y => Alexandroff.nhdsWithin_compl_coe_neBot y) x
-#align alexandroff.nhds_within_compl_ne_bot Alexandroff.nhdsWithin_compl_neBot
+    (x : OnePoint X) : NeBot (𝓝[≠] x) :=
+  OnePoint.rec _ OnePoint.nhdsWithin_compl_infty_neBot
+    (fun y => OnePoint.nhdsWithin_compl_coe_neBot y) x
+#align alexandroff.nhds_within_compl_ne_bot OnePoint.nhdsWithin_compl_neBot
 
-theorem nhds_infty_eq : 𝓝 (∞ : Alexandroff X) = map coe (coclosedCompact X) ⊔ pure ∞ := by
+theorem nhds_infty_eq : 𝓝 (∞ : OnePoint X) = map coe (coclosedCompact X) ⊔ pure ∞ := by
   rw [← nhds_within_compl_infty_eq, nhdsWithin_compl_singleton_sup_pure]
-#align alexandroff.nhds_infty_eq Alexandroff.nhds_infty_eq
+#align alexandroff.nhds_infty_eq OnePoint.nhds_infty_eq
 
 theorem hasBasis_nhds_infty :
-    (𝓝 (∞ : Alexandroff X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
+    (𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
       coe '' sᶜ ∪ {∞} :=
   by
   rw [nhds_infty_eq]
   exact (has_basis_coclosed_compact.map _).sup_pure _
-#align alexandroff.has_basis_nhds_infty Alexandroff.hasBasis_nhds_infty
+#align alexandroff.has_basis_nhds_infty OnePoint.hasBasis_nhds_infty
 
 @[simp]
-theorem comap_coe_nhds_infty : comap (coe : X → Alexandroff X) (𝓝 ∞) = coclosedCompact X := by
+theorem comap_coe_nhds_infty : comap (coe : X → OnePoint X) (𝓝 ∞) = coclosedCompact X := by
   simp [nhds_infty_eq, comap_sup, comap_map coe_injective]
-#align alexandroff.comap_coe_nhds_infty Alexandroff.comap_coe_nhds_infty
+#align alexandroff.comap_coe_nhds_infty OnePoint.comap_coe_nhds_infty
 
-theorem le_nhds_infty {f : Filter (Alexandroff X)} :
+theorem le_nhds_infty {f : Filter (OnePoint X)} :
     f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → coe '' sᶜ ∪ {∞} ∈ f := by
   simp only [has_basis_nhds_infty.ge_iff, and_imp]
-#align alexandroff.le_nhds_infty Alexandroff.le_nhds_infty
+#align alexandroff.le_nhds_infty OnePoint.le_nhds_infty
 
-theorem ultrafilter_le_nhds_infty {f : Ultrafilter (Alexandroff X)} :
-    (f : Filter (Alexandroff X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → coe '' s ∉ f := by
+theorem ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
+    (f : Filter (OnePoint X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → coe '' s ∉ f := by
   simp only [le_nhds_infty, ← compl_image_coe, Ultrafilter.mem_coe,
     Ultrafilter.compl_mem_iff_not_mem]
-#align alexandroff.ultrafilter_le_nhds_infty Alexandroff.ultrafilter_le_nhds_infty
+#align alexandroff.ultrafilter_le_nhds_infty OnePoint.ultrafilter_le_nhds_infty
 
-theorem tendsto_nhds_infty' {α : Type _} {f : Alexandroff X → α} {l : Filter α} :
+theorem tendsto_nhds_infty' {α : Type _} {f : OnePoint X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ coe) (coclosedCompact X) l := by
   simp [nhds_infty_eq, and_comm']
-#align alexandroff.tendsto_nhds_infty' Alexandroff.tendsto_nhds_infty'
+#align alexandroff.tendsto_nhds_infty' OnePoint.tendsto_nhds_infty'
 
-theorem tendsto_nhds_infty {α : Type _} {f : Alexandroff X → α} {l : Filter α} :
+theorem tendsto_nhds_infty {α : Type _} {f : OnePoint X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔
       ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ coe) (tᶜ) s :=
   tendsto_nhds_infty'.trans <| by
     simp only [tendsto_pure_left, has_basis_coclosed_compact.tendsto_left_iff, forall_and,
       and_assoc', exists_prop]
-#align alexandroff.tendsto_nhds_infty Alexandroff.tendsto_nhds_infty
+#align alexandroff.tendsto_nhds_infty OnePoint.tendsto_nhds_infty
 
-theorem continuousAt_infty' {Y : Type _} [TopologicalSpace Y] {f : Alexandroff X → Y} :
+theorem continuousAt_infty' {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} :
     ContinuousAt f ∞ ↔ Tendsto (f ∘ coe) (coclosedCompact X) (𝓝 (f ∞)) :=
   tendsto_nhds_infty'.trans <| and_iff_right (tendsto_pure_nhds _ _)
-#align alexandroff.continuous_at_infty' Alexandroff.continuousAt_infty'
+#align alexandroff.continuous_at_infty' OnePoint.continuousAt_infty'
 
-theorem continuousAt_infty {Y : Type _} [TopologicalSpace Y] {f : Alexandroff X → Y} :
+theorem continuousAt_infty {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} :
     ContinuousAt f ∞ ↔
       ∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ coe) (tᶜ) s :=
   continuousAt_infty'.trans <| by
     simp only [has_basis_coclosed_compact.tendsto_left_iff, exists_prop, and_assoc']
-#align alexandroff.continuous_at_infty Alexandroff.continuousAt_infty
+#align alexandroff.continuous_at_infty OnePoint.continuousAt_infty
 
-theorem continuousAt_coe {Y : Type _} [TopologicalSpace Y] {f : Alexandroff X → Y} {x : X} :
+theorem continuousAt_coe {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
     ContinuousAt f x ↔ ContinuousAt (f ∘ coe) x := by
   rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]
-#align alexandroff.continuous_at_coe Alexandroff.continuousAt_coe
+#align alexandroff.continuous_at_coe OnePoint.continuousAt_coe
 
 /-- If `X` is not a compact space, then the natural embedding `X → alexandroff X` has dense range.
 -/
-theorem denseRange_coe [NoncompactSpace X] : DenseRange (coe : X → Alexandroff X) :=
+theorem denseRange_coe [NoncompactSpace X] : DenseRange (coe : X → OnePoint X) :=
   by
   rw [DenseRange, ← compl_infty]
   exact dense_compl_singleton _
-#align alexandroff.dense_range_coe Alexandroff.denseRange_coe
+#align alexandroff.dense_range_coe OnePoint.denseRange_coe
 
-theorem denseEmbedding_coe [NoncompactSpace X] : DenseEmbedding (coe : X → Alexandroff X) :=
+theorem denseEmbedding_coe [NoncompactSpace X] : DenseEmbedding (coe : X → OnePoint X) :=
   { openEmbedding_coe with dense := denseRange_coe }
-#align alexandroff.dense_embedding_coe Alexandroff.denseEmbedding_coe
+#align alexandroff.dense_embedding_coe OnePoint.denseEmbedding_coe
 
 @[simp]
-theorem specializes_coe {x y : X} : (x : Alexandroff X) ⤳ y ↔ x ⤳ y :=
+theorem specializes_coe {x y : X} : (x : OnePoint X) ⤳ y ↔ x ⤳ y :=
   openEmbedding_coe.to_inducing.specializes_iff
-#align alexandroff.specializes_coe Alexandroff.specializes_coe
+#align alexandroff.specializes_coe OnePoint.specializes_coe
 
 @[simp]
-theorem inseparable_coe {x y : X} : Inseparable (x : Alexandroff X) y ↔ Inseparable x y :=
+theorem inseparable_coe {x y : X} : Inseparable (x : OnePoint X) y ↔ Inseparable x y :=
   openEmbedding_coe.to_inducing.inseparable_iff
-#align alexandroff.inseparable_coe Alexandroff.inseparable_coe
+#align alexandroff.inseparable_coe OnePoint.inseparable_coe
 
-theorem not_specializes_infty_coe {x : X} : ¬Specializes ∞ (x : Alexandroff X) :=
+theorem not_specializes_infty_coe {x : X} : ¬Specializes ∞ (x : OnePoint X) :=
   isClosed_infty.not_specializes rfl (coe_ne_infty x)
-#align alexandroff.not_specializes_infty_coe Alexandroff.not_specializes_infty_coe
+#align alexandroff.not_specializes_infty_coe OnePoint.not_specializes_infty_coe
 
-theorem not_inseparable_infty_coe {x : X} : ¬Inseparable ∞ (x : Alexandroff X) := fun h =>
+theorem not_inseparable_infty_coe {x : X} : ¬Inseparable ∞ (x : OnePoint X) := fun h =>
   not_specializes_infty_coe h.Specializes
-#align alexandroff.not_inseparable_infty_coe Alexandroff.not_inseparable_infty_coe
+#align alexandroff.not_inseparable_infty_coe OnePoint.not_inseparable_infty_coe
 
-theorem not_inseparable_coe_infty {x : X} : ¬Inseparable (x : Alexandroff X) ∞ := fun h =>
+theorem not_inseparable_coe_infty {x : X} : ¬Inseparable (x : OnePoint X) ∞ := fun h =>
   not_specializes_infty_coe h.specializes'
-#align alexandroff.not_inseparable_coe_infty Alexandroff.not_inseparable_coe_infty
+#align alexandroff.not_inseparable_coe_infty OnePoint.not_inseparable_coe_infty
 
