topology.paracompact ⟷ Mathlib.Topology.Compactness.Paracompact

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

@@ -85,7 +85,7 @@ theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a,
     ParacompactSpace.locallyFinite_refinement (range u) coe
       (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
   simp only [SetCoe.exists, Subtype.coe_mk, exists_range_iff', Union_eq_univ_iff, exists_prop] at
-    this 
+    this
   choose α t hto hXt htf ind hind; choose t_inv ht_inv using hXt; choose U hxU hU using htf
   -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
   refine' ⟨fun i => ⋃ (a : α) (ha : ind a = i), t a, _, _, _, _⟩
@@ -113,7 +113,7 @@ theorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s
       _ with
     ⟨v, vo, vc, vf, vu⟩
   refine' ⟨v ∘ some, fun i => vo _, _, vf.comp_injective (Option.some_injective _), fun i => vu _⟩
-  · simp only [Union_option, ← compl_subset_iff_union] at vc 
+  · simp only [Union_option, ← compl_subset_iff_union] at vc
     exact subset.trans (subset_compl_comm.1 <| vu Option.none) vc
   · simpa only [Union_option, Option.elim', ← compl_subset_iff_union, compl_compl]
 #align precise_refinement_set precise_refinement_set
@@ -127,7 +127,7 @@ instance (priority := 100) paracompact_of_compact [CompactSpace X] : Paracompact
   -- the proof is trivial: we choose a finite subcover using compactness, and use it
   refine' ⟨fun ι s ho hu => _⟩
   rcases is_compact_univ.elim_finite_subcover _ ho hu.ge with ⟨T, hT⟩
-  have := hT; simp only [subset_def, mem_Union] at this 
+  have := hT; simp only [subset_def, mem_Union] at this
   choose i hiT hi using fun x => this x (mem_univ x)
   refine'
     ⟨(T : Set ι), fun t => s t, fun t => ho _, _, locallyFinite_of_finite _, fun t =>

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

@@ -163,16 +163,53 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
     ∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
       (∀ a, c a ∈ s ∧ p (c a) (r a)) ∧
         (s ⊆ ⋃ a, B (c a) (r a)) ∧ LocallyFinite fun a => B (c a) (r a) :=
-  by classical
+  by
+  classical
+  -- For technical reasons we prepend two empty sets to the sequence `compact_exhaustion.choice X`
+  set K' : CompactExhaustion X := CompactExhaustion.choice X
+  set K : CompactExhaustion X := K'.shiftr.shiftr
+  set Kdiff := fun n => K (n + 1) \ interior (K n)
+  -- Now we restate some properties of `compact_exhaustion` for `K`/`Kdiff`
+  have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) :=
+    by
+    intro x
+    simpa only [K'.find_shiftr] using
+      diff_subset_diff_right interior_subset (K'.shiftr.mem_diff_shiftr_find x)
+  have Kdiffc : ∀ n, IsCompact (Kdiff n ∩ s) := fun n =>
+    ((K.is_compact _).diffₓ isOpen_interior).inter_right hs
+  -- Next we choose a finite covering `B (c n i) (r n i)` of each
+  -- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
+  have : ∀ (n) (x : Kdiff (n + 1) ∩ s), K nᶜ ∈ 𝓝 (x : X) := fun n x =>
+    IsOpen.mem_nhds (K.is_closed n).isOpen_compl fun hx' => x.2.1.2 <| K.subset_interior_succ _ hx'
+  haveI : ∀ (n) (x : Kdiff n ∩ s), Nonempty (ι x) := fun n x => (hB x x.2.2).Nonempty
+  choose! r hrp hr using fun n (x : Kdiff (n + 1) ∩ s) => (hB x x.2.2).mem_iff.1 (this n x)
+  have hxr : ∀ (n x) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x := fun n x hx =>
+    (hB x hx.2).mem_of_mem (hrp _ ⟨x, hx⟩)
+  choose T hT using fun n => (Kdiffc (n + 1)).elim_nhds_subcover' _ (hxr n)
+  set T' : ∀ n, Set ↥(Kdiff (n + 1) ∩ s) := fun n => T n
+  -- Finally, we take the union of all these coverings
+  refine' ⟨Σ n, T' n, fun a => a.2, fun a => r a.1 a.2, _, _, _⟩
+  · rintro ⟨n, x, hx⟩; exact ⟨x.2.2, hrp _ _⟩
+  · refine' fun x hx => mem_Union.2 _
+    rcases mem_Union₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
+    exact ⟨⟨_, ⟨c, hc⟩, hcT⟩, hcx⟩
+  · intro x
+    refine'
+      ⟨interior (K (K'.find x + 3)),
+        IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
+    have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σ n, T' n)).Finite :=
+      (finite_le_nat _).biUnion fun k hk => finite_range _
+    apply this.subset; rintro ⟨k, c, hc⟩
+    simp only [mem_Union, mem_set_of_eq, mem_image, Subtype.coe_mk]
+    rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩
+    refine' ⟨k, _, ⟨c, hc⟩, rfl⟩
+    have := (mem_compl_iff _ _).1 (hr k c hxB)
+    contrapose! this with hnk
+    exact K.subset hnk (interior_subset hxK)
 #align refinement_of_locally_compact_sigma_compact_of_nhds_basis_set refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set
 -/
 
 #print refinement_of_locallyCompact_sigmaCompact_of_nhds_basis /-
--- For technical reasons we prepend two empty sets to the sequence `compact_exhaustion.choice X`
--- Now we restate some properties of `compact_exhaustion` for `K`/`Kdiff`
--- Next we choose a finite covering `B (c n i) (r n i)` of each
--- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
--- Finally, we take the union of all these coverings
 /-- Let `X` be a locally compact sigma compact Hausdorff topological space. Suppose that for each
 `x` the sets `B x : ι x → set X` with the predicate `p x : ι x → Prop` form a basis of the filter
 `𝓝 x`. Then there exists a locally finite covering `λ i, B (c i) (r i)` of `X` such that each `r i`

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

@@ -163,53 +163,16 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
     ∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
       (∀ a, c a ∈ s ∧ p (c a) (r a)) ∧
         (s ⊆ ⋃ a, B (c a) (r a)) ∧ LocallyFinite fun a => B (c a) (r a) :=
-  by
-  classical
-  -- For technical reasons we prepend two empty sets to the sequence `compact_exhaustion.choice X`
-  set K' : CompactExhaustion X := CompactExhaustion.choice X
-  set K : CompactExhaustion X := K'.shiftr.shiftr
-  set Kdiff := fun n => K (n + 1) \ interior (K n)
-  -- Now we restate some properties of `compact_exhaustion` for `K`/`Kdiff`
-  have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) :=
-    by
-    intro x
-    simpa only [K'.find_shiftr] using
-      diff_subset_diff_right interior_subset (K'.shiftr.mem_diff_shiftr_find x)
-  have Kdiffc : ∀ n, IsCompact (Kdiff n ∩ s) := fun n =>
-    ((K.is_compact _).diffₓ isOpen_interior).inter_right hs
-  -- Next we choose a finite covering `B (c n i) (r n i)` of each
-  -- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
-  have : ∀ (n) (x : Kdiff (n + 1) ∩ s), K nᶜ ∈ 𝓝 (x : X) := fun n x =>
-    IsOpen.mem_nhds (K.is_closed n).isOpen_compl fun hx' => x.2.1.2 <| K.subset_interior_succ _ hx'
-  haveI : ∀ (n) (x : Kdiff n ∩ s), Nonempty (ι x) := fun n x => (hB x x.2.2).Nonempty
-  choose! r hrp hr using fun n (x : Kdiff (n + 1) ∩ s) => (hB x x.2.2).mem_iff.1 (this n x)
-  have hxr : ∀ (n x) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x := fun n x hx =>
-    (hB x hx.2).mem_of_mem (hrp _ ⟨x, hx⟩)
-  choose T hT using fun n => (Kdiffc (n + 1)).elim_nhds_subcover' _ (hxr n)
-  set T' : ∀ n, Set ↥(Kdiff (n + 1) ∩ s) := fun n => T n
-  -- Finally, we take the union of all these coverings
-  refine' ⟨Σ n, T' n, fun a => a.2, fun a => r a.1 a.2, _, _, _⟩
-  · rintro ⟨n, x, hx⟩; exact ⟨x.2.2, hrp _ _⟩
-  · refine' fun x hx => mem_Union.2 _
-    rcases mem_Union₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
-    exact ⟨⟨_, ⟨c, hc⟩, hcT⟩, hcx⟩
-  · intro x
-    refine'
-      ⟨interior (K (K'.find x + 3)),
-        IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
-    have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σ n, T' n)).Finite :=
-      (finite_le_nat _).biUnion fun k hk => finite_range _
-    apply this.subset; rintro ⟨k, c, hc⟩
-    simp only [mem_Union, mem_set_of_eq, mem_image, Subtype.coe_mk]
-    rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩
-    refine' ⟨k, _, ⟨c, hc⟩, rfl⟩
-    have := (mem_compl_iff _ _).1 (hr k c hxB)
-    contrapose! this with hnk
-    exact K.subset hnk (interior_subset hxK)
+  by classical
 #align refinement_of_locally_compact_sigma_compact_of_nhds_basis_set refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set
 -/
 
