topology.locally_finite ⟷ Mathlib.Topology.LocallyFinite

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

@@ -127,7 +127,7 @@ theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
   intro i
   by_cases hx : x ∈ closure (f i)
   · exact hc i _ hx
-  · rw [mem_closure_iff_nhdsWithin_neBot, not_ne_bot] at hx 
+  · rw [mem_closure_iff_nhdsWithin_neBot, not_ne_bot] at hx
     rw [hx]
     exact tendsto_bot
 #align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_iUnion'
@@ -188,7 +188,7 @@ theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)
   by
   refine' IsOpen.mem_nhds _ (mem_Inter₂.2 fun i => id)
   suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
-    rwa [← isOpen_compl_iff, compl_Union, Inter_subtype] at this 
+    rwa [← isOpen_compl_iff, compl_Union, Inter_subtype] at this
   exact (hf.comp_injective Subtype.coe_injective).isClosed_iUnion_of_finite fun i => hc _
 #align locally_finite.Inter_compl_mem_nhds LocallyFinite.iInter_compl_mem_nhds
 -/

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathbin.Topology.ContinuousOn
-import Mathbin.Order.Filter.SmallSets
+import Topology.ContinuousOn
+import Order.Filter.SmallSets
 
 #align_import topology.locally_finite from "leanprover-community/mathlib"@"55d771df074d0dd020139ee1cd4b95521422df9f"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -189,7 +189,7 @@ theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)
   refine' IsOpen.mem_nhds _ (mem_Inter₂.2 fun i => id)
   suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
     rwa [← isOpen_compl_iff, compl_Union, Inter_subtype] at this 
-  exact (hf.comp_injective Subtype.coe_injective).isClosed_iUnion fun i => hc _
+  exact (hf.comp_injective Subtype.coe_injective).isClosed_iUnion_of_finite fun i => hc _
 #align locally_finite.Inter_compl_mem_nhds LocallyFinite.iInter_compl_mem_nhds
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -81,7 +81,7 @@ theorem locallyFinite_iff_smallSets :
     LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, {i | (f i ∩ s).Nonempty}.Finite :=
   forall_congr' fun x =>
     Iff.symm <|
-      eventually_small_sets' fun s t hst ht =>
+      eventually_smallSets' fun s t hst ht =>
         ht.Subset fun i hi => hi.mono <| inter_subset_inter_right _ hst
 #align locally_finite_iff_small_sets locallyFinite_iff_smallSets
 -/

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.locally_finite
-! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.ContinuousOn
 import Mathbin.Order.Filter.SmallSets
 
+#align_import topology.locally_finite from "leanprover-community/mathlib"@"55d771df074d0dd020139ee1cd4b95521422df9f"
+
 /-!
 ### Locally finite families of sets
 

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

@@ -42,33 +42,44 @@ def LocallyFinite (f : ι → Set X) :=
 #align locally_finite LocallyFinite
 -/
 
+#print locallyFinite_of_finite /-
 theorem locallyFinite_of_finite [Finite ι] (f : ι → Set X) : LocallyFinite f := fun x =>
   ⟨univ, univ_mem, toFinite _⟩
 #align locally_finite_of_finite locallyFinite_of_finite
+-/
 
 namespace LocallyFinite
 
+#print LocallyFinite.point_finite /-
 theorem point_finite (hf : LocallyFinite f) (x : X) : {b | x ∈ f b}.Finite :=
   let ⟨t, hxt, ht⟩ := hf x
   ht.Subset fun b hb => ⟨x, hb, mem_of_mem_nhds hxt⟩
 #align locally_finite.point_finite LocallyFinite.point_finite
+-/
 
+#print LocallyFinite.subset /-
 protected theorem subset (hf : LocallyFinite f) (hg : ∀ i, g i ⊆ f i) : LocallyFinite g := fun a =>
   let ⟨t, ht₁, ht₂⟩ := hf a
   ⟨t, ht₁, ht₂.Subset fun i hi => hi.mono <| inter_subset_inter (hg i) Subset.rfl⟩
 #align locally_finite.subset LocallyFinite.subset
+-/
 
+#print LocallyFinite.comp_injOn /-
 theorem comp_injOn {g : ι' → ι} (hf : LocallyFinite f) (hg : InjOn g {i | (f (g i)).Nonempty}) :
     LocallyFinite (f ∘ g) := fun x =>
   let ⟨t, htx, htf⟩ := hf x
   ⟨t, htx, htf.Preimage <| hg.mono fun i hi => hi.out.mono <| inter_subset_left _ _⟩
 #align locally_finite.comp_inj_on LocallyFinite.comp_injOn
+-/
 
+#print LocallyFinite.comp_injective /-
 theorem comp_injective {g : ι' → ι} (hf : LocallyFinite f) (hg : Injective g) :
     LocallyFinite (f ∘ g) :=
   hf.comp_injOn (hg.InjOn _)
 #align locally_finite.comp_injective LocallyFinite.comp_injective
+-/
 
+#print locallyFinite_iff_smallSets /-
 theorem locallyFinite_iff_smallSets :
     LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, {i | (f i ∩ s).Nonempty}.Finite :=
   forall_congr' fun x =>
@@ -76,18 +87,24 @@ theorem locallyFinite_iff_smallSets :
       eventually_small_sets' fun s t hst ht =>
         ht.Subset fun i hi => hi.mono <| inter_subset_inter_right _ hst
 #align locally_finite_iff_small_sets locallyFinite_iff_smallSets
+-/
 
+#print LocallyFinite.eventually_smallSets /-
 protected theorem eventually_smallSets (hf : LocallyFinite f) (x : X) :
     ∀ᶠ s in (𝓝 x).smallSets, {i | (f i ∩ s).Nonempty}.Finite :=
   locallyFinite_iff_smallSets.mp hf x
 #align locally_finite.eventually_small_sets LocallyFinite.eventually_smallSets
+-/
 
+#print LocallyFinite.exists_mem_basis /-
 theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X}
     {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ (i : _) (hi : p i), {j | (f j ∩ s i).Nonempty}.Finite :=
   let ⟨i, hpi, hi⟩ := hb.smallSets.eventually_iff.mp (hf.eventually_smallSets x)
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
+-/
 
+#print LocallyFinite.nhdsWithin_iUnion /-
 protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
     𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a :=
   by
@@ -102,7 +119,9 @@ protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
     _ ≤ ⨆ i, 𝓝[f i ∩ U] a := (iSup₂_le_iSup _ _)
     _ ≤ ⨆ i, 𝓝[f i] a := iSup_mono fun i => nhdsWithin_mono _ <| inter_subset_left _ _
 #align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_iUnion
+-/
 
+#print LocallyFinite.continuousOn_iUnion' /-
 theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : ContinuousOn g (⋃ i, f i) :=
   by
@@ -115,22 +134,30 @@ theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
     rw [hx]
     exact tendsto_bot
 #align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_iUnion'
+-/
 
+#print LocallyFinite.continuousOn_iUnion /-
 theorem continuousOn_iUnion {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
     (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) :=
   hf.continuousOn_iUnion' fun i x hx => h_cont i x <| (h_cl i).closure_subset hx
 #align locally_finite.continuous_on_Union LocallyFinite.continuousOn_iUnion
+-/
 
+#print LocallyFinite.continuous' /-
 protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : Continuous g :=
   continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion' hc
 #align locally_finite.continuous' LocallyFinite.continuous'
+-/
 
+#print LocallyFinite.continuous /-
 protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) : Continuous g :=
   continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion h_cl h_cont
 #align locally_finite.continuous LocallyFinite.continuous
+-/
 
+#print LocallyFinite.closure /-
 protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closure (f i) :=
   by
   intro x
@@ -140,17 +167,23 @@ protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closur
     (hi.mono is_open_interior.closure_inter).of_closure.mono
       (inter_subset_inter_right _ interior_subset)
 #align locally_finite.closure LocallyFinite.closure
+-/
 
+#print LocallyFinite.closure_iUnion /-
 theorem closure_iUnion (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) :=
   by
   ext x
   simp only [mem_closure_iff_nhdsWithin_neBot, h.nhds_within_Union, supr_ne_bot, mem_Union]
 #align locally_finite.closure_Union LocallyFinite.closure_iUnion
+-/
 
+#print LocallyFinite.isClosed_iUnion /-
 theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) : IsClosed (⋃ i, f i) :=
   by simp only [← closure_eq_iff_isClosed, hf.closure_Union, (hc _).closure_eq]
 #align locally_finite.is_closed_Union LocallyFinite.isClosed_iUnion
+-/
 
+#print LocallyFinite.iInter_compl_mem_nhds /-
 /-- If `f : β → set α` is a locally finite family of closed sets, then for any `x : α`, the
 intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
 theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
@@ -161,6 +194,7 @@ theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)
     rwa [← isOpen_compl_iff, compl_Union, Inter_subtype] at this 
   exact (hf.comp_injective Subtype.coe_injective).isClosed_iUnion fun i => hc _
 #align locally_finite.Inter_compl_mem_nhds LocallyFinite.iInter_compl_mem_nhds
+-/
 
 #print LocallyFinite.exists_forall_eventually_eq_prod /-
 /-- Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose
@@ -213,40 +247,52 @@ theorem exists_forall_eventually_atTop_eventuallyEq {f : ℕ → X → α}
 #align locally_finite.exists_forall_eventually_at_top_eventually_eq LocallyFinite.exists_forall_eventually_atTop_eventuallyEq
 -/
 
+#print LocallyFinite.preimage_continuous /-
 theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g) :
     LocallyFinite fun i => g ⁻¹' f i := fun x =>
   let ⟨s, hsx, hs⟩ := hf (g x)
   ⟨g ⁻¹' s, hg.ContinuousAt hsx, hs.Subset fun i ⟨y, hy⟩ => ⟨g y, hy⟩⟩
 #align locally_finite.preimage_continuous LocallyFinite.preimage_continuous
+-/
 
 end LocallyFinite
 
+#print Equiv.locallyFinite_comp_iff /-
 @[simp]
 theorem Equiv.locallyFinite_comp_iff (e : ι' ≃ ι) : LocallyFinite (f ∘ e) ↔ LocallyFinite f :=
   ⟨fun h => by simpa only [(· ∘ ·), e.apply_symm_apply] using h.comp_injective e.symm.injective,
     fun h => h.comp_injective e.Injective⟩
 #align equiv.locally_finite_comp_iff Equiv.locallyFinite_comp_iff
+-/
 
+#print locallyFinite_sum /-
 theorem locallyFinite_sum {f : Sum ι ι' → Set X} :
     LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr) := by
   simp only [locallyFinite_iff_smallSets, ← forall_and, ← finite_preimage_inl_and_inr,
     preimage_set_of_eq, (· ∘ ·), eventually_and]
 #align locally_finite_sum locallyFinite_sum
+-/
 
+#print LocallyFinite.sum_elim /-
 theorem LocallyFinite.sum_elim {g : ι' → Set X} (hf : LocallyFinite f) (hg : LocallyFinite g) :
     LocallyFinite (Sum.elim f g) :=
   locallyFinite_sum.mpr ⟨hf, hg⟩
 #align locally_finite.sum_elim LocallyFinite.sum_elim
+-/
 
+#print locallyFinite_option /-
 theorem locallyFinite_option {f : Option ι → Set X} : LocallyFinite f ↔ LocallyFinite (f ∘ some) :=
   by
   simp only [← (Equiv.optionEquivSumPUnit.{u} ι).symm.locallyFinite_comp_iff, locallyFinite_sum,
     locallyFinite_of_finite, and_true_iff]
   rfl
 #align locally_finite_option locallyFinite_option
+-/
 
+#print LocallyFinite.option_elim' /-
 theorem LocallyFinite.option_elim' (hf : LocallyFinite f) (s : Set X) :
     LocallyFinite (Option.elim' s f) :=
   locallyFinite_option.2 hf
 #align locally_finite.option_elim LocallyFinite.option_elim'
+-/
 