-theorem inseparable_iff {x y : Alexandroff X} :
+theorem inseparable_iff {x y : OnePoint X} :
     Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
-  induction x using Alexandroff.rec <;> induction y using Alexandroff.rec <;>
+  induction x using OnePoint.rec <;> induction y using OnePoint.rec <;>
     simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe]
-#align alexandroff.inseparable_iff Alexandroff.inseparable_iff
+#align alexandroff.inseparable_iff OnePoint.inseparable_iff
 
 /-!
 ### Compactness and separation properties
@@ -473,10 +471,10 @@ Finally, if the original space `X` is *not* compact and is a preconnected space,
 
 
 /-- For any topological space `X`, its one point compactification is a compact space. -/
-instance : CompactSpace (Alexandroff X)
+instance : CompactSpace (OnePoint X)
     where isCompact_univ :=
     by
-    have : tendsto (coe : X → Alexandroff X) (cocompact X) (𝓝 ∞) :=
+    have : tendsto (coe : X → OnePoint X) (cocompact X) (𝓝 ∞) :=
       by
       rw [nhds_infty_eq]
       exact (tendsto_map.mono_left cocompact_le_coclosed_compact).mono_right le_sup_left
@@ -484,26 +482,26 @@ instance : CompactSpace (Alexandroff X)
     exact insert_none_range_some X
 
 /-- The one point compactification of a `t0_space` space is a `t0_space`. -/
-instance [T0Space X] : T0Space (Alexandroff X) :=
+instance [T0Space X] : T0Space (OnePoint X) :=
   by
   refine' ⟨fun x y hxy => _⟩
   rcases inseparable_iff.1 hxy with (⟨rfl, rfl⟩ | ⟨x, rfl, y, rfl, h⟩)
   exacts[rfl, congr_arg coe h.eq]
 
 /-- The one point compactification of a `t1_space` space is a `t1_space`. -/
-instance [T1Space X] : T1Space (Alexandroff X)
+instance [T1Space X] : T1Space (OnePoint X)
     where t1 z := by
-    induction z using Alexandroff.rec
+    induction z using OnePoint.rec
     · exact is_closed_infty
     · rw [← image_singleton, is_closed_image_coe]
       exact ⟨isClosed_singleton, isCompact_singleton⟩
 
 /-- The one point compactification of a locally compact Hausdorff space is a normal (hence,
 Hausdorff and regular) topological space. -/
-instance [LocallyCompactSpace X] [T2Space X] : NormalSpace (Alexandroff X) :=
+instance [LocallyCompactSpace X] [T2Space X] : NormalSpace (OnePoint X) :=
   by
   have key :
-    ∀ z : X, ∃ u v : Set (Alexandroff X), IsOpen u ∧ IsOpen v ∧ ↑z ∈ u ∧ ∞ ∈ v ∧ Disjoint u v :=
+    ∀ z : X, ∃ u v : Set (OnePoint X), IsOpen u ∧ IsOpen v ∧ ↑z ∈ u ∧ ∞ ∈ v ∧ Disjoint u v :=
     by
     intro z
     rcases exists_open_with_compact_closure z with ⟨u, hu, huy', Hu⟩
@@ -512,7 +510,7 @@ instance [LocallyCompactSpace X] [T2Space X] : NormalSpace (Alexandroff X) :=
         is_open_compl_image_coe.2 ⟨isClosed_closure, Hu⟩, mem_image_of_mem _ huy',
         mem_compl infty_not_mem_image_coe, (image_subset _ subset_closure).disjoint_compl_right⟩
   refine' @normalOfCompactT2 _ _ _ ⟨fun x y hxy => _⟩
-  induction x using Alexandroff.rec <;> induction y using Alexandroff.rec
+  induction x using OnePoint.rec <;> induction y using OnePoint.rec
   · exact (hxy rfl).elim
   · rcases key y with ⟨u, v, hu, hv, hxu, hyv, huv⟩
     exact ⟨v, u, hv, hu, hyv, hxu, huv.symm⟩
@@ -520,7 +518,7 @@ instance [LocallyCompactSpace X] [T2Space X] : NormalSpace (Alexandroff X) :=
   · exact separated_by_openEmbedding open_embedding_coe (mt coe_eq_coe.mpr hxy)
 
 /-- If `X` is not a compact space, then `alexandroff X` is a connected space. -/
-instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (Alexandroff X)
+instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (OnePoint X)
     where
   to_preconnectedSpace := denseEmbedding_coe.to_denseInducing.PreconnectedSpace
   to_nonempty := inferInstance
@@ -528,16 +526,16 @@ instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (Alexandroff
 /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
 `cofinite_topology (alexandroff X)` to `alexandroff X` is not continuous. -/
 theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
-    ¬Continuous (@CofiniteTopology.of (Alexandroff X)).symm :=
+    ¬Continuous (@CofiniteTopology.of (OnePoint X)).symm :=
   by
   inhabit X
   simp only [continuous_iff_continuousAt, ContinuousAt, not_forall]
   use CofiniteTopology.of ↑(default : X)
   simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using
-    (finite_singleton ((default : X) : Alexandroff X)).infinite_compl
-#align alexandroff.not_continuous_cofinite_topology_of_symm Alexandroff.not_continuous_cofiniteTopology_of_symm
+    (finite_singleton ((default : X) : OnePoint X)).infinite_compl
+#align alexandroff.not_continuous_cofinite_topology_of_symm OnePoint.not_continuous_cofiniteTopology_of_symm
 
-end Alexandroff
+end OnePoint
 
 #print Continuous.homeoOfEquivCompactToT2.t1_counterexample /-
 /-- A concrete counterexample shows that  `continuous.homeo_of_equiv_compact_to_t2`
@@ -550,9 +548,9 @@ Let `α = alexandroff ℕ` be the one-point compactification of `ℕ`, and let `
 theorem Continuous.homeoOfEquivCompactToT2.t1_counterexample :
     ∃ (α β : Type)(Iα : TopologicalSpace α)(Iβ : TopologicalSpace β),
       CompactSpace α ∧ T1Space β ∧ ∃ f : α ≃ β, Continuous f ∧ ¬Continuous f.symm :=
-  ⟨Alexandroff ℕ, CofiniteTopology (Alexandroff ℕ), inferInstance, inferInstance, inferInstance,
+  ⟨OnePoint ℕ, CofiniteTopology (OnePoint ℕ), inferInstance, inferInstance, inferInstance,
     inferInstance, CofiniteTopology.of, CofiniteTopology.continuous_of,
-    Alexandroff.not_continuous_cofiniteTopology_of_symm⟩
+    OnePoint.not_continuous_cofiniteTopology_of_symm⟩
 #align continuous.homeo_of_equiv_compact_to_t2.t1_counterexample Continuous.homeoOfEquivCompactToT2.t1_counterexample
 -/
 

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

@@ -44,7 +44,7 @@ one-point compactification, compactness
 
 open Set Filter
 
-open Classical Topology Filter
+open scoped Classical Topology Filter
 
 /-!
 ### Definition and basic properties

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

@@ -133,65 +133,29 @@ protected def rec (C : Alexandroff X → Sort _) (h₁ : C ∞) (h₂ : ∀ x :
 #align alexandroff.rec Alexandroff.rec
 -/
 
-/- warning: alexandroff.is_compl_range_coe_infty -> Alexandroff.isCompl_range_coe_infty is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}}, IsCompl.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Set.completeBooleanAlgebra.{u1} (Alexandroff.{u1} X))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Set.completeBooleanAlgebra.{u1} (Alexandroff.{u1} X)))))) (Set.range.{u1, succ u1} (Alexandroff.{u1} X) X ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) X (Alexandroff.{u1} X) (HasLiftT.mk.{succ u1, succ u1} X (Alexandroff.{u1} X) (CoeTCₓ.coe.{succ u1, succ u1} X (Alexandroff.{u1} X) (Alexandroff.hasCoeT.{u1} X))))) (Singleton.singleton.{u1, u1} (Alexandroff.{u1} X) (Set.{u1} (Alexandroff.{u1} X)) (Set.hasSingleton.{u1} (Alexandroff.{u1} X)) (Alexandroff.infty.{u1} X))
-but is expected to have type
-  forall {X : Type.{u1}}, IsCompl.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Set.instCompleteBooleanAlgebraSet.{u1} (Alexandroff.{u1} X))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (Alexandroff.{u1} X)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Set.instCompleteBooleanAlgebraSet.{u1} (Alexandroff.{u1} X)))))) (Set.range.{u1, succ u1} (Alexandroff.{u1} X) X (Alexandroff.some.{u1} X)) (Singleton.singleton.{u1, u1} (Alexandroff.{u1} X) (Set.{u1} (Alexandroff.{u1} X)) (Set.instSingletonSet.{u1} (Alexandroff.{u1} X)) (Alexandroff.infty.{u1} X))
-Case conversion may be inaccurate. Consider using '#align alexandroff.is_compl_range_coe_infty Alexandroff.isCompl_range_coe_inftyₓ'. -/
 theorem isCompl_range_coe_infty : IsCompl (range (coe : X → Alexandroff X)) {∞} :=
   isCompl_range_some_none X
 #align alexandroff.is_compl_range_coe_infty Alexandroff.isCompl_range_coe_infty
 
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 @[simp]
 theorem range_coe_union_infty : range (coe : X → Alexandroff X) ∪ {∞} = univ :=
   range_some_union_none X
 #align alexandroff.range_coe_union_infty Alexandroff.range_coe_union_infty
 
