topology.uniform_space.compact_convergence ⟷ Mathlib.Topology.UniformSpace.CompactConvergence

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

@@ -318,7 +318,7 @@ theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
         {fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V} ⊆ X :=
   by
   simp only [has_basis_compact_convergence_uniformity_aux.mem_iff, exists_prop, Prod.exists,
-    and_assoc']
+    and_assoc]
 #align continuous_map.mem_compact_convergence_uniformity ContinuousMap.mem_compactConvergenceUniformity
 
 #print ContinuousMap.compactConvergenceUniformSpace /-

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

@@ -210,8 +210,8 @@ convergence.
 
 The topology of compact convergence is thus at least as fine as the compact-open topology. -/
 theorem compactConvNhd_subset_compact_open (hK : IsCompact K) {U : Set β} (hU : IsOpen U)
-    (hf : f ∈ CompactOpen.gen K U) :
-    ∃ V ∈ 𝓤 β, IsOpen V ∧ compactConvNhd K V f ⊆ CompactOpen.gen K U :=
+    (hf : f ∈ compactOpen.gen K U) :
+    ∃ V ∈ 𝓤 β, IsOpen V ∧ compactConvNhd K V f ⊆ compactOpen.gen K U :=
   by
   obtain ⟨V, hV₁, hV₂, hV₃⟩ := lebesgue_number_of_compact_open (hK.image f.continuous) hU hf
   refine' ⟨V, hV₁, hV₂, _⟩
@@ -227,8 +227,8 @@ the compact-open topology is at least as fine as the topology of compact converg
 theorem iInter_compact_open_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
     ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (hC : ∀ i, IsCompact (C i)) (U :
       ι → Set β) (hU : ∀ i, IsOpen (U i)),
-      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
-        (⋂ i, CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f :=
+      (f ∈ ⋂ i, compactOpen.gen (C i) (U i)) ∧
+        (⋂ i, compactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f :=
   by
   obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
   obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
@@ -263,7 +263,7 @@ theorem iInter_compact_open_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V
     ⟨t, t.fintype_coe_sort, C, fun i => hK.inter_right isClosed_closure, fun i =>
       ball (f ((i : K) : α)) W, fun i => is_open_ball _ hW₄, by simp [compact_open.gen, hfC],
       fun g hg x hx => hW₃ (mem_comp_rel.mpr _)⟩
-  simp only [mem_Inter, compact_open.gen, mem_set_of_eq, image_subset_iff] at hg 
+  simp only [mem_Inter, compact_open.gen, mem_set_of_eq, image_subset_iff] at hg
   obtain ⟨y, hy⟩ := mem_Union.mp (hC hx)
   exact ⟨f y, (mem_ball_symmetry hW₂).mp (hfC y hy), mem_preimage.mp (hg y hy)⟩
 #align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.iInter_compact_open_gen_subset_compactConvNhd
@@ -284,7 +284,7 @@ theorem compactOpen_eq_compact_convergence :
     haveI := hι
     exact
       ⟨⋂ i, compact_open.gen (C i) (U i), h₂.trans hXf,
-        isOpen_iInter_of_finite fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
+        isOpen_iInter_of_finite fun i => ContinuousMap.isOpen_setOf_mapsTo (hC i) (hU i), h₁⟩
   · simp only [TopologicalSpace.le_generateFrom_iff_subset_isOpen, and_imp, exists_prop,
       forall_exists_index, set_of_subset_set_of]
     rintro - K hK U hU rfl f hf
@@ -428,7 +428,7 @@ See also `tendsto_iff_tendsto_locally_uniformly`, especially for T2 spaces. -/
 theorem tendstoLocallyUniformly_of_tendsto (hα : ∀ x : α, ∃ n, IsCompact n ∧ n ∈ 𝓝 x)
     (h : Tendsto F p (𝓝 f)) : TendstoLocallyUniformly (fun i a => F i a) f p :=
   by
-  rw [tendsto_iff_forall_compact_tendsto_uniformly_on] at h 
+  rw [tendsto_iff_forall_compact_tendsto_uniformly_on] at h
   intro V hV x
   obtain ⟨n, hn₁, hn₂⟩ := hα x
   exact ⟨n, hn₂, h n hn₁ V hV⟩

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

@@ -92,37 +92,28 @@ variable (K : Set α) (V : Set (β × β)) (f : C(α, β))
 
 namespace ContinuousMap
 
-#print ContinuousMap.compactConvNhd /-
 /-- Given `K ⊆ α`, `V ⊆ β × β`, and `f : C(α, β)`, we define `compact_conv_nhd K V f` to be the set
 of `g : C(α, β)` that are `V`-close to `f` on `K`. -/
 def compactConvNhd : Set C(α, β) :=
   {g | ∀ x ∈ K, (f x, g x) ∈ V}
 #align continuous_map.compact_conv_nhd ContinuousMap.compactConvNhd
--/
 
 variable {K V}
 
-#print ContinuousMap.self_mem_compactConvNhd /-
 theorem self_mem_compactConvNhd (hV : V ∈ 𝓤 β) : f ∈ compactConvNhd K V f := fun x hx =>
   refl_mem_uniformity hV
 #align continuous_map.self_mem_compact_conv_nhd ContinuousMap.self_mem_compactConvNhd
--/
 
-#print ContinuousMap.compactConvNhd_mono /-
 @[mono]
 theorem compactConvNhd_mono {V' : Set (β × β)} (hV' : V' ⊆ V) :
     compactConvNhd K V' f ⊆ compactConvNhd K V f := fun x hx a ha => hV' (hx a ha)
 #align continuous_map.compact_conv_nhd_mono ContinuousMap.compactConvNhd_mono
--/
 
-#print ContinuousMap.compactConvNhd_mem_comp /-
 theorem compactConvNhd_mem_comp {g₁ g₂ : C(α, β)} {V' : Set (β × β)}
     (hg₁ : g₁ ∈ compactConvNhd K V f) (hg₂ : g₂ ∈ compactConvNhd K V' g₁) :
     g₂ ∈ compactConvNhd K (V ○ V') f := fun x hx => ⟨g₁ x, hg₁ x hx, hg₂ x hx⟩
 #align continuous_map.compact_conv_nhd_mem_comp ContinuousMap.compactConvNhd_mem_comp
--/
 
-#print ContinuousMap.compactConvNhd_nhd_basis /-
 /-- A key property of `compact_conv_nhd`. It allows us to apply
 `topological_space.nhds_mk_of_nhds_filter_basis` below. -/
 theorem compactConvNhd_nhd_basis (hV : V ∈ 𝓤 β) :
@@ -134,46 +125,36 @@ theorem compactConvNhd_nhd_basis (hV : V ∈ 𝓤 β) :
     ⟨V', h₁, subset.trans (subset_comp_self_of_mem_uniformity h₁) h₂, fun g hg g' hg' =>
       compact_conv_nhd_mono f h₂ (compact_conv_nhd_mem_comp f hg hg')⟩
 #align continuous_map.compact_conv_nhd_nhd_basis ContinuousMap.compactConvNhd_nhd_basis
--/
 
-#print ContinuousMap.compactConvNhd_subset_inter /-
 theorem compactConvNhd_subset_inter (K₁ K₂ : Set α) (V₁ V₂ : Set (β × β)) :
     compactConvNhd (K₁ ∪ K₂) (V₁ ∩ V₂) f ⊆ compactConvNhd K₁ V₁ f ∩ compactConvNhd K₂ V₂ f :=
   fun g hg =>
   ⟨fun x hx => mem_of_mem_inter_left (hg x (mem_union_left K₂ hx)), fun x hx =>
     mem_of_mem_inter_right (hg x (mem_union_right K₁ hx))⟩
 #align continuous_map.compact_conv_nhd_subset_inter ContinuousMap.compactConvNhd_subset_inter
--/
 
-#print ContinuousMap.compactConvNhd_compact_entourage_nonempty /-
-theorem compactConvNhd_compact_entourage_nonempty :
+theorem compact_conv_nhd_compact_entourage_nonempty :
     {KV : Set α × Set (β × β) | IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β}.Nonempty :=
   ⟨⟨∅, univ⟩, isCompact_empty, Filter.univ_mem⟩
-#align continuous_map.compact_conv_nhd_compact_entourage_nonempty ContinuousMap.compactConvNhd_compact_entourage_nonempty
--/
+#align continuous_map.compact_conv_nhd_compact_entourage_nonempty ContinuousMap.compact_conv_nhd_compact_entourage_nonempty
 
-#print ContinuousMap.compactConvNhd_filter_isBasis /-
 theorem compactConvNhd_filter_isBasis :
     Filter.IsBasis (fun KV : Set α × Set (β × β) => IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β) fun KV =>
       compactConvNhd KV.1 KV.2 f :=
-  { Nonempty := compactConvNhd_compact_entourage_nonempty
+  { Nonempty := compact_conv_nhd_compact_entourage_nonempty
     inter := by
       rintro ⟨K₁, V₁⟩ ⟨K₂, V₂⟩ ⟨hK₁, hV₁⟩ ⟨hK₂, hV₂⟩
       exact
         ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, Filter.inter_mem hV₁ hV₂⟩,
           compact_conv_nhd_subset_inter f K₁ K₂ V₁ V₂⟩ }
 #align continuous_map.compact_conv_nhd_filter_is_basis ContinuousMap.compactConvNhd_filter_isBasis
--/
 
-#print ContinuousMap.compactConvergenceFilterBasis /-
 /-- A filter basis for the neighbourhood filter of a point in the compact-convergence topology. -/
 def compactConvergenceFilterBasis (f : C(α, β)) : FilterBasis C(α, β) :=
   (compactConvNhd_filter_isBasis f).FilterBasis
 #align continuous_map.compact_convergence_filter_basis ContinuousMap.compactConvergenceFilterBasis
--/
 
-#print ContinuousMap.mem_compactConvergence_nhd_filter /-
-theorem mem_compactConvergence_nhd_filter (Y : Set C(α, β)) :
+theorem mem_compact_convergence_nhd_filter (Y : Set C(α, β)) :
     Y ∈ (compactConvergenceFilterBasis f).filterₓ ↔
       ∃ (K : Set α) (V : Set (β × β)) (hK : IsCompact K) (hV : V ∈ 𝓤 β), compactConvNhd K V f ⊆ Y :=
   by
@@ -182,20 +163,16 @@ theorem mem_compactConvergence_nhd_filter (Y : Set C(α, β)) :
     exact ⟨K, V, hK, hV, hY⟩
   · rintro ⟨K, V, hK, hV, hY⟩
     exact ⟨compact_conv_nhd K V f, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩
-#align continuous_map.mem_compact_convergence_nhd_filter ContinuousMap.mem_compactConvergence_nhd_filter
--/
+#align continuous_map.mem_compact_convergence_nhd_filter ContinuousMap.mem_compact_convergence_nhd_filter
 
-#print ContinuousMap.compactConvergenceTopology /-
 /-- The compact-convergence topology. In fact, see `compact_open_eq_compact_convergence` this is
 the same as the compact-open topology. This definition is thus an auxiliary convenience definition
 and is unlikely to be of direct use. -/
 def compactConvergenceTopology : TopologicalSpace C(α, β) :=
   TopologicalSpace.mkOfNhds fun f => (compactConvergenceFilterBasis f).filterₓ
 #align continuous_map.compact_convergence_topology ContinuousMap.compactConvergenceTopology
--/
 
-#print ContinuousMap.nhds_compactConvergence /-
-theorem nhds_compactConvergence :
+theorem nhds_compact_convergence :
     @nhds _ compactConvergenceTopology f = (compactConvergenceFilterBasis f).filterₓ :=
   by
   rw [TopologicalSpace.nhds_mkOfNhds_filterBasis] <;> rintro g - ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩
@@ -204,23 +181,19 @@ theorem nhds_compactConvergence :
     exact
       ⟨compact_conv_nhd K V' g, ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, compact_conv_nhd_mono g h₁, fun g' hg' =>
         ⟨compact_conv_nhd K V' g', ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, h₂ g' hg'⟩⟩
-#align continuous_map.nhds_compact_convergence ContinuousMap.nhds_compactConvergence
--/
+#align continuous_map.nhds_compact_convergence ContinuousMap.nhds_compact_convergence
 
-#print ContinuousMap.hasBasis_nhds_compactConvergence /-
-theorem hasBasis_nhds_compactConvergence :
+theorem hasBasis_nhds_compact_convergence :
     HasBasis (@nhds _ compactConvergenceTopology f)
       (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
       compactConvNhd p.1 p.2 f :=
-  (nhds_compactConvergence f).symm ▸ (compactConvNhd_filter_isBasis f).HasBasis
-#align continuous_map.has_basis_nhds_compact_convergence ContinuousMap.hasBasis_nhds_compactConvergence
--/
+  (nhds_compact_convergence f).symm ▸ (compactConvNhd_filter_isBasis f).HasBasis
+#align continuous_map.has_basis_nhds_compact_convergence ContinuousMap.hasBasis_nhds_compact_convergence
 
-#print ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn' /-
 /-- This is an auxiliary lemma and is unlikely to be of direct use outside of this file. See
 `tendsto_iff_forall_compact_tendsto_uniformly_on` below for the useful version where the topology
 is picked up via typeclass inference. -/
-theorem tendsto_iff_forall_compact_tendstoUniformlyOn' {ι : Type u₃} {p : Filter ι}
+theorem tendsto_iff_forall_compact_tendsto_uniformly_on' {ι : Type u₃} {p : Filter ι}
     {F : ι → C(α, β)} :
     Filter.Tendsto F p (@nhds _ compactConvergenceTopology f) ↔
       ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K :=
@@ -230,15 +203,13 @@ theorem tendsto_iff_forall_compact_tendstoUniformlyOn' {ι : Type u₃} {p : Fil
   refine' forall_congr' fun K => _
   rw [forall_swap]
   exact forall₃_congr fun hK V hV => Iff.rfl
-#align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on' ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn'
--/
+#align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on' ContinuousMap.tendsto_iff_forall_compact_tendsto_uniformly_on'
 
-#print ContinuousMap.compactConvNhd_subset_compactOpen /-
 /-- Any point of `compact_open.gen K U` is also an interior point wrt the topology of compact
 convergence.
 
 The topology of compact convergence is thus at least as fine as the compact-open topology. -/
-theorem compactConvNhd_subset_compactOpen (hK : IsCompact K) {U : Set β} (hU : IsOpen U)
+theorem compactConvNhd_subset_compact_open (hK : IsCompact K) {U : Set β} (hU : IsOpen U)
     (hf : f ∈ CompactOpen.gen K U) :
     ∃ V ∈ 𝓤 β, IsOpen V ∧ compactConvNhd K V f ⊆ CompactOpen.gen K U :=
   by
@@ -246,16 +217,14 @@ theorem compactConvNhd_subset_compactOpen (hK : IsCompact K) {U : Set β} (hU :
   refine' ⟨V, hV₁, hV₂, _⟩
   rintro g hg _ ⟨x, hx, rfl⟩
   exact hV₃ (f x) ⟨x, hx, rfl⟩ (hg x hx)
-#align continuous_map.compact_conv_nhd_subset_compact_open ContinuousMap.compactConvNhd_subset_compactOpen
--/
+#align continuous_map.compact_conv_nhd_subset_compact_open ContinuousMap.compactConvNhd_subset_compact_open
 
-#print ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd /-
 /-- The point `f` in `compact_conv_nhd K V f` is also an interior point wrt the compact-open
 topology.
 