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

@@ -101,7 +101,6 @@ protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
     _ = ⨆ i ∈ {j | (f j ∩ U).Nonempty}, 𝓝[f i ∩ U] a := (nhdsWithin_biUnion hfin _ _)
     _ ≤ ⨆ i, 𝓝[f i ∩ U] a := (iSup₂_le_iSup _ _)
     _ ≤ ⨆ i, 𝓝[f i] a := iSup_mono fun i => nhdsWithin_mono _ <| inter_subset_left _ _
-    
 #align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_iUnion
 
 theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
@@ -187,7 +186,6 @@ theorem exists_forall_eventually_eq_prod {π : X → Sort _} {f : ℕ → ∀ x
     f n y = f (N x + 1) y := hN _ _ hn _ hy
     _ = f (max (N x + 1) (N y + 1)) y := (hN _ _ (le_max_left _ _) _ hy).symm
     _ = f (N y + 1) y := hN _ _ (le_max_right _ _) _ (mem_of_mem_nhds <| hUx y)
-    
 #align locally_finite.exists_forall_eventually_eq_prod LocallyFinite.exists_forall_eventually_eq_prod
 -/
 

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

@@ -38,7 +38,7 @@ variable {ι : Type u} {ι' α X Y : Type _} [TopologicalSpace X] [TopologicalSp
 /-- A family of sets in `set X` is locally finite if at every point `x : X`,
 there is a neighborhood of `x` which meets only finitely many sets in the family. -/
 def LocallyFinite (f : ι → Set X) :=
-  ∀ x : X, ∃ t ∈ 𝓝 x, { i | (f i ∩ t).Nonempty }.Finite
+  ∀ x : X, ∃ t ∈ 𝓝 x, {i | (f i ∩ t).Nonempty}.Finite
 #align locally_finite LocallyFinite
 -/
 
@@ -48,7 +48,7 @@ theorem locallyFinite_of_finite [Finite ι] (f : ι → Set X) : LocallyFinite f
 
 namespace LocallyFinite
 
-theorem point_finite (hf : LocallyFinite f) (x : X) : { b | x ∈ f b }.Finite :=
+theorem point_finite (hf : LocallyFinite f) (x : X) : {b | x ∈ f b}.Finite :=
   let ⟨t, hxt, ht⟩ := hf x
   ht.Subset fun b hb => ⟨x, hb, mem_of_mem_nhds hxt⟩
 #align locally_finite.point_finite LocallyFinite.point_finite
@@ -58,7 +58,7 @@ protected theorem subset (hf : LocallyFinite f) (hg : ∀ i, g i ⊆ f i) : Loca
   ⟨t, ht₁, ht₂.Subset fun i hi => hi.mono <| inter_subset_inter (hg i) Subset.rfl⟩
 #align locally_finite.subset LocallyFinite.subset
 
-theorem comp_injOn {g : ι' → ι} (hf : LocallyFinite f) (hg : InjOn g { i | (f (g i)).Nonempty }) :
+theorem comp_injOn {g : ι' → ι} (hf : LocallyFinite f) (hg : InjOn g {i | (f (g i)).Nonempty}) :
     LocallyFinite (f ∘ g) := fun x =>
   let ⟨t, htx, htf⟩ := hf x
   ⟨t, htx, htf.Preimage <| hg.mono fun i hi => hi.out.mono <| inter_subset_left _ _⟩
@@ -70,7 +70,7 @@ theorem comp_injective {g : ι' → ι} (hf : LocallyFinite f) (hg : Injective g
 #align locally_finite.comp_injective LocallyFinite.comp_injective
 
 theorem locallyFinite_iff_smallSets :
-    LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite :=
+    LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, {i | (f i ∩ s).Nonempty}.Finite :=
   forall_congr' fun x =>
     Iff.symm <|
       eventually_small_sets' fun s t hst ht =>
@@ -78,12 +78,12 @@ theorem locallyFinite_iff_smallSets :
 #align locally_finite_iff_small_sets locallyFinite_iff_smallSets
 
 protected theorem eventually_smallSets (hf : LocallyFinite f) (x : X) :
-    ∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite :=
+    ∀ᶠ s in (𝓝 x).smallSets, {i | (f i ∩ s).Nonempty}.Finite :=
   locallyFinite_iff_smallSets.mp hf x
 #align locally_finite.eventually_small_sets LocallyFinite.eventually_smallSets
 
 theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X}
-    {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ (i : _) (hi : p i), { j | (f j ∩ s i).Nonempty }.Finite :=
+    {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ (i : _) (hi : p i), {j | (f j ∩ s i).Nonempty}.Finite :=
   let ⟨i, hpi, hi⟩ := hb.smallSets.eventually_iff.mp (hf.eventually_smallSets x)
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
@@ -96,9 +96,9 @@ protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
   calc
     𝓝[⋃ i, f i] a = 𝓝[⋃ i, f i ∩ U] a := by
       rw [← Union_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]
-    _ = 𝓝[⋃ i ∈ { j | (f j ∩ U).Nonempty }, f i ∩ U] a := by
+    _ = 𝓝[⋃ i ∈ {j | (f j ∩ U).Nonempty}, f i ∩ U] a := by
       simp only [mem_set_of_eq, Union_nonempty_self]
-    _ = ⨆ i ∈ { j | (f j ∩ U).Nonempty }, 𝓝[f i ∩ U] a := (nhdsWithin_biUnion hfin _ _)
+    _ = ⨆ i ∈ {j | (f j ∩ U).Nonempty}, 𝓝[f i ∩ U] a := (nhdsWithin_biUnion hfin _ _)
     _ ≤ ⨆ i, 𝓝[f i ∩ U] a := (iSup₂_le_iSup _ _)
     _ ≤ ⨆ i, 𝓝[f i] a := iSup_mono fun i => nhdsWithin_mono _ <| inter_subset_left _ _
     
@@ -171,7 +171,7 @@ interval `[N, +∞)` and a neighbourhood of `x`.
 
 We formulate the conclusion in terms of the product of filter `filter.at_top` and `𝓝 x`. -/
 theorem exists_forall_eventually_eq_prod {π : X → Sort _} {f : ℕ → ∀ x : X, π x}
-    (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
+    (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x}) :
     ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in atTop ×ᶠ 𝓝 x, f p.1 p.2 = F p.2 :=
   by
   choose U hUx hU using hf
@@ -197,7 +197,7 @@ that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. T
 function `F : Π a, β a` such that for any `x`, for sufficiently large values of `n`, we have
 `f n y = F y` in a neighbourhood of `x`. -/
 theorem exists_forall_eventually_atTop_eventually_eq' {π : X → Sort _} {f : ℕ → ∀ x : X, π x}
-    (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
+    (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x}) :
     ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ n : ℕ in atTop, ∀ᶠ y : X in 𝓝 x, f n y = F y :=
   hf.exists_forall_eventually_eq_prod.imp fun F hF x => (hF x).curry
 #align locally_finite.exists_forall_eventually_at_top_eventually_eq' LocallyFinite.exists_forall_eventually_atTop_eventually_eq'
@@ -209,7 +209,7 @@ that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. T
 function `F :  α → β` such that for any `x`, for sufficiently large values of `n`, we have
 `f n =ᶠ[𝓝 x] F`. -/
 theorem exists_forall_eventually_atTop_eventuallyEq {f : ℕ → X → α}
-    (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
+    (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x}) :
     ∃ F : X → α, ∀ x, ∀ᶠ n : ℕ in atTop, f n =ᶠ[𝓝 x] F :=
   hf.exists_forall_eventually_atTop_eventually_eq'
 #align locally_finite.exists_forall_eventually_at_top_eventually_eq LocallyFinite.exists_forall_eventually_atTop_eventuallyEq

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

@@ -83,7 +83,7 @@ protected theorem eventually_smallSets (hf : LocallyFinite f) (x : X) :
 #align locally_finite.eventually_small_sets LocallyFinite.eventually_smallSets
 
 theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X}
-    {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ (i : _)(hi : p i), { j | (f j ∩ s i).Nonempty }.Finite :=
+    {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ (i : _) (hi : p i), { j | (f j ∩ s i).Nonempty }.Finite :=
   let ⟨i, hpi, hi⟩ := hb.smallSets.eventually_iff.mp (hf.eventually_smallSets x)
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
@@ -112,7 +112,7 @@ theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
   intro i
   by_cases hx : x ∈ closure (f i)
   · exact hc i _ hx
-  · rw [mem_closure_iff_nhdsWithin_neBot, not_ne_bot] at hx
+  · rw [mem_closure_iff_nhdsWithin_neBot, not_ne_bot] at hx 
     rw [hx]
     exact tendsto_bot
 #align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_iUnion'
@@ -159,7 +159,7 @@ theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)
   by
   refine' IsOpen.mem_nhds _ (mem_Inter₂.2 fun i => id)
   suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
-    rwa [← isOpen_compl_iff, compl_Union, Inter_subtype] at this
+    rwa [← isOpen_compl_iff, compl_Union, Inter_subtype] at this 
   exact (hf.comp_injective Subtype.coe_injective).isClosed_iUnion fun i => hc _
 #align locally_finite.Inter_compl_mem_nhds LocallyFinite.iInter_compl_mem_nhds
 

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

@@ -27,7 +27,7 @@ In this file we give the definition and prove basic properties of locally finite
 -- locally finite family [General Topology (Bourbaki, 1995)]
 open Set Function Filter
 
-open Topology Filter
+open scoped Topology Filter
 
 universe u
 

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

@@ -42,69 +42,33 @@ def LocallyFinite (f : ι → Set X) :=
 #align locally_finite LocallyFinite
 -/
 
-/- warning: locally_finite_of_finite -> locallyFinite_of_finite is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_3 : Finite.{succ u1} ι] (f : ι -> (Set.{u2} X)), LocallyFinite.{u1, u2} ι X _inst_1 f
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] [_inst_3 : Finite.{succ u2} ι] (f : ι -> (Set.{u1} X)), LocallyFinite.{u2, u1} ι X _inst_1 f
-Case conversion may be inaccurate. Consider using '#align locally_finite_of_finite locallyFinite_of_finiteₓ'. -/
 theorem locallyFinite_of_finite [Finite ι] (f : ι → Set X) : LocallyFinite f := fun x =>
   ⟨univ, univ_mem, toFinite _⟩
 #align locally_finite_of_finite locallyFinite_of_finite
 
 namespace LocallyFinite
 
-/- warning: locally_finite.point_finite -> LocallyFinite.point_finite is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (x : X), Set.Finite.{u1} ι (setOf.{u1} ι (fun (b : ι) => Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (f b))))
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (x : X), Set.Finite.{u2} ι (setOf.{u2} ι (fun (b : ι) => Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x (f b))))
-Case conversion may be inaccurate. Consider using '#align locally_finite.point_finite LocallyFinite.point_finiteₓ'. -/
 theorem point_finite (hf : LocallyFinite f) (x : X) : { b | x ∈ f b }.Finite :=
   let ⟨t, hxt, ht⟩ := hf x
   ht.Subset fun b hb => ⟨x, hb, mem_of_mem_nhds hxt⟩
 #align locally_finite.point_finite LocallyFinite.point_finite
 
-/- warning: locally_finite.subset -> LocallyFinite.subset is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)} {g : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), HasSubset.Subset.{u2} (Set.{u2} X) (Set.hasSubset.{u2} X) (g i) (f i)) -> (LocallyFinite.{u1, u2} ι X _inst_1 g)
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)} {g : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (i : ι), HasSubset.Subset.{u1} (Set.{u1} X) (Set.instHasSubsetSet.{u1} X) (g i) (f i)) -> (LocallyFinite.{u2, u1} ι X _inst_1 g)
-Case conversion may be inaccurate. Consider using '#align locally_finite.subset LocallyFinite.subsetₓ'. -/
 protected theorem subset (hf : LocallyFinite f) (hg : ∀ i, g i ⊆ f i) : LocallyFinite g := fun a =>
   let ⟨t, ht₁, ht₂⟩ := hf a
   ⟨t, ht₁, ht₂.Subset fun i hi => hi.mono <| inter_subset_inter (hg i) Subset.rfl⟩
 #align locally_finite.subset LocallyFinite.subset
 