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 @[simp]
 theorem range_coe_inter_infty : range (coe : X → Alexandroff X) ∩ {∞} = ∅ :=
   range_some_inter_none X
 #align alexandroff.range_coe_inter_infty Alexandroff.range_coe_inter_infty
 
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 @[simp]
 theorem compl_range_coe : range (coe : X → Alexandroff X)ᶜ = {∞} :=
   compl_range_some X
 #align alexandroff.compl_range_coe Alexandroff.compl_range_coe
 
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 theorem compl_infty : ({∞}ᶜ : Set (Alexandroff X)) = range (coe : X → Alexandroff X) :=
   (@isCompl_range_coe_infty X).symm.compl_eq
 #align alexandroff.compl_infty Alexandroff.compl_infty
 
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 theorem compl_image_coe (s : Set X) : (coe '' s : Set (Alexandroff X))ᶜ = coe '' sᶜ ∪ {∞} := by
   rw [coe_injective.compl_image_eq, compl_range_coe]
 #align alexandroff.compl_image_coe Alexandroff.compl_image_coe
@@ -271,89 +235,41 @@ instance : TopologicalSpace (Alexandroff X)
 
 variable {s : Set (Alexandroff X)} {t : Set X}
 
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 theorem isOpen_def :
     IsOpen s ↔ (∞ ∈ s → IsCompact ((coe ⁻¹' s : Set X)ᶜ)) ∧ IsOpen (coe ⁻¹' s : Set X) :=
   Iff.rfl
 #align alexandroff.is_open_def Alexandroff.isOpen_def
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.is_open_iff_of_mem' Alexandroff.isOpen_iff_of_mem'ₓ'. -/
 theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
     IsOpen s ↔ IsCompact ((coe ⁻¹' s : Set X)ᶜ) ∧ IsOpen (coe ⁻¹' s : Set X) := by
   simp [is_open_def, h]
 #align alexandroff.is_open_iff_of_mem' Alexandroff.isOpen_iff_of_mem'
 
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 theorem isOpen_iff_of_mem (h : ∞ ∈ s) :
     IsOpen s ↔ IsClosed ((coe ⁻¹' s : Set X)ᶜ) ∧ IsCompact ((coe ⁻¹' s : Set X)ᶜ) := by
   simp only [is_open_iff_of_mem' h, isClosed_compl_iff, and_comm]
 #align alexandroff.is_open_iff_of_mem Alexandroff.isOpen_iff_of_mem
 
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 theorem isOpen_iff_of_not_mem (h : ∞ ∉ s) : IsOpen s ↔ IsOpen (coe ⁻¹' s : Set X) := by
   simp [is_open_def, h]
 #align alexandroff.is_open_iff_of_not_mem Alexandroff.isOpen_iff_of_not_mem
 
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 theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed (coe ⁻¹' s : Set X) :=
   by
   have : ∞ ∉ sᶜ := fun H => H h
   rw [← isOpen_compl_iff, is_open_iff_of_not_mem this, ← isOpen_compl_iff, preimage_compl]
 #align alexandroff.is_closed_iff_of_mem Alexandroff.isClosed_iff_of_mem
 
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 theorem isClosed_iff_of_not_mem (h : ∞ ∉ s) :
     IsClosed s ↔ IsClosed (coe ⁻¹' s : Set X) ∧ IsCompact (coe ⁻¹' s : Set X) := by
   rw [← isOpen_compl_iff, is_open_iff_of_mem (mem_compl h), ← preimage_compl, compl_compl]
 #align alexandroff.is_closed_iff_of_not_mem Alexandroff.isClosed_iff_of_not_mem
 
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 @[simp]
 theorem isOpen_image_coe {s : Set X} : IsOpen (coe '' s : Set (Alexandroff X)) ↔ IsOpen s := by
   rw [is_open_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective]
 #align alexandroff.is_open_image_coe Alexandroff.isOpen_image_coe
 
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 theorem isOpen_compl_image_coe {s : Set X} :
     IsOpen ((coe '' s : Set (Alexandroff X))ᶜ) ↔ IsClosed s ∧ IsCompact s :=
   by
@@ -361,24 +277,12 @@ theorem isOpen_compl_image_coe {s : Set X} :
   exact infty_not_mem_image_coe
 #align alexandroff.is_open_compl_image_coe Alexandroff.isOpen_compl_image_coe
 
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 @[simp]
 theorem isClosed_image_coe {s : Set X} :
     IsClosed (coe '' s : Set (Alexandroff X)) ↔ IsClosed s ∧ IsCompact s := by
   rw [← isOpen_compl_iff, is_open_compl_image_coe]
 #align alexandroff.is_closed_image_coe Alexandroff.isClosed_image_coe
 
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 /-- An open set in `alexandroff X` constructed from a closed compact set in `X` -/
 def opensOfCompl (s : Set X) (h₁ : IsClosed s) (h₂ : IsCompact s) :
     TopologicalSpace.Opens (Alexandroff X) :=
@@ -392,103 +296,43 @@ theorem infty_mem_opensOfCompl {s : Set X} (h₁ : IsClosed s) (h₂ : IsCompact
 #align alexandroff.infty_mem_opens_of_compl Alexandroff.infty_mem_opensOfCompl
 -/
 
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 @[continuity]
 theorem continuous_coe : Continuous (coe : X → Alexandroff X) :=
   continuous_def.mpr fun s hs => hs.right
 #align alexandroff.continuous_coe Alexandroff.continuous_coe
 
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 theorem isOpenMap_coe : IsOpenMap (coe : X → Alexandroff X) := fun s => isOpen_image_coe.2
 #align alexandroff.is_open_map_coe Alexandroff.isOpenMap_coe
 
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 theorem openEmbedding_coe : OpenEmbedding (coe : X → Alexandroff X) :=
   openEmbedding_of_continuous_injective_open continuous_coe coe_injective isOpenMap_coe
 #align alexandroff.open_embedding_coe Alexandroff.openEmbedding_coe
 
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 theorem isOpen_range_coe : IsOpen (range (coe : X → Alexandroff X)) :=
   openEmbedding_coe.open_range
 #align alexandroff.is_open_range_coe Alexandroff.isOpen_range_coe
 
-/- warning: alexandroff.is_closed_infty -> Alexandroff.isClosed_infty is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align alexandroff.is_closed_infty Alexandroff.isClosed_inftyₓ'. -/
 theorem isClosed_infty : IsClosed ({∞} : Set (Alexandroff X)) := by
   rw [← compl_range_coe, isClosed_compl_iff]; exact is_open_range_coe
 #align alexandroff.is_closed_infty Alexandroff.isClosed_infty
 
-/- warning: alexandroff.nhds_coe_eq -> Alexandroff.nhds_coe_eq is a dubious translation:
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 theorem nhds_coe_eq (x : X) : 𝓝 ↑x = map (coe : X → Alexandroff X) (𝓝 x) :=
   (openEmbedding_coe.map_nhds_eq x).symm
 #align alexandroff.nhds_coe_eq Alexandroff.nhds_coe_eq
 
-/- warning: alexandroff.nhds_within_coe_image -> Alexandroff.nhdsWithin_coe_image is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align alexandroff.nhds_within_coe_image Alexandroff.nhdsWithin_coe_imageₓ'. -/
 theorem nhdsWithin_coe_image (s : Set X) (x : X) :
     𝓝[coe '' s] (x : Alexandroff X) = map coe (𝓝[s] x) :=
   (openEmbedding_coe.toEmbedding.map_nhdsWithin_eq _ _).symm
 #align alexandroff.nhds_within_coe_image Alexandroff.nhdsWithin_coe_image
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.nhds_within_coe Alexandroff.nhdsWithin_coeₓ'. -/
 theorem nhdsWithin_coe (s : Set (Alexandroff X)) (x : X) : 𝓝[s] ↑x = map coe (𝓝[coe ⁻¹' s] x) :=
   (openEmbedding_coe.map_nhdsWithin_preimage_eq _ _).symm
 #align alexandroff.nhds_within_coe Alexandroff.nhdsWithin_coe
 