 #print refinement_of_locallyCompact_sigmaCompact_of_nhds_basis /-
+-- For technical reasons we prepend two empty sets to the sequence `compact_exhaustion.choice X`
+-- Now we restate some properties of `compact_exhaustion` for `K`/`Kdiff`
+-- Next we choose a finite covering `B (c n i) (r n i)` of each
+-- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
+-- Finally, we take the union of all these coverings
 /-- Let `X` be a locally compact sigma compact Hausdorff topological space. Suppose that for each
 `x` the sets `B x : ι x → set X` with the predicate `p x : ι x → Prop` form a basis of the filter
 `𝓝 x`. Then there exists a locally finite covering `λ i, B (c i) (r i)` of `X` such that each `r i`

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

@@ -3,9 +3,9 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Reid Barton, Yury Kudryashov
 -/
-import Mathbin.Topology.SubsetProperties
-import Mathbin.Topology.Separation
-import Mathbin.Data.Option.Basic
+import Topology.SubsetProperties
+import Topology.Separation
+import Data.Option.Basic
 
 #align_import topology.paracompact from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
 

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

@@ -256,10 +256,10 @@ instance (priority := 100) paracompact_of_locallyCompact_sigmaCompact [LocallyCo
 #align paracompact_of_locally_compact_sigma_compact paracompact_of_locallyCompact_sigmaCompact
 -/
 
-#print normal_of_paracompact_t2 /-
+#print T4Space.of_paracompactSpace_t2Space /-
 /- Dieudonné‘s theorem: a paracompact Hausdorff space is normal. Formalization is based on the proof
 at [ncatlab](https://ncatlab.org/nlab/show/paracompact+Hausdorff+spaces+are+normal). -/
-theorem normal_of_paracompact_t2 [T2Space X] [ParacompactSpace X] : NormalSpace X :=
+theorem T4Space.of_paracompactSpace_t2Space [T2Space X] [ParacompactSpace X] : NormalSpace X :=
   by
   -- First we show how to go from points to a set on one side.
   have :
@@ -289,6 +289,6 @@ theorem normal_of_paracompact_t2 [T2Space X] [ParacompactSpace X] : NormalSpace
   · exact ⟨u, v, hu, hv, singleton_subset_iff.1 hxu, htv, huv.symm⟩
   · simp_rw [singleton_subset_iff]
     exact t2_separation (hst.symm.ne_of_mem hy hx)
-#align normal_of_paracompact_t2 normal_of_paracompact_t2
+#align normal_of_paracompact_t2 T4Space.of_paracompactSpace_t2Space
 -/
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Reid Barton, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.paracompact
-! leanprover-community/mathlib commit 0a0ec35061ed9960bf0e7ffb0335f44447b58977
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.SubsetProperties
 import Mathbin.Topology.Separation
 import Mathbin.Data.Option.Basic
 
+#align_import topology.paracompact from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
+
 /-!
 # Paracompact topological spaces
 

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

@@ -168,48 +168,47 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
         (s ⊆ ⋃ a, B (c a) (r a)) ∧ LocallyFinite fun a => B (c a) (r a) :=
   by
   classical
-    -- For technical reasons we prepend two empty sets to the sequence `compact_exhaustion.choice X`
-    set K' : CompactExhaustion X := CompactExhaustion.choice X
-    set K : CompactExhaustion X := K'.shiftr.shiftr
-    set Kdiff := fun n => K (n + 1) \ interior (K n)
-    -- Now we restate some properties of `compact_exhaustion` for `K`/`Kdiff`
-    have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) :=
-      by
-      intro x
-      simpa only [K'.find_shiftr] using
-        diff_subset_diff_right interior_subset (K'.shiftr.mem_diff_shiftr_find x)
-    have Kdiffc : ∀ n, IsCompact (Kdiff n ∩ s) := fun n =>
-      ((K.is_compact _).diffₓ isOpen_interior).inter_right hs
-    -- Next we choose a finite covering `B (c n i) (r n i)` of each
-    -- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
-    have : ∀ (n) (x : Kdiff (n + 1) ∩ s), K nᶜ ∈ 𝓝 (x : X) := fun n x =>
-      IsOpen.mem_nhds (K.is_closed n).isOpen_compl fun hx' =>
-        x.2.1.2 <| K.subset_interior_succ _ hx'
-    haveI : ∀ (n) (x : Kdiff n ∩ s), Nonempty (ι x) := fun n x => (hB x x.2.2).Nonempty
-    choose! r hrp hr using fun n (x : Kdiff (n + 1) ∩ s) => (hB x x.2.2).mem_iff.1 (this n x)
-    have hxr : ∀ (n x) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x := fun n x hx =>
-      (hB x hx.2).mem_of_mem (hrp _ ⟨x, hx⟩)
-    choose T hT using fun n => (Kdiffc (n + 1)).elim_nhds_subcover' _ (hxr n)
-    set T' : ∀ n, Set ↥(Kdiff (n + 1) ∩ s) := fun n => T n
-    -- Finally, we take the union of all these coverings
-    refine' ⟨Σ n, T' n, fun a => a.2, fun a => r a.1 a.2, _, _, _⟩
-    · rintro ⟨n, x, hx⟩; exact ⟨x.2.2, hrp _ _⟩
-    · refine' fun x hx => mem_Union.2 _
-      rcases mem_Union₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
-      exact ⟨⟨_, ⟨c, hc⟩, hcT⟩, hcx⟩
-    · intro x
-      refine'
-        ⟨interior (K (K'.find x + 3)),
-          IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
-      have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σ n, T' n)).Finite :=
-        (finite_le_nat _).biUnion fun k hk => finite_range _
-      apply this.subset; rintro ⟨k, c, hc⟩
-      simp only [mem_Union, mem_set_of_eq, mem_image, Subtype.coe_mk]
-      rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩
-      refine' ⟨k, _, ⟨c, hc⟩, rfl⟩
-      have := (mem_compl_iff _ _).1 (hr k c hxB)
-      contrapose! this with hnk
-      exact K.subset hnk (interior_subset hxK)
+  -- For technical reasons we prepend two empty sets to the sequence `compact_exhaustion.choice X`
+  set K' : CompactExhaustion X := CompactExhaustion.choice X
+  set K : CompactExhaustion X := K'.shiftr.shiftr
+  set Kdiff := fun n => K (n + 1) \ interior (K n)
+  -- Now we restate some properties of `compact_exhaustion` for `K`/`Kdiff`
+  have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) :=
+    by
+    intro x
+    simpa only [K'.find_shiftr] using
+      diff_subset_diff_right interior_subset (K'.shiftr.mem_diff_shiftr_find x)
+  have Kdiffc : ∀ n, IsCompact (Kdiff n ∩ s) := fun n =>
+    ((K.is_compact _).diffₓ isOpen_interior).inter_right hs
+  -- Next we choose a finite covering `B (c n i) (r n i)` of each
+  -- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
+  have : ∀ (n) (x : Kdiff (n + 1) ∩ s), K nᶜ ∈ 𝓝 (x : X) := fun n x =>
+    IsOpen.mem_nhds (K.is_closed n).isOpen_compl fun hx' => x.2.1.2 <| K.subset_interior_succ _ hx'
+  haveI : ∀ (n) (x : Kdiff n ∩ s), Nonempty (ι x) := fun n x => (hB x x.2.2).Nonempty
+  choose! r hrp hr using fun n (x : Kdiff (n + 1) ∩ s) => (hB x x.2.2).mem_iff.1 (this n x)
+  have hxr : ∀ (n x) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x := fun n x hx =>
+    (hB x hx.2).mem_of_mem (hrp _ ⟨x, hx⟩)
+  choose T hT using fun n => (Kdiffc (n + 1)).elim_nhds_subcover' _ (hxr n)
+  set T' : ∀ n, Set ↥(Kdiff (n + 1) ∩ s) := fun n => T n
+  -- Finally, we take the union of all these coverings
+  refine' ⟨Σ n, T' n, fun a => a.2, fun a => r a.1 a.2, _, _, _⟩
+  · rintro ⟨n, x, hx⟩; exact ⟨x.2.2, hrp _ _⟩
+  · refine' fun x hx => mem_Union.2 _
+    rcases mem_Union₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
+    exact ⟨⟨_, ⟨c, hc⟩, hcT⟩, hcx⟩
+  · intro x
+    refine'
+      ⟨interior (K (K'.find x + 3)),
+        IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
+    have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σ n, T' n)).Finite :=
+      (finite_le_nat _).biUnion fun k hk => finite_range _
+    apply this.subset; rintro ⟨k, c, hc⟩
+    simp only [mem_Union, mem_set_of_eq, mem_image, Subtype.coe_mk]
+    rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩
+    refine' ⟨k, _, ⟨c, hc⟩, rfl⟩
+    have := (mem_compl_iff _ _).1 (hr k c hxB)
+    contrapose! this with hnk
+    exact K.subset hnk (interior_subset hxK)
 #align refinement_of_locally_compact_sigma_compact_of_nhds_basis_set refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set
 -/
 