-/- warning: locally_finite.comp_inj_on -> LocallyFinite.comp_injOn is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {ι' : Type.{u2}} {X : Type.{u3}} [_inst_1 : TopologicalSpace.{u3} X] {f : ι -> (Set.{u3} X)} {g : ι' -> ι}, (LocallyFinite.{u1, u3} ι X _inst_1 f) -> (Set.InjOn.{u2, u1} ι' ι g (setOf.{u2} ι' (fun (i : ι') => Set.Nonempty.{u3} X (f (g i))))) -> (LocallyFinite.{u2, u3} ι' X _inst_1 (Function.comp.{succ u2, succ u1, succ u3} ι' ι (Set.{u3} X) f g))
-but is expected to have type
-  forall {ι : Type.{u3}} {ι' : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)} {g : ι' -> ι}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (Set.InjOn.{u1, u3} ι' ι g (setOf.{u1} ι' (fun (i : ι') => Set.Nonempty.{u2} X (f (g i))))) -> (LocallyFinite.{u1, u2} ι' X _inst_1 (Function.comp.{succ u1, succ u3, succ u2} ι' ι (Set.{u2} X) f g))
-Case conversion may be inaccurate. Consider using '#align locally_finite.comp_inj_on LocallyFinite.comp_injOnₓ'. -/
 theorem comp_injOn {g : ι' → ι} (hf : LocallyFinite f) (hg : InjOn g { i | (f (g i)).Nonempty }) :
     LocallyFinite (f ∘ g) := fun x =>
   let ⟨t, htx, htf⟩ := hf x
   ⟨t, htx, htf.Preimage <| hg.mono fun i hi => hi.out.mono <| inter_subset_left _ _⟩
 #align locally_finite.comp_inj_on LocallyFinite.comp_injOn
 
-/- warning: locally_finite.comp_injective -> LocallyFinite.comp_injective is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {ι' : Type.{u2}} {X : Type.{u3}} [_inst_1 : TopologicalSpace.{u3} X] {f : ι -> (Set.{u3} X)} {g : ι' -> ι}, (LocallyFinite.{u1, u3} ι X _inst_1 f) -> (Function.Injective.{succ u2, succ u1} ι' ι g) -> (LocallyFinite.{u2, u3} ι' X _inst_1 (Function.comp.{succ u2, succ u1, succ u3} ι' ι (Set.{u3} X) f g))
-but is expected to have type
-  forall {ι : Type.{u3}} {ι' : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)} {g : ι' -> ι}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (Function.Injective.{succ u1, succ u3} ι' ι g) -> (LocallyFinite.{u1, u2} ι' X _inst_1 (Function.comp.{succ u1, succ u3, succ u2} ι' ι (Set.{u2} X) f g))
-Case conversion may be inaccurate. Consider using '#align locally_finite.comp_injective LocallyFinite.comp_injectiveₓ'. -/
 theorem comp_injective {g : ι' → ι} (hf : LocallyFinite f) (hg : Injective g) :
     LocallyFinite (f ∘ g) :=
   hf.comp_injOn (hg.InjOn _)
 #align locally_finite.comp_injective LocallyFinite.comp_injective
 
-/- warning: locally_finite_iff_small_sets -> locallyFinite_iff_smallSets is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, Iff (LocallyFinite.{u1, u2} ι X _inst_1 f) (forall (x : X), Filter.Eventually.{u2} (Set.{u2} X) (fun (s : Set.{u2} X) => Set.Finite.{u1} ι (setOf.{u1} ι (fun (i : ι) => Set.Nonempty.{u2} X (Inter.inter.{u2} (Set.{u2} X) (Set.hasInter.{u2} X) (f i) s)))) (Filter.smallSets.{u2} X (nhds.{u2} X _inst_1 x)))
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, Iff (LocallyFinite.{u2, u1} ι X _inst_1 f) (forall (x : X), Filter.Eventually.{u1} (Set.{u1} X) (fun (s : Set.{u1} X) => Set.Finite.{u2} ι (setOf.{u2} ι (fun (i : ι) => Set.Nonempty.{u1} X (Inter.inter.{u1} (Set.{u1} X) (Set.instInterSet.{u1} X) (f i) s)))) (Filter.smallSets.{u1} X (nhds.{u1} X _inst_1 x)))
-Case conversion may be inaccurate. Consider using '#align locally_finite_iff_small_sets locallyFinite_iff_smallSetsₓ'. -/
 theorem locallyFinite_iff_smallSets :
     LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite :=
   forall_congr' fun x =>
@@ -113,35 +77,17 @@ theorem locallyFinite_iff_smallSets :
         ht.Subset fun i hi => hi.mono <| inter_subset_inter_right _ hst
 #align locally_finite_iff_small_sets locallyFinite_iff_smallSets
 
-/- warning: locally_finite.eventually_small_sets -> LocallyFinite.eventually_smallSets is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (x : X), Filter.Eventually.{u2} (Set.{u2} X) (fun (s : Set.{u2} X) => Set.Finite.{u1} ι (setOf.{u1} ι (fun (i : ι) => Set.Nonempty.{u2} X (Inter.inter.{u2} (Set.{u2} X) (Set.hasInter.{u2} X) (f i) s)))) (Filter.smallSets.{u2} X (nhds.{u2} X _inst_1 x)))
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (x : X), Filter.Eventually.{u1} (Set.{u1} X) (fun (s : Set.{u1} X) => Set.Finite.{u2} ι (setOf.{u2} ι (fun (i : ι) => Set.Nonempty.{u1} X (Inter.inter.{u1} (Set.{u1} X) (Set.instInterSet.{u1} X) (f i) s)))) (Filter.smallSets.{u1} X (nhds.{u1} X _inst_1 x)))
-Case conversion may be inaccurate. Consider using '#align locally_finite.eventually_small_sets LocallyFinite.eventually_smallSetsₓ'. -/
 protected theorem eventually_smallSets (hf : LocallyFinite f) (x : X) :
     ∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite :=
   locallyFinite_iff_smallSets.mp hf x
 #align locally_finite.eventually_small_sets LocallyFinite.eventually_smallSets
 
-/- warning: locally_finite.exists_mem_basis -> LocallyFinite.exists_mem_basis is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)} {ι' : Sort.{u3}}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall {p : ι' -> Prop} {s : ι' -> (Set.{u2} X)} {x : X}, (Filter.HasBasis.{u2, u3} X ι' (nhds.{u2} X _inst_1 x) p s) -> (Exists.{u3} ι' (fun (i : ι') => Exists.{0} (p i) (fun (hi : p i) => Set.Finite.{u1} ι (setOf.{u1} ι (fun (j : ι) => Set.Nonempty.{u2} X (Inter.inter.{u2} (Set.{u2} X) (Set.hasInter.{u2} X) (f j) (s i))))))))
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)} {ι' : Sort.{u3}}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall {p : ι' -> Prop} {s : ι' -> (Set.{u1} X)} {x : X}, (Filter.HasBasis.{u1, u3} X ι' (nhds.{u1} X _inst_1 x) p s) -> (Exists.{u3} ι' (fun (i : ι') => And (p i) (Set.Finite.{u2} ι (setOf.{u2} ι (fun (j : ι) => Set.Nonempty.{u1} X (Inter.inter.{u1} (Set.{u1} X) (Set.instInterSet.{u1} X) (f j) (s i))))))))
-Case conversion may be inaccurate. Consider using '#align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basisₓ'. -/
 theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X}
     {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ (i : _)(hi : p i), { j | (f j ∩ s i).Nonempty }.Finite :=
   let ⟨i, hpi, hi⟩ := hb.smallSets.eventually_iff.mp (hf.eventually_smallSets x)
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
 
-/- warning: locally_finite.nhds_within_Union -> LocallyFinite.nhdsWithin_iUnion is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (a : X), Eq.{succ u2} (Filter.{u2} X) (nhdsWithin.{u2} X _inst_1 a (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i))) (iSup.{u2, succ u1} (Filter.{u2} X) (ConditionallyCompleteLattice.toHasSup.{u2} (Filter.{u2} X) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} X) (Filter.completeLattice.{u2} X))) ι (fun (i : ι) => nhdsWithin.{u2} X _inst_1 a (f i))))
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (a : X), Eq.{succ u1} (Filter.{u1} X) (nhdsWithin.{u1} X _inst_1 a (Set.iUnion.{u1, succ u2} X ι (fun (i : ι) => f i))) (iSup.{u1, succ u2} (Filter.{u1} X) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} X) (Filter.instCompleteLatticeFilter.{u1} X))) ι (fun (i : ι) => nhdsWithin.{u1} X _inst_1 a (f i))))
-Case conversion may be inaccurate. Consider using '#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_iUnionₓ'. -/
 protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
     𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a :=
   by
@@ -158,12 +104,6 @@ protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
     
 #align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_iUnion
 
-/- warning: locally_finite.continuous_on_Union' -> LocallyFinite.continuousOn_iUnion' is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u3} X Y _inst_1 _inst_2 g (f i) x)) -> (ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i)))
-but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u1} X Y _inst_1 _inst_2 g (f i) x)) -> (ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (Set.iUnion.{u2, succ u3} X ι (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_iUnion'ₓ'. -/
 theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : ContinuousOn g (⋃ i, f i) :=
   by
@@ -177,45 +117,21 @@ theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
     exact tendsto_bot
 #align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_iUnion'
 
-/- warning: locally_finite.continuous_on_Union -> LocallyFinite.continuousOn_iUnion is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (f i)) -> (ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i)))
-but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (f i)) -> (ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (Set.iUnion.{u2, succ u3} X ι (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_iUnionₓ'. -/
 theorem continuousOn_iUnion {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
     (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) :=
   hf.continuousOn_iUnion' fun i x hx => h_cont i x <| (h_cl i).closure_subset hx
 #align locally_finite.continuous_on_Union LocallyFinite.continuousOn_iUnion
 
-/- warning: locally_finite.continuous' -> LocallyFinite.continuous' is a dubious translation:
-lean 3 declaration is
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 protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : Continuous g :=
   continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion' hc
 #align locally_finite.continuous' LocallyFinite.continuous'
 
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 protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) : Continuous g :=
   continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion h_cl h_cont
 #align locally_finite.continuous LocallyFinite.continuous
 
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 protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closure (f i) :=
   by
   intro x
@@ -226,34 +142,16 @@ protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closur
       (inter_subset_inter_right _ interior_subset)
 #align locally_finite.closure LocallyFinite.closure
 
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 theorem closure_iUnion (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) :=
   by
   ext x
   simp only [mem_closure_iff_nhdsWithin_neBot, h.nhds_within_Union, supr_ne_bot, mem_Union]
 #align locally_finite.closure_Union LocallyFinite.closure_iUnion
 
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 theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) : IsClosed (⋃ i, f i) :=
   by simp only [← closure_eq_iff_isClosed, hf.closure_Union, (hc _).closure_eq]
 #align locally_finite.is_closed_Union LocallyFinite.isClosed_iUnion
 
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 /-- If `f : β → set α` is a locally finite family of closed sets, then for any `x : α`, the
 intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
 theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
@@ -317,12 +215,6 @@ theorem exists_forall_eventually_atTop_eventuallyEq {f : ℕ → X → α}
 #align locally_finite.exists_forall_eventually_at_top_eventually_eq LocallyFinite.exists_forall_eventually_atTop_eventuallyEq
 -/
 