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 theorem comap_coe_nhds (x : X) : comap (coe : X → Alexandroff X) (𝓝 x) = 𝓝 x :=
   (openEmbedding_coe.to_inducing.nhds_eq_comap x).symm
 #align alexandroff.comap_coe_nhds Alexandroff.comap_coe_nhds
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.nhds_within_compl_coe_ne_bot Alexandroff.nhdsWithin_compl_coe_neBotₓ'. -/
 /-- If `x` is not an isolated point of `X`, then `x : alexandroff X` is not an isolated point
 of `alexandroff X`. -/
 instance nhdsWithin_compl_coe_neBot (x : X) [h : NeBot (𝓝[≠] x)] :
@@ -496,12 +340,6 @@ instance nhdsWithin_compl_coe_neBot (x : X) [h : NeBot (𝓝[≠] x)] :
   simpa [nhds_within_coe, preimage, coe_eq_coe] using h.map coe
 #align alexandroff.nhds_within_compl_coe_ne_bot Alexandroff.nhdsWithin_compl_coe_neBot
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.nhds_within_compl_infty_eq Alexandroff.nhdsWithin_compl_infty_eqₓ'. -/
 theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : Alexandroff X) = map coe (coclosedCompact X) :=
   by
   refine' (nhdsWithin_basis_open ∞ _).ext (has_basis_coclosed_compact.map _) _ _
@@ -513,45 +351,21 @@ theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : Alexandroff X) = map coe (c
     simp [compl_image_coe, ← diff_eq, subset_preimage_image]
 #align alexandroff.nhds_within_compl_infty_eq Alexandroff.nhdsWithin_compl_infty_eq
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.nhds_within_compl_infty_ne_bot Alexandroff.nhdsWithin_compl_infty_neBotₓ'. -/
 /-- If `X` is a non-compact space, then `∞` is not an isolated point of `alexandroff X`. -/
 instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : Alexandroff X)) := by
   rw [nhds_within_compl_infty_eq]; infer_instance
 #align alexandroff.nhds_within_compl_infty_ne_bot Alexandroff.nhdsWithin_compl_infty_neBot
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.nhds_within_compl_ne_bot Alexandroff.nhdsWithin_compl_neBotₓ'. -/
 instance (priority := 900) nhdsWithin_compl_neBot [∀ x : X, NeBot (𝓝[≠] x)] [NoncompactSpace X]
     (x : Alexandroff X) : NeBot (𝓝[≠] x) :=
   Alexandroff.rec _ Alexandroff.nhdsWithin_compl_infty_neBot
     (fun y => Alexandroff.nhdsWithin_compl_coe_neBot y) x
 #align alexandroff.nhds_within_compl_ne_bot Alexandroff.nhdsWithin_compl_neBot
 
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 theorem nhds_infty_eq : 𝓝 (∞ : Alexandroff X) = map coe (coclosedCompact X) ⊔ pure ∞ := by
   rw [← nhds_within_compl_infty_eq, nhdsWithin_compl_singleton_sup_pure]
 #align alexandroff.nhds_infty_eq Alexandroff.nhds_infty_eq
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.has_basis_nhds_infty Alexandroff.hasBasis_nhds_inftyₓ'. -/
 theorem hasBasis_nhds_infty :
     (𝓝 (∞ : Alexandroff X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
       coe '' sᶜ ∪ {∞} :=
@@ -560,57 +374,27 @@ theorem hasBasis_nhds_infty :
   exact (has_basis_coclosed_compact.map _).sup_pure _
 #align alexandroff.has_basis_nhds_infty Alexandroff.hasBasis_nhds_infty
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.comap_coe_nhds_infty Alexandroff.comap_coe_nhds_inftyₓ'. -/
 @[simp]
 theorem comap_coe_nhds_infty : comap (coe : X → Alexandroff X) (𝓝 ∞) = coclosedCompact X := by
   simp [nhds_infty_eq, comap_sup, comap_map coe_injective]
 #align alexandroff.comap_coe_nhds_infty Alexandroff.comap_coe_nhds_infty
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.le_nhds_infty Alexandroff.le_nhds_inftyₓ'. -/
 theorem le_nhds_infty {f : Filter (Alexandroff X)} :
     f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → coe '' sᶜ ∪ {∞} ∈ f := by
   simp only [has_basis_nhds_infty.ge_iff, and_imp]
 #align alexandroff.le_nhds_infty Alexandroff.le_nhds_infty
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.ultrafilter_le_nhds_infty Alexandroff.ultrafilter_le_nhds_inftyₓ'. -/
 theorem ultrafilter_le_nhds_infty {f : Ultrafilter (Alexandroff X)} :
     (f : Filter (Alexandroff X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → coe '' s ∉ f := by
   simp only [le_nhds_infty, ← compl_image_coe, Ultrafilter.mem_coe,
     Ultrafilter.compl_mem_iff_not_mem]
 #align alexandroff.ultrafilter_le_nhds_infty Alexandroff.ultrafilter_le_nhds_infty
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.tendsto_nhds_infty' Alexandroff.tendsto_nhds_infty'ₓ'. -/
 theorem tendsto_nhds_infty' {α : Type _} {f : Alexandroff X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ coe) (coclosedCompact X) l := by
   simp [nhds_infty_eq, and_comm']
 #align alexandroff.tendsto_nhds_infty' Alexandroff.tendsto_nhds_infty'
 
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-Case conversion may be inaccurate. Consider using '#align alexandroff.tendsto_nhds_infty Alexandroff.tendsto_nhds_inftyₓ'. -/
 theorem tendsto_nhds_infty {α : Type _} {f : Alexandroff X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔
       ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ coe) (tᶜ) s :=
@@ -619,23 +403,11 @@ theorem tendsto_nhds_infty {α : Type _} {f : Alexandroff X → α} {l : Filter
       and_assoc', exists_prop]
 #align alexandroff.tendsto_nhds_infty Alexandroff.tendsto_nhds_infty
 
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 theorem continuousAt_infty' {Y : Type _} [TopologicalSpace Y] {f : Alexandroff X → Y} :
     ContinuousAt f ∞ ↔ Tendsto (f ∘ coe) (coclosedCompact X) (𝓝 (f ∞)) :=
   tendsto_nhds_infty'.trans <| and_iff_right (tendsto_pure_nhds _ _)
 #align alexandroff.continuous_at_infty' Alexandroff.continuousAt_infty'
 
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 theorem continuousAt_infty {Y : Type _} [TopologicalSpace Y] {f : Alexandroff X → Y} :
     ContinuousAt f ∞ ↔
       ∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ coe) (tᶜ) s :=
@@ -643,23 +415,11 @@ theorem continuousAt_infty {Y : Type _} [TopologicalSpace Y] {f : Alexandroff X
     simp only [has_basis_coclosed_compact.tendsto_left_iff, exists_prop, and_assoc']
 #align alexandroff.continuous_at_infty Alexandroff.continuousAt_infty
 
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 theorem continuousAt_coe {Y : Type _} [TopologicalSpace Y] {f : Alexandroff X → Y} {x : X} :
     ContinuousAt f x ↔ ContinuousAt (f ∘ coe) x := by
   rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]
 #align alexandroff.continuous_at_coe Alexandroff.continuousAt_coe
 
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 /-- If `X` is not a compact space, then the natural embedding `X → alexandroff X` has dense range.
 -/
 theorem denseRange_coe [NoncompactSpace X] : DenseRange (coe : X → Alexandroff X) :=
@@ -668,74 +428,32 @@ theorem denseRange_coe [NoncompactSpace X] : DenseRange (coe : X → Alexandroff
   exact dense_compl_singleton _
 #align alexandroff.dense_range_coe Alexandroff.denseRange_coe
 
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 theorem denseEmbedding_coe [NoncompactSpace X] : DenseEmbedding (coe : X → Alexandroff X) :=
   { openEmbedding_coe with dense := denseRange_coe }
 #align alexandroff.dense_embedding_coe Alexandroff.denseEmbedding_coe
 
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 @[simp]
 theorem specializes_coe {x y : X} : (x : Alexandroff X) ⤳ y ↔ x ⤳ y :=
   openEmbedding_coe.to_inducing.specializes_iff
 #align alexandroff.specializes_coe Alexandroff.specializes_coe
 
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 @[simp]
 theorem inseparable_coe {x y : X} : Inseparable (x : Alexandroff X) y ↔ Inseparable x y :=
   openEmbedding_coe.to_inducing.inseparable_iff
 #align alexandroff.inseparable_coe Alexandroff.inseparable_coe
 
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 theorem not_specializes_infty_coe {x : X} : ¬Specializes ∞ (x : Alexandroff X) :=
   isClosed_infty.not_specializes rfl (coe_ne_infty x)
 #align alexandroff.not_specializes_infty_coe Alexandroff.not_specializes_infty_coe
 
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 theorem not_inseparable_infty_coe {x : X} : ¬Inseparable ∞ (x : Alexandroff X) := fun h =>
   not_specializes_infty_coe h.Specializes
 #align alexandroff.not_inseparable_infty_coe Alexandroff.not_inseparable_infty_coe
 
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 theorem not_inseparable_coe_infty {x : X} : ¬Inseparable (x : Alexandroff X) ∞ := fun h =>
   not_specializes_infty_coe h.specializes'
 #align alexandroff.not_inseparable_coe_infty Alexandroff.not_inseparable_coe_infty
 
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 theorem inseparable_iff {x y : Alexandroff X} :
     Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
   induction x using Alexandroff.rec <;> induction y using Alexandroff.rec <;>
@@ -807,12 +525,6 @@ instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (Alexandroff
   to_preconnectedSpace := denseEmbedding_coe.to_denseInducing.PreconnectedSpace
   to_nonempty := inferInstance
 
-/- warning: alexandroff.not_continuous_cofinite_topology_of_symm -> Alexandroff.not_continuous_cofiniteTopology_of_symm is a dubious translation:
-lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : Infinite.{succ u1} X] [_inst_3 : DiscreteTopology.{u1} X _inst_1], Not (Continuous.{u1, u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X) (CofiniteTopology.topologicalSpace.{u1} (Alexandroff.{u1} X)) (Alexandroff.topologicalSpace.{u1} X _inst_1) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (fun (_x : Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) => (CofiniteTopology.{u1} (Alexandroff.{u1} X)) -> (Alexandroff.{u1} X)) (Equiv.hasCoeToFun.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (Equiv.symm.{succ u1, succ u1} (Alexandroff.{u1} X) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (CofiniteTopology.of.{u1} (Alexandroff.{u1} X)))))
-but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : Infinite.{succ u1} X] [_inst_3 : DiscreteTopology.{u1} X _inst_1], Not (Continuous.{u1, u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X) (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (fun (_x : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => Alexandroff.{u1} X) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (Equiv.symm.{succ u1, succ u1} (Alexandroff.{u1} X) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (CofiniteTopology.of.{u1} (Alexandroff.{u1} X)))))
-Case conversion may be inaccurate. Consider using '#align alexandroff.not_continuous_cofinite_topology_of_symm Alexandroff.not_continuous_cofiniteTopology_of_symmₓ'. -/
 /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
 `cofinite_topology (alexandroff X)` to `alexandroff X` is not continuous. -/
 theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :

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

@@ -228,10 +228,7 @@ theorem infty_not_mem_image_coe {s : Set X} : ∞ ∉ (coe : X → Alexandroff X
 
 #print Alexandroff.coe_preimage_infty /-
 @[simp]
-theorem coe_preimage_infty : (coe : X → Alexandroff X) ⁻¹' {∞} = ∅ :=
-  by
-  ext
-  simp
+theorem coe_preimage_infty : (coe : X → Alexandroff X) ⁻¹' {∞} = ∅ := by ext; simp
 #align alexandroff.coe_preimage_infty Alexandroff.coe_preimage_infty
 -/
 
@@ -441,10 +438,8 @@ lean 3 declaration is
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X], IsClosed.{u1} (Alexandroff.{u1} X) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (Singleton.singleton.{u1, u1} (Alexandroff.{u1} X) (Set.{u1} (Alexandroff.{u1} X)) (Set.instSingletonSet.{u1} (Alexandroff.{u1} X)) (Alexandroff.infty.{u1} X))
 Case conversion may be inaccurate. Consider using '#align alexandroff.is_closed_infty Alexandroff.isClosed_inftyₓ'. -/
-theorem isClosed_infty : IsClosed ({∞} : Set (Alexandroff X)) :=
-  by
-  rw [← compl_range_coe, isClosed_compl_iff]
-  exact is_open_range_coe
+theorem isClosed_infty : IsClosed ({∞} : Set (Alexandroff X)) := by
+  rw [← compl_range_coe, isClosed_compl_iff]; exact is_open_range_coe
 #align alexandroff.is_closed_infty Alexandroff.isClosed_infty
 
 /- warning: alexandroff.nhds_coe_eq -> Alexandroff.nhds_coe_eq is a dubious translation:
@@ -525,10 +520,8 @@ but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : NoncompactSpace.{u1} X _inst_1], Filter.NeBot.{u1} (Alexandroff.{u1} X) (nhdsWithin.{u1} (Alexandroff.{u1} X) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (Alexandroff.infty.{u1} X) (HasCompl.compl.{u1} (Set.{u1} (Alexandroff.{u1} X)) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Set.instBooleanAlgebraSet.{u1} (Alexandroff.{u1} X))) (Singleton.singleton.{u1, u1} (Alexandroff.{u1} X) (Set.{u1} (Alexandroff.{u1} X)) (Set.instSingletonSet.{u1} (Alexandroff.{u1} X)) (Alexandroff.infty.{u1} X))))
 Case conversion may be inaccurate. Consider using '#align alexandroff.nhds_within_compl_infty_ne_bot Alexandroff.nhdsWithin_compl_infty_neBotₓ'. -/
 /-- If `X` is a non-compact space, then `∞` is not an isolated point of `alexandroff X`. -/
-instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : Alexandroff X)) :=
-  by
-  rw [nhds_within_compl_infty_eq]
-  infer_instance
+instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : Alexandroff X)) := by
+  rw [nhds_within_compl_infty_eq]; infer_instance
 #align alexandroff.nhds_within_compl_infty_ne_bot Alexandroff.nhdsWithin_compl_infty_neBot
 
 /- warning: alexandroff.nhds_within_compl_ne_bot -> Alexandroff.nhdsWithin_compl_neBot is a dubious translation:

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -818,7 +818,7 @@ instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (Alexandroff
 lean 3 declaration is
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : Infinite.{succ u1} X] [_inst_3 : DiscreteTopology.{u1} X _inst_1], Not (Continuous.{u1, u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X) (CofiniteTopology.topologicalSpace.{u1} (Alexandroff.{u1} X)) (Alexandroff.topologicalSpace.{u1} X _inst_1) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (fun (_x : Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) => (CofiniteTopology.{u1} (Alexandroff.{u1} X)) -> (Alexandroff.{u1} X)) (Equiv.hasCoeToFun.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (Equiv.symm.{succ u1, succ u1} (Alexandroff.{u1} X) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (CofiniteTopology.of.{u1} (Alexandroff.{u1} X)))))
 but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : Infinite.{succ u1} X] [_inst_3 : DiscreteTopology.{u1} X _inst_1], Not (Continuous.{u1, u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X) (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (fun (_x : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => Alexandroff.{u1} X) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (Equiv.symm.{succ u1, succ u1} (Alexandroff.{u1} X) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (CofiniteTopology.of.{u1} (Alexandroff.{u1} X)))))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : Infinite.{succ u1} X] [_inst_3 : DiscreteTopology.{u1} X _inst_1], Not (Continuous.{u1, u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X) (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (fun (_x : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => Alexandroff.{u1} X) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (Equiv.symm.{succ u1, succ u1} (Alexandroff.{u1} X) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (CofiniteTopology.of.{u1} (Alexandroff.{u1} X)))))
 Case conversion may be inaccurate. Consider using '#align alexandroff.not_continuous_cofinite_topology_of_symm Alexandroff.not_continuous_cofiniteTopology_of_symmₓ'. -/
 /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
 `cofinite_topology (alexandroff X)` to `alexandroff X` is not continuous. -/

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

@@ -580,7 +580,7 @@ theorem comap_coe_nhds_infty : comap (coe : X → Alexandroff X) (𝓝 ∞) = co
 
 /- warning: alexandroff.le_nhds_infty -> Alexandroff.le_nhds_infty is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : Filter.{u1} (Alexandroff.{u1} X)}, Iff (LE.le.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Preorder.toLE.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.partialOrder.{u1} (Alexandroff.{u1} X)))) f (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.topologicalSpace.{u1} X _inst_1) (Alexandroff.infty.{u1} X))) (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (IsCompact.{u1} X _inst_1 s) -> (Membership.Mem.{u1, u1} (Set.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (Filter.hasMem.{u1} (Alexandroff.{u1} X)) (Union.union.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Set.hasUnion.{u1} (Alexandroff.{u1} X)) (Set.image.{u1, u1} X (Alexandroff.{u1} X) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) X (Alexandroff.{u1} X) (HasLiftT.mk.{succ u1, succ u1} X (Alexandroff.{u1} X) (CoeTCₓ.coe.{succ u1, succ u1} X (Alexandroff.{u1} X) (Alexandroff.hasCoeT.{u1} X)))) (HasCompl.compl.{u1} (Set.{u1} X) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} X) (Set.booleanAlgebra.{u1} X)) s)) (Singleton.singleton.{u1, u1} (Alexandroff.{u1} X) (Set.{u1} (Alexandroff.{u1} X)) (Set.hasSingleton.{u1} (Alexandroff.{u1} X)) (Alexandroff.infty.{u1} X))) f))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : Filter.{u1} (Alexandroff.{u1} X)}, Iff (LE.le.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Preorder.toHasLe.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.partialOrder.{u1} (Alexandroff.{u1} X)))) f (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.topologicalSpace.{u1} X _inst_1) (Alexandroff.infty.{u1} X))) (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (IsCompact.{u1} X _inst_1 s) -> (Membership.Mem.{u1, u1} (Set.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (Filter.hasMem.{u1} (Alexandroff.{u1} X)) (Union.union.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Set.hasUnion.{u1} (Alexandroff.{u1} X)) (Set.image.{u1, u1} X (Alexandroff.{u1} X) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) X (Alexandroff.{u1} X) (HasLiftT.mk.{succ u1, succ u1} X (Alexandroff.{u1} X) (CoeTCₓ.coe.{succ u1, succ u1} X (Alexandroff.{u1} X) (Alexandroff.hasCoeT.{u1} X)))) (HasCompl.compl.{u1} (Set.{u1} X) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} X) (Set.booleanAlgebra.{u1} X)) s)) (Singleton.singleton.{u1, u1} (Alexandroff.{u1} X) (Set.{u1} (Alexandroff.{u1} X)) (Set.hasSingleton.{u1} (Alexandroff.{u1} X)) (Alexandroff.infty.{u1} X))) f))
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : Filter.{u1} (Alexandroff.{u1} X)}, Iff (LE.le.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Preorder.toLE.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.instPartialOrderFilter.{u1} (Alexandroff.{u1} X)))) f (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (Alexandroff.infty.{u1} X))) (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (IsCompact.{u1} X _inst_1 s) -> (Membership.mem.{u1, u1} (Set.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (instMembershipSetFilter.{u1} (Alexandroff.{u1} X)) (Union.union.{u1} (Set.{u1} (Alexandroff.{u1} X)) (Set.instUnionSet.{u1} (Alexandroff.{u1} X)) (Set.image.{u1, u1} X (Alexandroff.{u1} X) (Alexandroff.some.{u1} X) (HasCompl.compl.{u1} (Set.{u1} X) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} X) (Set.instBooleanAlgebraSet.{u1} X)) s)) (Singleton.singleton.{u1, u1} (Alexandroff.{u1} X) (Set.{u1} (Alexandroff.{u1} X)) (Set.instSingletonSet.{u1} (Alexandroff.{u1} X)) (Alexandroff.infty.{u1} X))) f))
 Case conversion may be inaccurate. Consider using '#align alexandroff.le_nhds_infty Alexandroff.le_nhds_inftyₓ'. -/
@@ -591,7 +591,7 @@ theorem le_nhds_infty {f : Filter (Alexandroff X)} :
 