 Since `compact_conv_nhd K V f` are a neighbourhood basis at `f` for each `f`, it follows that
 the compact-open topology is at least as fine as the topology of compact convergence. -/
-theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
+theorem iInter_compact_open_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
     ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (hC : ∀ i, IsCompact (C i)) (U :
       ι → Set β) (hU : ∀ i, IsOpen (U i)),
       (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
@@ -297,12 +266,10 @@ theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V
   simp only [mem_Inter, compact_open.gen, mem_set_of_eq, image_subset_iff] at hg 
   obtain ⟨y, hy⟩ := mem_Union.mp (hC hx)
   exact ⟨f y, (mem_ball_symmetry hW₂).mp (hfC y hy), mem_preimage.mp (hg y hy)⟩
-#align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd
--/
+#align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.iInter_compact_open_gen_subset_compactConvNhd
 
-#print ContinuousMap.compactOpen_eq_compactConvergence /-
 /-- The compact-open topology is equal to the compact-convergence topology. -/
-theorem compactOpen_eq_compactConvergence :
+theorem compactOpen_eq_compact_convergence :
     ContinuousMap.compactOpen = (compactConvergenceTopology : TopologicalSpace C(α, β)) :=
   by
   rw [compact_convergence_topology, ContinuousMap.compactOpen]
@@ -323,18 +290,14 @@ theorem compactOpen_eq_compactConvergence :
     rintro - K hK U hU rfl f hf
     obtain ⟨V, hV, hV', hVf⟩ := compact_conv_nhd_subset_compact_open f hK hU hf
     exact Filter.mem_of_superset (FilterBasis.mem_filter_of_mem _ ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩) hVf
-#align continuous_map.compact_open_eq_compact_convergence ContinuousMap.compactOpen_eq_compactConvergence
--/
+#align continuous_map.compact_open_eq_compact_convergence ContinuousMap.compactOpen_eq_compact_convergence
 
-#print ContinuousMap.compactConvergenceUniformity /-
 /-- The filter on `C(α, β) × C(α, β)` which underlies the uniform space structure on `C(α, β)`. -/
 def compactConvergenceUniformity : Filter (C(α, β) × C(α, β)) :=
   ⨅ KV ∈ {KV : Set α × Set (β × β) | IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β},
     𝓟 {fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ KV.1 → (fg.1 x, fg.2 x) ∈ KV.2}
 #align continuous_map.compact_convergence_uniformity ContinuousMap.compactConvergenceUniformity
--/
 
-#print ContinuousMap.hasBasis_compactConvergenceUniformity_aux /-
 theorem hasBasis_compactConvergenceUniformity_aux :
     HasBasis (@compactConvergenceUniformity α β _ _)
       (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
@@ -347,9 +310,7 @@ theorem hasBasis_compactConvergenceUniformity_aux :
     forall_and, mem_inter_iff, mem_union]
   exact fun f g => forall_imp fun x => by tauto
 #align continuous_map.has_basis_compact_convergence_uniformity_aux ContinuousMap.hasBasis_compactConvergenceUniformity_aux
--/
 
-#print ContinuousMap.mem_compactConvergenceUniformity /-
 /-- An intermediate lemma. Usually `mem_compact_convergence_entourage_iff` is more useful. -/
 theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
     X ∈ @compactConvergenceUniformity α β _ _ ↔
@@ -359,7 +320,6 @@ theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
   simp only [has_basis_compact_convergence_uniformity_aux.mem_iff, exists_prop, Prod.exists,
     and_assoc']
 #align continuous_map.mem_compact_convergence_uniformity ContinuousMap.mem_compactConvergenceUniformity
--/
 
 #print ContinuousMap.compactConvergenceUniformSpace /-
 /-- Note that we ensure the induced topology is definitionally the compact-open topology. -/

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

@@ -3,8 +3,8 @@ Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
-import Mathbin.Topology.CompactOpen
-import Mathbin.Topology.UniformSpace.UniformConvergence
+import Topology.CompactOpen
+import Topology.UniformSpace.UniformConvergence
 
 #align_import topology.uniform_space.compact_convergence from "leanprover-community/mathlib"@"1ead22342e1a078bd44744ace999f85756555d35"
 

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

@@ -317,7 +317,7 @@ theorem compactOpen_eq_compactConvergence :
     haveI := hι
     exact
       ⟨⋂ i, compact_open.gen (C i) (U i), h₂.trans hXf,
-        isOpen_iInter fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
+        isOpen_iInter_of_finite fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
   · simp only [TopologicalSpace.le_generateFrom_iff_subset_isOpen, and_imp, exists_prop,
       forall_exists_index, set_of_subset_set_of]
     rintro - K hK U hU rfl f hf

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

@@ -484,7 +484,8 @@ the `←` direction is true unconditionally. See `tendsto_locally_uniformly_of_t
 `tendsto_of_tendsto_locally_uniformly` for versions requiring weaker hypotheses. -/
 theorem tendsto_iff_tendstoLocallyUniformly [LocallyCompactSpace α] :
     Tendsto F p (𝓝 f) ↔ TendstoLocallyUniformly (fun i a => F i a) f p :=
-  ⟨tendstoLocallyUniformly_of_tendsto exists_compact_mem_nhds, tendsto_of_tendstoLocallyUniformly⟩
+  ⟨tendstoLocallyUniformly_of_tendsto WeaklyLocallyCompactSpace.exists_compact_mem_nhds,
+    tendsto_of_tendstoLocallyUniformly⟩
 #align continuous_map.tendsto_iff_tendsto_locally_uniformly ContinuousMap.tendsto_iff_tendstoLocallyUniformly
 -/
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module topology.uniform_space.compact_convergence
-! leanprover-community/mathlib commit 1ead22342e1a078bd44744ace999f85756555d35
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Topology.CompactOpen
 import Mathbin.Topology.UniformSpace.UniformConvergence
 
+#align_import topology.uniform_space.compact_convergence from "leanprover-community/mathlib"@"1ead22342e1a078bd44744ace999f85756555d35"
+
 /-!
 # Compact convergence (uniform convergence on compact sets)
 

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

@@ -125,6 +125,7 @@ theorem compactConvNhd_mem_comp {g₁ g₂ : C(α, β)} {V' : Set (β × β)}
 #align continuous_map.compact_conv_nhd_mem_comp ContinuousMap.compactConvNhd_mem_comp
 -/
 
+#print ContinuousMap.compactConvNhd_nhd_basis /-
 /-- A key property of `compact_conv_nhd`. It allows us to apply
 `topological_space.nhds_mk_of_nhds_filter_basis` below. -/
 theorem compactConvNhd_nhd_basis (hV : V ∈ 𝓤 β) :
@@ -136,13 +137,16 @@ theorem compactConvNhd_nhd_basis (hV : V ∈ 𝓤 β) :
     ⟨V', h₁, subset.trans (subset_comp_self_of_mem_uniformity h₁) h₂, fun g hg g' hg' =>
       compact_conv_nhd_mono f h₂ (compact_conv_nhd_mem_comp f hg hg')⟩
 #align continuous_map.compact_conv_nhd_nhd_basis ContinuousMap.compactConvNhd_nhd_basis
+-/
 
+#print ContinuousMap.compactConvNhd_subset_inter /-
 theorem compactConvNhd_subset_inter (K₁ K₂ : Set α) (V₁ V₂ : Set (β × β)) :
     compactConvNhd (K₁ ∪ K₂) (V₁ ∩ V₂) f ⊆ compactConvNhd K₁ V₁ f ∩ compactConvNhd K₂ V₂ f :=
   fun g hg =>
   ⟨fun x hx => mem_of_mem_inter_left (hg x (mem_union_left K₂ hx)), fun x hx =>
     mem_of_mem_inter_right (hg x (mem_union_right K₁ hx))⟩
 #align continuous_map.compact_conv_nhd_subset_inter ContinuousMap.compactConvNhd_subset_inter
+-/
 
 #print ContinuousMap.compactConvNhd_compact_entourage_nonempty /-
 theorem compactConvNhd_compact_entourage_nonempty :
@@ -215,6 +219,7 @@ theorem hasBasis_nhds_compactConvergence :
 #align continuous_map.has_basis_nhds_compact_convergence ContinuousMap.hasBasis_nhds_compactConvergence
 -/
 
+#print ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn' /-
 /-- This is an auxiliary lemma and is unlikely to be of direct use outside of this file. See
 `tendsto_iff_forall_compact_tendsto_uniformly_on` below for the useful version where the topology
 is picked up via typeclass inference. -/
@@ -229,7 +234,9 @@ theorem tendsto_iff_forall_compact_tendstoUniformlyOn' {ι : Type u₃} {p : Fil
   rw [forall_swap]
   exact forall₃_congr fun hK V hV => Iff.rfl
 #align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on' ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn'
+-/
 
+#print ContinuousMap.compactConvNhd_subset_compactOpen /-
 /-- Any point of `compact_open.gen K U` is also an interior point wrt the topology of compact
 convergence.
 
@@ -243,6 +250,7 @@ theorem compactConvNhd_subset_compactOpen (hK : IsCompact K) {U : Set β} (hU :
   rintro g hg _ ⟨x, hx, rfl⟩
   exact hV₃ (f x) ⟨x, hx, rfl⟩ (hg x hx)
 #align continuous_map.compact_conv_nhd_subset_compact_open ContinuousMap.compactConvNhd_subset_compactOpen
+-/
 
 #print ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd /-
 /-- The point `f` in `compact_conv_nhd K V f` is also an interior point wrt the compact-open
@@ -329,6 +337,7 @@ def compactConvergenceUniformity : Filter (C(α, β) × C(α, β)) :=
 #align continuous_map.compact_convergence_uniformity ContinuousMap.compactConvergenceUniformity
 -/
 
+#print ContinuousMap.hasBasis_compactConvergenceUniformity_aux /-
 theorem hasBasis_compactConvergenceUniformity_aux :
     HasBasis (@compactConvergenceUniformity α β _ _)
       (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
@@ -341,7 +350,9 @@ theorem hasBasis_compactConvergenceUniformity_aux :
     forall_and, mem_inter_iff, mem_union]
   exact fun f g => forall_imp fun x => by tauto
 #align continuous_map.has_basis_compact_convergence_uniformity_aux ContinuousMap.hasBasis_compactConvergenceUniformity_aux
+-/
 
+#print ContinuousMap.mem_compactConvergenceUniformity /-
 /-- An intermediate lemma. Usually `mem_compact_convergence_entourage_iff` is more useful. -/
 theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
     X ∈ @compactConvergenceUniformity α β _ _ ↔
@@ -351,6 +362,7 @@ theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
   simp only [has_basis_compact_convergence_uniformity_aux.mem_iff, exists_prop, Prod.exists,
     and_assoc']
 #align continuous_map.mem_compact_convergence_uniformity ContinuousMap.mem_compactConvergenceUniformity
+-/
 
 #print ContinuousMap.compactConvergenceUniformSpace /-
 /-- Note that we ensure the induced topology is definitionally the compact-open topology. -/
@@ -398,19 +410,24 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
 #align continuous_map.compact_convergence_uniform_space ContinuousMap.compactConvergenceUniformSpace
 -/
 
+#print ContinuousMap.mem_compactConvergence_entourage_iff /-
 theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
     X ∈ 𝓤 C(α, β) ↔
       ∃ (K : Set α) (V : Set (β × β)) (hK : IsCompact K) (hV : V ∈ 𝓤 β),
         {fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V} ⊆ X :=
   mem_compactConvergenceUniformity X
 #align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff
+-/
 
+#print ContinuousMap.hasBasis_compactConvergenceUniformity /-
 theorem hasBasis_compactConvergenceUniformity :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
       {fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2} :=
   hasBasis_compactConvergenceUniformity_aux
 #align continuous_map.has_basis_compact_convergence_uniformity ContinuousMap.hasBasis_compactConvergenceUniformity
+-/
 
+#print Filter.HasBasis.compactConvergenceUniformity /-
 theorem Filter.HasBasis.compactConvergenceUniformity {ι : Type _} {pi : ι → Prop}
     {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s) :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × ι => IsCompact p.1 ∧ pi p.2) fun p =>
@@ -423,14 +440,18 @@ theorem Filter.HasBasis.compactConvergenceUniformity {ι : Type _} {pi : ι →
   · rintro ⟨t, i⟩ ⟨ht, hi⟩
     exact ⟨(t, s i), ⟨ht, h.mem_of_mem hi⟩, subset.rfl⟩
 #align filter.has_basis.compact_convergence_uniformity Filter.HasBasis.compactConvergenceUniformity
+-/
 
 variable {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f}
 
+#print ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn /-
 theorem tendsto_iff_forall_compact_tendstoUniformlyOn :
     Tendsto F p (𝓝 f) ↔ ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K := by
   rw [compact_open_eq_compact_convergence, tendsto_iff_forall_compact_tendsto_uniformly_on']
 #align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn
+-/
 
+#print ContinuousMap.tendsto_of_tendstoLocallyUniformly /-
 /-- Locally uniform convergence implies convergence in the compact-open topology. -/
 theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a => F i a) f p) :
     Tendsto F p (𝓝 f) :=
@@ -440,7 +461,9 @@ theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a
   rw [← tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK]
   exact h.tendsto_locally_uniformly_on
 #align continuous_map.tendsto_of_tendsto_locally_uniformly ContinuousMap.tendsto_of_tendstoLocallyUniformly
+-/
 
+#print ContinuousMap.tendstoLocallyUniformly_of_tendsto /-
 /-- If every point has a compact neighbourhood, then convergence in the compact-open topology
 implies locally uniform convergence.
 
@@ -453,7 +476,9 @@ theorem tendstoLocallyUniformly_of_tendsto (hα : ∀ x : α, ∃ n, IsCompact n
   obtain ⟨n, hn₁, hn₂⟩ := hα x
   exact ⟨n, hn₂, h n hn₁ V hV⟩
 #align continuous_map.tendsto_locally_uniformly_of_tendsto ContinuousMap.tendstoLocallyUniformly_of_tendsto
+-/
 
+#print ContinuousMap.tendsto_iff_tendstoLocallyUniformly /-
 /-- Convergence in the compact-open topology is the same as locally uniform convergence on a locally
 compact space.
 
