statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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isClosedEmbedding_toCloseds [T2Space α] [CompleteSpace α] :
IsClosedEmbedding (toCloseds (α := α)) | Compacts.isClosedEmbedding_toCloseds.comp isClosedEmbedding_toCompacts | theorem | TopologicalSpace.NonemptyCompacts.isClosedEmbedding_toCloseds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"CompleteSpace",
"T2Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_toCompacts : IsUniformEmbedding (toCompacts (α := α)) | where
injective := toCompacts_injective
comap_uniformity := Filter.comap_comap | theorem | TopologicalSpace.NonemptyCompacts.isUniformEmbedding_toCompacts | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Filter.comap_comap",
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_toCompacts : UniformContinuous (toCompacts (α := α)) | isUniformEmbedding_toCompacts.uniformContinuous | theorem | TopologicalSpace.NonemptyCompacts.uniformContinuous_toCompacts | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_subsets_of_totallyBounded {t : Set α} (ht : TotallyBounded t) :
TotallyBounded {K : NonemptyCompacts α | ↑K ⊆ t} | totallyBounded_preimage isUniformEmbedding_coe.isUniformInducing ht.powerset_hausdorff | theorem | TopologicalSpace.NonemptyCompacts.totallyBounded_subsets_of_totallyBounded | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"Set",
"TotallyBounded",
"totallyBounded_preimage"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_singleton : IsUniformEmbedding ({·} : α → NonemptyCompacts α) | isUniformEmbedding_coe.of_comp_iff.mp UniformSpace.hausdorff.isUniformEmbedding_singleton | theorem | TopologicalSpace.NonemptyCompacts.isUniformEmbedding_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding",
"UniformSpace.hausdorff.isUniformEmbedding_singleton"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_singleton : UniformContinuous ({·} : α → NonemptyCompacts α) | isUniformEmbedding_singleton.uniformContinuous | theorem | TopologicalSpace.NonemptyCompacts.uniformContinuous_singleton | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_sup :
UniformContinuous (fun x : NonemptyCompacts α × NonemptyCompacts α => x.1 ⊔ x.2) | isUniformEmbedding_coe.uniformContinuous_iff.mpr <|
UniformSpace.hausdorff.uniformContinuous_union.comp <|
uniformContinuous_coe.prodMap uniformContinuous_coe | theorem | TopologicalSpace.NonemptyCompacts.uniformContinuous_sup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformContinuous.sup_nonemptyCompacts
{f g : α → NonemptyCompacts β} (hf : UniformContinuous f) (hg : UniformContinuous g) :
UniformContinuous (fun x => f x ⊔ g x) | uniformContinuous_sup.comp <| hf.prodMk hg | theorem | UniformContinuous.sup_nonemptyCompacts | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_prod :
UniformContinuous (fun x : NonemptyCompacts α × NonemptyCompacts β => x.1 ×ˢ x.2) | isUniformEmbedding_coe.uniformContinuous_iff.mpr <|
UniformSpace.hausdorff.uniformContinuous_prod.comp <|
uniformContinuous_coe.prodMap uniformContinuous_coe | theorem | TopologicalSpace.NonemptyCompacts.uniformContinuous_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformContinuous.prod_nonemptyCompacts
{f : α → NonemptyCompacts β} {g : α → NonemptyCompacts γ} (hf : UniformContinuous f)
(hg : UniformContinuous g) : UniformContinuous (fun x => f x ×ˢ g x) | uniformContinuous_prod.comp (hf.prodMk hg) | theorem | UniformContinuous.prod_nonemptyCompacts | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformContinuous.nonemptyCompacts_map {f : α → β} (hf : UniformContinuous f) :
UniformContinuous (NonemptyCompacts.map f hf.continuous) | isUniformEmbedding_coe.uniformContinuous_iff.mpr <| hf.image_hausdorff.comp uniformContinuous_coe | theorem | UniformContinuous.nonemptyCompacts_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.IsUniformInducing.nonemptyCompacts_map {f : α → β} (hf : IsUniformInducing f) :