 /- warning: alexandroff.ultrafilter_le_nhds_infty -> Alexandroff.ultrafilter_le_nhds_infty is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : Ultrafilter.{u1} (Alexandroff.{u1} X)}, Iff (LE.le.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Preorder.toLE.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.partialOrder.{u1} (Alexandroff.{u1} X)))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (HasLiftT.mk.{succ u1, succ u1} (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (CoeTCₓ.coe.{succ u1, succ u1} (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (Ultrafilter.Filter.hasCoeT.{u1} (Alexandroff.{u1} X)))) f) (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.topologicalSpace.{u1} X _inst_1) (Alexandroff.infty.{u1} X))) (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (IsCompact.{u1} X _inst_1 s) -> (Not (Membership.Mem.{u1, u1} (Set.{u1} (Alexandroff.{u1} X)) (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Ultrafilter.hasMem.{u1} (Alexandroff.{u1} X)) (Set.image.{u1, u1} X (Alexandroff.{u1} X) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) X (Alexandroff.{u1} X) (HasLiftT.mk.{succ u1, succ u1} X (Alexandroff.{u1} X) (CoeTCₓ.coe.{succ u1, succ u1} X (Alexandroff.{u1} X) (Alexandroff.hasCoeT.{u1} X)))) s) f)))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : Ultrafilter.{u1} (Alexandroff.{u1} X)}, Iff (LE.le.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Preorder.toHasLe.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.partialOrder.{u1} (Alexandroff.{u1} X)))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (HasLiftT.mk.{succ u1, succ u1} (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (CoeTCₓ.coe.{succ u1, succ u1} (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Filter.{u1} (Alexandroff.{u1} X)) (Ultrafilter.Filter.hasCoeT.{u1} (Alexandroff.{u1} X)))) f) (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.topologicalSpace.{u1} X _inst_1) (Alexandroff.infty.{u1} X))) (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (IsCompact.{u1} X _inst_1 s) -> (Not (Membership.Mem.{u1, u1} (Set.{u1} (Alexandroff.{u1} X)) (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Ultrafilter.hasMem.{u1} (Alexandroff.{u1} X)) (Set.image.{u1, u1} X (Alexandroff.{u1} X) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) X (Alexandroff.{u1} X) (HasLiftT.mk.{succ u1, succ u1} X (Alexandroff.{u1} X) (CoeTCₓ.coe.{succ u1, succ u1} X (Alexandroff.{u1} X) (Alexandroff.hasCoeT.{u1} X)))) s) f)))
 but is expected to have type
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : Ultrafilter.{u1} (Alexandroff.{u1} X)}, Iff (LE.le.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Preorder.toLE.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (PartialOrder.toPreorder.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.instPartialOrderFilter.{u1} (Alexandroff.{u1} X)))) (Ultrafilter.toFilter.{u1} (Alexandroff.{u1} X) f) (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (Alexandroff.infty.{u1} X))) (forall (s : Set.{u1} X), (IsClosed.{u1} X _inst_1 s) -> (IsCompact.{u1} X _inst_1 s) -> (Not (Membership.mem.{u1, u1} (Set.{u1} (Alexandroff.{u1} X)) (Ultrafilter.{u1} (Alexandroff.{u1} X)) (Ultrafilter.instMembershipSetUltrafilter.{u1} (Alexandroff.{u1} X)) (Set.image.{u1, u1} X (Alexandroff.{u1} X) (Alexandroff.some.{u1} X) s) f)))
 Case conversion may be inaccurate. Consider using '#align alexandroff.ultrafilter_le_nhds_infty Alexandroff.ultrafilter_le_nhds_inftyₓ'. -/

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

@@ -262,7 +262,7 @@ instance : TopologicalSpace (Alexandroff X)
     refine' ⟨_, hs.inter ht⟩
     rintro ⟨hms', hmt'⟩
     simpa [compl_inter] using (hms hms').union (hmt hmt')
-  isOpen_unionₛ S ho :=
+  isOpen_sUnion S ho :=
     by
     suffices IsOpen (coe ⁻¹' ⋃₀ S : Set X) by
       refine' ⟨_, this⟩
@@ -270,7 +270,7 @@ instance : TopologicalSpace (Alexandroff X)
       refine' isCompact_of_isClosed_subset ((ho s hsS).1 hs) this.is_closed_compl _
       exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS)
     rw [preimage_sUnion]
-    exact isOpen_bunionᵢ fun s hs => (ho s hs).2
+    exact isOpen_biUnion fun s hs => (ho s hs).2
 
 variable {s : Set (Alexandroff X)} {t : Set X}
 

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

@@ -769,7 +769,7 @@ instance : CompactSpace (Alexandroff X)
       by
       rw [nhds_infty_eq]
       exact (tendsto_map.mono_left cocompact_le_coclosed_compact).mono_right le_sup_left
-    convert ← this.is_compact_insert_range_of_cocompact continuous_coe
+    convert← this.is_compact_insert_range_of_cocompact continuous_coe
     exact insert_none_range_some X
 
 /-- The one point compactification of a `t0_space` space is a `t0_space`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -818,7 +818,7 @@ instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (Alexandroff
 lean 3 declaration is
   forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : Infinite.{succ u1} X] [_inst_3 : DiscreteTopology.{u1} X _inst_1], Not (Continuous.{u1, u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X) (CofiniteTopology.topologicalSpace.{u1} (Alexandroff.{u1} X)) (Alexandroff.topologicalSpace.{u1} X _inst_1) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (fun (_x : Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) => (CofiniteTopology.{u1} (Alexandroff.{u1} X)) -> (Alexandroff.{u1} X)) (Equiv.hasCoeToFun.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (Equiv.symm.{succ u1, succ u1} (Alexandroff.{u1} X) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (CofiniteTopology.of.{u1} (Alexandroff.{u1} X)))))
 but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : Infinite.{succ u1} X] [_inst_3 : DiscreteTopology.{u1} X _inst_1], Not (Continuous.{u1, u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X) (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (fun (_x : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => Alexandroff.{u1} X) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (Equiv.symm.{succ u1, succ u1} (Alexandroff.{u1} X) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (CofiniteTopology.of.{u1} (Alexandroff.{u1} X)))))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_2 : Infinite.{succ u1} X] [_inst_3 : DiscreteTopology.{u1} X _inst_1], Not (Continuous.{u1, u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X) (CofiniteTopology.instTopologicalSpaceCofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (fun (_x : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : CofiniteTopology.{u1} (Alexandroff.{u1} X)) => Alexandroff.{u1} X) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (Alexandroff.{u1} X)) (Equiv.symm.{succ u1, succ u1} (Alexandroff.{u1} X) (CofiniteTopology.{u1} (Alexandroff.{u1} X)) (CofiniteTopology.of.{u1} (Alexandroff.{u1} X)))))
 Case conversion may be inaccurate. Consider using '#align alexandroff.not_continuous_cofinite_topology_of_symm Alexandroff.not_continuous_cofiniteTopology_of_symmₓ'. -/
 /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
 `cofinite_topology (alexandroff X)` to `alexandroff X` is not continuous. -/

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

@@ -545,9 +545,9 @@ instance (priority := 900) nhdsWithin_compl_neBot [∀ x : X, NeBot (𝓝[≠] x
 
 /- warning: alexandroff.nhds_infty_eq -> Alexandroff.nhds_infty_eq is a dubious translation:
 lean 3 declaration is
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X], Eq.{succ u1} (Filter.{u1} (Alexandroff.{u1} X)) (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.topologicalSpace.{u1} X _inst_1) (Alexandroff.infty.{u1} X)) (HasSup.sup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.completeLattice.{u1} (Alexandroff.{u1} X)))))) (Filter.map.{u1, u1} X (Alexandroff.{u1} X) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) X (Alexandroff.{u1} X) (HasLiftT.mk.{succ u1, succ u1} X (Alexandroff.{u1} X) (CoeTCₓ.coe.{succ u1, succ u1} X (Alexandroff.{u1} X) (Alexandroff.hasCoeT.{u1} X)))) (Filter.coclosedCompact.{u1} X _inst_1)) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} (Alexandroff.{u1} X) (Alexandroff.infty.{u1} X)))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X], Eq.{succ u1} (Filter.{u1} (Alexandroff.{u1} X)) (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.topologicalSpace.{u1} X _inst_1) (Alexandroff.infty.{u1} X)) (Sup.sup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.completeLattice.{u1} (Alexandroff.{u1} X)))))) (Filter.map.{u1, u1} X (Alexandroff.{u1} X) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) X (Alexandroff.{u1} X) (HasLiftT.mk.{succ u1, succ u1} X (Alexandroff.{u1} X) (CoeTCₓ.coe.{succ u1, succ u1} X (Alexandroff.{u1} X) (Alexandroff.hasCoeT.{u1} X)))) (Filter.coclosedCompact.{u1} X _inst_1)) (Pure.pure.{u1, u1} Filter.{u1} Filter.hasPure.{u1} (Alexandroff.{u1} X) (Alexandroff.infty.{u1} X)))
 but is expected to have type
-  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X], Eq.{succ u1} (Filter.{u1} (Alexandroff.{u1} X)) (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (Alexandroff.infty.{u1} X)) (HasSup.sup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (SemilatticeSup.toHasSup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.instCompleteLatticeFilter.{u1} (Alexandroff.{u1} X)))))) (Filter.map.{u1, u1} X (Alexandroff.{u1} X) (Alexandroff.some.{u1} X) (Filter.coclosedCompact.{u1} X _inst_1)) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} (Alexandroff.{u1} X) (Alexandroff.infty.{u1} X)))
+  forall {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X], Eq.{succ u1} (Filter.{u1} (Alexandroff.{u1} X)) (nhds.{u1} (Alexandroff.{u1} X) (Alexandroff.instTopologicalSpaceAlexandroff.{u1} X _inst_1) (Alexandroff.infty.{u1} X)) (Sup.sup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (SemilatticeSup.toSup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Lattice.toSemilatticeSup.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (ConditionallyCompleteLattice.toLattice.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Alexandroff.{u1} X)) (Filter.instCompleteLatticeFilter.{u1} (Alexandroff.{u1} X)))))) (Filter.map.{u1, u1} X (Alexandroff.{u1} X) (Alexandroff.some.{u1} X) (Filter.coclosedCompact.{u1} X _inst_1)) (Pure.pure.{u1, u1} Filter.{u1} Filter.instPureFilter.{u1} (Alexandroff.{u1} X) (Alexandroff.infty.{u1} X)))
 Case conversion may be inaccurate. Consider using '#align alexandroff.nhds_infty_eq Alexandroff.nhds_infty_eqₓ'. -/
 theorem nhds_infty_eq : 𝓝 (∞ : Alexandroff X) = map coe (coclosedCompact X) ⊔ pure ∞ := by
   rw [← nhds_within_compl_infty_eq, nhdsWithin_compl_singleton_sup_pure]