@@ -464,11 +489,13 @@ theorem tendsto_iff_tendstoLocallyUniformly [LocallyCompactSpace α] :
     Tendsto F p (𝓝 f) ↔ TendstoLocallyUniformly (fun i a => F i a) f p :=
   ⟨tendstoLocallyUniformly_of_tendsto exists_compact_mem_nhds, tendsto_of_tendstoLocallyUniformly⟩
 #align continuous_map.tendsto_iff_tendsto_locally_uniformly ContinuousMap.tendsto_iff_tendstoLocallyUniformly
+-/
 
 section CompactDomain
 
 variable [CompactSpace α]
 
+#print ContinuousMap.hasBasis_compactConvergenceUniformity_of_compact /-
 theorem hasBasis_compactConvergenceUniformity_of_compact :
     HasBasis (𝓤 C(α, β)) (fun V : Set (β × β) => V ∈ 𝓤 β) fun V =>
       {fg : C(α, β) × C(α, β) | ∀ x, (fg.1 x, fg.2 x) ∈ V} :=
@@ -476,7 +503,9 @@ theorem hasBasis_compactConvergenceUniformity_of_compact :
     (fun p hp => ⟨p.2, hp.2, fun fg hfg x hx => hfg x⟩) fun V hV =>
     ⟨⟨univ, V⟩, ⟨isCompact_univ, hV⟩, fun fg hfg x => hfg x (mem_univ x)⟩
 #align continuous_map.has_basis_compact_convergence_uniformity_of_compact ContinuousMap.hasBasis_compactConvergenceUniformity_of_compact
+-/
 
+#print ContinuousMap.tendsto_iff_tendstoUniformly /-
 /-- Convergence in the compact-open topology is the same as uniform convergence for sequences of
 continuous functions on a compact space. -/
 theorem tendsto_iff_tendstoUniformly :
@@ -485,6 +514,7 @@ theorem tendsto_iff_tendstoUniformly :
   rw [tendsto_iff_forall_compact_tendsto_uniformly_on, ← tendstoUniformlyOn_univ]
   exact ⟨fun h => h univ isCompact_univ, fun h K hK => h.mono (subset_univ K)⟩
 #align continuous_map.tendsto_iff_tendsto_uniformly ContinuousMap.tendsto_iff_tendstoUniformly
+-/
 
 end CompactDomain
 

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

@@ -285,7 +285,6 @@ theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V
       _ ⊆ closure (f '' U x) := f.continuous.continuous_on.image_closure
       _ ⊆ closure (ball (f x) Z) := by apply closure_mono; simp
       _ ⊆ ball (f x) W := hZW
-      
   refine'
     ⟨t, t.fintype_coe_sort, C, fun i => hK.inter_right isClosed_closure, fun i =>
       ball (f ((i : K) : α)) W, fun i => is_open_ball _ hW₄, by simp [compact_open.gen, hfC],

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

@@ -99,7 +99,7 @@ namespace ContinuousMap
 /-- Given `K ⊆ α`, `V ⊆ β × β`, and `f : C(α, β)`, we define `compact_conv_nhd K V f` to be the set
 of `g : C(α, β)` that are `V`-close to `f` on `K`. -/
 def compactConvNhd : Set C(α, β) :=
-  { g | ∀ x ∈ K, (f x, g x) ∈ V }
+  {g | ∀ x ∈ K, (f x, g x) ∈ V}
 #align continuous_map.compact_conv_nhd ContinuousMap.compactConvNhd
 -/
 
@@ -146,7 +146,7 @@ theorem compactConvNhd_subset_inter (K₁ K₂ : Set α) (V₁ V₂ : Set (β ×
 
 #print ContinuousMap.compactConvNhd_compact_entourage_nonempty /-
 theorem compactConvNhd_compact_entourage_nonempty :
-    { KV : Set α × Set (β × β) | IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β }.Nonempty :=
+    {KV : Set α × Set (β × β) | IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β}.Nonempty :=
   ⟨⟨∅, univ⟩, isCompact_empty, Filter.univ_mem⟩
 #align continuous_map.compact_conv_nhd_compact_entourage_nonempty ContinuousMap.compactConvNhd_compact_entourage_nonempty
 -/
@@ -325,15 +325,15 @@ theorem compactOpen_eq_compactConvergence :
 #print ContinuousMap.compactConvergenceUniformity /-
 /-- The filter on `C(α, β) × C(α, β)` which underlies the uniform space structure on `C(α, β)`. -/
 def compactConvergenceUniformity : Filter (C(α, β) × C(α, β)) :=
-  ⨅ KV ∈ { KV : Set α × Set (β × β) | IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β },
-    𝓟 { fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ KV.1 → (fg.1 x, fg.2 x) ∈ KV.2 }
+  ⨅ KV ∈ {KV : Set α × Set (β × β) | IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β},
+    𝓟 {fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ KV.1 → (fg.1 x, fg.2 x) ∈ KV.2}
 #align continuous_map.compact_convergence_uniformity ContinuousMap.compactConvergenceUniformity
 -/
 
 theorem hasBasis_compactConvergenceUniformity_aux :
     HasBasis (@compactConvergenceUniformity α β _ _)
       (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
-      { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } :=
+      {fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2} :=
   by
   refine' Filter.hasBasis_biInf_principal _ compact_conv_nhd_compact_entourage_nonempty
   rintro ⟨K₁, V₁⟩ ⟨hK₁, hV₁⟩ ⟨K₂, V₂⟩ ⟨hK₂, hV₂⟩
@@ -347,7 +347,7 @@ theorem hasBasis_compactConvergenceUniformity_aux :
 theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
     X ∈ @compactConvergenceUniformity α β _ _ ↔
       ∃ (K : Set α) (V : Set (β × β)) (hK : IsCompact K) (hV : V ∈ 𝓤 β),
-        { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
+        {fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V} ⊆ X :=
   by
   simp only [has_basis_compact_convergence_uniformity_aux.mem_iff, exists_prop, Prod.exists,
     and_assoc']
@@ -369,7 +369,7 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
       Filter.tendsto_principal, Prod.snd_swap, Filter.tendsto_iInf]
     intro K V hK hV
     obtain ⟨V', hV', hsymm, hsub⟩ := symm_of_uniformity hV
-    let X := { fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ K → (fg.1 x, fg.2 x) ∈ V' }
+    let X := {fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ K → (fg.1 x, fg.2 x) ∈ V'}
     have hX : X ∈ compact_convergence_uniformity :=
       (mem_compact_convergence_uniformity X).mpr ⟨K, V', hK, hV', by simp⟩
     exact Filter.eventually_of_mem hX fun fg hfg x hx => hsub (hsymm _ _ (hfg x hx))
@@ -379,7 +379,7 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
     obtain ⟨V', hV', hcomp⟩ := comp_mem_uniformity_sets hV
     let h := fun s : Set (C(α, β) × C(α, β)) => s ○ s
     suffices
-      h { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V' } ∈
+      h {fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V'} ∈
         compact_convergence_uniformity.lift' h
       by
       apply Filter.mem_of_superset this
@@ -402,20 +402,20 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
 theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
     X ∈ 𝓤 C(α, β) ↔
       ∃ (K : Set α) (V : Set (β × β)) (hK : IsCompact K) (hV : V ∈ 𝓤 β),
-        { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
+        {fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V} ⊆ X :=
   mem_compactConvergenceUniformity X
 #align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff
 
 theorem hasBasis_compactConvergenceUniformity :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
-      { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } :=
+      {fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2} :=
   hasBasis_compactConvergenceUniformity_aux
 #align continuous_map.has_basis_compact_convergence_uniformity ContinuousMap.hasBasis_compactConvergenceUniformity
 
 theorem Filter.HasBasis.compactConvergenceUniformity {ι : Type _} {pi : ι → Prop}
     {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s) :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × ι => IsCompact p.1 ∧ pi p.2) fun p =>
-      { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2 } :=
+      {fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2} :=
   by
   refine' has_basis_compact_convergence_uniformity.to_has_basis _ _
   · rintro ⟨t₁, t₂⟩ ⟨h₁, h₂⟩
@@ -472,7 +472,7 @@ variable [CompactSpace α]
 
 theorem hasBasis_compactConvergenceUniformity_of_compact :
     HasBasis (𝓤 C(α, β)) (fun V : Set (β × β) => V ∈ 𝓤 β) fun V =>
-      { fg : C(α, β) × C(α, β) | ∀ x, (fg.1 x, fg.2 x) ∈ V } :=
+      {fg : C(α, β) × C(α, β) | ∀ x, (fg.1 x, fg.2 x) ∈ V} :=
   hasBasis_compactConvergenceUniformity.to_hasBasis
     (fun p hp => ⟨p.2, hp.2, fun fg hfg x hx => hfg x⟩) fun V hV =>
     ⟨⟨univ, V⟩, ⟨isCompact_univ, hV⟩, fun fg hfg x => hfg x (mem_univ x)⟩

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

@@ -174,7 +174,7 @@ def compactConvergenceFilterBasis (f : C(α, β)) : FilterBasis C(α, β) :=
 #print ContinuousMap.mem_compactConvergence_nhd_filter /-
 theorem mem_compactConvergence_nhd_filter (Y : Set C(α, β)) :
     Y ∈ (compactConvergenceFilterBasis f).filterₓ ↔
-      ∃ (K : Set α)(V : Set (β × β))(hK : IsCompact K)(hV : V ∈ 𝓤 β), compactConvNhd K V f ⊆ Y :=
+      ∃ (K : Set α) (V : Set (β × β)) (hK : IsCompact K) (hV : V ∈ 𝓤 β), compactConvNhd K V f ⊆ Y :=
   by
   constructor
   · rintro ⟨X, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩
@@ -251,8 +251,8 @@ topology.
 Since `compact_conv_nhd K V f` are a neighbourhood basis at `f` for each `f`, it follows that
 the compact-open topology is at least as fine as the topology of compact convergence. -/
 theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
-    ∃ (ι : Sort (u₁ + 1))(_ : Fintype ι)(C : ι → Set α)(hC : ∀ i, IsCompact (C i))(U :
-      ι → Set β)(hU : ∀ i, IsOpen (U i)),
+    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (hC : ∀ i, IsCompact (C i)) (U :
+      ι → Set β) (hU : ∀ i, IsOpen (U i)),
       (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
         (⋂ i, CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f :=
   by
@@ -290,7 +290,7 @@ theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V
     ⟨t, t.fintype_coe_sort, C, fun i => hK.inter_right isClosed_closure, fun i =>
       ball (f ((i : K) : α)) W, fun i => is_open_ball _ hW₄, by simp [compact_open.gen, hfC],
       fun g hg x hx => hW₃ (mem_comp_rel.mpr _)⟩
-  simp only [mem_Inter, compact_open.gen, mem_set_of_eq, image_subset_iff] at hg
+  simp only [mem_Inter, compact_open.gen, mem_set_of_eq, image_subset_iff] at hg 
   obtain ⟨y, hy⟩ := mem_Union.mp (hC hx)
   exact ⟨f y, (mem_ball_symmetry hW₂).mp (hfC y hy), mem_preimage.mp (hg y hy)⟩
 #align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd
@@ -346,7 +346,7 @@ theorem hasBasis_compactConvergenceUniformity_aux :
 /-- An intermediate lemma. Usually `mem_compact_convergence_entourage_iff` is more useful. -/
 theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
     X ∈ @compactConvergenceUniformity α β _ _ ↔
-      ∃ (K : Set α)(V : Set (β × β))(hK : IsCompact K)(hV : V ∈ 𝓤 β),
+      ∃ (K : Set α) (V : Set (β × β)) (hK : IsCompact K) (hV : V ∈ 𝓤 β),
         { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
   by
   simp only [has_basis_compact_convergence_uniformity_aux.mem_iff, exists_prop, Prod.exists,
@@ -401,7 +401,7 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
 
 theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
     X ∈ 𝓤 C(α, β) ↔
-      ∃ (K : Set α)(V : Set (β × β))(hK : IsCompact K)(hV : V ∈ 𝓤 β),
+      ∃ (K : Set α) (V : Set (β × β)) (hK : IsCompact K) (hV : V ∈ 𝓤 β),
         { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
   mem_compactConvergenceUniformity X
 #align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff
@@ -449,7 +449,7 @@ See also `tendsto_iff_tendsto_locally_uniformly`, especially for T2 spaces. -/
 theorem tendstoLocallyUniformly_of_tendsto (hα : ∀ x : α, ∃ n, IsCompact n ∧ n ∈ 𝓝 x)
     (h : Tendsto F p (𝓝 f)) : TendstoLocallyUniformly (fun i a => F i a) f p :=
   by
-  rw [tendsto_iff_forall_compact_tendsto_uniformly_on] at h
+  rw [tendsto_iff_forall_compact_tendsto_uniformly_on] at h 
   intro V hV x
   obtain ⟨n, hn₁, hn₂⟩ := hα x
   exact ⟨n, hn₂, h n hn₁ V hV⟩

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

@@ -85,7 +85,7 @@ of the uniform space structure on `C(α, β)` definitionally equal to the compac
 
 universe u₁ u₂ u₃
 
-open Filter uniformity Topology
+open scoped Filter uniformity Topology
 
 open UniformSpace Set Filter
 

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

@@ -125,12 +125,6 @@ theorem compactConvNhd_mem_comp {g₁ g₂ : C(α, β)} {V' : Set (β × β)}
 #align continuous_map.compact_conv_nhd_mem_comp ContinuousMap.compactConvNhd_mem_comp
 -/
 
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-Case conversion may be inaccurate. Consider using '#align continuous_map.compact_conv_nhd_nhd_basis ContinuousMap.compactConvNhd_nhd_basisₓ'. -/
 /-- A key property of `compact_conv_nhd`. It allows us to apply
 `topological_space.nhds_mk_of_nhds_filter_basis` below. -/
 theorem compactConvNhd_nhd_basis (hV : V ∈ 𝓤 β) :
@@ -143,12 +137,6 @@ theorem compactConvNhd_nhd_basis (hV : V ∈ 𝓤 β) :
       compact_conv_nhd_mono f h₂ (compact_conv_nhd_mem_comp f hg hg')⟩
 #align continuous_map.compact_conv_nhd_nhd_basis ContinuousMap.compactConvNhd_nhd_basis
 
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-Case conversion may be inaccurate. Consider using '#align continuous_map.compact_conv_nhd_subset_inter ContinuousMap.compactConvNhd_subset_interₓ'. -/
 theorem compactConvNhd_subset_inter (K₁ K₂ : Set α) (V₁ V₂ : Set (β × β)) :
     compactConvNhd (K₁ ∪ K₂) (V₁ ∩ V₂) f ⊆ compactConvNhd K₁ V₁ f ∩ compactConvNhd K₂ V₂ f :=
   fun g hg =>
@@ -227,12 +215,6 @@ theorem hasBasis_nhds_compactConvergence :
 #align continuous_map.has_basis_nhds_compact_convergence ContinuousMap.hasBasis_nhds_compactConvergence
 -/
 
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-Case conversion may be inaccurate. Consider using '#align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on' ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn'ₓ'. -/
 /-- This is an auxiliary lemma and is unlikely to be of direct use outside of this file. See
 `tendsto_iff_forall_compact_tendsto_uniformly_on` below for the useful version where the topology
 is picked up via typeclass inference. -/
@@ -248,12 +230,6 @@ theorem tendsto_iff_forall_compact_tendstoUniformlyOn' {ι : Type u₃} {p : Fil
   exact forall₃_congr fun hK V hV => Iff.rfl
 #align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on' ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn'
 
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 /-- Any point of `compact_open.gen K U` is also an interior point wrt the topology of compact
 convergence.
 