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

@@ -69,7 +69,7 @@ finite refinement `t : α → set X` indexed on the same type such that each `
 class ParacompactSpace (X : Type v) [TopologicalSpace X] : Prop where
   locallyFinite_refinement :
     ∀ (α : Type v) (s : α → Set X) (ho : ∀ a, IsOpen (s a)) (hc : (⋃ a, s a) = univ),
-      ∃ (β : Type v)(t : β → Set X)(ho : ∀ b, IsOpen (t b))(hc : (⋃ b, t b) = univ),
+      ∃ (β : Type v) (t : β → Set X) (ho : ∀ b, IsOpen (t b)) (hc : (⋃ b, t b) = univ),
         LocallyFinite t ∧ ∀ b, ∃ a, t b ⊆ s a
 #align paracompact_space ParacompactSpace
 -/
@@ -88,7 +88,7 @@ theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a,
     ParacompactSpace.locallyFinite_refinement (range u) coe
       (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
   simp only [SetCoe.exists, Subtype.coe_mk, exists_range_iff', Union_eq_univ_iff, exists_prop] at
-    this
+    this 
   choose α t hto hXt htf ind hind; choose t_inv ht_inv using hXt; choose U hxU hU using htf
   -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
   refine' ⟨fun i => ⋃ (a : α) (ha : ind a = i), t a, _, _, _, _⟩
@@ -116,7 +116,7 @@ theorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s
       _ with
     ⟨v, vo, vc, vf, vu⟩
   refine' ⟨v ∘ some, fun i => vo _, _, vf.comp_injective (Option.some_injective _), fun i => vu _⟩
-  · simp only [Union_option, ← compl_subset_iff_union] at vc
+  · simp only [Union_option, ← compl_subset_iff_union] at vc 
     exact subset.trans (subset_compl_comm.1 <| vu Option.none) vc
   · simpa only [Union_option, Option.elim', ← compl_subset_iff_union, compl_compl]
 #align precise_refinement_set precise_refinement_set
@@ -130,7 +130,7 @@ instance (priority := 100) paracompact_of_compact [CompactSpace X] : Paracompact
   -- the proof is trivial: we choose a finite subcover using compactness, and use it
   refine' ⟨fun ι s ho hu => _⟩
   rcases is_compact_univ.elim_finite_subcover _ ho hu.ge with ⟨T, hT⟩
-  have := hT; simp only [subset_def, mem_Union] at this
+  have := hT; simp only [subset_def, mem_Union] at this 
   choose i hiT hi using fun x => this x (mem_univ x)
   refine'
     ⟨(T : Set ι), fun t => s t, fun t => ho _, _, locallyFinite_of_finite _, fun t =>
@@ -163,7 +163,7 @@ to choose `α = X`. This fact is not yet formalized in `mathlib`. -/
 theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyCompactSpace X]
     [SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
     {s : Set X} (hs : IsClosed s) (hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)) :
-    ∃ (α : Type v)(c : α → X)(r : ∀ a, ι (c a)),
+    ∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
       (∀ a, c a ∈ s ∧ p (c a) (r a)) ∧
         (s ⊆ ⋃ a, B (c a) (r a)) ∧ LocallyFinite fun a => B (c a) (r a) :=
   by
@@ -192,7 +192,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
     choose T hT using fun n => (Kdiffc (n + 1)).elim_nhds_subcover' _ (hxr n)
     set T' : ∀ n, Set ↥(Kdiff (n + 1) ∩ s) := fun n => T n
     -- Finally, we take the union of all these coverings
-    refine' ⟨Σn, T' n, fun a => a.2, fun a => r a.1 a.2, _, _, _⟩
+    refine' ⟨Σ n, T' n, fun a => a.2, fun a => r a.1 a.2, _, _, _⟩
     · rintro ⟨n, x, hx⟩; exact ⟨x.2.2, hrp _ _⟩
     · refine' fun x hx => mem_Union.2 _
       rcases mem_Union₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
@@ -201,7 +201,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
       refine'
         ⟨interior (K (K'.find x + 3)),
           IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
-      have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σn, T' n)).Finite :=
+      have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σ n, T' n)).Finite :=
         (finite_le_nat _).biUnion fun k hk => finite_range _
       apply this.subset; rintro ⟨k, c, hc⟩
       simp only [mem_Union, mem_set_of_eq, mem_image, Subtype.coe_mk]
@@ -236,7 +236,7 @@ to choose `α = X`. This fact is not yet formalized in `mathlib`. -/
 theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis [LocallyCompactSpace X]
     [SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
     (hB : ∀ x, (𝓝 x).HasBasis (p x) (B x)) :
-    ∃ (α : Type v)(c : α → X)(r : ∀ a, ι (c a)),
+    ∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
       (∀ a, p (c a) (r a)) ∧ (⋃ a, B (c a) (r a)) = univ ∧ LocallyFinite fun a => B (c a) (r a) :=
   let ⟨α, c, r, hp, hU, hfin⟩ :=
     refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set isClosed_univ fun x _ => hB x

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

@@ -56,7 +56,7 @@ compact space, paracompact space, locally finite covering
 
 open Set Filter Function
 
-open Filter Topology
+open scoped Filter Topology
 
 universe u v
 

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

@@ -193,8 +193,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
     set T' : ∀ n, Set ↥(Kdiff (n + 1) ∩ s) := fun n => T n
     -- Finally, we take the union of all these coverings
     refine' ⟨Σn, T' n, fun a => a.2, fun a => r a.1 a.2, _, _, _⟩
-    · rintro ⟨n, x, hx⟩
-      exact ⟨x.2.2, hrp _ _⟩
+    · rintro ⟨n, x, hx⟩; exact ⟨x.2.2, hrp _ _⟩
     · refine' fun x hx => mem_Union.2 _
       rcases mem_Union₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
       exact ⟨⟨_, ⟨c, hc⟩, hcT⟩, hcx⟩
@@ -204,8 +203,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
           IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
       have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σn, T' n)).Finite :=
         (finite_le_nat _).biUnion fun k hk => finite_range _
-      apply this.subset
-      rintro ⟨k, c, hc⟩
+      apply this.subset; rintro ⟨k, c, hc⟩
       simp only [mem_Union, mem_set_of_eq, mem_image, Subtype.coe_mk]
       rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩
       refine' ⟨k, _, ⟨c, hc⟩, rfl⟩
@@ -278,7 +276,7 @@ theorem normal_of_paracompact_t2 [T2Space X] [ParacompactSpace X] : NormalSpace
     /- For each `x ∈ s` we choose open disjoint `u x ∋ x` and `v x ⊇ t`. The sets `u x` form an
         open covering of `s`. We choose a locally finite refinement `u' : s → set X`, then `⋃ i, u' i`
         and `(closure (⋃ i, u' i))ᶜ` are disjoint open neighborhoods of `s` and `t`. -/
-    intro s t hs ht H
+    intro s t hs ht H;
     choose u v hu hv hxu htv huv using SetCoe.forall'.1 H
     rcases precise_refinement_set hs u hu fun x hx => mem_Union.2 ⟨⟨x, hx⟩, hxu _⟩ with
       ⟨u', hu'o, hcov', hu'fin, hsub⟩