IsUniformInducing (NonemptyCompacts.map f hf.uniformContinuous.continuous) | .of_comp hf.uniformContinuous.nonemptyCompacts_map uniformContinuous_coe <|
hf.image_hausdorff.comp isUniformEmbedding_coe.isUniformInducing | theorem | IsUniformInducing.nonemptyCompacts_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.IsUniformEmbedding.nonemptyCompacts_map {f : α → β} (hf : IsUniformEmbedding f) :
IsUniformEmbedding (NonemptyCompacts.map f hf.uniformContinuous.continuous) | where
__ := hf.isUniformInducing.nonemptyCompacts_map
injective := map_injective hf.uniformContinuous.continuous hf.injective | theorem | IsUniformEmbedding.nonemptyCompacts_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discreteUniformity_iff : DiscreteUniformity (NonemptyCompacts α) ↔ DiscreteUniformity α | ⟨fun _ => isUniformEmbedding_singleton.discreteUniformity, fun _ => inferInstance⟩ | theorem | TopologicalSpace.NonemptyCompacts.discreteUniformity_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"DiscreteUniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completeSpace_iff : CompleteSpace (NonemptyCompacts α) ↔ CompleteSpace α | by
refine ⟨fun _ => ⟨fun {f} hf => ?_⟩, fun _ => inferInstance⟩
obtain ⟨K, hK⟩ := CompleteSpace.complete <| hf.map uniformContinuous_singleton
obtain ⟨x, hx⟩ := K.nonempty
exists x
rw [(nhds_basis_opens x).ge_iff]
intro U ⟨hxU, hU⟩
filter_upwards [hK <| (isOpen_inter_nonempty_of_isOpen hU).mem_nhds ⟨x, hx... | theorem | TopologicalSpace.NonemptyCompacts.completeSpace_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"CompleteSpace",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.TopologicalSpace.Compacts.completeSpace_iff :
CompleteSpace (Compacts α) ↔ CompleteSpace α | where
mp _ :=
NonemptyCompacts.completeSpace_iff.mp <|
NonemptyCompacts.isUniformEmbedding_toCompacts.completeSpace
isClosedEmbedding_toCompacts.isClosed_range.isComplete
mpr _ := inferInstance | theorem | TopologicalSpace.Compacts.completeSpace_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Closeds.lean | [
"Mathlib.Topology.UniformSpace.Compact"
] | [
"CompleteSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lebesgue_number_lemma {ι : Sort*} {U : ι → Set α} (hK : IsCompact K)
(hopen : ∀ i, IsOpen (U i)) (hcover : K ⊆ ⋃ i, U i) :
∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ i, ball x V ⊆ U i | by
have : ∀ x ∈ K, ∃ i, ∃ V ∈ 𝓤 α, ball x (V ○ V) ⊆ U i := fun x hx ↦ by
obtain ⟨i, hi⟩ := mem_iUnion.1 (hcover hx)
rw [← (hopen i).mem_nhds_iff, nhds_eq_comap_uniformity, ← lift'_comp_uniformity] at hi
exact ⟨i, (((basis_sets _).lift' <| monotone_id.relComp monotone_id).comap _).mem_iff.1 hi⟩
choose i... | theorem | lebesgue_number_lemma | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"comap",
"lift'_comp_uniformity",
"mem_nhds_iff",
"monotone_id",
"nhds_eq_comap_uniformity"
] | Let `c : ι → Set α` be an open cover of a compact set `s`. Then there exists an entourage
`n` such that for each `x ∈ s` its `n`-neighborhood is contained in some `c i`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lebesgue_number_lemma_nhds' {U : (x : α) → x ∈ K → Set α} (hK : IsCompact K)
(hU : ∀ x hx, U x hx ∈ 𝓝 x) : ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ y : K, ball x V ⊆ U y y.2 | by
rcases lebesgue_number_lemma (U := fun x : K => interior (U x x.2)) hK (fun _ => isOpen_interior)
(fun x hx => mem_iUnion.2 ⟨⟨x, hx⟩, mem_interior_iff_mem_nhds.2 (hU x hx)⟩) with ⟨V, V_uni, hV⟩
exact ⟨V, V_uni, fun x hx => (hV x hx).imp fun _ hy => hy.trans interior_subset⟩ | theorem | lebesgue_number_lemma_nhds' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"interior",
"isOpen_interior",
"lebesgue_number_lemma"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lebesgue_number_lemma_nhds {U : α → Set α} (hK : IsCompact K) (hU : ∀ x ∈ K, U x ∈ 𝓝 x) :
∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ y, ball x V ⊆ U y | by
rcases lebesgue_number_lemma (U := fun x => interior (U x)) hK (fun _ => isOpen_interior)