@@ -354,12 +330,6 @@ def compactConvergenceUniformity : Filter (C(α, β) × C(α, β)) :=
 #align continuous_map.compact_convergence_uniformity ContinuousMap.compactConvergenceUniformity
 -/
 
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-Case conversion may be inaccurate. Consider using '#align continuous_map.has_basis_compact_convergence_uniformity_aux ContinuousMap.hasBasis_compactConvergenceUniformity_auxₓ'. -/
 theorem hasBasis_compactConvergenceUniformity_aux :
     HasBasis (@compactConvergenceUniformity α β _ _)
       (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
@@ -373,9 +343,6 @@ theorem hasBasis_compactConvergenceUniformity_aux :
   exact fun f g => forall_imp fun x => by tauto
 #align continuous_map.has_basis_compact_convergence_uniformity_aux ContinuousMap.hasBasis_compactConvergenceUniformity_aux
 
-/- warning: continuous_map.mem_compact_convergence_uniformity -> ContinuousMap.mem_compactConvergenceUniformity is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align continuous_map.mem_compact_convergence_uniformity ContinuousMap.mem_compactConvergenceUniformityₓ'. -/
 /-- An intermediate lemma. Usually `mem_compact_convergence_entourage_iff` is more useful. -/
 theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
     X ∈ @compactConvergenceUniformity α β _ _ ↔
@@ -432,9 +399,6 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
 #align continuous_map.compact_convergence_uniform_space ContinuousMap.compactConvergenceUniformSpace
 -/
 
-/- warning: continuous_map.mem_compact_convergence_entourage_iff -> ContinuousMap.mem_compactConvergence_entourage_iff is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iffₓ'. -/
 theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
     X ∈ 𝓤 C(α, β) ↔
       ∃ (K : Set α)(V : Set (β × β))(hK : IsCompact K)(hV : V ∈ 𝓤 β),
@@ -442,21 +406,12 @@ theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β)))
   mem_compactConvergenceUniformity X
 #align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff
 
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-Case conversion may be inaccurate. Consider using '#align continuous_map.has_basis_compact_convergence_uniformity ContinuousMap.hasBasis_compactConvergenceUniformityₓ'. -/
 theorem hasBasis_compactConvergenceUniformity :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
       { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } :=
   hasBasis_compactConvergenceUniformity_aux
 #align continuous_map.has_basis_compact_convergence_uniformity ContinuousMap.hasBasis_compactConvergenceUniformity
 
-/- warning: filter.has_basis.compact_convergence_uniformity -> Filter.HasBasis.compactConvergenceUniformity is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align filter.has_basis.compact_convergence_uniformity Filter.HasBasis.compactConvergenceUniformityₓ'. -/
 theorem Filter.HasBasis.compactConvergenceUniformity {ι : Type _} {pi : ι → Prop}
     {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s) :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × ι => IsCompact p.1 ∧ pi p.2) fun p =>
@@ -472,23 +427,11 @@ theorem Filter.HasBasis.compactConvergenceUniformity {ι : Type _} {pi : ι →
 
 variable {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f}
 
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 theorem tendsto_iff_forall_compact_tendstoUniformlyOn :
     Tendsto F p (𝓝 f) ↔ ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K := by
   rw [compact_open_eq_compact_convergence, tendsto_iff_forall_compact_tendsto_uniformly_on']
 #align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn
 
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 /-- Locally uniform convergence implies convergence in the compact-open topology. -/
 theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a => F i a) f p) :
     Tendsto F p (𝓝 f) :=
@@ -499,12 +442,6 @@ theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a
   exact h.tendsto_locally_uniformly_on
 #align continuous_map.tendsto_of_tendsto_locally_uniformly ContinuousMap.tendsto_of_tendstoLocallyUniformly
 
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 /-- If every point has a compact neighbourhood, then convergence in the compact-open topology
 implies locally uniform convergence.
 
@@ -518,12 +455,6 @@ theorem tendstoLocallyUniformly_of_tendsto (hα : ∀ x : α, ∃ n, IsCompact n
   exact ⟨n, hn₂, h n hn₁ V hV⟩
 #align continuous_map.tendsto_locally_uniformly_of_tendsto ContinuousMap.tendstoLocallyUniformly_of_tendsto
 
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 /-- Convergence in the compact-open topology is the same as locally uniform convergence on a locally
 compact space.
 
@@ -539,12 +470,6 @@ section CompactDomain
 
 variable [CompactSpace α]
 
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 theorem hasBasis_compactConvergenceUniformity_of_compact :
     HasBasis (𝓤 C(α, β)) (fun V : Set (β × β) => V ∈ 𝓤 β) fun V =>
       { fg : C(α, β) × C(α, β) | ∀ x, (fg.1 x, fg.2 x) ∈ V } :=
@@ -553,12 +478,6 @@ theorem hasBasis_compactConvergenceUniformity_of_compact :
     ⟨⟨univ, V⟩, ⟨isCompact_univ, hV⟩, fun fg hfg x => hfg x (mem_univ x)⟩
 #align continuous_map.has_basis_compact_convergence_uniformity_of_compact ContinuousMap.hasBasis_compactConvergenceUniformity_of_compact
 
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 /-- Convergence in the compact-open topology is the same as uniform convergence for sequences of
 continuous functions on a compact space. -/
 theorem tendsto_iff_tendstoUniformly :

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

@@ -307,9 +307,7 @@ theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V
     calc
       f '' (K ∩ closure (U x)) ⊆ f '' closure (U x) := image_subset _ (inter_subset_right _ _)
       _ ⊆ closure (f '' U x) := f.continuous.continuous_on.image_closure
-      _ ⊆ closure (ball (f x) Z) := by
-        apply closure_mono
-        simp
+      _ ⊆ closure (ball (f x) Z) := by apply closure_mono; simp
       _ ⊆ ball (f x) W := hZW
       
   refine'

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

@@ -376,10 +376,7 @@ theorem hasBasis_compactConvergenceUniformity_aux :
 #align continuous_map.has_basis_compact_convergence_uniformity_aux ContinuousMap.hasBasis_compactConvergenceUniformity_aux
 
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 Case conversion may be inaccurate. Consider using '#align continuous_map.mem_compact_convergence_uniformity ContinuousMap.mem_compactConvergenceUniformityₓ'. -/
 /-- An intermediate lemma. Usually `mem_compact_convergence_entourage_iff` is more useful. -/
 theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
@@ -438,10 +435,7 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
 -/
 
 /- warning: continuous_map.mem_compact_convergence_entourage_iff -> ContinuousMap.mem_compactConvergence_entourage_iff is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iffₓ'. -/
 theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
     X ∈ 𝓤 C(α, β) ↔
@@ -463,10 +457,7 @@ theorem hasBasis_compactConvergenceUniformity :
 #align continuous_map.has_basis_compact_convergence_uniformity ContinuousMap.hasBasis_compactConvergenceUniformity
 
 /- warning: filter.has_basis.compact_convergence_uniformity -> Filter.HasBasis.compactConvergenceUniformity is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align filter.has_basis.compact_convergence_uniformity Filter.HasBasis.compactConvergenceUniformityₓ'. -/
 theorem Filter.HasBasis.compactConvergenceUniformity {ι : Type _} {pi : ι → Prop}
     {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s) :

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

@@ -268,13 +268,13 @@ theorem compactConvNhd_subset_compactOpen (hK : IsCompact K) {U : Set β} (hU :
   exact hV₃ (f x) ⟨x, hx, rfl⟩ (hg x hx)
 #align continuous_map.compact_conv_nhd_subset_compact_open ContinuousMap.compactConvNhd_subset_compactOpen
 
-#print ContinuousMap.interᵢ_compactOpen_gen_subset_compactConvNhd /-
+#print ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd /-
 /-- The point `f` in `compact_conv_nhd K V f` is also an interior point wrt the compact-open
 topology.
 
 Since `compact_conv_nhd K V f` are a neighbourhood basis at `f` for each `f`, it follows that
 the compact-open topology is at least as fine as the topology of compact convergence. -/
-theorem interᵢ_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
+theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
     ∃ (ι : Sort (u₁ + 1))(_ : Fintype ι)(C : ι → Set α)(hC : ∀ i, IsCompact (C i))(U :
       ι → Set β)(hU : ∀ i, IsOpen (U i)),
       (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
@@ -319,7 +319,7 @@ theorem interᵢ_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV :
   simp only [mem_Inter, compact_open.gen, mem_set_of_eq, image_subset_iff] at hg
   obtain ⟨y, hy⟩ := mem_Union.mp (hC hx)
   exact ⟨f y, (mem_ball_symmetry hW₂).mp (hfC y hy), mem_preimage.mp (hg y hy)⟩
-#align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.interᵢ_compactOpen_gen_subset_compactConvNhd
+#align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd
 -/
 
 #print ContinuousMap.compactOpen_eq_compactConvergence /-
@@ -339,7 +339,7 @@ theorem compactOpen_eq_compactConvergence :
     haveI := hι
     exact
       ⟨⋂ i, compact_open.gen (C i) (U i), h₂.trans hXf,
-        isOpen_interᵢ fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
+        isOpen_iInter fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
   · simp only [TopologicalSpace.le_generateFrom_iff_subset_isOpen, and_imp, exists_prop,
       forall_exists_index, set_of_subset_set_of]
     rintro - K hK U hU rfl f hf
@@ -367,7 +367,7 @@ theorem hasBasis_compactConvergenceUniformity_aux :
       (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
       { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } :=
   by
-  refine' Filter.hasBasis_binfᵢ_principal _ compact_conv_nhd_compact_entourage_nonempty
+  refine' Filter.hasBasis_biInf_principal _ compact_conv_nhd_compact_entourage_nonempty
   rintro ⟨K₁, V₁⟩ ⟨hK₁, hV₁⟩ ⟨K₂, V₂⟩ ⟨hK₂, hV₂⟩
   refine' ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, Filter.inter_mem hV₁ hV₂⟩, _⟩
   simp only [le_eq_subset, Prod.forall, set_of_subset_set_of, ge_iff_le, Order.Preimage, ←
@@ -399,12 +399,12 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
   refl :=
     by
     simp only [compact_convergence_uniformity, and_imp, Filter.le_principal_iff, Prod.forall,
-      Filter.mem_principal, mem_set_of_eq, le_infᵢ_iff, idRel_subset]
+      Filter.mem_principal, mem_set_of_eq, le_iInf_iff, idRel_subset]
     exact fun K V hK hV f x hx => refl_mem_uniformity hV
   symm :=
     by
     simp only [compact_convergence_uniformity, and_imp, Prod.forall, mem_set_of_eq, Prod.fst_swap,
-      Filter.tendsto_principal, Prod.snd_swap, Filter.tendsto_infᵢ]
+      Filter.tendsto_principal, Prod.snd_swap, Filter.tendsto_iInf]
     intro K V hK hV
     obtain ⟨V', hV', hsymm, hsub⟩ := symm_of_uniformity hV
     let X := { fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ K → (fg.1 x, fg.2 x) ∈ V' }

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

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -91,7 +91,6 @@ open scoped Uniformity Topology UniformConvergence
 open UniformSpace Set Filter
 
 variable {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β]
-
 variable (K : Set α) (V : Set (β × β)) (f : C(α, β))
 
 namespace ContinuousMap

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

Also a few extra golfs and variations.

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

@@ -164,7 +164,7 @@ and replace the topology with `compactOpen` to avoid non-defeq diamonds,
 see Note [forgetful inheritance].  -/
 instance compactConvergenceUniformSpace : UniformSpace C(α, β) :=
   .replaceTopology (.comap toUniformOnFunIsCompact inferInstance) <| by
-    refine eq_of_nhds_eq_nhds fun f ↦ eq_of_forall_le_iff fun l ↦ ?_
+    refine TopologicalSpace.ext_nhds fun f ↦ eq_of_forall_le_iff fun l ↦ ?_
     simp_rw [← tendsto_id', tendsto_iff_forall_compact_tendstoUniformlyOn,
       nhds_induced, tendsto_comap_iff, UniformOnFun.tendsto_iff_tendstoUniformlyOn]
     rfl

@@ -79,8 +79,6 @@ so that the resulting instance uses the compact-open topology.
 
 ## TODO
 
-* When `α` is compact and `β` is a metric space,
-  the compact-convergence topology (and thus also the compact-open topology) is metrisable.
 * Results about uniformly continuous functions `γ → C(α, β)`
   and uniform limits of sequences `ι → γ → C(α, β)`.
 -/
@@ -223,6 +221,34 @@ theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β)))
   simp [hasBasis_compactConvergenceUniformity.mem_iff, and_assoc]
 #align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff
 
+/-- If `K` is a compact exhaustion of `α`
+and `V i` bounded by `p i` is a basis of entourages of `β`,
+then `fun (n, i) ↦ {(f, g) | ∀ x ∈ K n, (f x, g x) ∈ V i}` bounded by `p i`
+is a basis of entourages of `C(α, β)`. -/
+theorem _root_.CompactExhaustion.hasBasis_compactConvergenceUniformity {ι : Type*}
+    {p : ι → Prop} {V : ι → Set (β × β)} (K : CompactExhaustion α) (hb : (𝓤 β).HasBasis p V) :
+    HasBasis (𝓤 C(α, β)) (fun i : ℕ × ι ↦ p i.2) fun i ↦
+      {fg | ∀ x ∈ K i.1, (fg.1 x, fg.2 x) ∈ V i.2} :=
+  (UniformOnFun.hasBasis_uniformity_of_covering_of_basis {K | IsCompact K} K.isCompact
+    (Monotone.directed_le K.subset) (fun _ ↦ K.exists_superset_of_isCompact) hb).comap _
+
+theorem _root_.CompactExhaustion.hasAntitoneBasis_compactConvergenceUniformity
+    {V : ℕ → Set (β × β)} (K : CompactExhaustion α) (hb : (𝓤 β).HasAntitoneBasis V) :
+    HasAntitoneBasis (𝓤 C(α, β)) fun n ↦ {fg | ∀ x ∈ K n, (fg.1 x, fg.2 x) ∈ V n} :=
+  (UniformOnFun.hasAntitoneBasis_uniformity {K | IsCompact K} K.isCompact
+    K.subset (fun _ ↦ K.exists_superset_of_isCompact) hb).comap _
+
+/-- If `α` is a weakly locally compact σ-compact space
+(e.g., a proper pseudometric space or a compact spaces)
+and the uniformity on `β` is pseudometrizable,
+then the uniformity on `C(α, β)` is pseudometrizable too.
+-/
+instance [WeaklyLocallyCompactSpace α] [SigmaCompactSpace α] [IsCountablyGenerated (𝓤 β)] :
+    IsCountablyGenerated (𝓤 (C(α, β))) :=
+  let ⟨_V, hV⟩ := exists_antitone_basis (𝓤 β)
+  ((CompactExhaustion.choice α).hasAntitoneBasis_compactConvergenceUniformity
+    hV).isCountablyGenerated
+
 variable {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f}
 
 /-- Locally uniform convergence implies convergence in the compact-open topology. -/

@@ -1,10 +1,10 @@
 /-
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Oliver Nash
+Authors: Oliver Nash, Yury Kudryashov
 -/
 import Mathlib.Topology.CompactOpen
-import Mathlib.Topology.UniformSpace.UniformConvergence
+import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
 
 #align_import topology.uniform_space.compact_convergence from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
 
@@ -12,76 +12,83 @@ import Mathlib.Topology.UniformSpace.UniformConvergence
 # Compact convergence (uniform convergence on compact sets)
 