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 theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g) :
     LocallyFinite fun i => g ⁻¹' f i := fun x =>
   let ⟨s, hsx, hs⟩ := hf (g x)
@@ -331,47 +223,23 @@ theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuou
 
 end LocallyFinite
 
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 @[simp]
 theorem Equiv.locallyFinite_comp_iff (e : ι' ≃ ι) : LocallyFinite (f ∘ e) ↔ LocallyFinite f :=
   ⟨fun h => by simpa only [(· ∘ ·), e.apply_symm_apply] using h.comp_injective e.symm.injective,
     fun h => h.comp_injective e.Injective⟩
 #align equiv.locally_finite_comp_iff Equiv.locallyFinite_comp_iff
 
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 theorem locallyFinite_sum {f : Sum ι ι' → Set X} :
     LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr) := by
   simp only [locallyFinite_iff_smallSets, ← forall_and, ← finite_preimage_inl_and_inr,
     preimage_set_of_eq, (· ∘ ·), eventually_and]
 #align locally_finite_sum locallyFinite_sum
 
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 theorem LocallyFinite.sum_elim {g : ι' → Set X} (hf : LocallyFinite f) (hg : LocallyFinite g) :
     LocallyFinite (Sum.elim f g) :=
   locallyFinite_sum.mpr ⟨hf, hg⟩
 #align locally_finite.sum_elim LocallyFinite.sum_elim
 
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 theorem locallyFinite_option {f : Option ι → Set X} : LocallyFinite f ↔ LocallyFinite (f ∘ some) :=
   by
   simp only [← (Equiv.optionEquivSumPUnit.{u} ι).symm.locallyFinite_comp_iff, locallyFinite_sum,
@@ -379,12 +247,6 @@ theorem locallyFinite_option {f : Option ι → Set X} : LocallyFinite f ↔ Loc
   rfl
 #align locally_finite_option locallyFinite_option
 
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 theorem LocallyFinite.option_elim' (hf : LocallyFinite f) (s : Set X) :
     LocallyFinite (Option.elim' s f) :=
   locallyFinite_option.2 hf

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

@@ -335,7 +335,7 @@ end LocallyFinite
 lean 3 declaration is
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 but is expected to have type
-  forall {ι : Type.{u2}} {ι' : Type.{u3}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)} (e : Equiv.{succ u3, succ u2} ι' ι), Iff (LocallyFinite.{u3, u1} ι' X _inst_1 (Function.comp.{succ u3, succ u2, succ u1} ι' ι (Set.{u1} X) f (FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (Equiv.{succ u3, succ u2} ι' ι) ι' (fun (_x : ι') => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : ι') => ι) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} ι' ι) e))) (LocallyFinite.{u2, u1} ι X _inst_1 f)
+  forall {ι : Type.{u2}} {ι' : Type.{u3}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)} (e : Equiv.{succ u3, succ u2} ι' ι), Iff (LocallyFinite.{u3, u1} ι' X _inst_1 (Function.comp.{succ u3, succ u2, succ u1} ι' ι (Set.{u1} X) f (FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (Equiv.{succ u3, succ u2} ι' ι) ι' (fun (_x : ι') => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.812 : ι') => ι) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} ι' ι) e))) (LocallyFinite.{u2, u1} ι X _inst_1 f)
 Case conversion may be inaccurate. Consider using '#align equiv.locally_finite_comp_iff Equiv.locallyFinite_comp_iffₓ'. -/
 @[simp]
 theorem Equiv.locallyFinite_comp_iff (e : ι' ≃ ι) : LocallyFinite (f ∘ e) ↔ LocallyFinite f :=

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

@@ -136,35 +136,35 @@ theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
 
-/- warning: locally_finite.nhds_within_Union -> LocallyFinite.nhdsWithin_unionᵢ is a dubious translation:
+/- warning: locally_finite.nhds_within_Union -> LocallyFinite.nhdsWithin_iUnion is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (a : X), Eq.{succ u2} (Filter.{u2} X) (nhdsWithin.{u2} X _inst_1 a (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i))) (supᵢ.{u2, succ u1} (Filter.{u2} X) (ConditionallyCompleteLattice.toHasSup.{u2} (Filter.{u2} X) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} X) (Filter.completeLattice.{u2} X))) ι (fun (i : ι) => nhdsWithin.{u2} X _inst_1 a (f i))))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (a : X), Eq.{succ u2} (Filter.{u2} X) (nhdsWithin.{u2} X _inst_1 a (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i))) (iSup.{u2, succ u1} (Filter.{u2} X) (ConditionallyCompleteLattice.toHasSup.{u2} (Filter.{u2} X) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} X) (Filter.completeLattice.{u2} X))) ι (fun (i : ι) => nhdsWithin.{u2} X _inst_1 a (f i))))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (a : X), Eq.{succ u1} (Filter.{u1} X) (nhdsWithin.{u1} X _inst_1 a (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => f i))) (supᵢ.{u1, succ u2} (Filter.{u1} X) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} X) (Filter.instCompleteLatticeFilter.{u1} X))) ι (fun (i : ι) => nhdsWithin.{u1} X _inst_1 a (f i))))
-Case conversion may be inaccurate. Consider using '#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_unionᵢₓ'. -/
-protected theorem nhdsWithin_unionᵢ (hf : LocallyFinite f) (a : X) :
+  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (a : X), Eq.{succ u1} (Filter.{u1} X) (nhdsWithin.{u1} X _inst_1 a (Set.iUnion.{u1, succ u2} X ι (fun (i : ι) => f i))) (iSup.{u1, succ u2} (Filter.{u1} X) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} X) (Filter.instCompleteLatticeFilter.{u1} X))) ι (fun (i : ι) => nhdsWithin.{u1} X _inst_1 a (f i))))
+Case conversion may be inaccurate. Consider using '#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_iUnionₓ'. -/
+protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
     𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a :=
   by
   rcases hf a with ⟨U, haU, hfin⟩
-  refine' le_antisymm _ (supᵢ_le fun i => nhdsWithin_mono _ (subset_Union _ _))
+  refine' le_antisymm _ (iSup_le fun i => nhdsWithin_mono _ (subset_Union _ _))
   calc
     𝓝[⋃ i, f i] a = 𝓝[⋃ i, f i ∩ U] a := by
       rw [← Union_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]
     _ = 𝓝[⋃ i ∈ { j | (f j ∩ U).Nonempty }, f i ∩ U] a := by
       simp only [mem_set_of_eq, Union_nonempty_self]
-    _ = ⨆ i ∈ { j | (f j ∩ U).Nonempty }, 𝓝[f i ∩ U] a := (nhdsWithin_bunionᵢ hfin _ _)
-    _ ≤ ⨆ i, 𝓝[f i ∩ U] a := (supᵢ₂_le_supᵢ _ _)
-    _ ≤ ⨆ i, 𝓝[f i] a := supᵢ_mono fun i => nhdsWithin_mono _ <| inter_subset_left _ _
+    _ = ⨆ i ∈ { j | (f j ∩ U).Nonempty }, 𝓝[f i ∩ U] a := (nhdsWithin_biUnion hfin _ _)
+    _ ≤ ⨆ i, 𝓝[f i ∩ U] a := (iSup₂_le_iSup _ _)
+    _ ≤ ⨆ i, 𝓝[f i] a := iSup_mono fun i => nhdsWithin_mono _ <| inter_subset_left _ _
     
-#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_unionᵢ
+#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_iUnion
 
-/- warning: locally_finite.continuous_on_Union' -> LocallyFinite.continuousOn_unionᵢ' is a dubious translation:
+/- warning: locally_finite.continuous_on_Union' -> LocallyFinite.continuousOn_iUnion' is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u3} X Y _inst_1 _inst_2 g (f i) x)) -> (ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)))
+  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u3} X Y _inst_1 _inst_2 g (f i) x)) -> (ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i)))
 but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u1} X Y _inst_1 _inst_2 g (f i) x)) -> (ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (Set.unionᵢ.{u2, succ u3} X ι (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_unionᵢ'ₓ'. -/
-theorem continuousOn_unionᵢ' {g : X → Y} (hf : LocallyFinite f)
+  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u1} X Y _inst_1 _inst_2 g (f i) x)) -> (ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (Set.iUnion.{u2, succ u3} X ι (fun (i : ι) => f i)))
+Case conversion may be inaccurate. Consider using '#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_iUnion'ₓ'. -/
+theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : ContinuousOn g (⋃ i, f i) :=
   by
   rintro x -
@@ -175,39 +175,39 @@ theorem continuousOn_unionᵢ' {g : X → Y} (hf : LocallyFinite f)
   · rw [mem_closure_iff_nhdsWithin_neBot, not_ne_bot] at hx
     rw [hx]
     exact tendsto_bot
-#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_unionᵢ'
+#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_iUnion'
 
-/- warning: locally_finite.continuous_on_Union -> LocallyFinite.continuousOn_unionᵢ is a dubious translation:
+/- warning: locally_finite.continuous_on_Union -> LocallyFinite.continuousOn_iUnion is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (f i)) -> (ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)))
+  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (f i)) -> (ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i)))
 but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (f i)) -> (ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (Set.unionᵢ.{u2, succ u3} X ι (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_unionᵢₓ'. -/
-theorem continuousOn_unionᵢ {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
+  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (f i)) -> (ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (Set.iUnion.{u2, succ u3} X ι (fun (i : ι) => f i)))
+Case conversion may be inaccurate. Consider using '#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_iUnionₓ'. -/
+theorem continuousOn_iUnion {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
     (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) :=
-  hf.continuousOn_unionᵢ' fun i x hx => h_cont i x <| (h_cl i).closure_subset hx
-#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_unionᵢ
+  hf.continuousOn_iUnion' fun i x hx => h_cont i x <| (h_cl i).closure_subset hx
+#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_iUnion
 
 /- warning: locally_finite.continuous' -> LocallyFinite.continuous' is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u3} X Y _inst_1 _inst_2 g (f i) x)) -> (Continuous.{u2, u3} X Y _inst_1 _inst_2 g)
+  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u3} X Y _inst_1 _inst_2 g (f i) x)) -> (Continuous.{u2, u3} X Y _inst_1 _inst_2 g)
 but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.unionᵢ.{u2, succ u3} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u1} X Y _inst_1 _inst_2 g (f i) x)) -> (Continuous.{u2, u1} X Y _inst_1 _inst_2 g)
+  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.iUnion.{u2, succ u3} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u1} X Y _inst_1 _inst_2 g (f i) x)) -> (Continuous.{u2, u1} X Y _inst_1 _inst_2 g)
 Case conversion may be inaccurate. Consider using '#align locally_finite.continuous' LocallyFinite.continuous'ₓ'. -/
 protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : Continuous g :=
-  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ' hc
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion' hc
 #align locally_finite.continuous' LocallyFinite.continuous'
 
 /- warning: locally_finite.continuous -> LocallyFinite.continuous is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (f i)) -> (Continuous.{u2, u3} X Y _inst_1 _inst_2 g)
+  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (f i)) -> (Continuous.{u2, u3} X Y _inst_1 _inst_2 g)
 but is expected to have type
-  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.unionᵢ.{u2, succ u3} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (f i)) -> (Continuous.{u2, u1} X Y _inst_1 _inst_2 g)
+  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.iUnion.{u2, succ u3} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (f i)) -> (Continuous.{u2, u1} X Y _inst_1 _inst_2 g)
 Case conversion may be inaccurate. Consider using '#align locally_finite.continuous LocallyFinite.continuousₓ'. -/
 protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) : Continuous g :=
-  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ h_cl h_cont
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion h_cl h_cont
 #align locally_finite.continuous LocallyFinite.continuous
 
 /- warning: locally_finite.closure -> LocallyFinite.closure is a dubious translation:
@@ -226,44 +226,44 @@ protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closur
       (inter_subset_inter_right _ interior_subset)
 #align locally_finite.closure LocallyFinite.closure
 