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

@@ -92,7 +92,7 @@ theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a,
   choose α t hto hXt htf ind hind; choose t_inv ht_inv using hXt; choose U hxU hU using htf
   -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
   refine' ⟨fun i => ⋃ (a : α) (ha : ind a = i), t a, _, _, _, _⟩
-  · exact fun a => isOpen_unionᵢ fun a => isOpen_unionᵢ fun ha => hto a
+  · exact fun a => isOpen_iUnion fun a => isOpen_iUnion fun ha => hto a
   · simp only [eq_univ_iff_forall, mem_Union]
     exact fun x => ⟨ind (t_inv x), _, rfl, ht_inv _⟩
   · refine' fun x => ⟨U x, hxU x, ((hU x).image ind).Subset _⟩
@@ -203,7 +203,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
         ⟨interior (K (K'.find x + 3)),
           IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
       have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σn, T' n)).Finite :=
-        (finite_le_nat _).bunionᵢ fun k hk => finite_range _
+        (finite_le_nat _).biUnion fun k hk => finite_range _
       apply this.subset
       rintro ⟨k, c, hc⟩
       simp only [mem_Union, mem_set_of_eq, mem_image, Subtype.coe_mk]
@@ -283,7 +283,7 @@ theorem normal_of_paracompact_t2 [T2Space X] [ParacompactSpace X] : NormalSpace
     rcases precise_refinement_set hs u hu fun x hx => mem_Union.2 ⟨⟨x, hx⟩, hxu _⟩ with
       ⟨u', hu'o, hcov', hu'fin, hsub⟩
     refine'
-      ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_unionᵢ hu'o, is_closed_closure.is_open_compl, hcov',
+      ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_iUnion hu'o, is_closed_closure.is_open_compl, hcov',
         _, disjoint_compl_right.mono le_rfl (compl_le_compl subset_closure)⟩
     rw [hu'fin.closure_Union, compl_Union, subset_Inter_iff]
     refine' fun i x hxt hxu =>

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

All of these changes appear to be oversights to me.

@@ -287,7 +287,7 @@ instance (priority := 100) paracompact_of_locallyCompact_sigmaCompact [WeaklyLoc
   exact ⟨β, t, fun x ↦ (hto x).1.2, htc, htf, fun b ↦ ⟨i <| c b, (hto b).2⟩⟩
 #align paracompact_of_locally_compact_sigma_compact paracompact_of_locallyCompact_sigmaCompact
 
-/- **Dieudonné's theorem**: a paracompact Hausdorff space is normal.
+/-- **Dieudonné's theorem**: a paracompact Hausdorff space is normal.
 Formalization is based on the proof
 at [ncatlab](https://ncatlab.org/nlab/show/paracompact+Hausdorff+spaces+are+normal). -/
 instance (priority := 100) T4Space.of_paracompactSpace_t2Space [T2Space X] [ParacompactSpace X] :

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

@@ -119,7 +119,7 @@ theorem ClosedEmbedding.paracompactSpace [ParacompactSpace Y] {e : X → Y} (he
     simp only [← hU] at hu ⊢
     have heU : range e ⊆ ⋃ i, U i := by
       simpa only [range_subset_iff, mem_iUnion, iUnion_eq_univ_iff] using hu
-    rcases precise_refinement_set he.closed_range U hUo heU with ⟨V, hVo, heV, hVf, hVU⟩
+    rcases precise_refinement_set he.isClosed_range U hUo heU with ⟨V, hVo, heV, hVf, hVU⟩
     refine ⟨α, fun a ↦ e ⁻¹' (V a), fun a ↦ (hVo a).preimage he.continuous, ?_,
       hVf.preimage_continuous he.continuous, fun a ↦ ⟨a, preimage_mono (hVU a)⟩⟩
     simpa only [range_subset_iff, mem_iUnion, iUnion_eq_univ_iff] using heV

... as a @[deprecated] alias

Also add dates to some theorems deprecated in #10816, deprecate Set.exists_range_iff', and use simp-normal form in the definition of ParacompactSpace.

@@ -61,9 +61,9 @@ finite refinement `t : α → Set X` indexed on the same type such that each `
 class ParacompactSpace (X : Type v) [TopologicalSpace X] : Prop where
   /-- Every open cover of a paracompact space assumes a locally finite refinement. -/
   locallyFinite_refinement :
-    ∀ (α : Type v) (s : α → Set X) (_ : ∀ a, IsOpen (s a)) (_ : ⋃ a, s a = univ),
-      ∃ (β : Type v) (t : β → Set X) (_ : ∀ b, IsOpen (t b)) (_ : ⋃ b, t b = univ),
-        LocallyFinite t ∧ ∀ b, ∃ a, t b ⊆ s a
+    ∀ (α : Type v) (s : α → Set X), (∀ a, IsOpen (s a)) → (⋃ a, s a = univ) →
+      ∃ (β : Type v) (t : β → Set X),
+        (∀ b, IsOpen (t b)) ∧ (⋃ b, t b = univ) ∧ LocallyFinite t ∧ ∀ b, ∃ a, t b ⊆ s a
 #align paracompact_space ParacompactSpace
 
 variable {ι : Type u} {X : Type v} {Y : Type w} [TopologicalSpace X] [TopologicalSpace Y]
@@ -75,8 +75,8 @@ theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a,
     LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
   -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
   have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
-    (SetCoe.forall.2 <| forall_mem_range.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
-  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
+    (forall_subtype_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
+  simp only [exists_subtype_range_iff, iUnion_eq_univ_iff] at this
   choose α t hto hXt htf ind hind using this
   choose t_inv ht_inv using hXt
   choose U hxU hU using htf

ball for "bounded forall" and bex for "bounded exists" are from experience very confusing abbreviations. This PR renames them to forall_mem and exists_mem in the few Set lemma names that mention them.

Also deprecate ball_image_of_ball, mem_image_elim, mem_image_elim_on since those lemmas are duplicates of the renamed lemmas (apart from argument order and implicitness, which I am also fixing by making the binder in the RHS of forall_mem_image semi-implicit), have obscure names and are completely unused.

@@ -75,7 +75,7 @@ theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a,
     LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
   -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
   have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
-    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
+    (SetCoe.forall.2 <| forall_mem_range.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
   simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
   choose α t hto hXt htf ind hind using this
   choose t_inv ht_inv using hXt

Classifies by adding issue number (#10888) to porting notes claiming added proof.

@@ -99,7 +99,7 @@ indexed by the same type. -/
 theorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s) (u : ι → Set X)
     (uo : ∀ i, IsOpen (u i)) (us : s ⊆ ⋃ i, u i) :
     ∃ v : ι → Set X, (∀ i, IsOpen (v i)) ∧ (s ⊆ ⋃ i, v i) ∧ LocallyFinite v ∧ ∀ i, v i ⊆ u i := by
-  -- Porting note: Added proof of uc
+  -- Porting note (#10888): added proof of uc
   have uc : (iUnion fun i => Option.elim' sᶜ u i) = univ := by
     apply Subset.antisymm (subset_univ _)
     · simp_rw [← compl_union_self s, Option.elim', iUnion_option]

• 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

@@ -287,17 +287,18 @@ instance (priority := 100) paracompact_of_locallyCompact_sigmaCompact [WeaklyLoc
   exact ⟨β, t, fun x ↦ (hto x).1.2, htc, htf, fun b ↦ ⟨i <| c b, (hto b).2⟩⟩
 #align paracompact_of_locally_compact_sigma_compact paracompact_of_locallyCompact_sigmaCompact
 
-/- Dieudonné's theorem: a paracompact Hausdorff space is normal. Formalization is based on the proof
+/- **Dieudonné's theorem**: a paracompact Hausdorff space is normal.
+Formalization is based on the proof
 at [ncatlab](https://ncatlab.org/nlab/show/paracompact+Hausdorff+spaces+are+normal). -/
-theorem normal_of_paracompact_t2 [T2Space X] [ParacompactSpace X] : NormalSpace X := by
+instance (priority := 100) T4Space.of_paracompactSpace_t2Space [T2Space X] [ParacompactSpace X] :
+    T4Space X := by
   -- First we show how to go from points to a set on one side.
-  have : ∀ s t : Set X, IsClosed s → IsClosed t →
+  have : ∀ s t : Set X, IsClosed s →
       (∀ x ∈ s, ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
-      ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v := by
+      ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v := fun s t hs H ↦ by
     /- For each `x ∈ s` we choose open disjoint `u x ∋ x` and `v x ⊇ t`. The sets `u x` form an
         open covering of `s`. We choose a locally finite refinement `u' : s → Set X`, then
         `⋃ i, u' i` and `(closure (⋃ i, u' i))ᶜ` are disjoint open neighborhoods of `s` and `t`. -/
-    intro s t hs _ H
     choose u v hu hv hxu htv huv using SetCoe.forall'.1 H
     rcases precise_refinement_set hs u hu fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, hxu _⟩ with
       ⟨u', hu'o, hcov', hu'fin, hsub⟩
@@ -308,10 +309,10 @@ theorem normal_of_paracompact_t2 [T2Space X] [ParacompactSpace X] : NormalSpace
       absurd (htv i hxt) (closure_minimal _ (isClosed_compl_iff.2 <| hv _) hxu)
     exact fun y hyu hyv ↦ (huv i).le_bot ⟨hsub _ hyu, hyv⟩
   -- Now we apply the lemma twice: first to `s` and `t`, then to `t` and each point of `s`.
-  refine' ⟨fun s t hs ht hst ↦ this s t hs ht fun x hx ↦ _⟩
-  rcases this t {x} ht isClosed_singleton fun y hy ↦ (by
+  refine { normal := fun s t hs ht hst ↦ this s t hs fun x hx ↦ ?_ }
+  rcases this t {x} ht fun y hy ↦ (by
     simp_rw [singleton_subset_iff]
     exact t2_separation (hst.symm.ne_of_mem hy hx))
     with ⟨v, u, hv, hu, htv, hxu, huv⟩
   exact ⟨u, v, hu, hv, singleton_subset_iff.1 hxu, htv, huv.symm⟩
-#align normal_of_paracompact_t2 normal_of_paracompact_t2
+#align normal_of_paracompact_t2 T4Space.of_paracompactSpace_t2Space