 Given a topological space `α` and a uniform space `β` (e.g., a metric space or a topological group),
-the space of continuous maps `C(α, β)` carries a natural uniform space structure. We define this
-uniform space structure in this file and also prove the following properties of the topology it
-induces on `C(α, β)`:
-
- 1. Given a sequence of continuous functions `Fₙ : α → β` together with some continuous `f : α → β`,
-    then `Fₙ` converges to `f` as a sequence in `C(α, β)` iff `Fₙ` converges to `f` uniformly on
-    each compact subset `K` of `α`.
- 2. Given `Fₙ` and `f` as above and suppose `α` is locally compact, then `Fₙ` converges to `f` iff
-    `Fₙ` converges to `f` locally uniformly.
- 3. The topology coincides with the compact-open topology.
-
-Property 1 is essentially true by definition, 2 follows from basic results about uniform
-convergence, but 3 requires a little work and uses the Lebesgue number lemma.
-
-## The uniform space structure
-
-Given subsets `K ⊆ α` and `V ⊆ β × β`, let `E(K, V) ⊆ C(α, β) × C(α, β)` be the set of pairs of
-continuous functions `α → β` which are `V`-close on `K`:
-$$
-  E(K, V) = \{ (f, g) | ∀ (x ∈ K), (f x, g x) ∈ V \}.
-$$
-Fixing some `f ∈ C(α, β)`, let `N(K, V, f) ⊆ C(α, β)` be the set of continuous functions `α → β`
-which are `V`-close to `f` on `K`:
-$$
-  N(K, V, f) = \{ g | ∀ (x ∈ K), (f x, g x) ∈ V \}.
-$$
-Using this notation we can describe the uniform space structure and the topology it induces.
-Specifically:
-  * A subset `X ⊆ C(α, β) × C(α, β)` is an entourage for the uniform space structure on `C(α, β)`
-    iff there exists a compact `K` and entourage `V` such that `E(K, V) ⊆ X`.
-  * A subset `Y ⊆ C(α, β)` is a neighbourhood of `f` iff there exists a compact `K` and entourage
-    `V` such that `N(K, V, f) ⊆ Y`.
-
-The topology on `C(α, β)` thus has a natural subbasis (the compact-open subbasis) and a natural
-neighbourhood basis (the compact-convergence neighbourhood basis).
-
-## Main definitions / results
-
- * `ContinuousMap.compactOpen_eq_compactConvergence`: the compact-open topology is equal to the
-   compact-convergence topology.
- * `ContinuousMap.compactConvergenceUniformSpace`: the uniform space structure on `C(α, β)`.
- * `ContinuousMap.mem_compactConvergence_entourage_iff`: a characterisation of the entourages
-    of `C(α, β)`.
- * `ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn`: a sequence of functions `Fₙ` in
-    `C(α, β)` converges to some `f` iff `Fₙ` converges to `f` uniformly on each compact subset
-    `K` of `α`.
- * `ContinuousMap.tendsto_iff_tendstoLocallyUniformly`: on a locally compact space, a sequence of
-    functions `Fₙ` in `C(α, β)` converges to some `f` iff `Fₙ` converges to `f` locally uniformly.
- * `ContinuousMap.tendsto_iff_tendstoUniformly`: on a compact space, a sequence of functions `Fₙ` in
-    `C(α, β)` converges to some `f` iff `Fₙ` converges to `f` uniformly.
+the space of continuous maps `C(α, β)` carries a natural uniform space structure.
+We define this uniform space structure in this file
+and also prove its basic properties.
+
+## Main definitions
+
+- `ContinuousMap.toUniformOnFunIsCompact`:
+  natural embedding of `C(α, β)`
+  into the space `α →ᵤ[{K | IsCompact K}] β` of all maps `α → β`
+  with the uniform space structure of uniform convergence on compacts.
+
+- `ContinuousMap.compactConvergenceUniformSpace`:
+  the `UniformSpace` structure on `C(α, β)` induced by the map above.
+
+## Main results
+
+* `ContinuousMap.mem_compactConvergence_entourage_iff`:
+  a characterisation of the entourages of `C(α, β)`.
+
+  The entourages are generated by the following sets.
+  Given `K : Set α` and `V : Set (β × β)`,
+  let `E(K, V) : Set (C(α, β) × C(α, β))` be the set of pairs of continuous functions `α → β`
+  which are `V`-close on `K`:
+  $$
+    E(K, V) = \{ (f, g) | ∀ (x ∈ K), (f x, g x) ∈ V \}.
+  $$
+  Then the sets `E(K, V)` for all compact sets `K` and all entourages `V`
+  form a basis of entourages of `C(α, β)`.
+
+  As usual, this basis of entourages provides a basis of neighbourhoods
+  by fixing `f`, see `nhds_basis_uniformity'`.
+
+* `Filter.HasBasis.compactConvergenceUniformity`:
+  a similar statement that uses a basis of entourages of `β` instead of all entourages.
+  It is useful, e.g., if `β` is a metric space.
+
+* `ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn`:
+  a sequence of functions `Fₙ` in `C(α, β)` converges in the compact-open topology to some `f`
+  iff `Fₙ` converges to `f` uniformly on each compact subset `K` of `α`.
+
+* Topology induced by the uniformity described above agrees with the compact-open topology.
+  This is essentially the same as `ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn`.
+
+  This fact is not available as a separate theorem.
+  Instead, we override the projection of `ContinuousMap.compactConvergenceUniformity`
+  to `TopologicalSpace` to be `ContinuousMap.compactOpen` and prove that they agree,
+  see Note [forgetful inheritance] and implementation notes below.
+
+* `ContinuousMap.tendsto_iff_tendstoLocallyUniformly`:
+  on a weakly locally compact space,
+  a sequence of functions `Fₙ` in `C(α, β)` converges to some `f`
+  iff `Fₙ` converges to `f` locally uniformly.
+
+* `ContinuousMap.tendsto_iff_tendstoUniformly`:
+  on a compact space, a sequence of functions `Fₙ` in `C(α, β)` converges to some `f`
+  iff `Fₙ` converges to `f` uniformly.
 
 ## Implementation details
 
-We use the forgetful inheritance pattern (see Note [forgetful inheritance]) to make the topology
-of the uniform space structure on `C(α, β)` definitionally equal to the compact-open topology.
+For technical reasons (see Note [forgetful inheritance]),
+instead of defining a `UniformSpace C(α, β)` structure
+and proving in a theorem that it agrees with the compact-open topology,
+we override the projection right in the definition,
+so that the resulting instance uses the compact-open topology.
 
 ## TODO
 
- * When `β` is a metric space, there is natural basis for the compact-convergence topology
-   parameterised by triples `(K, ε, f)` for a real number `ε > 0`.
- * When `α` is compact and `β` is a metric space, the compact-convergence topology (and thus also
-   the compact-open topology) is metrisable.
- * Results about uniformly continuous functions `γ → C(α, β)` and uniform limits of sequences
-   `ι → γ → C(α, β)`.
+* When `α` is compact and `β` is a metric space,
+  the compact-convergence topology (and thus also the compact-open topology) is metrisable.
+* Results about uniformly continuous functions `γ → C(α, β)`
+  and uniform limits of sequences `ι → γ → C(α, β)`.
 -/
 
 
 universe u₁ u₂ u₃
 
-open Filter Uniformity Topology
+open scoped Uniformity Topology UniformConvergence
 
 open UniformSpace Set Filter
 
@@ -91,297 +98,132 @@ variable (K : Set α) (V : Set (β × β)) (f : C(α, β))
 