-/- warning: locally_finite.closure_Union -> LocallyFinite.closure_unionᵢ is a dubious translation:
+/- warning: locally_finite.closure_Union -> LocallyFinite.closure_iUnion is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (closure.{u2} X _inst_1 (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i))) (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => closure.{u2} X _inst_1 (f i))))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (closure.{u2} X _inst_1 (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i))) (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => closure.{u2} X _inst_1 (f i))))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (Eq.{succ u1} (Set.{u1} X) (closure.{u1} X _inst_1 (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => f i))) (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => closure.{u1} X _inst_1 (f i))))
-Case conversion may be inaccurate. Consider using '#align locally_finite.closure_Union LocallyFinite.closure_unionᵢₓ'. -/
-theorem closure_unionᵢ (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) :=
+  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (Eq.{succ u1} (Set.{u1} X) (closure.{u1} X _inst_1 (Set.iUnion.{u1, succ u2} X ι (fun (i : ι) => f i))) (Set.iUnion.{u1, succ u2} X ι (fun (i : ι) => closure.{u1} X _inst_1 (f i))))
+Case conversion may be inaccurate. Consider using '#align locally_finite.closure_Union LocallyFinite.closure_iUnionₓ'. -/
+theorem closure_iUnion (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) :=
   by
   ext x
   simp only [mem_closure_iff_nhdsWithin_neBot, h.nhds_within_Union, supr_ne_bot, mem_Union]
-#align locally_finite.closure_Union LocallyFinite.closure_unionᵢ
+#align locally_finite.closure_Union LocallyFinite.closure_iUnion
 
-/- warning: locally_finite.is_closed_Union -> LocallyFinite.isClosed_unionᵢ is a dubious translation:
+/- warning: locally_finite.is_closed_Union -> LocallyFinite.isClosed_iUnion is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (IsClosed.{u2} X _inst_1 (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (IsClosed.{u2} X _inst_1 (Set.iUnion.{u2, succ u1} X ι (fun (i : ι) => f i)))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u1} X _inst_1 (f i)) -> (IsClosed.{u1} X _inst_1 (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align locally_finite.is_closed_Union LocallyFinite.isClosed_unionᵢₓ'. -/
-theorem isClosed_unionᵢ (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) : IsClosed (⋃ i, f i) :=
+  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u1} X _inst_1 (f i)) -> (IsClosed.{u1} X _inst_1 (Set.iUnion.{u1, succ u2} X ι (fun (i : ι) => f i)))
+Case conversion may be inaccurate. Consider using '#align locally_finite.is_closed_Union LocallyFinite.isClosed_iUnionₓ'. -/
+theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) : IsClosed (⋃ i, f i) :=
   by simp only [← closure_eq_iff_isClosed, hf.closure_Union, (hc _).closure_eq]
-#align locally_finite.is_closed_Union LocallyFinite.isClosed_unionᵢ
+#align locally_finite.is_closed_Union LocallyFinite.isClosed_iUnion
 
-/- warning: locally_finite.Inter_compl_mem_nhds -> LocallyFinite.interᵢ_compl_mem_nhds is a dubious translation:
+/- warning: locally_finite.Inter_compl_mem_nhds -> LocallyFinite.iInter_compl_mem_nhds is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (x : X), Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) (Set.interᵢ.{u2, succ u1} X ι (fun (i : ι) => Set.interᵢ.{u2, 0} X (Not (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (f i))) (fun (hi : Not (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (f i))) => HasCompl.compl.{u2} (Set.{u2} X) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X)) (f i)))) (nhds.{u2} X _inst_1 x))
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (x : X), Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) (Set.iInter.{u2, succ u1} X ι (fun (i : ι) => Set.iInter.{u2, 0} X (Not (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (f i))) (fun (hi : Not (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (f i))) => HasCompl.compl.{u2} (Set.{u2} X) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X)) (f i)))) (nhds.{u2} X _inst_1 x))
 but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u1} X _inst_1 (f i)) -> (forall (x : X), Membership.mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (instMembershipSetFilter.{u1} X) (Set.interᵢ.{u1, succ u2} X ι (fun (i : ι) => Set.interᵢ.{u1, 0} X (Not (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x (f i))) (fun (hi : Not (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x (f i))) => HasCompl.compl.{u1} (Set.{u1} X) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} X) (Set.instBooleanAlgebraSet.{u1} X)) (f i)))) (nhds.{u1} X _inst_1 x))
-Case conversion may be inaccurate. Consider using '#align locally_finite.Inter_compl_mem_nhds LocallyFinite.interᵢ_compl_mem_nhdsₓ'. -/
+  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u1} X _inst_1 (f i)) -> (forall (x : X), Membership.mem.{u1, u1} (Set.{u1} X) (Filter.{u1} X) (instMembershipSetFilter.{u1} X) (Set.iInter.{u1, succ u2} X ι (fun (i : ι) => Set.iInter.{u1, 0} X (Not (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x (f i))) (fun (hi : Not (Membership.mem.{u1, u1} X (Set.{u1} X) (Set.instMembershipSet.{u1} X) x (f i))) => HasCompl.compl.{u1} (Set.{u1} X) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} X) (Set.instBooleanAlgebraSet.{u1} X)) (f i)))) (nhds.{u1} X _inst_1 x))
+Case conversion may be inaccurate. Consider using '#align locally_finite.Inter_compl_mem_nhds LocallyFinite.iInter_compl_mem_nhdsₓ'. -/
 /-- If `f : β → set α` is a locally finite family of closed sets, then for any `x : α`, the
 intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
-theorem interᵢ_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
+theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
     (⋂ (i) (hi : x ∉ f i), f iᶜ) ∈ 𝓝 x :=
   by
   refine' IsOpen.mem_nhds _ (mem_Inter₂.2 fun i => id)
   suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
     rwa [← isOpen_compl_iff, compl_Union, Inter_subtype] at this
-  exact (hf.comp_injective Subtype.coe_injective).isClosed_unionᵢ fun i => hc _
-#align locally_finite.Inter_compl_mem_nhds LocallyFinite.interᵢ_compl_mem_nhds
+  exact (hf.comp_injective Subtype.coe_injective).isClosed_iUnion fun i => hc _
+#align locally_finite.Inter_compl_mem_nhds LocallyFinite.iInter_compl_mem_nhds
 
 #print LocallyFinite.exists_forall_eventually_eq_prod /-
 /-- Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose

mathlib commit https://github.com/leanprover-community/mathlib/commit/0148d455199ed64bf8eb2f493a1e7eb9211ce170

@@ -136,6 +136,12 @@ theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
 
+/- warning: locally_finite.nhds_within_Union -> LocallyFinite.nhdsWithin_unionᵢ is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (a : X), Eq.{succ u2} (Filter.{u2} X) (nhdsWithin.{u2} X _inst_1 a (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i))) (supᵢ.{u2, succ u1} (Filter.{u2} X) (ConditionallyCompleteLattice.toHasSup.{u2} (Filter.{u2} X) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Filter.{u2} X) (Filter.completeLattice.{u2} X))) ι (fun (i : ι) => nhdsWithin.{u2} X _inst_1 a (f i))))
+but is expected to have type
+  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (a : X), Eq.{succ u1} (Filter.{u1} X) (nhdsWithin.{u1} X _inst_1 a (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => f i))) (supᵢ.{u1, succ u2} (Filter.{u1} X) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} X) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} X) (Filter.instCompleteLatticeFilter.{u1} X))) ι (fun (i : ι) => nhdsWithin.{u1} X _inst_1 a (f i))))
+Case conversion may be inaccurate. Consider using '#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_unionᵢₓ'. -/
 protected theorem nhdsWithin_unionᵢ (hf : LocallyFinite f) (a : X) :
     𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a :=
   by
@@ -146,13 +152,19 @@ protected theorem nhdsWithin_unionᵢ (hf : LocallyFinite f) (a : X) :
       rw [← Union_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]
     _ = 𝓝[⋃ i ∈ { j | (f j ∩ U).Nonempty }, f i ∩ U] a := by
       simp only [mem_set_of_eq, Union_nonempty_self]
-    _ = ⨆ i ∈ { j | (f j ∩ U).Nonempty }, 𝓝[f i ∩ U] a := (nhdsWithin_bUnion hfin _ _)
+    _ = ⨆ i ∈ { j | (f j ∩ U).Nonempty }, 𝓝[f i ∩ U] a := (nhdsWithin_bunionᵢ hfin _ _)
     _ ≤ ⨆ i, 𝓝[f i ∩ U] a := (supᵢ₂_le_supᵢ _ _)
     _ ≤ ⨆ i, 𝓝[f i] a := supᵢ_mono fun i => nhdsWithin_mono _ <| inter_subset_left _ _
     
 #align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_unionᵢ
 
-theorem continuousOn_Union' {g : X → Y} (hf : LocallyFinite f)
+/- warning: locally_finite.continuous_on_Union' -> LocallyFinite.continuousOn_unionᵢ' is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u3} X Y _inst_1 _inst_2 g (f i) x)) -> (ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)))
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u1} X Y _inst_1 _inst_2 g (f i) x)) -> (ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (Set.unionᵢ.{u2, succ u3} X ι (fun (i : ι) => f i)))
+Case conversion may be inaccurate. Consider using '#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_unionᵢ'ₓ'. -/
+theorem continuousOn_unionᵢ' {g : X → Y} (hf : LocallyFinite f)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : ContinuousOn g (⋃ i, f i) :=
   by
   rintro x -
@@ -163,18 +175,36 @@ theorem continuousOn_Union' {g : X → Y} (hf : LocallyFinite f)
   · rw [mem_closure_iff_nhdsWithin_neBot, not_ne_bot] at hx
     rw [hx]
     exact tendsto_bot
-#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_Union'
+#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_unionᵢ'
 
+/- warning: locally_finite.continuous_on_Union -> LocallyFinite.continuousOn_unionᵢ is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (f i)) -> (ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)))
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (f i)) -> (ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (Set.unionᵢ.{u2, succ u3} X ι (fun (i : ι) => f i)))
+Case conversion may be inaccurate. Consider using '#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_unionᵢₓ'. -/
 theorem continuousOn_unionᵢ {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
     (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) :=
-  hf.continuousOn_Union' fun i x hx => h_cont i x <| (h_cl i).closure_subset hx
+  hf.continuousOn_unionᵢ' fun i x hx => h_cont i x <| (h_cl i).closure_subset hx
 #align locally_finite.continuous_on_Union LocallyFinite.continuousOn_unionᵢ
 
+/- warning: locally_finite.continuous' -> LocallyFinite.continuous' is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι) (x : X), (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u3} X Y _inst_1 _inst_2 g (f i) x)) -> (Continuous.{u2, u3} X Y _inst_1 _inst_2 g)
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.unionᵢ.{u2, succ u3} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι) (x : X), (Membership.mem.{u2, u2} X (Set.{u2} X) (Set.instMembershipSet.{u2} X) x (closure.{u2} X _inst_1 (f i))) -> (ContinuousWithinAt.{u2, u1} X Y _inst_1 _inst_2 g (f i) x)) -> (Continuous.{u2, u1} X Y _inst_1 _inst_2 g)
+Case conversion may be inaccurate. Consider using '#align locally_finite.continuous' LocallyFinite.continuous'ₓ'. -/
 protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : Continuous g :=
-  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_Union' hc
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ' hc
 #align locally_finite.continuous' LocallyFinite.continuous'
 