(fun x hx => mem_iUnion.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x hx)⟩) with ⟨V, V_uni, hV⟩
exact ⟨V, V_uni, fun x hx => (hV x hx).imp fun _ hy => hy.trans interior_subset⟩ | theorem | lebesgue_number_lemma_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"interior",
"isOpen_interior",
"lebesgue_number_lemma"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lebesgue_number_lemma_nhdsWithin' {U : (x : α) → x ∈ K → Set α} (hK : IsCompact K)
(hU : ∀ x hx, U x hx ∈ 𝓝[K] x) : ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ y : K, ball x V ∩ K ⊆ U y y.2 | (lebesgue_number_lemma_nhds' hK (fun x hx => Filter.mem_inf_principal'.1 (hU x hx))).imp
fun _ ⟨V_uni, hV⟩ => ⟨V_uni, fun x hx => (hV x hx).imp fun _ hy => (inter_subset _ _ _).2 hy⟩ | theorem | lebesgue_number_lemma_nhdsWithin' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"lebesgue_number_lemma_nhds'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lebesgue_number_lemma_nhdsWithin {U : α → Set α} (hK : IsCompact K)
(hU : ∀ x ∈ K, U x ∈ 𝓝[K] x) : ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ y, ball x V ∩ K ⊆ U y | (lebesgue_number_lemma_nhds hK (fun x hx => Filter.mem_inf_principal'.1 (hU x hx))).imp
fun _ ⟨V_uni, hV⟩ => ⟨V_uni, fun x hx => (hV x hx).imp fun _ hy => (inter_subset _ _ _).2 hy⟩ | theorem | lebesgue_number_lemma_nhdsWithin | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"lebesgue_number_lemma_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.lebesgue_number_lemma {ι' ι : Sort*} {p : ι' → Prop}
{V : ι' → Set (α × α)} {U : ι → Set α} (hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K)
(hopen : ∀ j, IsOpen (U j)) (hcover : K ⊆ ⋃ j, U j) :
∃ i, p i ∧ ∀ x ∈ K, ∃ j, ball x (V i) ⊆ U j | by
refine (hbasis.exists_iff ?_).1 (lebesgue_number_lemma hK hopen hcover)
exact fun s t hst ht x hx ↦ (ht x hx).imp fun i hi ↦ Subset.trans (ball_mono hst _) hi | theorem | Filter.HasBasis.lebesgue_number_lemma | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"lebesgue_number_lemma"
] | Let `U : ι → Set α` be an open cover of a compact set `K`.
Then there exists an entourage `V`
such that for each `x ∈ K` its `V`-neighborhood is included in some `U i`.
Moreover, one can choose an entourage from a given basis. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.HasBasis.lebesgue_number_lemma_nhds' {ι' : Sort*} {p : ι' → Prop}
{V : ι' → Set (α × α)} {U : (x : α) → x ∈ K → Set α} (hbasis : (𝓤 α).HasBasis p V)
(hK : IsCompact K) (hU : ∀ x hx, U x hx ∈ 𝓝 x) :
∃ i, p i ∧ ∀ x ∈ K, ∃ y : K, ball x (V i) ⊆ U y y.2 | by
refine (hbasis.exists_iff ?_).1 (lebesgue_number_lemma_nhds' hK hU)
exact fun s t hst ht x hx ↦ (ht x hx).imp fun y hy ↦ Subset.trans (ball_mono hst _) hy | theorem | Filter.HasBasis.lebesgue_number_lemma_nhds' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"lebesgue_number_lemma_nhds'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.lebesgue_number_lemma_nhds {ι' : Sort*} {p : ι' → Prop}
{V : ι' → Set (α × α)} {U : α → Set α} (hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K)
(hU : ∀ x ∈ K, U x ∈ 𝓝 x) : ∃ i, p i ∧ ∀ x ∈ K, ∃ y, ball x (V i) ⊆ U y | by
refine (hbasis.exists_iff ?_).1 (lebesgue_number_lemma_nhds hK hU)
exact fun s t hst ht x hx ↦ (ht x hx).imp fun y hy ↦ Subset.trans (ball_mono hst _) hy | theorem | Filter.HasBasis.lebesgue_number_lemma_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"lebesgue_number_lemma_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.lebesgue_number_lemma_nhdsWithin' {ι' : Sort*} {p : ι' → Prop}
{V : ι' → Set (α × α)} {U : (x : α) → x ∈ K → Set α} (hbasis : (𝓤 α).HasBasis p V)
(hK : IsCompact K) (hU : ∀ x hx, U x hx ∈ 𝓝[K] x) :
∃ i, p i ∧ ∀ x ∈ K, ∃ y : K, ball x (V i) ∩ K ⊆ U y y.2 | by
refine (hbasis.exists_iff ?_).1 (lebesgue_number_lemma_nhdsWithin' hK hU)
exact fun s t hst ht x hx ↦ (ht x hx).imp
fun y hy ↦ Subset.trans (Set.inter_subset_inter_left K (ball_mono hst _)) hy | theorem | Filter.HasBasis.lebesgue_number_lemma_nhdsWithin' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"Set.inter_subset_inter_left",