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

Generalize coclosedCompact_eq_cocompact and relativelyCompact.

@@ -394,12 +394,12 @@ theorem denseEmbedding_coe [NoncompactSpace X] : DenseEmbedding ((↑) : X → O
   { openEmbedding_coe with dense := denseRange_coe }
 #align alexandroff.dense_embedding_coe OnePoint.denseEmbedding_coe
 
-@[simp]
+@[simp, norm_cast]
 theorem specializes_coe {x y : X} : (x : OnePoint X) ⤳ y ↔ x ⤳ y :=
   openEmbedding_coe.toInducing.specializes_iff
 #align alexandroff.specializes_coe OnePoint.specializes_coe
 
-@[simp]
+@[simp, norm_cast]
 theorem inseparable_coe {x y : X} : Inseparable (x : OnePoint X) y ↔ Inseparable x y :=
   openEmbedding_coe.toInducing.inseparable_iff
 #align alexandroff.inseparable_coe OnePoint.inseparable_coe
@@ -457,21 +457,24 @@ instance [T1Space X] : T1Space (OnePoint X) where
     · rw [← image_singleton, isClosed_image_coe]
       exact ⟨isClosed_singleton, isCompact_singleton⟩
 
-/-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
-(hence, Hausdorff and regular) topological space. -/
-instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
-  have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
+/-- The one point compactification of a locally compact R₁ space is a normal topological space. -/
+instance [LocallyCompactSpace X] [R1Space X] : NormalSpace (OnePoint X) := by
+  suffices R1Space (OnePoint X) by infer_instance
+  have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z ↦ by
     rw [nhds_infty_eq, disjoint_sup_right, nhds_coe_eq, coclosedCompact_eq_cocompact,
       disjoint_map coe_injective, ← principal_singleton, disjoint_principal_right, compl_infty]
     exact ⟨disjoint_nhds_cocompact z, range_mem_map⟩
-  suffices T2Space (OnePoint X) by infer_instance
-  refine t2Space_iff_disjoint_nhds.2 fun x y hxy => ?_
+  refine ⟨fun x y ↦ ?_⟩
   induction x using OnePoint.rec <;> induction y using OnePoint.rec
-  · exact (hxy rfl).elim
-  · exact (key _).symm
-  · exact key _
-  · rwa [nhds_coe_eq, nhds_coe_eq, disjoint_map coe_injective, disjoint_nhds_nhds,
-      ← coe_injective.ne_iff]
+  · exact .inl le_rfl
+  · exact .inr (key _).symm
+  · exact .inr (key _)
+  · rw [nhds_coe_eq, nhds_coe_eq, disjoint_map coe_injective, specializes_coe]
+    apply specializes_or_disjoint_nhds
+
+/-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
+(hence, Hausdorff and regular) topological space. -/
+example [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := inferInstance
 
 /-- If `X` is not a compact space, then `OnePoint X` is a connected space. -/
 instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (OnePoint X) where

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

@@ -273,7 +273,7 @@ theorem openEmbedding_coe : OpenEmbedding ((↑) : X → OnePoint X) :=
 #align alexandroff.open_embedding_coe OnePoint.openEmbedding_coe
 
 theorem isOpen_range_coe : IsOpen (range ((↑) : X → OnePoint X)) :=
-  openEmbedding_coe.open_range
+  openEmbedding_coe.isOpen_range
 #align alexandroff.is_open_range_coe OnePoint.isOpen_range_coe
 
 theorem isClosed_infty : IsClosed ({∞} : Set (OnePoint X)) := by

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

@@ -94,12 +94,12 @@ theorem coe_eq_coe {x y : X} : (x : OnePoint X) = y ↔ x = y :=
 
 @[simp]
 theorem coe_ne_infty (x : X) : (x : OnePoint X) ≠ ∞ :=
-  fun.
+  nofun
 #align alexandroff.coe_ne_infty OnePoint.coe_ne_infty
 
 @[simp]
 theorem infty_ne_coe (x : X) : ∞ ≠ (x : OnePoint X) :=
-  fun.
+  nofun
 #align alexandroff.infty_ne_coe OnePoint.infty_ne_coe
 
 /-- Recursor for `OnePoint` using the preferred forms `∞` and `↑x`. -/

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".

@@ -114,7 +114,7 @@ theorem isCompl_range_coe_infty : IsCompl (range ((↑) : X → OnePoint X)) {
   isCompl_range_some_none X
 #align alexandroff.is_compl_range_coe_infty OnePoint.isCompl_range_coe_infty
 
--- porting note: moved @[simp] to a new lemma
+-- Porting note: moved @[simp] to a new lemma
 theorem range_coe_union_infty : range ((↑) : X → OnePoint X) ∪ {∞} = univ :=
   range_some_union_none X
 #align alexandroff.range_coe_union_infty OnePoint.range_coe_union_infty

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

This follows on from #6964.

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

@@ -464,7 +464,7 @@ instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
     rw [nhds_infty_eq, disjoint_sup_right, nhds_coe_eq, coclosedCompact_eq_cocompact,
       disjoint_map coe_injective, ← principal_singleton, disjoint_principal_right, compl_infty]
     exact ⟨disjoint_nhds_cocompact z, range_mem_map⟩
-  suffices : T2Space (OnePoint X); infer_instance
+  suffices T2Space (OnePoint X) by infer_instance
   refine t2Space_iff_disjoint_nhds.2 fun x y hxy => ?_
   induction x using OnePoint.rec <;> induction y using OnePoint.rec
   · exact (hxy rfl).elim

As discussed on Zulip.

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

@@ -196,7 +196,7 @@ instance : TopologicalSpace (OnePoint X) where
     suffices IsOpen ((↑) ⁻¹' ⋃₀ S : Set X) by
       refine' ⟨_, this⟩
       rintro ⟨s, hsS : s ∈ S, hs : ∞ ∈ s⟩
-      refine' isCompact_of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl _
+      refine' IsCompact.of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl _
       exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS)
     rw [preimage_sUnion]
     exact isOpen_biUnion fun s hs => (ho s hs).2

• Rename NormalSpace to T4Space.
• Add NormalSpace, a version without the T1Space assumption.
• Adjust some theorems.
• Supersedes thus closes #6892.
• Add some instance cycles, see #2030

@@ -457,14 +457,14 @@ instance [T1Space X] : T1Space (OnePoint X) where
     · rw [← image_singleton, isClosed_image_coe]
       exact ⟨isClosed_singleton, isCompact_singleton⟩
 
-/-- The one point compactification of a locally compact Hausdorff space is a normal (hence,
-Hausdorff and regular) topological space. -/
-instance [LocallyCompactSpace X] [T2Space X] : NormalSpace (OnePoint X) := by
+/-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
+(hence, Hausdorff and regular) topological space. -/
+instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
   have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
     rw [nhds_infty_eq, disjoint_sup_right, nhds_coe_eq, coclosedCompact_eq_cocompact,
       disjoint_map coe_injective, ← principal_singleton, disjoint_principal_right, compl_infty]
     exact ⟨disjoint_nhds_cocompact z, range_mem_map⟩
-  suffices : T2Space (OnePoint X); exact normalOfCompactT2
+  suffices : T2Space (OnePoint X); infer_instance
   refine t2Space_iff_disjoint_nhds.2 fun x y hxy => ?_
   induction x using OnePoint.rec <;> induction y using OnePoint.rec
   · exact (hxy rfl).elim

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

This has nice performance benefits.