+/- warning: locally_finite.continuous -> LocallyFinite.continuous is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} {Y : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u3} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u3} X Y _inst_1 _inst_2 g (f i)) -> (Continuous.{u2, u3} X Y _inst_1 _inst_2 g)
+but is expected to have type
+  forall {ι : Type.{u3}} {X : Type.{u2}} {Y : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} X] [_inst_2 : TopologicalSpace.{u1} Y] {f : ι -> (Set.{u2} X)} {g : X -> Y}, (LocallyFinite.{u3, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (Set.unionᵢ.{u2, succ u3} X ι (fun (i : ι) => f i)) (Set.univ.{u2} X)) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (i : ι), ContinuousOn.{u2, u1} X Y _inst_1 _inst_2 g (f i)) -> (Continuous.{u2, u1} X Y _inst_1 _inst_2 g)
+Case conversion may be inaccurate. Consider using '#align locally_finite.continuous LocallyFinite.continuousₓ'. -/
 protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) : Continuous g :=
   continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ h_cl h_cont

mathlib commit https://github.com/leanprover-community/mathlib/commit/55d771df074d0dd020139ee1cd4b95521422df9f

@@ -4,11 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.locally_finite
-! leanprover-community/mathlib commit fac369018417f980cec5fcdafc766a69f88d8cfe
+! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathbin.Topology.Basic
+import Mathbin.Topology.ContinuousOn
 import Mathbin.Order.Filter.SmallSets
 
 /-!
@@ -136,6 +136,50 @@ theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
 
+protected theorem nhdsWithin_unionᵢ (hf : LocallyFinite f) (a : X) :
+    𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a :=
+  by
+  rcases hf a with ⟨U, haU, hfin⟩
+  refine' le_antisymm _ (supᵢ_le fun i => nhdsWithin_mono _ (subset_Union _ _))
+  calc
+    𝓝[⋃ i, f i] a = 𝓝[⋃ i, f i ∩ U] a := by
+      rw [← Union_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]
+    _ = 𝓝[⋃ i ∈ { j | (f j ∩ U).Nonempty }, f i ∩ U] a := by
+      simp only [mem_set_of_eq, Union_nonempty_self]
+    _ = ⨆ i ∈ { j | (f j ∩ U).Nonempty }, 𝓝[f i ∩ U] a := (nhdsWithin_bUnion hfin _ _)
+    _ ≤ ⨆ i, 𝓝[f i ∩ U] a := (supᵢ₂_le_supᵢ _ _)
+    _ ≤ ⨆ i, 𝓝[f i] a := supᵢ_mono fun i => nhdsWithin_mono _ <| inter_subset_left _ _
+    
+#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_unionᵢ
+
+theorem continuousOn_Union' {g : X → Y} (hf : LocallyFinite f)
+    (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : ContinuousOn g (⋃ i, f i) :=
+  by
+  rintro x -
+  rw [ContinuousWithinAt, hf.nhds_within_Union, tendsto_supr]
+  intro i
+  by_cases hx : x ∈ closure (f i)
+  · exact hc i _ hx
+  · rw [mem_closure_iff_nhdsWithin_neBot, not_ne_bot] at hx
+    rw [hx]
+    exact tendsto_bot
+#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_Union'
+
+theorem continuousOn_unionᵢ {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
+    (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) :=
+  hf.continuousOn_Union' fun i x hx => h_cont i x <| (h_cl i).closure_subset hx
+#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_unionᵢ
+
+protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
+    (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : Continuous g :=
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_Union' hc
+#align locally_finite.continuous' LocallyFinite.continuous'
+
+protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
+    (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) : Continuous g :=
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ h_cl h_cont
+#align locally_finite.continuous LocallyFinite.continuous
+
 /- warning: locally_finite.closure -> LocallyFinite.closure is a dubious translation:
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (LocallyFinite.{u1, u2} ι X _inst_1 (fun (i : ι) => closure.{u2} X _inst_1 (f i)))
@@ -152,26 +196,6 @@ protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closur
       (inter_subset_inter_right _ interior_subset)
 #align locally_finite.closure LocallyFinite.closure
 
-/- warning: locally_finite.is_closed_Union -> LocallyFinite.isClosed_unionᵢ is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (IsClosed.{u2} X _inst_1 (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)))
-but is expected to have type
-  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u1} X _inst_1 (f i)) -> (IsClosed.{u1} X _inst_1 (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => f i)))
-Case conversion may be inaccurate. Consider using '#align locally_finite.is_closed_Union LocallyFinite.isClosed_unionᵢₓ'. -/
-theorem isClosed_unionᵢ (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) : IsClosed (⋃ i, f i) :=
-  by
-  simp only [← isOpen_compl_iff, compl_Union, isOpen_iff_mem_nhds, mem_Inter]
-  intro a ha
-  replace ha : ∀ i, f iᶜ ∈ 𝓝 a := fun i => (hc i).isOpen_compl.mem_nhds (ha i)
-  rcases hf a with ⟨t, h_nhds, h_fin⟩
-  have : (t ∩ ⋂ i ∈ { i | (f i ∩ t).Nonempty }, f iᶜ) ∈ 𝓝 a :=
-    inter_mem h_nhds ((bInter_mem h_fin).2 fun i _ => ha i)
-  filter_upwards [this]
-  simp only [mem_inter_iff, mem_Inter]
-  rintro b ⟨hbt, hn⟩ i hfb
-  exact hn i ⟨b, hfb, hbt⟩ hfb
-#align locally_finite.is_closed_Union LocallyFinite.isClosed_unionᵢ
-
 /- warning: locally_finite.closure_Union -> LocallyFinite.closure_unionᵢ is a dubious translation:
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (Eq.{succ u2} (Set.{u2} X) (closure.{u2} X _inst_1 (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i))) (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => closure.{u2} X _inst_1 (f i))))
@@ -179,12 +203,21 @@ but is expected to have type
   forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (Eq.{succ u1} (Set.{u1} X) (closure.{u1} X _inst_1 (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => f i))) (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => closure.{u1} X _inst_1 (f i))))
 Case conversion may be inaccurate. Consider using '#align locally_finite.closure_Union LocallyFinite.closure_unionᵢₓ'. -/
 theorem closure_unionᵢ (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) :=
-  Subset.antisymm
-    (closure_minimal (unionᵢ_mono fun _ => subset_closure) <|
-      h.closure.isClosed_unionᵢ fun _ => isClosed_closure)
-    (unionᵢ_subset fun i => closure_mono <| subset_unionᵢ _ _)
+  by
+  ext x
+  simp only [mem_closure_iff_nhdsWithin_neBot, h.nhds_within_Union, supr_ne_bot, mem_Union]
 #align locally_finite.closure_Union LocallyFinite.closure_unionᵢ
 
+/- warning: locally_finite.is_closed_Union -> LocallyFinite.isClosed_unionᵢ is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (IsClosed.{u2} X _inst_1 (Set.unionᵢ.{u2, succ u1} X ι (fun (i : ι) => f i)))
+but is expected to have type
+  forall {ι : Type.{u2}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)}, (LocallyFinite.{u2, u1} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u1} X _inst_1 (f i)) -> (IsClosed.{u1} X _inst_1 (Set.unionᵢ.{u1, succ u2} X ι (fun (i : ι) => f i)))
+Case conversion may be inaccurate. Consider using '#align locally_finite.is_closed_Union LocallyFinite.isClosed_unionᵢₓ'. -/
+theorem isClosed_unionᵢ (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) : IsClosed (⋃ i, f i) :=
+  by simp only [← closure_eq_iff_isClosed, hf.closure_Union, (hc _).closure_eq]
+#align locally_finite.is_closed_Union LocallyFinite.isClosed_unionᵢ
+
 /- warning: locally_finite.Inter_compl_mem_nhds -> LocallyFinite.interᵢ_compl_mem_nhds is a dubious translation:
 lean 3 declaration is
   forall {ι : Type.{u1}} {X : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} X] {f : ι -> (Set.{u2} X)}, (LocallyFinite.{u1, u2} ι X _inst_1 f) -> (forall (i : ι), IsClosed.{u2} X _inst_1 (f i)) -> (forall (x : X), Membership.Mem.{u2, u2} (Set.{u2} X) (Filter.{u2} X) (Filter.hasMem.{u2} X) (Set.interᵢ.{u2, succ u1} X ι (fun (i : ι) => Set.interᵢ.{u2, 0} X (Not (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (f i))) (fun (hi : Not (Membership.Mem.{u2, u2} X (Set.{u2} X) (Set.hasMem.{u2} X) x (f i))) => HasCompl.compl.{u2} (Set.{u2} X) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} X) (Set.booleanAlgebra.{u2} X)) (f i)))) (nhds.{u2} X _inst_1 x))

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

@@ -272,7 +272,7 @@ end LocallyFinite
 lean 3 declaration is
   forall {ι : Type.{u1}} {ι' : Type.{u2}} {X : Type.{u3}} [_inst_1 : TopologicalSpace.{u3} X] {f : ι -> (Set.{u3} X)} (e : Equiv.{succ u2, succ u1} ι' ι), Iff (LocallyFinite.{u2, u3} ι' X _inst_1 (Function.comp.{succ u2, succ u1, succ u3} ι' ι (Set.{u3} X) f (coeFn.{max 1 (max (succ u2) (succ u1)) (succ u1) (succ u2), max (succ u2) (succ u1)} (Equiv.{succ u2, succ u1} ι' ι) (fun (_x : Equiv.{succ u2, succ u1} ι' ι) => ι' -> ι) (Equiv.hasCoeToFun.{succ u2, succ u1} ι' ι) e))) (LocallyFinite.{u1, u3} ι X _inst_1 f)
 but is expected to have type
-  forall {ι : Type.{u2}} {ι' : Type.{u3}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)} (e : Equiv.{succ u3, succ u2} ι' ι), Iff (LocallyFinite.{u3, u1} ι' X _inst_1 (Function.comp.{succ u3, succ u2, succ u1} ι' ι (Set.{u1} X) f (FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (Equiv.{succ u3, succ u2} ι' ι) ι' (fun (_x : ι') => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.805 : ι') => ι) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} ι' ι) e))) (LocallyFinite.{u2, u1} ι X _inst_1 f)
+  forall {ι : Type.{u2}} {ι' : Type.{u3}} {X : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} X] {f : ι -> (Set.{u1} X)} (e : Equiv.{succ u3, succ u2} ι' ι), Iff (LocallyFinite.{u3, u1} ι' X _inst_1 (Function.comp.{succ u3, succ u2, succ u1} ι' ι (Set.{u1} X) f (FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (Equiv.{succ u3, succ u2} ι' ι) ι' (fun (_x : ι') => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : ι') => ι) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} ι' ι) e))) (LocallyFinite.{u2, u1} ι X _inst_1 f)
 Case conversion may be inaccurate. Consider using '#align equiv.locally_finite_comp_iff Equiv.locallyFinite_comp_iffₓ'. -/
 @[simp]
 theorem Equiv.locallyFinite_comp_iff (e : ι' ≃ ι) : LocallyFinite (f ∘ e) ↔ LocallyFinite f :=

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

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>

@@ -157,10 +157,10 @@ theorem exists_forall_eventually_eq_prod {π : X → Sort*} {f : ℕ → ∀ x :
     ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in atTop ×ˢ 𝓝 x, f p.1 p.2 = F p.2 := by
   choose U hUx hU using hf
   choose N hN using fun x => (hU x).bddAbove
-  replace hN : ∀ (x), ∀ n > N x, ∀ y ∈ U x, f (n + 1) y = f n y
-  exact fun x n hn y hy => by_contra fun hne => hn.lt.not_le <| hN x ⟨y, hne, hy⟩
-  replace hN : ∀ (x), ∀ n ≥ N x + 1, ∀ y ∈ U x, f n y = f (N x + 1) y
-  exact fun x n hn y hy => Nat.le_induction rfl (fun k hle => (hN x _ hle _ hy).trans) n hn
+  replace hN : ∀ (x), ∀ n > N x, ∀ y ∈ U x, f (n + 1) y = f n y :=
+    fun x n hn y hy => by_contra fun hne => hn.lt.not_le <| hN x ⟨y, hne, hy⟩
+  replace hN : ∀ (x), ∀ n ≥ N x + 1, ∀ y ∈ U x, f n y = f (N x + 1) y :=
+    fun x n hn y hy => Nat.le_induction rfl (fun k hle => (hN x _ hle _ hy).trans) n hn
   refine ⟨fun x => f (N x + 1) x, fun x => ?_⟩
   filter_upwards [Filter.prod_mem_prod (eventually_gt_atTop (N x)) (hUx x)]
   rintro ⟨n, y⟩ ⟨hn : N x < n, hy : y ∈ U x⟩

From sphere-eversion; I'm just submitting this upstream.