@@ -34,7 +34,7 @@ We also prove the following facts.
 * Every paracompact Hausdorff space is normal. This statement is not an instance to avoid loops in
   the instance graph.
 
-* Every `EMetricSpace` is a paracompact space, see instance `EMetricSpace.ParacompactSpace` in
+* Every `EMetricSpace` is a paracompact space, see instance `EMetric.instParacompactSpace` in
   `Topology/EMetricSpace/Paracompact`.
 
 ## TODO

@@ -193,7 +193,7 @@ dealing with a covering of the whole space.
 
 In most cases (namely, if `B c r ∪ B c r'` is again a set of the form `B c r''`) it is possible
 to choose `α = X`. This fact is not yet formalized in `mathlib`. -/
-theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyCompactSpace X]
+theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [WeaklyLocallyCompactSpace X]
     [SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
     {s : Set X} (hs : IsClosed s) (hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)) :
     ∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
@@ -263,7 +263,7 @@ dealing with a covering of a closed set.
 
 In most cases (namely, if `B c r ∪ B c r'` is again a set of the form `B c r''`) it is possible
 to choose `α = X`. This fact is not yet formalized in `mathlib`. -/
-theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis [LocallyCompactSpace X]
+theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis [WeaklyLocallyCompactSpace X]
     [SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
     (hB : ∀ x, (𝓝 x).HasBasis (p x) (B x)) :
     ∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
@@ -276,7 +276,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis [LocallyCompactS
 -- See note [lower instance priority]
 /-- A locally compact sigma compact Hausdorff space is paracompact. See also
 `refinement_of_locallyCompact_sigmaCompact_of_nhds_basis` for a more precise statement. -/
-instance (priority := 100) paracompact_of_locallyCompact_sigmaCompact [LocallyCompactSpace X]
+instance (priority := 100) paracompact_of_locallyCompact_sigmaCompact [WeaklyLocallyCompactSpace X]
     [SigmaCompactSpace X] [T2Space X] : ParacompactSpace X := by
   refine' ⟨fun α s ho hc ↦ _⟩
   choose i hi using iUnion_eq_univ_iff.1 hc

Later I'm going to split files like Lipschitz into 2: one in EMetricSpace/ and one in MetricSpace/.

@@ -35,7 +35,7 @@ We also prove the following facts.
   the instance graph.
 
 * Every `EMetricSpace` is a paracompact space, see instance `EMetricSpace.ParacompactSpace` in
-  `Topology/MetricSpace/EMetricParacompact`.
+  `Topology/EMetricSpace/Paracompact`.
 
 ## TODO
 

@@ -205,8 +205,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
     set K : CompactExhaustion X := K'.shiftr.shiftr
     set Kdiff := fun n ↦ K (n + 1) \ interior (K n)
     -- Now we restate some properties of `CompactExhaustion` for `K`/`Kdiff`
-    have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) := by
-      intro x
+    have hKcov : ∀ x, x ∈ Kdiff (K'.find x + 1) := fun x ↦ by
       simpa only [K'.find_shiftr] using
         diff_subset_diff_right interior_subset (K'.shiftr.mem_diff_shiftr_find x)
     have Kdiffc : ∀ n, IsCompact (Kdiff n ∩ s) :=
@@ -214,8 +213,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
     -- Next we choose a finite covering `B (c n i) (r n i)` of each
     -- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
     have : ∀ (n) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 (x : X) :=
-      fun n x ↦ IsOpen.mem_nhds (K.isClosed n).isOpen_compl
-        fun hx' ↦ x.2.1.2 <| K.subset_interior_succ _ hx'
+      fun n x ↦ (K.isClosed n).compl_mem_nhds fun hx' ↦ x.2.1.2 <| K.subset_interior_succ _ hx'
     -- Porting note: Commented out `haveI` for now.
     --haveI : ∀ (n) (x : ↑(Kdiff n ∩ s)), Nonempty (ι x) := fun n x ↦ (hB x x.2.2).nonempty
     choose! r hrp hr using fun n (x : ↑(Kdiff (n + 1) ∩ s)) ↦ (hB x x.2.2).mem_iff.1 (this n x)

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Reid Barton, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.paracompact
-! leanprover-community/mathlib commit 2705404e701abc6b3127da906f40bae062a169c9
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.Homeomorph
 import Mathlib.Data.Option.Basic
 
+#align_import topology.paracompact from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
+
 /-!
 # Paracompact topological spaces
 

@@ -64,8 +64,8 @@ finite refinement `t : α → Set X` indexed on the same type such that each `
 class ParacompactSpace (X : Type v) [TopologicalSpace X] : Prop where
   /-- Every open cover of a paracompact space assumes a locally finite refinement. -/
   locallyFinite_refinement :
-    ∀ (α : Type v) (s : α → Set X) (_ : ∀ a, IsOpen (s a)) (_ : (⋃ a, s a) = univ),
-      ∃ (β : Type v) (t : β → Set X) (_ : ∀ b, IsOpen (t b)) (_ : (⋃ b, t b) = univ),
+    ∀ (α : Type v) (s : α → Set X) (_ : ∀ a, IsOpen (s a)) (_ : ⋃ a, s a = univ),
+      ∃ (β : Type v) (t : β → Set X) (_ : ∀ b, IsOpen (t b)) (_ : ⋃ b, t b = univ),
         LocallyFinite t ∧ ∀ b, ∃ a, t b ⊆ s a
 #align paracompact_space ParacompactSpace
 
@@ -74,7 +74,7 @@ variable {ι : Type u} {X : Type v} {Y : Type w} [TopologicalSpace X] [Topologic
 /-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
 one indexed on the same type with each open set contained in the corresponding original one. -/
 theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
-    (uc : (⋃ i, u i) = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ (⋃ i, v i) = univ ∧
+    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧
     LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
   -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
   have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
@@ -237,7 +237,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
       refine'
         ⟨interior (K (K'.find x + 3)),
           IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
-      have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σn, T' n)).Finite :=
+      have : (⋃ k ≤ K'.find x + 2, range (Sigma.mk k) : Set (Σn, T' n)).Finite :=
         (finite_le_nat _).biUnion fun k _ ↦ finite_range _
       apply this.subset
       rintro ⟨k, c, hc⟩
@@ -272,7 +272,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis [LocallyCompactS
     [SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
     (hB : ∀ x, (𝓝 x).HasBasis (p x) (B x)) :
     ∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
-      (∀ a, p (c a) (r a)) ∧ (⋃ a, B (c a) (r a)) = univ ∧ LocallyFinite fun a ↦ B (c a) (r a) :=
+      (∀ a, p (c a) (r a)) ∧ ⋃ a, B (c a) (r a) = univ ∧ LocallyFinite fun a ↦ B (c a) (r a) :=
   let ⟨α, c, r, hp, hU, hfin⟩ :=
     refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set isClosed_univ fun x _ ↦ hB x
   ⟨α, c, r, fun a ↦ (hp a).2, univ_subset_iff.1 hU, hfin⟩