 namespace ContinuousMap
 
-/-- Given `K ⊆ α`, `V ⊆ β × β`, and `f : C(α, β)`, we define `ContinuousMap.compactConvNhd K V f`
-to be the set of `g : C(α, β)` that are `V`-close to `f` on `K`. -/
-def compactConvNhd : Set C(α, β) :=
-  { g | ∀ x ∈ K, (f x, g x) ∈ V }
-#align continuous_map.compact_conv_nhd ContinuousMap.compactConvNhd
-
-variable {K V}
-
-theorem self_mem_compactConvNhd (hV : V ∈ 𝓤 β) : f ∈ compactConvNhd K V f := fun _x _hx =>
-  refl_mem_uniformity hV
-#align continuous_map.self_mem_compact_conv_nhd ContinuousMap.self_mem_compactConvNhd
-
-@[mono]
-theorem compactConvNhd_mono {V' : Set (β × β)} (hV' : V' ⊆ V) :
-    compactConvNhd K V' f ⊆ compactConvNhd K V f := fun _x hx a ha => hV' (hx a ha)
-#align continuous_map.compact_conv_nhd_mono ContinuousMap.compactConvNhd_mono
-
-theorem compactConvNhd_mem_comp {g₁ g₂ : C(α, β)} {V' : Set (β × β)}
-    (hg₁ : g₁ ∈ compactConvNhd K V f) (hg₂ : g₂ ∈ compactConvNhd K V' g₁) :
-    g₂ ∈ compactConvNhd K (V ○ V') f := fun x hx => ⟨g₁ x, hg₁ x hx, hg₂ x hx⟩
-#align continuous_map.compact_conv_nhd_mem_comp ContinuousMap.compactConvNhd_mem_comp
-
-/-- A key property of `ContinuousMap.compactConvNhd`. It allows us to apply
-`TopologicalSpace.nhds_mkOfNhds_filterBasis` below. -/
-theorem compactConvNhd_nhd_basis (hV : V ∈ 𝓤 β) :
-    ∃ V' ∈ 𝓤 β,
-      V' ⊆ V ∧ ∀ g ∈ compactConvNhd K V' f, compactConvNhd K V' g ⊆ compactConvNhd K V f := by
-  obtain ⟨V', h₁, h₂⟩ := comp_mem_uniformity_sets hV
-  exact
-    ⟨V', h₁, Subset.trans (subset_comp_self_of_mem_uniformity h₁) h₂, fun g hg g' hg' =>
-      compactConvNhd_mono f h₂ (compactConvNhd_mem_comp f hg hg')⟩
-#align continuous_map.compact_conv_nhd_nhd_basis ContinuousMap.compactConvNhd_nhd_basis
-
-theorem compactConvNhd_subset_inter (K₁ K₂ : Set α) (V₁ V₂ : Set (β × β)) :
-    compactConvNhd (K₁ ∪ K₂) (V₁ ∩ V₂) f ⊆ compactConvNhd K₁ V₁ f ∩ compactConvNhd K₂ V₂ f :=
-  fun _g hg =>
-  ⟨fun x hx => mem_of_mem_inter_left (hg x (mem_union_left K₂ hx)), fun x hx =>
-    mem_of_mem_inter_right (hg x (mem_union_right K₁ hx))⟩
-#align continuous_map.compact_conv_nhd_subset_inter ContinuousMap.compactConvNhd_subset_inter
-
-theorem compactConvNhd_compact_entourage_nonempty :
-    { KV : Set α × Set (β × β) | IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β }.Nonempty :=
-  ⟨⟨∅, univ⟩, isCompact_empty, Filter.univ_mem⟩
-#align continuous_map.compact_conv_nhd_compact_entourage_nonempty ContinuousMap.compactConvNhd_compact_entourage_nonempty
-
-theorem compactConvNhd_filter_isBasis :
-    Filter.IsBasis (fun KV : Set α × Set (β × β) => IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β) fun KV =>
-      compactConvNhd KV.1 KV.2 f :=
-  { nonempty := compactConvNhd_compact_entourage_nonempty
-    inter := by
-      rintro ⟨K₁, V₁⟩ ⟨K₂, V₂⟩ ⟨hK₁, hV₁⟩ ⟨hK₂, hV₂⟩
-      exact
-        ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, Filter.inter_mem hV₁ hV₂⟩,
-          compactConvNhd_subset_inter f K₁ K₂ V₁ V₂⟩ }
-#align continuous_map.compact_conv_nhd_filter_is_basis ContinuousMap.compactConvNhd_filter_isBasis
-
-/-- A filter basis for the neighbourhood filter of a point in the compact-convergence topology. -/
-def compactConvergenceFilterBasis (f : C(α, β)) : FilterBasis C(α, β) :=
-  (compactConvNhd_filter_isBasis f).filterBasis
-#align continuous_map.compact_convergence_filter_basis ContinuousMap.compactConvergenceFilterBasis
-
-theorem mem_compactConvergence_nhd_filter (Y : Set C(α, β)) :
-    Y ∈ (compactConvergenceFilterBasis f).filter ↔
-    ∃ (K : Set α) (V : Set (β × β)), IsCompact K ∧ V ∈ 𝓤 β ∧ compactConvNhd K V f ⊆ Y := by
+/-- Compact-open topology on `C(α, β)` agrees with the topology of uniform convergence on compacts:
+a family of continuous functions `F i` tends to `f` in the compact-open topology
+if and only if the `F i` tends to `f` uniformly on all compact sets. -/
+theorem tendsto_iff_forall_compact_tendstoUniformlyOn
+    {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f} :
+    Tendsto F p (𝓝 f) ↔ ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K := by
+  rw [tendsto_nhds_compactOpen]
   constructor
-  · rintro ⟨X, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩
-    exact ⟨K, V, hK, hV, hY⟩
-  · rintro ⟨K, V, hK, hV, hY⟩
-    exact ⟨compactConvNhd K V f, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩
-#align continuous_map.mem_compact_convergence_nhd_filter ContinuousMap.mem_compactConvergence_nhd_filter
-
-/-- The compact-convergence topology. In fact, see `ContinuousMap.compactOpen_eq_compactConvergence`
-this is the same as the compact-open topology. This definition is thus an auxiliary convenience
-definition and is unlikely to be of direct use. -/
-def compactConvergenceTopology : TopologicalSpace C(α, β) :=
-  TopologicalSpace.mkOfNhds fun f => (compactConvergenceFilterBasis f).filter
-#align continuous_map.compact_convergence_topology ContinuousMap.compactConvergenceTopology
-
-theorem nhds_compactConvergence :
-    @nhds _ compactConvergenceTopology f = (compactConvergenceFilterBasis f).filter := by
-  rw [TopologicalSpace.nhds_mkOfNhds_filterBasis] <;> rintro g - ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩
-  · exact self_mem_compactConvNhd g hV
-  · obtain ⟨V', hV', _, h₂⟩ := compactConvNhd_nhd_basis g hV
-    exact ⟨compactConvNhd K V' g, ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, fun g' hg' =>
-      ⟨compactConvNhd K V' g', ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, h₂ g' hg'⟩⟩
-#align continuous_map.nhds_compact_convergence ContinuousMap.nhds_compactConvergence
-
-theorem hasBasis_nhds_compactConvergence :
-    HasBasis (@nhds _ compactConvergenceTopology f)
-      (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
-      compactConvNhd p.1 p.2 f :=
-  (nhds_compactConvergence f).symm ▸ (compactConvNhd_filter_isBasis f).hasBasis
-#align continuous_map.has_basis_nhds_compact_convergence ContinuousMap.hasBasis_nhds_compactConvergence
-
-/-- This is an auxiliary lemma and is unlikely to be of direct use outside of this file. See
-`ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn` below for the useful version where the
-topology is picked up via typeclass inference. -/
-theorem tendsto_iff_forall_compact_tendstoUniformlyOn' {ι : Type u₃} {p : Filter ι}
-    {F : ι → C(α, β)} :
-    Filter.Tendsto F p (@nhds _ compactConvergenceTopology f) ↔
-      ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K := by
-  simp only [(hasBasis_nhds_compactConvergence f).tendsto_right_iff, TendstoUniformlyOn, and_imp,
-    Prod.forall]
-  refine' forall_congr' fun K => _
-  rw [forall_swap]
-  exact forall₃_congr fun _hK V _hV => Iff.rfl
-#align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on' ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn'
-
-/-- Any point of `ContinuousMap.CompactOpen.gen K U` is also an interior point wrt the topology of
-compact convergence.
-
-The topology of compact convergence is thus at least as fine as the compact-open topology. -/
-theorem compactConvNhd_subset_compactOpen (hK : IsCompact K) {U : Set β} (hU : IsOpen U)
-    (hf : f ∈ CompactOpen.gen K U) :
-    ∃ V ∈ 𝓤 β, IsOpen V ∧ compactConvNhd K V f ⊆ CompactOpen.gen K U := by
-  obtain ⟨V, hV₁, hV₂, hV₃⟩ := lebesgue_number_of_compact_open (hK.image f.continuous) hU hf
-  refine' ⟨V, hV₁, hV₂, _⟩
-  rintro g hg _ ⟨x, hx, rfl⟩
-  exact hV₃ (f x) ⟨x, hx, rfl⟩ (hg x hx)
-#align continuous_map.compact_conv_nhd_subset_compact_open ContinuousMap.compactConvNhd_subset_compactOpen
-
-/-- The point `f` in `ContinuousMap.compactConvNhd K V f` is also an interior point wrt the
-compact-open topology.
-
-Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
-that the compact-open topology is at least as fine as the topology of compact convergence. -/
-theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
-    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
-      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
-      (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
-        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
-  obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
-  obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
-  let U : α → Set α := fun x => f ⁻¹' ball (f x) Z
-  have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
-  have hUK : K ⊆ ⋃ x : K, U (x : K) := by
-    intro x hx
-    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
-    exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
-  obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
-  let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
-  have hC : K ⊆ ⋃ i, C i := by
-    rw [← K.inter_iUnion, subset_inter_iff]
-    refine' ⟨Subset.rfl, ht.trans _⟩
-    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
-    exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
-  have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W := by
-    simp only [← image_subset_iff, ← mem_preimage]
-    rintro ⟨⟨x, hx₁⟩, hx₂⟩
-    have hZW : closure (ball (f x) Z) ⊆ ball (f x) W := by
-      intro y hy
-      obtain ⟨z, hz₁, hz₂⟩ := UniformSpace.mem_closure_iff_ball.mp hy hZ₁
-      exact ball_mono hZ₃ _ (mem_ball_comp hz₂ ((mem_ball_symmetry hZ₂).mp hz₁))
-    calc
-      f '' (K ∩ closure (U x)) ⊆ f '' closure (U x) := image_subset _ (inter_subset_right _ _)
-      _ ⊆ closure (f '' U x) := f.continuous.continuousOn.image_closure
-      _ ⊆ closure (ball (f x) Z) := by
-        apply closure_mono
-        simp only [image_subset_iff]
-        rfl
-      _ ⊆ ball (f x) W := hZW
-
-  refine'
-    ⟨t, t.fintypeCoeSort, C, fun i => hK.inter_right isClosed_closure, fun i =>
-      ball (f ((i : K) : α)) W, fun i => isOpen_ball _ hW₄, by simp [CompactOpen.gen, hfC],
-      fun g hg x hx => hW₃ (mem_compRel.mpr _)⟩
-  simp only [mem_iInter, CompactOpen.gen, mem_setOf_eq, image_subset_iff] at hg
-  obtain ⟨y, hy⟩ := mem_iUnion.mp (hC hx)
-  exact ⟨f y, (mem_ball_symmetry hW₂).mp (hfC y hy), mem_preimage.mp (hg y hy)⟩
-#align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd
-
-/-- The compact-open topology is equal to the compact-convergence topology. -/
-theorem compactOpen_eq_compactConvergence :
-    ContinuousMap.compactOpen = (compactConvergenceTopology : TopologicalSpace C(α, β)) := by
-  rw [compactConvergenceTopology, ContinuousMap.compactOpen]
-  refine' le_antisymm _ _
-  · refine' fun X hX => isOpen_iff_forall_mem_open.mpr fun f hf => _
-    have hXf : X ∈ (compactConvergenceFilterBasis f).filter := by
-      rw [← nhds_compactConvergence]
-      exact @IsOpen.mem_nhds C(α, β) _ _ compactConvergenceTopology hX hf
-    obtain ⟨-, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hXf⟩ := hXf
-    obtain ⟨ι, hι, C, hC, U, hU, h₁, h₂⟩ := iInter_compactOpen_gen_subset_compactConvNhd f hK hV
-    haveI := hι
-    exact
-      ⟨⋂ i, CompactOpen.gen (C i) (U i), h₂.trans hXf,
-        isOpen_iInter_of_finite fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
-  · simp only [TopologicalSpace.le_generateFrom_iff_subset_isOpen, and_imp, exists_prop,
-      forall_exists_index, setOf_subset_setOf]
-    rintro - K hK U hU rfl f hf
-    obtain ⟨V, hV, _hV', hVf⟩ := compactConvNhd_subset_compactOpen f hK hU hf
-    exact Filter.mem_of_superset (FilterBasis.mem_filter_of_mem _ ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩) hVf
-#align continuous_map.compact_open_eq_compact_convergence ContinuousMap.compactOpen_eq_compactConvergence
-
-/-- The filter on `C(α, β) × C(α, β)` which underlies the uniform space structure on `C(α, β)`. -/
-def compactConvergenceUniformity : Filter (C(α, β) × C(α, β)) :=
-  ⨅ KV ∈ { KV : Set α × Set (β × β) | IsCompact KV.1 ∧ KV.2 ∈ 𝓤 β },
-    𝓟 { fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ KV.1 → (fg.1 x, fg.2 x) ∈ KV.2 }
-#align continuous_map.compact_convergence_uniformity ContinuousMap.compactConvergenceUniformity
-
-theorem hasBasis_compactConvergenceUniformity_aux :
-    HasBasis (@compactConvergenceUniformity α β _ _)
-      (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
-      { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } := by
-  refine' Filter.hasBasis_biInf_principal _ compactConvNhd_compact_entourage_nonempty
-  rintro ⟨K₁, V₁⟩ ⟨hK₁, hV₁⟩ ⟨K₂, V₂⟩ ⟨hK₂, hV₂⟩
-  refine' ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, Filter.inter_mem hV₁ hV₂⟩, _⟩
-  simp only [le_eq_subset, Prod.forall, setOf_subset_setOf, ge_iff_le, Order.Preimage, ←
-    forall_and, mem_inter_iff, mem_union]
-  exact fun f g => forall_imp fun x => by tauto
-#align continuous_map.has_basis_compact_convergence_uniformity_aux ContinuousMap.hasBasis_compactConvergenceUniformity_aux
-
-/-- An intermediate lemma. Usually `ContinuousMap.mem_compactConvergence_entourage_iff` is more
-useful. -/
-theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
-    X ∈ @compactConvergenceUniformity α β _ _ ↔
-      ∃ (K : Set α) (V : Set (β × β)), IsCompact K ∧ V ∈ 𝓤 β ∧
-        { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X := by
-  simp only [hasBasis_compactConvergenceUniformity_aux.mem_iff, exists_prop, Prod.exists,
-    and_assoc]
-#align continuous_map.mem_compact_convergence_uniformity ContinuousMap.mem_compactConvergenceUniformity
-
-/-- Note that we ensure the induced topology is definitionally the compact-open topology. -/
-instance compactConvergenceUniformSpace : UniformSpace C(α, β) where
-  uniformity := compactConvergenceUniformity
-  refl := by
-    simp only [compactConvergenceUniformity, and_imp, Filter.le_principal_iff, Prod.forall,
-      Filter.mem_principal, mem_setOf_eq, le_iInf_iff, idRel_subset]
-    exact fun K V _hK hV f x _hx => refl_mem_uniformity hV
-  symm := by
-    simp only [compactConvergenceUniformity, and_imp, Prod.forall, mem_setOf_eq, Prod.fst_swap,
-      Filter.tendsto_principal, Prod.snd_swap, Filter.tendsto_iInf]
-    intro K V hK hV
-    obtain ⟨V', hV', hsymm, hsub⟩ := symm_of_uniformity hV
-    let X := { fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ K → (fg.1 x, fg.2 x) ∈ V' }
-    have hX : X ∈ compactConvergenceUniformity :=
-      (mem_compactConvergenceUniformity X).mpr ⟨K, V', hK, hV', by simp⟩
-    exact Filter.eventually_of_mem hX fun fg hfg x hx => hsub (hsymm _ _ (hfg x hx))
-  comp X hX := by
-    obtain ⟨K, V, hK, hV, hX⟩ := (mem_compactConvergenceUniformity X).mp hX
-    obtain ⟨V', hV', hcomp⟩ := comp_mem_uniformity_sets hV
-    let h := fun s : Set (C(α, β) × C(α, β)) => s ○ s
-    suffices h { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V' } ∈
-        compactConvergenceUniformity.lift' h by
-      apply Filter.mem_of_superset this
-      rintro ⟨f, g⟩ ⟨z, hz₁, hz₂⟩
-      refine' hX fun x hx => hcomp _
-      exact ⟨z x, hz₁ x hx, hz₂ x hx⟩
-    apply Filter.mem_lift'
-    exact (mem_compactConvergenceUniformity _).mpr ⟨K, V', hK, hV', Subset.refl _⟩
-  isOpen_uniformity := by
-    rw [compactOpen_eq_compactConvergence]
-    refine' fun Y => forall₂_congr fun f hf => _
-    simp only [mem_compactConvergence_nhd_filter, mem_compactConvergenceUniformity, Prod.forall,
-      setOf_subset_setOf, compactConvNhd]
-    refine' exists₂_congr fun K V => and_congr_right' <| and_congr_right'
-      ⟨_, fun hY g hg => hY f g hg rfl⟩
-    rintro hY g₁ g₂ hg₁ rfl
-    exact hY hg₁
-#align continuous_map.compact_convergence_uniform_space ContinuousMap.compactConvergenceUniformSpace
+  · -- Let us prove that convergence in the compact-open topology
+    -- implies uniform convergence on compacts.
+    -- Consider a compact set `K`
+    intro h K hK
+    -- Since `K` is compact, it suffices to prove locally uniform convergence
+    rw [← tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK]
+    -- Now choose an entourage `U` in the codomain and a point `x ∈ K`.
+    intro U hU x _
+    -- Choose an open symmetric entourage `V` such that `V ○ V ⊆ U`.
+    rcases comp_open_symm_mem_uniformity_sets hU with ⟨V, hV, hVo, hVsymm, hVU⟩
+    -- Then choose a closed entourage `W ⊆ V`
+    rcases mem_uniformity_isClosed hV with ⟨W, hW, hWc, hWU⟩
+    -- Consider `s = {y ∈ K | (f x, f y) ∈ W}`
+    set s := K ∩ f ⁻¹' ball (f x) W
+    -- This is a neighbourhood of `x` within `K`, because `W` is an entourage.
+    have hnhds : s ∈ 𝓝[K] x := inter_mem_nhdsWithin _ <| f.continuousAt _ (ball_mem_nhds _ hW)
+    -- This set is compact because it is an intersection of `K`
+    -- with a closed set `{y | (f x, f y) ∈ W} = f ⁻¹' UniformSpace.ball (f x) W`
+    have hcomp : IsCompact s := hK.inter_right <| (isClosed_ball _ hWc).preimage f.continuous
+    -- `f` maps `s` to the open set `ball (f x) V = {z | (f x, z) ∈ V}`
+    have hmaps : MapsTo f s (ball (f x) V) := fun x hx ↦ hWU hx.2
+    use s, hnhds
+    -- Continuous maps `F i` in a neighbourhood of `f` map `s` to `ball (f x) V` as well.
+    refine (h s hcomp _ (isOpen_ball _ hVo) hmaps).mono fun g hg y hy ↦ ?_
+    -- Then for `y ∈ s` we have `(f y, f x) ∈ V` and `(f x, F i y) ∈ V`, thus `(f y, F i y) ∈ U`
+    exact hVU ⟨f x, hVsymm.mk_mem_comm.2 <| hmaps hy, hg hy⟩
+  · -- Now we prove that uniform convergence on compacts
+    -- implies convergence in the compact-open topology
+    -- Consider a compact set `K`, an open set `U`, and a continuous map `f` that maps `K` to `U`
+    intro h K hK U hU hf
+    -- Due to Lebesgue number lemma, there exists an entourage `V`
+    -- such that `U` includes the `V`-thickening of `f '' K`.
+    rcases lebesgue_number_of_compact_open (hK.image (map_continuous f)) hU hf.image_subset
+        with ⟨V, hV, -, hVf⟩
+    -- Then any continuous map that is uniformly `V`-close to `f` on `K`
+    -- maps `K` to `U` as well
+    filter_upwards [h K hK V hV] with g hg x hx using hVf _ (mem_image_of_mem f hx) (hg x hx)
+#align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn
 
-theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
-    X ∈ 𝓤 C(α, β) ↔
-      ∃ (K : Set α) (V : Set (β × β)), IsCompact K ∧ V ∈ 𝓤 β ∧
-        { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
-  mem_compactConvergenceUniformity X
-#align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff
+/-- Interpret a bundled continuous map as an element of `α →ᵤ[{K | IsCompact K}] β`.
+
+We use this map to induce the `UniformSpace` structure on `C(α, β)`. -/
+def toUniformOnFunIsCompact (f : C(α, β)) : α →ᵤ[{K | IsCompact K}] β :=
+  UniformOnFun.ofFun {K | IsCompact K} f
+
+@[simp]
+theorem toUniformOnFun_toFun (f : C(α, β)) :
+    UniformOnFun.toFun _ f.toUniformOnFunIsCompact = f := rfl
+
+open UniformSpace in
+/-- Uniform space structure on `C(α, β)`.
+
+The uniformity comes from `α →ᵤ[{K | IsCompact K}] β` (i.e., `UniformOnFun α β {K | IsCompact K}`)
+which defines topology of uniform convergence on compact sets.
+We use `ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn`
+to show that the induced topology agrees with the compact-open topology
+and replace the topology with `compactOpen` to avoid non-defeq diamonds,
+see Note [forgetful inheritance].  -/
+instance compactConvergenceUniformSpace : UniformSpace C(α, β) :=
+  .replaceTopology (.comap toUniformOnFunIsCompact inferInstance) <| by
+    refine eq_of_nhds_eq_nhds fun f ↦ eq_of_forall_le_iff fun l ↦ ?_
+    simp_rw [← tendsto_id', tendsto_iff_forall_compact_tendstoUniformlyOn,
+      nhds_induced, tendsto_comap_iff, UniformOnFun.tendsto_iff_tendstoUniformlyOn]
+    rfl
+#align continuous_map.compact_convergence_uniform_space ContinuousMap.compactConvergenceUniformSpace
 