@@ -233,3 +233,10 @@ theorem LocallyFinite.option_elim' (hf : LocallyFinite f) (s : Set X) :
     LocallyFinite (Option.elim' s f) :=
   locallyFinite_option.2 hf
 #align locally_finite.option_elim LocallyFinite.option_elim'
+
+theorem LocallyFinite.eventually_subset {s : ι → Set X}
+    (hs : LocallyFinite s) (hs' : ∀ i, IsClosed (s i)) (x : X) :
+    ∀ᶠ y in 𝓝 x, {i | y ∈ s i} ⊆ {i | x ∈ s i} := by
+  filter_upwards [hs.iInter_compl_mem_nhds hs' x] with y hy i hi
+  simp only [mem_iInter, mem_compl_iff] at hy
+  exact not_imp_not.mp (hy i) hi

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

This has nice performance benefits.

@@ -20,7 +20,7 @@ In this file we give the definition and prove basic properties of locally finite
 -- locally finite family [General Topology (Bourbaki, 1995)]
 open Set Function Filter Topology
 
-variable {ι ι' α X Y : Type _} [TopologicalSpace X] [TopologicalSpace Y] {f g : ι → Set X}
+variable {ι ι' α X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f g : ι → Set X}
 
 /-- A family of sets in `Set X` is locally finite if at every point `x : X`,
 there is a neighborhood of `x` which meets only finitely many sets in the family. -/
@@ -68,7 +68,7 @@ protected theorem eventually_smallSets (hf : LocallyFinite f) (x : X) :
   locallyFinite_iff_smallSets.mp hf x
 #align locally_finite.eventually_small_sets LocallyFinite.eventually_smallSets
 
-theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X}
+theorem exists_mem_basis {ι' : Sort*} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X}
     {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ i, p i ∧ { j | (f j ∩ s i).Nonempty }.Finite :=
   let ⟨i, hpi, hi⟩ := hb.smallSets.eventually_iff.mp (hf.eventually_smallSets x)
   ⟨i, hpi, hi Subset.rfl⟩
@@ -152,7 +152,7 @@ function `F : Π a, β a` such that for any `x`, we have `f n x = F x` on the pr
 interval `[N, +∞)` and a neighbourhood of `x`.
 
 We formulate the conclusion in terms of the product of filter `Filter.atTop` and `𝓝 x`. -/
-theorem exists_forall_eventually_eq_prod {π : X → Sort _} {f : ℕ → ∀ x : X, π x}
+theorem exists_forall_eventually_eq_prod {π : X → Sort*} {f : ℕ → ∀ x : X, π x}
     (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
     ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in atTop ×ˢ 𝓝 x, f p.1 p.2 = F p.2 := by
   choose U hUx hU using hf
@@ -174,7 +174,7 @@ theorem exists_forall_eventually_eq_prod {π : X → Sort _} {f : ℕ → ∀ x
 that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a
 function `F : Π a, β a` such that for any `x`, for sufficiently large values of `n`, we have
 `f n y = F y` in a neighbourhood of `x`. -/
-theorem exists_forall_eventually_atTop_eventually_eq' {π : X → Sort _} {f : ℕ → ∀ x : X, π x}
+theorem exists_forall_eventually_atTop_eventually_eq' {π : X → Sort*} {f : ℕ → ∀ x : X, π x}
     (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
     ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ n : ℕ in atTop, ∀ᶠ y : X in 𝓝 x, f n y = F y :=
   hf.exists_forall_eventually_eq_prod.imp fun _F hF x => (hF x).curry

@@ -59,7 +59,7 @@ theorem comp_injective {g : ι' → ι} (hf : LocallyFinite f) (hg : Injective g
 theorem _root_.locallyFinite_iff_smallSets :
     LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, { i | (f i ∩ s).Nonempty }.Finite :=
   forall_congr' fun _ => Iff.symm <|
-    eventually_small_sets' fun _s _t hst ht =>
+    eventually_smallSets' fun _s _t hst ht =>
       ht.subset fun _i hi => hi.mono <| inter_subset_inter_right _ hst
 #align locally_finite_iff_small_sets locallyFinite_iff_smallSets
 

A typo restricted LocallyFinite.prod_left to (f g : ι → Set Y) instead of (f : ι → Set X) (g : ι → Set Y).

@@ -199,7 +199,7 @@ theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuou
 theorem prod_right (hf : LocallyFinite f) (g : ι → Set Y) : LocallyFinite (fun i ↦ f i ×ˢ g i) :=
   (hf.preimage_continuous continuous_fst).subset fun _ ↦ prod_subset_preimage_fst _ _
 
-theorem prod_left {g : ι → Set Y} (hg : LocallyFinite g) (f : ι → Set Y) :
+theorem prod_left {g : ι → Set Y} (hg : LocallyFinite g) (f : ι → Set X) :
     LocallyFinite (fun i ↦ f i ×ˢ g i) :=
   (hg.preimage_continuous continuous_snd).subset fun _ ↦ prod_subset_preimage_snd _ _
 

• 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) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.locally_finite
-! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.ContinuousOn
 import Mathlib.Order.Filter.SmallSets
 
+#align_import topology.locally_finite from "leanprover-community/mathlib"@"55d771df074d0dd020139ee1cd4b95521422df9f"
+
 /-!
 ### Locally finite families of sets
 

From the sphere eversion project.

@@ -194,11 +194,18 @@ theorem exists_forall_eventually_atTop_eventuallyEq {f : ℕ → X → α}
 #align locally_finite.exists_forall_eventually_at_top_eventually_eq LocallyFinite.exists_forall_eventually_atTop_eventuallyEq
 
 theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g) :
-    LocallyFinite fun i => g ⁻¹' f i := fun x =>
+    LocallyFinite (g ⁻¹' f ·) := fun x =>
   let ⟨s, hsx, hs⟩ := hf (g x)
   ⟨g ⁻¹' s, hg.continuousAt hsx, hs.subset fun _ ⟨y, hy⟩ => ⟨g y, hy⟩⟩
 #align locally_finite.preimage_continuous LocallyFinite.preimage_continuous
 
+theorem prod_right (hf : LocallyFinite f) (g : ι → Set Y) : LocallyFinite (fun i ↦ f i ×ˢ g i) :=
+  (hf.preimage_continuous continuous_fst).subset fun _ ↦ prod_subset_preimage_fst _ _
+
+theorem prod_left {g : ι → Set Y} (hg : LocallyFinite g) (f : ι → Set Y) :
+    LocallyFinite (fun i ↦ f i ×ˢ g i) :=
+  (hg.preimage_continuous continuous_snd).subset fun _ ↦ prod_subset_preimage_snd _ _
+
 end LocallyFinite
 
 @[simp]

@@ -109,13 +109,13 @@ theorem continuousOn_iUnion {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i,
   hf.continuousOn_iUnion' fun i x hx ↦ h_cont i x <| (h_cl i).closure_subset hx
 #align locally_finite.continuous_on_Union LocallyFinite.continuousOn_iUnion
 
-protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
+protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = univ)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
     Continuous g :=
   continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion' hc
 #align locally_finite.continuous' LocallyFinite.continuous'
 
-protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
+protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = univ)
     (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) :
     Continuous g :=
   continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion h_cl h_cont

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

@@ -142,7 +142,7 @@ theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) :
 /-- If `f : β → Set α` is a locally finite family of closed sets, then for any `x : α`, the
 intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
 theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
-    (⋂ (i) (_ : x ∉ f i), f iᶜ) ∈ 𝓝 x := by
+    (⋂ (i) (_ : x ∉ f i), (f i)ᶜ) ∈ 𝓝 x := by
   refine' IsOpen.mem_nhds _ (mem_iInter₂.2 fun i => id)
   suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
     rwa [← isOpen_compl_iff, compl_iUnion, iInter_subtype] at this

@@ -142,7 +142,7 @@ theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) :
 /-- If `f : β → Set α` is a locally finite family of closed sets, then for any `x : α`, the
 intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
 theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
-    (⋂ (i) (_hi : x ∉ f i), f iᶜ) ∈ 𝓝 x := by
+    (⋂ (i) (_ : x ∉ f i), f iᶜ) ∈ 𝓝 x := by
   refine' IsOpen.mem_nhds _ (mem_iInter₂.2 fun i => id)
   suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
     rwa [← isOpen_compl_iff, compl_iUnion, iInter_subtype] at this

Currently, the following notations are changed from · ×ˢ · because Lean 4 can't deal with ambiguous notations. | Definition | Notation | | :

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -157,7 +157,7 @@ interval `[N, +∞)` and a neighbourhood of `x`.
 We formulate the conclusion in terms of the product of filter `Filter.atTop` and `𝓝 x`. -/
 theorem exists_forall_eventually_eq_prod {π : X → Sort _} {f : ℕ → ∀ x : X, π x}
     (hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
-    ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in atTop ×ᶠ 𝓝 x, f p.1 p.2 = F p.2 := by
+    ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in atTop ×ˢ 𝓝 x, f p.1 p.2 = F p.2 := by
   choose U hUx hU using hf
   choose N hN using fun x => (hU x).bddAbove
   replace hN : ∀ (x), ∀ n > N x, ∀ y ∈ U x, f (n + 1) y = f n y

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

@@ -139,7 +139,7 @@ theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) :
   simp only [← closure_eq_iff_isClosed, hf.closure_iUnion, (hc _).closure_eq]
 #align locally_finite.is_closed_Union LocallyFinite.isClosed_iUnion
 
-/-- If `f : β → set α` is a locally finite family of closed sets, then for any `x : α`, the
+/-- If `f : β → Set α` is a locally finite family of closed sets, then for any `x : α`, the
 intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
 theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
     (⋂ (i) (_hi : x ∉ f i), f iᶜ) ∈ 𝓝 x := by

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>

@@ -77,48 +77,48 @@ theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
 
-protected theorem nhdsWithin_unionᵢ (hf : LocallyFinite f) (a : X) :
+protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) :
     𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a := by
   rcases hf a with ⟨U, haU, hfin⟩
-  refine le_antisymm ?_ (Monotone.le_map_supᵢ fun _ _ ↦ nhdsWithin_mono _)
+  refine le_antisymm ?_ (Monotone.le_map_iSup fun _ _ ↦ nhdsWithin_mono _)
   calc
     𝓝[⋃ i, f i] a = 𝓝[⋃ i, f i ∩ U] a := by
-      rw [← unionᵢ_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]
+      rw [← iUnion_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]
     _ = 𝓝[⋃ i ∈ {j | (f j ∩ U).Nonempty}, (f i ∩ U)] a := by
-      simp only [mem_setOf_eq, unionᵢ_nonempty_self]
-    _ = ⨆ i ∈ {j | (f j ∩ U).Nonempty}, 𝓝[f i ∩ U] a := nhdsWithin_bunionᵢ hfin _ _
-    _ ≤ ⨆ i, 𝓝[f i ∩ U] a := supᵢ₂_le_supᵢ _ _
-    _ ≤ ⨆ i, 𝓝[f i] a := supᵢ_mono fun i ↦ nhdsWithin_mono _ <| inter_subset_left _ _
-#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_unionᵢ
+      simp only [mem_setOf_eq, iUnion_nonempty_self]
+    _ = ⨆ i ∈ {j | (f j ∩ U).Nonempty}, 𝓝[f i ∩ U] a := nhdsWithin_biUnion hfin _ _
+    _ ≤ ⨆ i, 𝓝[f i ∩ U] a := iSup₂_le_iSup _ _
+    _ ≤ ⨆ i, 𝓝[f i] a := iSup_mono fun i ↦ nhdsWithin_mono _ <| inter_subset_left _ _
+#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_iUnion
 