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

@@ -103,11 +103,11 @@ theorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s
     (uo : ∀ i, IsOpen (u i)) (us : s ⊆ ⋃ i, u i) :
     ∃ v : ι → Set X, (∀ i, IsOpen (v i)) ∧ (s ⊆ ⋃ i, v i) ∧ LocallyFinite v ∧ ∀ i, v i ⊆ u i := by
   -- Porting note: Added proof of uc
-  have uc : (iUnion fun i => Option.elim' (sᶜ) u i) = univ := by
+  have uc : (iUnion fun i => Option.elim' sᶜ u i) = univ := by
     apply Subset.antisymm (subset_univ _)
     · simp_rw [← compl_union_self s, Option.elim', iUnion_option]
-      apply union_subset_union_right (sᶜ) us
-  rcases precise_refinement (Option.elim' (sᶜ) u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)
+      apply union_subset_union_right sᶜ us
+  rcases precise_refinement (Option.elim' sᶜ u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)
       uc with
     ⟨v, vo, vc, vf, vu⟩
   refine' ⟨v ∘ some, fun i ↦ vo _, _, vf.comp_injective (Option.some_injective _), fun i ↦ vu _⟩
@@ -216,7 +216,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
       fun n ↦ ((K.isCompact _).diff isOpen_interior).inter_right hs
     -- Next we choose a finite covering `B (c n i) (r n i)` of each
     -- `Kdiff (n + 1) ∩ s` such that `B (c n i) (r n i) ∩ s` is disjoint with `K n`
-    have : ∀ (n) (x : ↑(Kdiff (n + 1) ∩ s)), K nᶜ ∈ 𝓝 (x : X) :=
+    have : ∀ (n) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 (x : X) :=
       fun n x ↦ IsOpen.mem_nhds (K.isClosed n).isOpen_compl
         fun hx' ↦ x.2.1.2 <| K.subset_interior_succ _ hx'
     -- Porting note: Commented out `haveI` for now.
@@ -306,7 +306,7 @@ theorem normal_of_paracompact_t2 [T2Space X] [ParacompactSpace X] : NormalSpace
     choose u v hu hv hxu htv huv using SetCoe.forall'.1 H
     rcases precise_refinement_set hs u hu fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, hxu _⟩ with
       ⟨u', hu'o, hcov', hu'fin, hsub⟩
-    refine' ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_iUnion hu'o, isClosed_closure.isOpen_compl,
+    refine' ⟨⋃ i, u' i, (closure (⋃ i, u' i))ᶜ, isOpen_iUnion hu'o, isClosed_closure.isOpen_compl,
       hcov', _, disjoint_compl_right.mono le_rfl (compl_le_compl subset_closure)⟩
     rw [hu'fin.closure_iUnion, compl_iUnion, subset_iInter_iff]
     refine' fun i x hxt hxu ↦

These are just three cases where a mathported proof had to be changed, due to unwanted behaviour with push_neg/contradiction. Since #5082 is supposed to fix some issues with these tactics, some of these workarounds can be removed.

Note: this was not in any way systematic, I simply grepped the line after a contra, then grepped for a not and then selected 3 likely candidates. There are potentially more situations like these.

@@ -245,10 +245,8 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
       rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩
       refine' ⟨k, _, ⟨c, hc⟩, rfl⟩
       have := (mem_compl_iff _ _).1 (hr k c hxB)
-      revert this
-      contrapose!
-      simp only [ge_iff_le, not_le]
-      exact fun hnk ↦ K.subset hnk (interior_subset hxK)
+      contrapose! this with hnk
+      exact K.subset hnk (interior_subset hxK)
 #align refinement_of_locally_compact_sigma_compact_of_nhds_basis_set refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set
 
 /-- Let `X` be a locally compact sigma compact Hausdorff topological space. Suppose that for each

I wrote a script to find lines that contain an odd number of backticks

@@ -30,7 +30,7 @@ We also prove the following facts.
 
 * A locally compact sigma compact Hausdorff space is paracompact, see instance
   `paracompact_of_locallyCompact_sigmaCompact`. Moreover, we can choose a locally finite
-  refinement with sets in a given collection of filter bases of `𝓝 x, `x : X`, see
+  refinement with sets in a given collection of filter bases of `𝓝 x`, `x : X`, see
   `refinement_of_locallyCompact_sigmaCompact_of_nhds_basis`. For example, in a proper metric space
   every open covering `⋃ i, s i` admits a refinement `⋃ i, Metric.ball (c i) (r i)`.
 

@@ -84,7 +84,7 @@ theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a,
   choose t_inv ht_inv using hXt
   choose U hxU hU using htf
   -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
-  refine' ⟨fun i ↦ ⋃ (a : α) (_ha : ind a = i), t a, _, _, _, _⟩
+  refine' ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, _, _, _, _⟩
   · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
   · simp only [eq_univ_iff_forall, mem_iUnion]
     exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩

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>

@@ -78,21 +78,21 @@ theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a,
     LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
   -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
   have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
-    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← unionₛ_range, Subtype.range_coe])
-  simp only [SetCoe.exists, exists_range_iff', unionᵢ_eq_univ_iff, exists_prop] at this
+    (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])
+  simp only [SetCoe.exists, exists_range_iff', iUnion_eq_univ_iff, exists_prop] at this
   choose α t hto hXt htf ind hind using this
   choose t_inv ht_inv using hXt
   choose U hxU hU using htf
   -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`
   refine' ⟨fun i ↦ ⋃ (a : α) (_ha : ind a = i), t a, _, _, _, _⟩
-  · exact fun a ↦ isOpen_unionᵢ fun a ↦ isOpen_unionᵢ fun _ ↦ hto a
-  · simp only [eq_univ_iff_forall, mem_unionᵢ]
+  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a
+  · simp only [eq_univ_iff_forall, mem_iUnion]
     exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩
   · refine' fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset _⟩
-    simp only [subset_def, mem_unionᵢ, mem_setOf_eq, Set.Nonempty, mem_inter_iff]
+    simp only [subset_def, mem_iUnion, mem_setOf_eq, Set.Nonempty, mem_inter_iff]
     rintro i ⟨y, ⟨a, rfl, hya⟩, hyU⟩
     exact mem_image_of_mem _ ⟨y, hya, hyU⟩
-  · simp only [subset_def, mem_unionᵢ]
+  · simp only [subset_def, mem_iUnion]
     rintro i x ⟨a, rfl, hxa⟩
     exact hind _ hxa
 #align precise_refinement precise_refinement
@@ -103,15 +103,15 @@ theorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s
     (uo : ∀ i, IsOpen (u i)) (us : s ⊆ ⋃ i, u i) :
     ∃ v : ι → Set X, (∀ i, IsOpen (v i)) ∧ (s ⊆ ⋃ i, v i) ∧ LocallyFinite v ∧ ∀ i, v i ⊆ u i := by
   -- Porting note: Added proof of uc
-  have uc : (unionᵢ fun i => Option.elim' (sᶜ) u i) = univ := by
+  have uc : (iUnion fun i => Option.elim' (sᶜ) u i) = univ := by
     apply Subset.antisymm (subset_univ _)
-    · simp_rw [← compl_union_self s, Option.elim', unionᵢ_option]
+    · simp_rw [← compl_union_self s, Option.elim', iUnion_option]
       apply union_subset_union_right (sᶜ) us
   rcases precise_refinement (Option.elim' (sᶜ) u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)
       uc with
     ⟨v, vo, vc, vf, vu⟩
   refine' ⟨v ∘ some, fun i ↦ vo _, _, vf.comp_injective (Option.some_injective _), fun i ↦ vu _⟩
-  · simp only [unionᵢ_option, ← compl_subset_iff_union] at vc
+  · simp only [iUnion_option, ← compl_subset_iff_union] at vc
     exact Subset.trans (subset_compl_comm.1 <| vu Option.none) vc
 #align precise_refinement_set precise_refinement_set
 