-theorem hasBasis_compactConvergenceUniformity :
-    HasBasis (𝓤 C(α, β)) (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
-      { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } :=
-  hasBasis_compactConvergenceUniformity_aux
-#align continuous_map.has_basis_compact_convergence_uniformity ContinuousMap.hasBasis_compactConvergenceUniformity
+theorem uniformEmbedding_toUniformOnFunIsCompact :
+    UniformEmbedding (toUniformOnFunIsCompact : C(α, β) → α →ᵤ[{K | IsCompact K}] β) where
+  comap_uniformity := rfl
+  inj := DFunLike.coe_injective
+
+-- The following definitions and theorems
+-- used to be a part of the construction of the `UniformSpace C(α, β)` structure
+-- before it was migrated to `UniformOnFun`
+#noalign continuous_map.compact_conv_nhd
+#noalign continuous_map.self_mem_compact_conv_nhd
+#noalign continuous_map.compact_conv_nhd_mono
+#noalign continuous_map.compact_conv_nhd_mem_comp
+#noalign continuous_map.compact_conv_nhd_nhd_basis
+#noalign continuous_map.compact_conv_nhd_subset_inter
+#noalign continuous_map.compact_conv_nhd_compact_entourage_nonempty
+#noalign continuous_map.compact_conv_nhd_filter_is_basis
+#noalign continuous_map.compact_convergence_filter_basis
+#noalign continuous_map.mem_compact_convergence_nhd_filter
+#noalign continuous_map.compact_convergence_topology
+#noalign continuous_map.nhds_compact_convergence
+#noalign continuous_map.has_basis_nhds_compact_convergence
+#noalign continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on'
+#noalign continuous_map.compact_conv_nhd_subset_compact_open
+#noalign continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd
+#noalign continuous_map.compact_open_eq_compact_convergence
+#noalign continuous_map.compact_convergence_uniformity
+#noalign continuous_map.has_basis_compact_convergence_uniformity_aux
+#noalign continuous_map.mem_compact_convergence_uniformity
 
 theorem _root_.Filter.HasBasis.compactConvergenceUniformity {ι : Type*} {pi : ι → Prop}
     {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s) :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × ι => IsCompact p.1 ∧ pi p.2) fun p =>
       { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2 } := by
-  refine' hasBasis_compactConvergenceUniformity.to_hasBasis _ _
-  · rintro ⟨t₁, t₂⟩ ⟨h₁, h₂⟩
-    rcases h.mem_iff.1 h₂ with ⟨i, hpi, hi⟩
-    exact ⟨(t₁, i), ⟨h₁, hpi⟩, fun fg hfg x hx => hi (hfg _ hx)⟩
-  · rintro ⟨t, i⟩ ⟨ht, hi⟩
-    exact ⟨(t, s i), ⟨ht, h.mem_of_mem hi⟩, Subset.rfl⟩
+  rw [← uniformEmbedding_toUniformOnFunIsCompact.comap_uniformity]
+  exact .comap _ <| UniformOnFun.hasBasis_uniformity_of_basis _ _ {K | IsCompact K}
+    ⟨∅, isCompact_empty⟩ (directedOn_of_sup_mem fun _ _ ↦ IsCompact.union) h
 #align filter.has_basis.compact_convergence_uniformity Filter.HasBasis.compactConvergenceUniformity
 
-variable {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f}
+theorem hasBasis_compactConvergenceUniformity :
+    HasBasis (𝓤 C(α, β)) (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
+      { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } :=
+  (basis_sets _).compactConvergenceUniformity
+#align continuous_map.has_basis_compact_convergence_uniformity ContinuousMap.hasBasis_compactConvergenceUniformity
 
-theorem tendsto_iff_forall_compact_tendstoUniformlyOn :
-    Tendsto F p (𝓝 f) ↔ ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K := by
-  rw [compactOpen_eq_compactConvergence, tendsto_iff_forall_compact_tendstoUniformlyOn']
-#align continuous_map.tendsto_iff_forall_compact_tendsto_uniformly_on ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn
+theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
+    X ∈ 𝓤 C(α, β) ↔
+      ∃ (K : Set α) (V : Set (β × β)), IsCompact K ∧ V ∈ 𝓤 β ∧
+        { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X := by
+  simp [hasBasis_compactConvergenceUniformity.mem_iff, and_assoc]
+#align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff
+
+variable {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f}
 
 /-- Locally uniform convergence implies convergence in the compact-open topology. -/
 theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a => F i a) f p) :

• add TopologicalSpace.nhds_mkOfNhds_of_hasBasis
• add Trans instance for Filter.mem_of_superset
• change assumptions of TopologicalSpace.nhds_mkOfNhds, golf
  • the new assumption is equivalent to the old one with t ⊆ s removed
  • but is formulated in terms of Filter.Eventually

@@ -173,10 +173,9 @@ theorem nhds_compactConvergence :
     @nhds _ compactConvergenceTopology f = (compactConvergenceFilterBasis f).filter := by
   rw [TopologicalSpace.nhds_mkOfNhds_filterBasis] <;> rintro g - ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩
   · exact self_mem_compactConvNhd g hV
-  · obtain ⟨V', hV', h₁, h₂⟩ := compactConvNhd_nhd_basis g hV
-    exact
-      ⟨compactConvNhd K V' g, ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, compactConvNhd_mono g h₁, fun g' hg' =>
-        ⟨compactConvNhd K V' g', ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, h₂ g' hg'⟩⟩
+  · obtain ⟨V', hV', _, h₂⟩ := compactConvNhd_nhd_basis g hV
+    exact ⟨compactConvNhd K V' g, ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, fun g' hg' =>
+      ⟨compactConvNhd K V' g', ⟨⟨K, V'⟩, ⟨hK, hV'⟩, rfl⟩, h₂ g' hg'⟩⟩
 #align continuous_map.nhds_compact_convergence ContinuousMap.nhds_compactConvergence
 
 theorem hasBasis_nhds_compactConvergence :

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -271,7 +271,7 @@ theorem compactOpen_eq_compactConvergence :
   · refine' fun X hX => isOpen_iff_forall_mem_open.mpr fun f hf => _
     have hXf : X ∈ (compactConvergenceFilterBasis f).filter := by
       rw [← nhds_compactConvergence]
-      exact @IsOpen.mem_nhds C(α, β) compactConvergenceTopology _ _ hX hf
+      exact @IsOpen.mem_nhds C(α, β) _ _ compactConvergenceTopology hX hf
     obtain ⟨-, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hXf⟩ := hXf
     obtain ⟨ι, hι, C, hC, U, hU, h₁, h₂⟩ := iInter_compactOpen_gen_subset_compactConvNhd f hK hV
     haveI := hι

Follow-up #9184

@@ -154,8 +154,7 @@ def compactConvergenceFilterBasis (f : C(α, β)) : FilterBasis C(α, β) :=
 
 theorem mem_compactConvergence_nhd_filter (Y : Set C(α, β)) :
     Y ∈ (compactConvergenceFilterBasis f).filter ↔
-    ∃ (K : Set α) (V : Set (β × β)) (_hK : IsCompact K) (_hV : V ∈ 𝓤 β),
-      compactConvNhd K V f ⊆ Y := by
+    ∃ (K : Set α) (V : Set (β × β)), IsCompact K ∧ V ∈ 𝓤 β ∧ compactConvNhd K V f ⊆ Y := by
   constructor
   · rintro ⟨X, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩
     exact ⟨K, V, hK, hV, hY⟩
@@ -308,15 +307,14 @@ theorem hasBasis_compactConvergenceUniformity_aux :
 useful. -/
 theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
     X ∈ @compactConvergenceUniformity α β _ _ ↔
-      ∃ (K : Set α) (V : Set (β × β)) (_hK : IsCompact K) (_hV : V ∈ 𝓤 β),
+      ∃ (K : Set α) (V : Set (β × β)), IsCompact K ∧ V ∈ 𝓤 β ∧
         { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X := by
   simp only [hasBasis_compactConvergenceUniformity_aux.mem_iff, exists_prop, Prod.exists,
     and_assoc]
 #align continuous_map.mem_compact_convergence_uniformity ContinuousMap.mem_compactConvergenceUniformity
 
 /-- Note that we ensure the induced topology is definitionally the compact-open topology. -/
-instance compactConvergenceUniformSpace : UniformSpace C(α, β)
-    where
+instance compactConvergenceUniformSpace : UniformSpace C(α, β) where
   uniformity := compactConvergenceUniformity
   refl := by
     simp only [compactConvergenceUniformity, and_imp, Filter.le_principal_iff, Prod.forall,
@@ -348,14 +346,15 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
     refine' fun Y => forall₂_congr fun f hf => _
     simp only [mem_compactConvergence_nhd_filter, mem_compactConvergenceUniformity, Prod.forall,
       setOf_subset_setOf, compactConvNhd]
-    refine' exists₄_congr fun K V _hK _hV => ⟨_, fun hY g hg => hY f g hg rfl⟩
+    refine' exists₂_congr fun K V => and_congr_right' <| and_congr_right'
+      ⟨_, fun hY g hg => hY f g hg rfl⟩
     rintro hY g₁ g₂ hg₁ rfl
     exact hY hg₁
 #align continuous_map.compact_convergence_uniform_space ContinuousMap.compactConvergenceUniformSpace
 
 theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
     X ∈ 𝓤 C(α, β) ↔
-      ∃ (K : Set α) (V : Set (β × β)) (_hK : IsCompact K) (_hV : V ∈ 𝓤 β),
+      ∃ (K : Set α) (V : Set (β × β)), IsCompact K ∧ V ∈ 𝓤 β ∧
         { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
   mem_compactConvergenceUniformity X
 #align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff

We define Alexandrov-discrete spaces as topological spaces where the intersection of a family of open sets is open.

This PR only gives a minimal API because the goal is to ensure that lemma names like isOpen_sInter are free to use for AlexandrovDiscrete. The existing lemmas are getting prefixed by Set.Finite or suffixed by _of_finite.

@@ -278,7 +278,7 @@ theorem compactOpen_eq_compactConvergence :
     haveI := hι
     exact
       ⟨⋂ i, CompactOpen.gen (C i) (U i), h₂.trans hXf,
-        isOpen_iInter fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
+        isOpen_iInter_of_finite fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
   · simp only [TopologicalSpace.le_generateFrom_iff_subset_isOpen, and_imp, exists_prop,
       forall_exists_index, setOf_subset_setOf]
     rintro - K hK U hU rfl f hf

@@ -394,29 +394,25 @@ theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a
   exact h.tendstoLocallyUniformlyOn
 #align continuous_map.tendsto_of_tendsto_locally_uniformly ContinuousMap.tendsto_of_tendstoLocallyUniformly
 
-/-- If every point has a compact neighbourhood, then convergence in the compact-open topology
-implies locally uniform convergence.
-
-See also `ContinuousMap.tendsto_iff_tendstoLocallyUniformly`, especially for T2 spaces. -/
-theorem tendstoLocallyUniformly_of_tendsto (hα : ∀ x : α, ∃ n, IsCompact n ∧ n ∈ 𝓝 x)
-    (h : Tendsto F p (𝓝 f)) : TendstoLocallyUniformly (fun i a => F i a) f p := by
+/-- In a weakly locally compact space,
+convergence in the compact-open topology is the same as locally uniform convergence.
+
+The right-to-left implication holds in any topological space,
+see `ContinuousMap.tendsto_of_tendstoLocallyUniformly`. -/
+theorem tendsto_iff_tendstoLocallyUniformly [WeaklyLocallyCompactSpace α] :
+    Tendsto F p (𝓝 f) ↔ TendstoLocallyUniformly (fun i a => F i a) f p := by
+  refine ⟨fun h V hV x ↦ ?_, tendsto_of_tendstoLocallyUniformly⟩
   rw [tendsto_iff_forall_compact_tendstoUniformlyOn] at h
-  intro V hV x
-  obtain ⟨n, hn₁, hn₂⟩ := hα x
+  obtain ⟨n, hn₁, hn₂⟩ := exists_compact_mem_nhds x
   exact ⟨n, hn₂, h n hn₁ V hV⟩
-#align continuous_map.tendsto_locally_uniformly_of_tendsto ContinuousMap.tendstoLocallyUniformly_of_tendsto
-
-/-- Convergence in the compact-open topology is the same as locally uniform convergence on a locally
-compact space.
-
-For non-T2 spaces, the assumption `LocallyCompactSpace α` is stronger than we need and in fact
-the `←` direction is true unconditionally. See `ContinuousMap.tendstoLocallyUniformly_of_tendsto`
-and `ContinuousMap.tendsto_of_tendstoLocallyUniformly` for versions requiring weaker hypotheses. -/
-theorem tendsto_iff_tendstoLocallyUniformly [LocallyCompactSpace α] :
-    Tendsto F p (𝓝 f) ↔ TendstoLocallyUniformly (fun i a => F i a) f p :=
-  ⟨tendstoLocallyUniformly_of_tendsto exists_compact_mem_nhds, tendsto_of_tendstoLocallyUniformly⟩
 #align continuous_map.tendsto_iff_tendsto_locally_uniformly ContinuousMap.tendsto_iff_tendstoLocallyUniformly
 
+@[deprecated tendsto_iff_tendstoLocallyUniformly]
+theorem tendstoLocallyUniformly_of_tendsto [WeaklyLocallyCompactSpace α] (h : Tendsto F p (𝓝 f)) :
+    TendstoLocallyUniformly (fun i a => F i a) f p :=
+  tendsto_iff_tendstoLocallyUniformly.1 h
+#align continuous_map.tendsto_locally_uniformly_of_tendsto ContinuousMap.tendstoLocallyUniformly_of_tendsto
+
 section CompactDomain
 
 variable [CompactSpace α]

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

This has nice performance benefits.

@@ -366,7 +366,7 @@ theorem hasBasis_compactConvergenceUniformity :
   hasBasis_compactConvergenceUniformity_aux
 #align continuous_map.has_basis_compact_convergence_uniformity ContinuousMap.hasBasis_compactConvergenceUniformity
 
-theorem _root_.Filter.HasBasis.compactConvergenceUniformity {ι : Type _} {pi : ι → Prop}
+theorem _root_.Filter.HasBasis.compactConvergenceUniformity {ι : Type*} {pi : ι → Prop}
     {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s) :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × ι => IsCompact p.1 ∧ pi p.2) fun p =>
       { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2 } := by

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module topology.uniform_space.compact_convergence
-! leanprover-community/mathlib commit dc6c365e751e34d100e80fe6e314c3c3e0fd2988
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.CompactOpen
 import Mathlib.Topology.UniformSpace.UniformConvergence
 
+#align_import topology.uniform_space.compact_convergence from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
+
 /-!
 # Compact convergence (uniform convergence on compact sets)
 

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

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

@@ -43,9 +43,9 @@ $$
 $$
 Using this notation we can describe the uniform space structure and the topology it induces.
 Specifically:
- *  A subset `X ⊆ C(α, β) × C(α, β)` is an entourage for the uniform space structure on `C(α, β)`
+  * A subset `X ⊆ C(α, β) × C(α, β)` is an entourage for the uniform space structure on `C(α, β)`
     iff there exists a compact `K` and entourage `V` such that `E(K, V) ⊆ X`.
- *  A subset `Y ⊆ C(α, β)` is a neighbourhood of `f` iff there exists a compact `K` and entourage
+  * A subset `Y ⊆ C(α, β)` is a neighbourhood of `f` iff there exists a compact `K` and entourage
     `V` such that `N(K, V, f) ⊆ Y`.
 