@@ -46,10 +46,10 @@ In this section we define `OnePoint X` to be the disjoint union of `X` and `∞`
 -/
 
 
-variable {X : Type _}
+variable {X : Type*}
 
 /-- The OnePoint extension of an arbitrary topological space `X` -/
-def OnePoint (X : Type _) :=
+def OnePoint (X : Type*) :=
   Option X
 #align alexandroff OnePoint
 
@@ -104,7 +104,7 @@ theorem infty_ne_coe (x : X) : ∞ ≠ (x : OnePoint X) :=
 
 /-- Recursor for `OnePoint` using the preferred forms `∞` and `↑x`. -/
 @[elab_as_elim]
-protected def rec {C : OnePoint X → Sort _} (h₁ : C ∞) (h₂ : ∀ x : X, C x) :
+protected def rec {C : OnePoint X → Sort*} (h₁ : C ∞) (h₂ : ∀ x : X, C x) :
     ∀ z : OnePoint X, C z
   | ∞ => h₁
   | (x : X) => h₂ x
@@ -354,12 +354,12 @@ theorem ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
     Ultrafilter.compl_mem_iff_not_mem]
 #align alexandroff.ultrafilter_le_nhds_infty OnePoint.ultrafilter_le_nhds_infty
 
-theorem tendsto_nhds_infty' {α : Type _} {f : OnePoint X → α} {l : Filter α} :
+theorem tendsto_nhds_infty' {α : Type*} {f : OnePoint X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ (↑)) (coclosedCompact X) l := by
   simp [nhds_infty_eq, and_comm]
 #align alexandroff.tendsto_nhds_infty' OnePoint.tendsto_nhds_infty'
 
-theorem tendsto_nhds_infty {α : Type _} {f : OnePoint X → α} {l : Filter α} :
+theorem tendsto_nhds_infty {α : Type*} {f : OnePoint X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔
       ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
   tendsto_nhds_infty'.trans <| by
@@ -367,18 +367,18 @@ theorem tendsto_nhds_infty {α : Type _} {f : OnePoint X → α} {l : Filter α}
       and_assoc, exists_prop]
 #align alexandroff.tendsto_nhds_infty OnePoint.tendsto_nhds_infty
 
-theorem continuousAt_infty' {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} :
+theorem continuousAt_infty' {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
     ContinuousAt f ∞ ↔ Tendsto (f ∘ (↑)) (coclosedCompact X) (𝓝 (f ∞)) :=
   tendsto_nhds_infty'.trans <| and_iff_right (tendsto_pure_nhds _ _)
 #align alexandroff.continuous_at_infty' OnePoint.continuousAt_infty'
 
-theorem continuousAt_infty {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} :
+theorem continuousAt_infty {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
     ContinuousAt f ∞ ↔
       ∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
   continuousAt_infty'.trans <| by simp only [hasBasis_coclosedCompact.tendsto_left_iff, and_assoc]
 #align alexandroff.continuous_at_infty OnePoint.continuousAt_infty
 
-theorem continuousAt_coe {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
+theorem continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
     ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x := by
   rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]; rfl
 #align alexandroff.continuous_at_coe OnePoint.continuousAt_coe

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Yourong Zang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yourong Zang, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.alexandroff
-! leanprover-community/mathlib commit dc6c365e751e34d100e80fe6e314c3c3e0fd2988
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Fintype.Option
 import Mathlib.Topology.Separation
 import Mathlib.Topology.Sets.Opens
 
+#align_import topology.alexandroff from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
+
 /-!
 # The OnePoint Compactification
 

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

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

@@ -494,7 +494,7 @@ theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X
 
 end OnePoint
 
-/-- A concrete counterexample shows that  `Continuous.homeoOfEquivCompactToT2`
+/-- A concrete counterexample shows that `Continuous.homeoOfEquivCompactToT2`
 cannot be generalized from `T2Space` to `T1Space`.
 
 Let `α = OnePoint ℕ` be the one-point compactification of `ℕ`, and let `β` be the same space

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

@@ -132,7 +132,7 @@ theorem range_coe_inter_infty : range ((↑) : X → OnePoint X) ∩ {∞} = ∅
 #align alexandroff.range_coe_inter_infty OnePoint.range_coe_inter_infty
 
 @[simp]
-theorem compl_range_coe : range ((↑) : X → OnePoint X)ᶜ = {∞} :=
+theorem compl_range_coe : (range ((↑) : X → OnePoint X))ᶜ = {∞} :=
   compl_range_some X
 #align alexandroff.compl_range_coe OnePoint.compl_range_coe
 
@@ -187,7 +187,7 @@ that `(↑)` has dense range, so it is a dense embedding.
 variable [TopologicalSpace X]
 
 instance : TopologicalSpace (OnePoint X) where
-  IsOpen s := (∞ ∈ s → IsCompact ((((↑) : X → OnePoint X) ⁻¹' s)ᶜ)) ∧
+  IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
     IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
   isOpen_univ := by simp
   isOpen_inter s t := by
@@ -207,17 +207,17 @@ instance : TopologicalSpace (OnePoint X) where
 variable {s : Set (OnePoint X)} {t : Set X}
 
 theorem isOpen_def :
-    IsOpen s ↔ (∞ ∈ s → IsCompact (((↑) ⁻¹' s : Set X)ᶜ)) ∧ IsOpen ((↑) ⁻¹' s : Set X) :=
+    IsOpen s ↔ (∞ ∈ s → IsCompact ((↑) ⁻¹' s : Set X)ᶜ) ∧ IsOpen ((↑) ⁻¹' s : Set X) :=
   Iff.rfl
 #align alexandroff.is_open_def OnePoint.isOpen_def
 
 theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
-    IsOpen s ↔ IsCompact (((↑) ⁻¹' s : Set X)ᶜ) ∧ IsOpen ((↑) ⁻¹' s : Set X) := by
+    IsOpen s ↔ IsCompact ((↑) ⁻¹' s : Set X)ᶜ ∧ IsOpen ((↑) ⁻¹' s : Set X) := by
   simp [isOpen_def, h]
 #align alexandroff.is_open_iff_of_mem' OnePoint.isOpen_iff_of_mem'
 
 theorem isOpen_iff_of_mem (h : ∞ ∈ s) :
-    IsOpen s ↔ IsClosed (((↑) ⁻¹' s : Set X)ᶜ) ∧ IsCompact (((↑) ⁻¹' s : Set X)ᶜ) := by
+    IsOpen s ↔ IsClosed ((↑) ⁻¹' s : Set X)ᶜ ∧ IsCompact ((↑) ⁻¹' s : Set X)ᶜ := by
   simp only [isOpen_iff_of_mem' h, isClosed_compl_iff, and_comm]
 #align alexandroff.is_open_iff_of_mem OnePoint.isOpen_iff_of_mem
 
@@ -241,7 +241,7 @@ theorem isOpen_image_coe {s : Set X} : IsOpen ((↑) '' s : Set (OnePoint X)) 
 #align alexandroff.is_open_image_coe OnePoint.isOpen_image_coe
 
 theorem isOpen_compl_image_coe {s : Set X} :
-    IsOpen (((↑) '' s : Set (OnePoint X))ᶜ) ↔ IsClosed s ∧ IsCompact s := by
+    IsOpen ((↑) '' s : Set (OnePoint X))ᶜ ↔ IsClosed s ∧ IsCompact s := by
   rw [isOpen_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective]
   exact infty_not_mem_image_coe
 #align alexandroff.is_open_compl_image_coe OnePoint.isOpen_compl_image_coe
@@ -364,7 +364,7 @@ theorem tendsto_nhds_infty' {α : Type _} {f : OnePoint X → α} {l : Filter α
 
 theorem tendsto_nhds_infty {α : Type _} {f : OnePoint X → α} {l : Filter α} :
     Tendsto f (𝓝 ∞) l ↔
-      ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) (tᶜ) s :=
+      ∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
   tendsto_nhds_infty'.trans <| by
     simp only [tendsto_pure_left, hasBasis_coclosedCompact.tendsto_left_iff, forall_and,
       and_assoc, exists_prop]
@@ -377,7 +377,7 @@ theorem continuousAt_infty' {Y : Type _} [TopologicalSpace Y] {f : OnePoint X 
 
 theorem continuousAt_infty {Y : Type _} [TopologicalSpace Y] {f : OnePoint X → Y} :
     ContinuousAt f ∞ ↔
-      ∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) (tᶜ) s :=
+      ∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
   continuousAt_infty'.trans <| by simp only [hasBasis_coclosedCompact.tendsto_left_iff, and_assoc]
 #align alexandroff.continuous_at_infty OnePoint.continuousAt_infty
 

We rename Alexandroff to Compactification.OnePoint to avoid future name conflicts (with e.g. Alexandroff topological spaces). See this zulip thread.

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>

@@ -195,14 +195,14 @@ instance : TopologicalSpace (Alexandroff X) where
     refine' ⟨_, hs.inter ht⟩
     rintro ⟨hms', hmt'⟩
     simpa [compl_inter] using (hms hms').union (hmt hmt')
-  isOpen_unionₛ S ho := by
+  isOpen_sUnion S ho := by
     suffices IsOpen ((↑) ⁻¹' ⋃₀ S : Set X) by
       refine' ⟨_, this⟩
       rintro ⟨s, hsS : s ∈ S, hs : ∞ ∈ s⟩
       refine' isCompact_of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl _
-      exact compl_subset_compl.mpr (preimage_mono <| subset_unionₛ_of_mem hsS)
-    rw [preimage_unionₛ]
-    exact isOpen_bunionᵢ fun s hs => (ho s hs).2
+      exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS)
+    rw [preimage_sUnion]
+    exact isOpen_biUnion fun s hs => (ho s hs).2
 
 variable {s : Set (Alexandroff X)} {t : Set X}