@@ -121,11 +121,11 @@ theorem ClosedEmbedding.paracompactSpace [ParacompactSpace Y] {e : X → Y} (he
     choose U hUo hU using fun a ↦ he.isOpen_iff.1 (ho a)
     simp only [← hU] at hu ⊢
     have heU : range e ⊆ ⋃ i, U i := by
-      simpa only [range_subset_iff, mem_unionᵢ, unionᵢ_eq_univ_iff] using hu
+      simpa only [range_subset_iff, mem_iUnion, iUnion_eq_univ_iff] using hu
     rcases precise_refinement_set he.closed_range U hUo heU with ⟨V, hVo, heV, hVf, hVU⟩
     refine ⟨α, fun a ↦ e ⁻¹' (V a), fun a ↦ (hVo a).preimage he.continuous, ?_,
       hVf.preimage_continuous he.continuous, fun a ↦ ⟨a, preimage_mono (hVU a)⟩⟩
-    simpa only [range_subset_iff, mem_unionᵢ, unionᵢ_eq_univ_iff] using heV
+    simpa only [range_subset_iff, mem_iUnion, iUnion_eq_univ_iff] using heV
 
 theorem Homeomorph.paracompactSpace_iff (e : X ≃ₜ Y) : ParacompactSpace X ↔ ParacompactSpace Y :=
   ⟨fun _ ↦ e.symm.closedEmbedding.paracompactSpace, fun _ ↦ e.closedEmbedding.paracompactSpace⟩
@@ -140,27 +140,27 @@ instance (priority := 200) [CompactSpace X] [ParacompactSpace Y] : ParacompactSp
   locallyFinite_refinement α s ho hu := by
     have : ∀ (x : X) (y : Y), ∃ (a : α) (U : Set X) (V : Set Y),
         IsOpen U ∧ IsOpen V ∧ x ∈ U ∧ y ∈ V ∧ U ×ˢ V ⊆ s a := fun x y ↦
-      (unionᵢ_eq_univ_iff.1 hu (x, y)).imp fun a ha ↦ isOpen_prod_iff.1 (ho a) x y ha
+      (iUnion_eq_univ_iff.1 hu (x, y)).imp fun a ha ↦ isOpen_prod_iff.1 (ho a) x y ha
     choose a U V hUo hVo hxU hyV hUV using this
     choose T hT using fun y ↦ CompactSpace.elim_nhds_subcover (U · y) fun x ↦
       (hUo x y).mem_nhds (hxU x y)
     set W : Y → Set Y := fun y ↦ ⋂ x ∈ T y, V x y
-    have hWo : ∀ y, IsOpen (W y) := fun y ↦ isOpen_binterᵢ_finset fun _ _ ↦ hVo _ _
-    have hW : ∀ y, y ∈ W y := fun _ ↦ mem_interᵢ₂.2 fun _ _ ↦ hyV _ _
-    rcases precise_refinement W hWo (unionᵢ_eq_univ_iff.2 fun y ↦ ⟨y, hW y⟩)
+    have hWo : ∀ y, IsOpen (W y) := fun y ↦ isOpen_biInter_finset fun _ _ ↦ hVo _ _
+    have hW : ∀ y, y ∈ W y := fun _ ↦ mem_iInter₂.2 fun _ _ ↦ hyV _ _
+    rcases precise_refinement W hWo (iUnion_eq_univ_iff.2 fun y ↦ ⟨y, hW y⟩)
       with ⟨E, hEo, hE, hEf, hEA⟩
     refine ⟨Σ y, T y, fun z ↦ U z.2.1 z.1 ×ˢ E z.1, fun _ ↦ (hUo _ _).prod (hEo _),
-      unionᵢ_eq_univ_iff.2 fun (x, y) ↦ ?_, fun (x, y) ↦ ?_, fun ⟨y, x, hx⟩ ↦ ?_⟩
-    · rcases unionᵢ_eq_univ_iff.1 hE y with ⟨b, hb⟩
-      rcases unionᵢ₂_eq_univ_iff.1 (hT b) x with ⟨a, ha, hx⟩
+      iUnion_eq_univ_iff.2 fun (x, y) ↦ ?_, fun (x, y) ↦ ?_, fun ⟨y, x, hx⟩ ↦ ?_⟩
+    · rcases iUnion_eq_univ_iff.1 hE y with ⟨b, hb⟩
+      rcases iUnion₂_eq_univ_iff.1 (hT b) x with ⟨a, ha, hx⟩
       exact ⟨⟨b, a, ha⟩, hx, hb⟩
     · rcases hEf y with ⟨t, ht, htf⟩
       refine ⟨univ ×ˢ t, prod_mem_nhds univ_mem ht, ?_⟩
-      refine (htf.bunionᵢ fun y _ ↦ finite_range (Sigma.mk y)).subset ?_
+      refine (htf.biUnion fun y _ ↦ finite_range (Sigma.mk y)).subset ?_
       rintro ⟨b, a, ha⟩ ⟨⟨c, d⟩, ⟨-, hd : d ∈ E b⟩, -, hdt : d ∈ t⟩
-      exact mem_unionᵢ₂.2 ⟨b, ⟨d, hd, hdt⟩, mem_range_self _⟩
+      exact mem_iUnion₂.2 ⟨b, ⟨d, hd, hdt⟩, mem_range_self _⟩
     · refine ⟨a x y, (Set.prod_mono Subset.rfl ?_).trans (hUV x y)⟩
-      exact (hEA _).trans (interᵢ₂_subset x hx)
+      exact (hEA _).trans (iInter₂_subset x hx)
 
 instance (priority := 200) [ParacompactSpace X] [CompactSpace Y] : ParacompactSpace (X × Y) :=
   (Homeomorph.prodComm X Y).paracompactSpace_iff.2 inferInstance
@@ -173,7 +173,7 @@ instance (priority := 100) paracompact_of_compact [CompactSpace X] : Paracompact
   rcases isCompact_univ.elim_finite_subcover _ ho hu.ge with ⟨T, hT⟩
   refine' ⟨(T : Set ι), fun t ↦ s t, fun t ↦ ho _, _, locallyFinite_of_finite _,
     fun t ↦ ⟨t, Subset.rfl⟩⟩
-  simpa only [unionᵢ_coe_set, ← univ_subset_iff]
+  simpa only [iUnion_coe_set, ← univ_subset_iff]
 #align paracompact_of_compact paracompact_of_compact
 
 /-- Let `X` be a locally compact sigma compact Hausdorff topological space, let `s` be a closed set
@@ -230,18 +230,18 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [LocallyComp
     refine' ⟨Σn, T' n, fun a ↦ a.2, fun a ↦ r a.1 a.2, _, _, _⟩
     · rintro ⟨n, x, hx⟩
       exact ⟨x.2.2, hrp _ _⟩
-    · refine' fun x hx ↦ mem_unionᵢ.2 _
-      rcases mem_unionᵢ₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
+    · refine' fun x hx ↦ mem_iUnion.2 _
+      rcases mem_iUnion₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩
       exact ⟨⟨_, ⟨c, hc⟩, hcT⟩, hcx⟩
     · intro x
       refine'
         ⟨interior (K (K'.find x + 3)),
           IsOpen.mem_nhds isOpen_interior (K.subset_interior_succ _ (hKcov x).1), _⟩
       have : (⋃ k ≤ K'.find x + 2, range <| Sigma.mk k : Set (Σn, T' n)).Finite :=
-        (finite_le_nat _).bunionᵢ fun k _ ↦ finite_range _
+        (finite_le_nat _).biUnion fun k _ ↦ finite_range _
       apply this.subset
       rintro ⟨k, c, hc⟩
-      simp only [mem_unionᵢ, mem_setOf_eq, mem_image, Subtype.coe_mk]
+      simp only [mem_iUnion, mem_setOf_eq, mem_image, Subtype.coe_mk]
       rintro ⟨x, hxB : x ∈ B c (r k c), hxK⟩
       refine' ⟨k, _, ⟨c, hc⟩, rfl⟩
       have := (mem_compl_iff _ _).1 (hr k c hxB)
@@ -286,7 +286,7 @@ theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis [LocallyCompactS
 instance (priority := 100) paracompact_of_locallyCompact_sigmaCompact [LocallyCompactSpace X]
     [SigmaCompactSpace X] [T2Space X] : ParacompactSpace X := by
   refine' ⟨fun α s ho hc ↦ _⟩
-  choose i hi using unionᵢ_eq_univ_iff.1 hc
+  choose i hi using iUnion_eq_univ_iff.1 hc
   have : ∀ x : X, (𝓝 x).HasBasis (fun t : Set X ↦ (x ∈ t ∧ IsOpen t) ∧ t ⊆ s (i x)) id :=
     fun x : X ↦ (nhds_basis_opens x).restrict_subset (IsOpen.mem_nhds (ho (i x)) (hi x))
   rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis this with
@@ -306,11 +306,11 @@ theorem normal_of_paracompact_t2 [T2Space X] [ParacompactSpace X] : NormalSpace
         `⋃ i, u' i` and `(closure (⋃ i, u' i))ᶜ` are disjoint open neighborhoods of `s` and `t`. -/
     intro s t hs _ H
     choose u v hu hv hxu htv huv using SetCoe.forall'.1 H
-    rcases precise_refinement_set hs u hu fun x hx ↦ mem_unionᵢ.2 ⟨⟨x, hx⟩, hxu _⟩ with
+    rcases precise_refinement_set hs u hu fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, hxu _⟩ with
       ⟨u', hu'o, hcov', hu'fin, hsub⟩
-    refine' ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_unionᵢ hu'o, isClosed_closure.isOpen_compl,
+    refine' ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_iUnion hu'o, isClosed_closure.isOpen_compl,
       hcov', _, disjoint_compl_right.mono le_rfl (compl_le_compl subset_closure)⟩
-    rw [hu'fin.closure_unionᵢ, compl_unionᵢ, subset_interᵢ_iff]
+    rw [hu'fin.closure_iUnion, compl_iUnion, subset_iInter_iff]
     refine' fun i x hxt hxu ↦
       absurd (htv i hxt) (closure_minimal _ (isClosed_compl_iff.2 <| hv _) hxu)
     exact fun y hyu hyv ↦ (huv i).le_bot ⟨hsub _ hyu, hyv⟩

This PR puts, with one exception, every single remaining by that lies all by itself on its own line to the previous line, thus matching the current behaviour of start-port.sh. The exception is when the by begins the second or later argument to a tuple or anonymous constructor; see https://github.com/leanprover-community/mathlib4/pull/3825#discussion_r1186702599.