 The topology on `C(α, β)` thus has a natural subbasis (the compact-open subbasis) and a natural

@@ -226,7 +226,7 @@ theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V
     ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
       (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
       (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
-        (⋂ i, CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f := by
+        ⋂ i, CompactOpen.gen (C i) (U i) ⊆ compactConvNhd K V f := by
   obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
   obtain ⟨Z, hZ₁, hZ₄, hZ₂, hZ₃⟩ := comp_open_symm_mem_uniformity_sets hW₁
   let U : α → Set α := fun x => f ⁻¹' ball (f x) Z

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

@@ -223,8 +223,8 @@ compact-open topology.
 Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
 that the compact-open topology is at least as fine as the topology of compact convergence. -/
 theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
-    ∃ (ι : Sort (u₁ + 1))(_ : Fintype ι)(C : ι → Set α)(_hC : ∀ i, IsCompact (C i))(U :
-      ι → Set β)(_hU : ∀ i, IsOpen (U i)),
+    ∃ (ι : Sort (u₁ + 1)) (_ : Fintype ι) (C : ι → Set α) (_hC : ∀ i, IsCompact (C i))
+      (U : ι → Set β) (_hU : ∀ i, IsOpen (U i)),
       (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
         (⋂ i, CompactOpen.gen (C i) (U i)) ⊆ compactConvNhd K V f := by
   obtain ⟨W, hW₁, hW₄, hW₂, hW₃⟩ := comp_open_symm_mem_uniformity_sets hV
@@ -311,7 +311,7 @@ theorem hasBasis_compactConvergenceUniformity_aux :
 useful. -/
 theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
     X ∈ @compactConvergenceUniformity α β _ _ ↔
-      ∃ (K : Set α)(V : Set (β × β))(_hK : IsCompact K)(_hV : V ∈ 𝓤 β),
+      ∃ (K : Set α) (V : Set (β × β)) (_hK : IsCompact K) (_hV : V ∈ 𝓤 β),
         { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X := by
   simp only [hasBasis_compactConvergenceUniformity_aux.mem_iff, exists_prop, Prod.exists,
     and_assoc]
@@ -358,7 +358,7 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
 
 theorem mem_compactConvergence_entourage_iff (X : Set (C(α, β) × C(α, β))) :
     X ∈ 𝓤 C(α, β) ↔
-      ∃ (K : Set α)(V : Set (β × β))(_hK : IsCompact K)(_hV : V ∈ 𝓤 β),
+      ∃ (K : Set α) (V : Set (β × β)) (_hK : IsCompact K) (_hV : V ∈ 𝓤 β),
         { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V } ⊆ X :=
   mem_compactConvergenceUniformity X
 #align continuous_map.mem_compact_convergence_entourage_iff ContinuousMap.mem_compactConvergence_entourage_iff

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>

@@ -222,7 +222,7 @@ compact-open topology.
 
 Since `ContinuousMap.compactConvNhd K V f` are a neighbourhood basis at `f` for each `f`, it follows
 that the compact-open topology is at least as fine as the topology of compact convergence. -/
-theorem interᵢ_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
+theorem iInter_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV : V ∈ 𝓤 β) :
     ∃ (ι : Sort (u₁ + 1))(_ : Fintype ι)(C : ι → Set α)(_hC : ∀ i, IsCompact (C i))(U :
       ι → Set β)(_hU : ∀ i, IsOpen (U i)),
       (f ∈ ⋂ i, CompactOpen.gen (C i) (U i)) ∧
@@ -233,15 +233,15 @@ theorem interᵢ_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV :
   have hU : ∀ x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ₄)
   have hUK : K ⊆ ⋃ x : K, U (x : K) := by
     intro x hx
-    simp only [exists_prop, mem_unionᵢ, unionᵢ_coe_set, mem_preimage]
+    simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]
     exact ⟨(⟨x, hx⟩ : K), by simp [hx, mem_ball_self (f x) hZ₁]⟩
   obtain ⟨t, ht⟩ := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK
   let C : t → Set α := fun i => K ∩ closure (U ((i : K) : α))
   have hC : K ⊆ ⋃ i, C i := by
-    rw [← K.inter_unionᵢ, subset_inter_iff]
+    rw [← K.inter_iUnion, subset_inter_iff]
     refine' ⟨Subset.rfl, ht.trans _⟩
-    simp only [SetCoe.forall, Subtype.coe_mk, unionᵢ_subset_iff]
-    exact fun x hx₁ hx₂ => subset_unionᵢ_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
+    simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]
+    exact fun x hx₁ hx₂ => subset_iUnion_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
   have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W := by
     simp only [← image_subset_iff, ← mem_preimage]
     rintro ⟨⟨x, hx₁⟩, hx₂⟩
@@ -262,10 +262,10 @@ theorem interᵢ_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV :
     ⟨t, t.fintypeCoeSort, C, fun i => hK.inter_right isClosed_closure, fun i =>
       ball (f ((i : K) : α)) W, fun i => isOpen_ball _ hW₄, by simp [CompactOpen.gen, hfC],
       fun g hg x hx => hW₃ (mem_compRel.mpr _)⟩
-  simp only [mem_interᵢ, CompactOpen.gen, mem_setOf_eq, image_subset_iff] at hg
-  obtain ⟨y, hy⟩ := mem_unionᵢ.mp (hC hx)
+  simp only [mem_iInter, CompactOpen.gen, mem_setOf_eq, image_subset_iff] at hg
+  obtain ⟨y, hy⟩ := mem_iUnion.mp (hC hx)
   exact ⟨f y, (mem_ball_symmetry hW₂).mp (hfC y hy), mem_preimage.mp (hg y hy)⟩
-#align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.interᵢ_compactOpen_gen_subset_compactConvNhd
+#align continuous_map.Inter_compact_open_gen_subset_compact_conv_nhd ContinuousMap.iInter_compactOpen_gen_subset_compactConvNhd
 
 /-- The compact-open topology is equal to the compact-convergence topology. -/
 theorem compactOpen_eq_compactConvergence :
@@ -277,11 +277,11 @@ theorem compactOpen_eq_compactConvergence :
       rw [← nhds_compactConvergence]
       exact @IsOpen.mem_nhds C(α, β) compactConvergenceTopology _ _ hX hf
     obtain ⟨-, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hXf⟩ := hXf
-    obtain ⟨ι, hι, C, hC, U, hU, h₁, h₂⟩ := interᵢ_compactOpen_gen_subset_compactConvNhd f hK hV
+    obtain ⟨ι, hι, C, hC, U, hU, h₁, h₂⟩ := iInter_compactOpen_gen_subset_compactConvNhd f hK hV
     haveI := hι
     exact
       ⟨⋂ i, CompactOpen.gen (C i) (U i), h₂.trans hXf,
-        isOpen_interᵢ fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
+        isOpen_iInter fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h₁⟩
   · simp only [TopologicalSpace.le_generateFrom_iff_subset_isOpen, and_imp, exists_prop,
       forall_exists_index, setOf_subset_setOf]
     rintro - K hK U hU rfl f hf
@@ -299,7 +299,7 @@ theorem hasBasis_compactConvergenceUniformity_aux :
     HasBasis (@compactConvergenceUniformity α β _ _)
       (fun p : Set α × Set (β × β) => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p =>
       { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2 } := by
-  refine' Filter.hasBasis_binfᵢ_principal _ compactConvNhd_compact_entourage_nonempty
+  refine' Filter.hasBasis_biInf_principal _ compactConvNhd_compact_entourage_nonempty
   rintro ⟨K₁, V₁⟩ ⟨hK₁, hV₁⟩ ⟨K₂, V₂⟩ ⟨hK₂, hV₂⟩
   refine' ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, Filter.inter_mem hV₁ hV₂⟩, _⟩
   simp only [le_eq_subset, Prod.forall, setOf_subset_setOf, ge_iff_le, Order.Preimage, ←
@@ -323,11 +323,11 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
   uniformity := compactConvergenceUniformity
   refl := by
     simp only [compactConvergenceUniformity, and_imp, Filter.le_principal_iff, Prod.forall,
-      Filter.mem_principal, mem_setOf_eq, le_infᵢ_iff, idRel_subset]
+      Filter.mem_principal, mem_setOf_eq, le_iInf_iff, idRel_subset]
     exact fun K V _hK hV f x _hx => refl_mem_uniformity hV
   symm := by
     simp only [compactConvergenceUniformity, and_imp, Prod.forall, mem_setOf_eq, Prod.fst_swap,
-      Filter.tendsto_principal, Prod.snd_swap, Filter.tendsto_infᵢ]
+      Filter.tendsto_principal, Prod.snd_swap, Filter.tendsto_iInf]
     intro K V hK hV
     obtain ⟨V', hV', hsymm, hsub⟩ := symm_of_uniformity hV
     let X := { fg : C(α, β) × C(α, β) | ∀ x : α, x ∈ K → (fg.1 x, fg.2 x) ∈ V' }

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

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

@@ -157,8 +157,8 @@ def compactConvergenceFilterBasis (f : C(α, β)) : FilterBasis C(α, β) :=
 
 theorem mem_compactConvergence_nhd_filter (Y : Set C(α, β)) :
     Y ∈ (compactConvergenceFilterBasis f).filter ↔
-      ∃ (K : Set α)(V : Set (β × β))(_hK : IsCompact K)(_hV : V ∈ 𝓤 β), compactConvNhd K V f ⊆ Y :=
-  by
+    ∃ (K : Set α) (V : Set (β × β)) (_hK : IsCompact K) (_hV : V ∈ 𝓤 β),
+      compactConvNhd K V f ⊆ Y := by
   constructor
   · rintro ⟨X, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hY⟩
     exact ⟨K, V, hK, hV, hY⟩
@@ -242,23 +242,20 @@ theorem interᵢ_compactOpen_gen_subset_compactConvNhd (hK : IsCompact K) (hV :
     refine' ⟨Subset.rfl, ht.trans _⟩
     simp only [SetCoe.forall, Subtype.coe_mk, unionᵢ_subset_iff]
     exact fun x hx₁ hx₂ => subset_unionᵢ_of_subset (⟨_, hx₂⟩ : t) (by simp [subset_closure])
-  have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W :=
-    by
+  have hfC : ∀ i : t, C i ⊆ f ⁻¹' ball (f ((i : K) : α)) W := by
     simp only [← image_subset_iff, ← mem_preimage]
     rintro ⟨⟨x, hx₁⟩, hx₂⟩
-    have hZW : closure (ball (f x) Z) ⊆ ball (f x) W :=
-      by
+    have hZW : closure (ball (f x) Z) ⊆ ball (f x) W := by
       intro y hy
       obtain ⟨z, hz₁, hz₂⟩ := UniformSpace.mem_closure_iff_ball.mp hy hZ₁
       exact ball_mono hZ₃ _ (mem_ball_comp hz₂ ((mem_ball_symmetry hZ₂).mp hz₁))
     calc
       f '' (K ∩ closure (U x)) ⊆ f '' closure (U x) := image_subset _ (inter_subset_right _ _)
       _ ⊆ closure (f '' U x) := f.continuous.continuousOn.image_closure
-      _ ⊆ closure (ball (f x) Z) :=
-        by
-          apply closure_mono
-          simp only [image_subset_iff]
-          rfl
+      _ ⊆ closure (ball (f x) Z) := by
+        apply closure_mono
+        simp only [image_subset_iff]
+        rfl
       _ ⊆ ball (f x) W := hZW
 
   refine'
@@ -276,8 +273,7 @@ theorem compactOpen_eq_compactConvergence :
   rw [compactConvergenceTopology, ContinuousMap.compactOpen]
   refine' le_antisymm _ _
   · refine' fun X hX => isOpen_iff_forall_mem_open.mpr fun f hf => _
-    have hXf : X ∈ (compactConvergenceFilterBasis f).filter :=
-      by
+    have hXf : X ∈ (compactConvergenceFilterBasis f).filter := by
       rw [← nhds_compactConvergence]
       exact @IsOpen.mem_nhds C(α, β) compactConvergenceTopology _ _ hX hf
     obtain ⟨-, ⟨⟨K, V⟩, ⟨hK, hV⟩, rfl⟩, hXf⟩ := hXf
@@ -325,13 +321,11 @@ theorem mem_compactConvergenceUniformity (X : Set (C(α, β) × C(α, β))) :
 instance compactConvergenceUniformSpace : UniformSpace C(α, β)
     where
   uniformity := compactConvergenceUniformity
-  refl :=
-    by
+  refl := by
     simp only [compactConvergenceUniformity, and_imp, Filter.le_principal_iff, Prod.forall,
       Filter.mem_principal, mem_setOf_eq, le_infᵢ_iff, idRel_subset]
     exact fun K V _hK hV f x _hx => refl_mem_uniformity hV
-  symm :=
-    by
+  symm := by
     simp only [compactConvergenceUniformity, and_imp, Prod.forall, mem_setOf_eq, Prod.fst_swap,
       Filter.tendsto_principal, Prod.snd_swap, Filter.tendsto_infᵢ]
     intro K V hK hV
@@ -340,15 +334,12 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β)
     have hX : X ∈ compactConvergenceUniformity :=
       (mem_compactConvergenceUniformity X).mpr ⟨K, V', hK, hV', by simp⟩
     exact Filter.eventually_of_mem hX fun fg hfg x hx => hsub (hsymm _ _ (hfg x hx))
-  comp X hX :=
-    by
+  comp X hX := by
     obtain ⟨K, V, hK, hV, hX⟩ := (mem_compactConvergenceUniformity X).mp hX
     obtain ⟨V', hV', hcomp⟩ := comp_mem_uniformity_sets hV
     let h := fun s : Set (C(α, β) × C(α, β)) => s ○ s
-    suffices
-      h { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V' } ∈
-        compactConvergenceUniformity.lift' h
-      by
+    suffices h { fg : C(α, β) × C(α, β) | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V' } ∈
+        compactConvergenceUniformity.lift' h by
       apply Filter.mem_of_superset this
       rintro ⟨f, g⟩ ⟨z, hz₁, hz₂⟩
       refine' hX fun x hx => hcomp _

Restore most of the mono attribute now that #1740 is merged.

I think I got all of the monos.

@@ -106,7 +106,7 @@ theorem self_mem_compactConvNhd (hV : V ∈ 𝓤 β) : f ∈ compactConvNhd K V
   refl_mem_uniformity hV
 #align continuous_map.self_mem_compact_conv_nhd ContinuousMap.self_mem_compactConvNhd
 
--- porting note: need to add @[mono] attribute
+@[mono]
 theorem compactConvNhd_mono {V' : Set (β × β)} (hV' : V' ⊆ V) :
     compactConvNhd K V' f ⊆ compactConvNhd K V f := fun _x hx a ha => hV' (hx a ha)
 #align continuous_map.compact_conv_nhd_mono ContinuousMap.compactConvNhd_mono