-theorem continuousOn_unionᵢ' {g : X → Y} (hf : LocallyFinite f)
+theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
     ContinuousOn g (⋃ i, f i) := by
   rintro x -
-  rw [ContinuousWithinAt, hf.nhdsWithin_unionᵢ, tendsto_supᵢ]
+  rw [ContinuousWithinAt, hf.nhdsWithin_iUnion, tendsto_iSup]
   intro i
   by_cases hx : x ∈ closure (f i)
   · exact hc i _ hx
   · rw [mem_closure_iff_nhdsWithin_neBot, not_neBot] at hx
     rw [hx]
     exact tendsto_bot
-#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_unionᵢ'
+#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_iUnion'
 
-theorem continuousOn_unionᵢ {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
+theorem continuousOn_iUnion {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
     (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) :=
-  hf.continuousOn_unionᵢ' fun i x hx ↦ h_cont i x <| (h_cl i).closure_subset hx
-#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_unionᵢ
+  hf.continuousOn_iUnion' fun i x hx ↦ h_cont i x <| (h_cl i).closure_subset hx
+#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_iUnion
 
 protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
     Continuous g :=
-  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ' hc
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion' hc
 #align locally_finite.continuous' LocallyFinite.continuous'
 
 protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
     (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) :
     Continuous g :=
-  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ h_cl h_cont
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_iUnion h_cl h_cont
 #align locally_finite.continuous LocallyFinite.continuous
 
 protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closure (f i) := by
@@ -129,25 +129,25 @@ protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closur
     (inter_subset_inter_right _ interior_subset)
 #align locally_finite.closure LocallyFinite.closure
 
-theorem closure_unionᵢ (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) := by
+theorem closure_iUnion (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) := by
   ext x
-  simp only [mem_closure_iff_nhdsWithin_neBot, h.nhdsWithin_unionᵢ, supᵢ_neBot, mem_unionᵢ]
-#align locally_finite.closure_Union LocallyFinite.closure_unionᵢ
+  simp only [mem_closure_iff_nhdsWithin_neBot, h.nhdsWithin_iUnion, iSup_neBot, mem_iUnion]
+#align locally_finite.closure_Union LocallyFinite.closure_iUnion
 
-theorem isClosed_unionᵢ (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) :
+theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) :
     IsClosed (⋃ i, f i) := by
-  simp only [← closure_eq_iff_isClosed, hf.closure_unionᵢ, (hc _).closure_eq]
-#align locally_finite.is_closed_Union LocallyFinite.isClosed_unionᵢ
+  simp only [← closure_eq_iff_isClosed, hf.closure_iUnion, (hc _).closure_eq]
+#align locally_finite.is_closed_Union LocallyFinite.isClosed_iUnion
 
 /-- If `f : β → set α` is a locally finite family of closed sets, then for any `x : α`, the
 intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
-theorem interᵢ_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
+theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :
     (⋂ (i) (_hi : x ∉ f i), f iᶜ) ∈ 𝓝 x := by
-  refine' IsOpen.mem_nhds _ (mem_interᵢ₂.2 fun i => id)
+  refine' IsOpen.mem_nhds _ (mem_iInter₂.2 fun i => id)
   suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
-    rwa [← isOpen_compl_iff, compl_unionᵢ, interᵢ_subtype] at this
-  exact (hf.comp_injective Subtype.val_injective).isClosed_unionᵢ fun i => hc _
-#align locally_finite.Inter_compl_mem_nhds LocallyFinite.interᵢ_compl_mem_nhds
+    rwa [← isOpen_compl_iff, compl_iUnion, iInter_subtype] at this
+  exact (hf.comp_injective Subtype.val_injective).isClosed_iUnion fun i => hc _
+#align locally_finite.Inter_compl_mem_nhds LocallyFinite.iInter_compl_mem_nhds
 
 /-- Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose
 that the family of sets `s n = {x | f (n + 1) x ≠ f n x}` is locally finite. Then there exists a

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

@@ -4,11 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.locally_finite
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 55d771df074d0dd020139ee1cd4b95521422df9f
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
-import Mathlib.Topology.Basic
+import Mathlib.Topology.ContinuousOn
 import Mathlib.Order.Filter.SmallSets
 
 /-!
@@ -77,36 +77,68 @@ theorem exists_mem_basis {ι' : Sort _} (hf : LocallyFinite f) {p : ι' → Prop
   ⟨i, hpi, hi Subset.rfl⟩
 #align locally_finite.exists_mem_basis LocallyFinite.exists_mem_basis
 
+protected theorem nhdsWithin_unionᵢ (hf : LocallyFinite f) (a : X) :
+    𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a := by
+  rcases hf a with ⟨U, haU, hfin⟩
+  refine le_antisymm ?_ (Monotone.le_map_supᵢ fun _ _ ↦ nhdsWithin_mono _)
+  calc
+    𝓝[⋃ i, f i] a = 𝓝[⋃ i, f i ∩ U] a := by
+      rw [← unionᵢ_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)]
+    _ = 𝓝[⋃ i ∈ {j | (f j ∩ U).Nonempty}, (f i ∩ U)] a := by
+      simp only [mem_setOf_eq, unionᵢ_nonempty_self]
+    _ = ⨆ i ∈ {j | (f j ∩ U).Nonempty}, 𝓝[f i ∩ U] a := nhdsWithin_bunionᵢ hfin _ _
+    _ ≤ ⨆ i, 𝓝[f i ∩ U] a := supᵢ₂_le_supᵢ _ _
+    _ ≤ ⨆ i, 𝓝[f i] a := supᵢ_mono fun i ↦ nhdsWithin_mono _ <| inter_subset_left _ _
+#align locally_finite.nhds_within_Union LocallyFinite.nhdsWithin_unionᵢ
+
+theorem continuousOn_unionᵢ' {g : X → Y} (hf : LocallyFinite f)
+    (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
+    ContinuousOn g (⋃ i, f i) := by
+  rintro x -
+  rw [ContinuousWithinAt, hf.nhdsWithin_unionᵢ, tendsto_supᵢ]
+  intro i
+  by_cases hx : x ∈ closure (f i)
+  · exact hc i _ hx
+  · rw [mem_closure_iff_nhdsWithin_neBot, not_neBot] at hx
+    rw [hx]
+    exact tendsto_bot
+#align locally_finite.continuous_on_Union' LocallyFinite.continuousOn_unionᵢ'
+
+theorem continuousOn_unionᵢ {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i))
+    (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) :=
+  hf.continuousOn_unionᵢ' fun i x hx ↦ h_cont i x <| (h_cl i).closure_subset hx
+#align locally_finite.continuous_on_Union LocallyFinite.continuousOn_unionᵢ
+
+protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
+    (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
+    Continuous g :=
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ' hc
+#align locally_finite.continuous' LocallyFinite.continuous'
+
+protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : (⋃ i, f i) = univ)
+    (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) :
+    Continuous g :=
+  continuous_iff_continuousOn_univ.2 <| h_cov ▸ hf.continuousOn_unionᵢ h_cl h_cont
+#align locally_finite.continuous LocallyFinite.continuous
+
 protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closure (f i) := by
   intro x
   rcases hf x with ⟨s, hsx, hsf⟩
   refine' ⟨interior s, interior_mem_nhds.2 hsx, hsf.subset fun i hi => _⟩
-  exact
-    (hi.mono isOpen_interior.closure_inter).of_closure.mono
-      (inter_subset_inter_right _ interior_subset)
+  exact (hi.mono isOpen_interior.closure_inter).of_closure.mono
+    (inter_subset_inter_right _ interior_subset)
 #align locally_finite.closure LocallyFinite.closure
 
+theorem closure_unionᵢ (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) := by
+  ext x
+  simp only [mem_closure_iff_nhdsWithin_neBot, h.nhdsWithin_unionᵢ, supᵢ_neBot, mem_unionᵢ]
+#align locally_finite.closure_Union LocallyFinite.closure_unionᵢ
+
 theorem isClosed_unionᵢ (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) :
     IsClosed (⋃ i, f i) := by
-  simp only [← isOpen_compl_iff, compl_unionᵢ, isOpen_iff_mem_nhds, mem_interᵢ]
-  intro a ha
-  replace ha : ∀ i, f iᶜ ∈ 𝓝 a := fun i => (hc i).compl_mem_nhds (ha i)
-  rcases hf a with ⟨t, h_nhds, h_fin⟩
-  have : (t ∩ ⋂ i ∈ { i | (f i ∩ t).Nonempty }, f iᶜ) ∈ 𝓝 a :=
-    inter_mem h_nhds ((binterᵢ_mem h_fin).2 fun i _ => ha i)
-  filter_upwards [this]
-  simp only [mem_inter_iff, mem_interᵢ]
-  rintro b ⟨hbt, hn⟩ i hfb
-  exact hn i ⟨b, hfb, hbt⟩ hfb
+  simp only [← closure_eq_iff_isClosed, hf.closure_unionᵢ, (hc _).closure_eq]
 #align locally_finite.is_closed_Union LocallyFinite.isClosed_unionᵢ
 
-theorem closure_unionᵢ (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) :=
-  Subset.antisymm
-    (closure_minimal (unionᵢ_mono fun _ => subset_closure) <|
-      h.closure.isClosed_unionᵢ fun _ => isClosed_closure)
-    (unionᵢ_subset fun _ => closure_mono <| subset_unionᵢ _ _)
-#align locally_finite.closure_Union LocallyFinite.closure_unionᵢ
-
 /-- If `f : β → set α` is a locally finite family of closed sets, then for any `x : α`, the
 intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/
 theorem interᵢ_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) :

This PR resyncs the first 28 entries of https://leanprover-community.github.io/mathlib-port-status/out-of-sync.html after sorting by diff size.

• resync Mathlib/Data/Bool/Count
• resync Mathlib/Order/Max
• resync Mathlib/Algebra/EuclideanDomain/Instances
• resync Mathlib/Data/List/Duplicate
• resync Mathlib/Data/Multiset/Nodup
• resync Mathlib/Data/Set/Pointwise/ListOfFn
• resync Mathlib/Dynamics/FixedPoints/Basic
• resync Mathlib/Order/OmegaCompletePartialOrder
• resync Mathlib/Order/PropInstances
• resync Mathlib/Topology/LocallyFinite
• resync Mathlib/Data/Bool/Set
• resync Mathlib/Data/Fintype/Card
• resync Mathlib/Data/Multiset/Bind
• resync Mathlib/Data/Rat/Floor
• resync Mathlib/Algebra/Order/Floor
• resync Mathlib/Data/Int/Basic
• resync Mathlib/Data/Int/Dvd/Basic
• resync Mathlib/Data/List/Sort
• resync Mathlib/Data/Nat/GCD/Basic
• resync Mathlib/Data/Set/Enumerate
• resync Mathlib/Data/Set/Intervals/OrdConnectedComponent
• resync Mathlib/GroupTheory/Subsemigroup/Basic
• resync Mathlib/Topology/Connected
• resync Mathlib/Topology/NhdsSet
• resync Mathlib/Algebra/BigOperators/Multiset/Lemmas
• resync Mathlib/Algebra/CharZero/Infinite
• resync Mathlib/Data/Multiset/Range
• resync Mathlib/Data/Set/Pointwise/Finite

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.locally_finite
-! leanprover-community/mathlib commit f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c
+! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/