Essentially this is s/\n *by$/ by/g, but with manual editing to satisfy the linter's max-100-char-line requirement. The Python style linter is also modified to catch these "isolated bys".

@@ -74,9 +74,8 @@ variable {ι : Type u} {X : Type v} {Y : Type w} [TopologicalSpace X] [Topologic
 /-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
 one indexed on the same type with each open set contained in the corresponding original one. -/
 theorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))
-    (uc : (⋃ i, u i) = univ) :
-    ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ (⋃ i, v i) = univ ∧ LocallyFinite v ∧ ∀ a, v a ⊆ u a :=
-  by
+    (uc : (⋃ i, u i) = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ (⋃ i, v i) = univ ∧
+    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by
   -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`
   have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))
     (SetCoe.forall.2 <| forall_range_iff.2 uo) (by rwa [← unionₛ_range, Subtype.range_coe])

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

@@ -8,8 +8,7 @@ Authors: Reid Barton, Yury Kudryashov
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathlib.Topology.SubsetProperties
-import Mathlib.Topology.Separation
+import Mathlib.Topology.Homeomorph
 import Mathlib.Data.Option.Basic
 
 /-!
@@ -55,7 +54,7 @@ open Set Filter Function
 
 open Filter Topology
 
-universe u v
+universe u v w
 
 /-- A topological space is called paracompact, if every open covering of this space admits a locally
 finite refinement. We use the same universe for all types in the definition to avoid creating a
@@ -70,7 +69,7 @@ class ParacompactSpace (X : Type v) [TopologicalSpace X] : Prop where
         LocallyFinite t ∧ ∀ b, ∃ a, t b ⊆ s a
 #align paracompact_space ParacompactSpace
 
-variable {ι : Type u} {X : Type v} [TopologicalSpace X]
+variable {ι : Type u} {X : Type v} {Y : Type w} [TopologicalSpace X] [TopologicalSpace Y]
 
 /-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,
 one indexed on the same type with each open set contained in the corresponding original one. -/
@@ -117,14 +116,62 @@ theorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s
     exact Subset.trans (subset_compl_comm.1 <| vu Option.none) vc
 #align precise_refinement_set precise_refinement_set
 
+theorem ClosedEmbedding.paracompactSpace [ParacompactSpace Y] {e : X → Y} (he : ClosedEmbedding e) :
+    ParacompactSpace X where
+  locallyFinite_refinement α s ho hu := by
+    choose U hUo hU using fun a ↦ he.isOpen_iff.1 (ho a)
+    simp only [← hU] at hu ⊢
+    have heU : range e ⊆ ⋃ i, U i := by
+      simpa only [range_subset_iff, mem_unionᵢ, unionᵢ_eq_univ_iff] using hu
+    rcases precise_refinement_set he.closed_range U hUo heU with ⟨V, hVo, heV, hVf, hVU⟩
+    refine ⟨α, fun a ↦ e ⁻¹' (V a), fun a ↦ (hVo a).preimage he.continuous, ?_,
+      hVf.preimage_continuous he.continuous, fun a ↦ ⟨a, preimage_mono (hVU a)⟩⟩
+    simpa only [range_subset_iff, mem_unionᵢ, unionᵢ_eq_univ_iff] using heV
+
+theorem Homeomorph.paracompactSpace_iff (e : X ≃ₜ Y) : ParacompactSpace X ↔ ParacompactSpace Y :=
+  ⟨fun _ ↦ e.symm.closedEmbedding.paracompactSpace, fun _ ↦ e.closedEmbedding.paracompactSpace⟩
+
+/-- The product of a compact space and a paracompact space is a paracompact space. The formalization
+is based on https://dantopology.wordpress.com/2009/10/24/compact-x-paracompact-is-paracompact/
+with some minor modifications.
+
+This version assumes that `X` in `X × Y` is compact and `Y` is paracompact, see next lemma for the
+other case. -/
+instance (priority := 200) [CompactSpace X] [ParacompactSpace Y] : ParacompactSpace (X × Y) where
+  locallyFinite_refinement α s ho hu := by
+    have : ∀ (x : X) (y : Y), ∃ (a : α) (U : Set X) (V : Set Y),
+        IsOpen U ∧ IsOpen V ∧ x ∈ U ∧ y ∈ V ∧ U ×ˢ V ⊆ s a := fun x y ↦
+      (unionᵢ_eq_univ_iff.1 hu (x, y)).imp fun a ha ↦ isOpen_prod_iff.1 (ho a) x y ha
+    choose a U V hUo hVo hxU hyV hUV using this
+    choose T hT using fun y ↦ CompactSpace.elim_nhds_subcover (U · y) fun x ↦
+      (hUo x y).mem_nhds (hxU x y)
+    set W : Y → Set Y := fun y ↦ ⋂ x ∈ T y, V x y
+    have hWo : ∀ y, IsOpen (W y) := fun y ↦ isOpen_binterᵢ_finset fun _ _ ↦ hVo _ _
+    have hW : ∀ y, y ∈ W y := fun _ ↦ mem_interᵢ₂.2 fun _ _ ↦ hyV _ _
+    rcases precise_refinement W hWo (unionᵢ_eq_univ_iff.2 fun y ↦ ⟨y, hW y⟩)
+      with ⟨E, hEo, hE, hEf, hEA⟩
+    refine ⟨Σ y, T y, fun z ↦ U z.2.1 z.1 ×ˢ E z.1, fun _ ↦ (hUo _ _).prod (hEo _),
+      unionᵢ_eq_univ_iff.2 fun (x, y) ↦ ?_, fun (x, y) ↦ ?_, fun ⟨y, x, hx⟩ ↦ ?_⟩
+    · rcases unionᵢ_eq_univ_iff.1 hE y with ⟨b, hb⟩
+      rcases unionᵢ₂_eq_univ_iff.1 (hT b) x with ⟨a, ha, hx⟩
+      exact ⟨⟨b, a, ha⟩, hx, hb⟩
+    · rcases hEf y with ⟨t, ht, htf⟩
+      refine ⟨univ ×ˢ t, prod_mem_nhds univ_mem ht, ?_⟩
+      refine (htf.bunionᵢ fun y _ ↦ finite_range (Sigma.mk y)).subset ?_
+      rintro ⟨b, a, ha⟩ ⟨⟨c, d⟩, ⟨-, hd : d ∈ E b⟩, -, hdt : d ∈ t⟩
+      exact mem_unionᵢ₂.2 ⟨b, ⟨d, hd, hdt⟩, mem_range_self _⟩
+    · refine ⟨a x y, (Set.prod_mono Subset.rfl ?_).trans (hUV x y)⟩
+      exact (hEA _).trans (interᵢ₂_subset x hx)
+
+instance (priority := 200) [ParacompactSpace X] [CompactSpace Y] : ParacompactSpace (X × Y) :=
+  (Homeomorph.prodComm X Y).paracompactSpace_iff.2 inferInstance
+
 -- See note [lower instance priority]
 /-- A compact space is paracompact. -/
 instance (priority := 100) paracompact_of_compact [CompactSpace X] : ParacompactSpace X := by
   -- the proof is trivial: we choose a finite subcover using compactness, and use it
   refine' ⟨fun ι s ho hu ↦ _⟩
   rcases isCompact_univ.elim_finite_subcover _ ho hu.ge with ⟨T, hT⟩
-  have := hT; simp only [subset_def, mem_unionᵢ] at this
-  choose i _ _ using fun x ↦ this x (mem_univ x)
   refine' ⟨(T : Set ι), fun t ↦ s t, fun t ↦ ho _, _, locallyFinite_of_finite _,
     fun t ↦ ⟨t, Subset.rfl⟩⟩
   simpa only [unionᵢ_coe_set, ← univ_subset_iff]

Co-authored-by: Lukas Miaskiwskyi <lukas.mias@gmail.com>