"lebesgue_number_lemma_nhdsWithin'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.lebesgue_number_lemma_nhdsWithin {ι' : Sort*} {p : ι' → Prop}
{V : ι' → Set (α × α)} {U : α → Set α} (hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K)
(hU : ∀ x ∈ K, U x ∈ 𝓝[K] x) : ∃ i, p i ∧ ∀ x ∈ K, ∃ y, ball x (V i) ∩ K ⊆ U y | by
refine (hbasis.exists_iff ?_).1 (lebesgue_number_lemma_nhdsWithin hK hU)
exact fun s t hst ht x hx ↦ (ht x hx).imp
fun y hy ↦ Subset.trans (Set.inter_subset_inter_left K (ball_mono hst _)) hy | theorem | Filter.HasBasis.lebesgue_number_lemma_nhdsWithin | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"Set.inter_subset_inter_left",
"lebesgue_number_lemma_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lebesgue_number_lemma_sUnion {S : Set (Set α)}
(hK : IsCompact K) (hopen : ∀ s ∈ S, IsOpen s) (hcover : K ⊆ ⋃₀ S) :
∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, ball x V ⊆ s | by
rw [sUnion_eq_iUnion] at hcover
simpa using lebesgue_number_lemma hK (by simpa) hcover | theorem | lebesgue_number_lemma_sUnion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"lebesgue_number_lemma"
] | Let `c : Set (Set α)` be an open cover of a compact set `s`. Then there exists an entourage
`n` such that for each `x ∈ s` its `n`-neighborhood is contained in some `t ∈ c`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.nhdsSet_basis_uniformity {p : ι → Prop} {V : ι → Set (α × α)}
(hbasis : (𝓤 α).HasBasis p V) (hK : IsCompact K) :
(𝓝ˢ K).HasBasis p fun i => ⋃ x ∈ K, ball x (V i) | where
mem_iff' U := by
constructor
· intro H
have HKU : K ⊆ ⋃ _ : Unit, interior U := by
simpa only [iUnion_const, subset_interior_iff_mem_nhdsSet] using H
obtain ⟨i, hpi, hi⟩ : ∃ i, p i ∧ ⋃ x ∈ K, ball x (V i) ⊆ interior U := by
simpa using hbasis.lebesgue_number_lemma hK (fun _ ↦... | theorem | IsCompact.nhdsSet_basis_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"Set",
"bUnion_mem_nhdsSet",
"interior",
"isOpen_interior",
"subset_interior_iff_mem_nhdsSet"
] | If `K` is a compact set in a uniform space and `{V i | p i}` is a basis of entourages,
then `{⋃ x ∈ K, UniformSpace.ball x (V i) | p i}` is a basis of `𝓝ˢ K`.
Here "`{s i | p i}` is a basis of a filter `l`" means `Filter.HasBasis l p s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Disjoint.exists_uniform_thickening {A B : Set α} (hA : IsCompact A) (hB : IsClosed B)
(h : Disjoint A B) : ∃ V ∈ 𝓤 α, Disjoint (⋃ x ∈ A, ball x V) (⋃ x ∈ B, ball x V) | by
have : Bᶜ ∈ 𝓝ˢ A := hB.isOpen_compl.mem_nhdsSet.mpr h.le_compl_right
rw [(hA.nhdsSet_basis_uniformity (Filter.basis_sets _)).mem_iff] at this
rcases this with ⟨U, hU, hUAB⟩
rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩
refine ⟨V, hV, Set.disjoint_left.mpr fun x => ?_⟩
simp only [mem_... | theorem | Disjoint.exists_uniform_thickening | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"Disjoint",
"Filter.basis_sets",
"IsClosed",
"IsCompact",
"Set",
"comp_symm_mem_uniformity_sets"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Disjoint.exists_uniform_thickening_of_basis {p : ι → Prop} {s : ι → Set (α × α)}
(hU : (𝓤 α).HasBasis p s) {A B : Set α} (hA : IsCompact A) (hB : IsClosed B)
(h : Disjoint A B) : ∃ i, p i ∧ Disjoint (⋃ x ∈ A, ball x (s i)) (⋃ x ∈ B, ball x (s i)) | by
rcases h.exists_uniform_thickening hA hB with ⟨V, hV, hVAB⟩
rcases hU.mem_iff.1 hV with ⟨i, hi, hiV⟩
exact ⟨i, hi, hVAB.mono (iUnion₂_mono fun a _ => ball_mono hiV a)
(iUnion₂_mono fun b _ => ball_mono hiV b)⟩ | theorem | Disjoint.exists_uniform_thickening_of_basis | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"Disjoint",
"IsClosed",
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lebesgue_number_of_compact_open {K U : Set α} (hK : IsCompact K)
(hU : IsOpen U) (hKU : K ⊆ U) : ∃ V ∈ 𝓤 α, IsOpen V ∧ ∀ x ∈ K, UniformSpace.ball x V ⊆ U | let ⟨V, ⟨hV, hVo⟩, hVU⟩ :=
(hK.nhdsSet_basis_uniformity uniformity_hasBasis_open).mem_iff.1 (hU.mem_nhdsSet.2 hKU)
⟨V, hV, hVo, iUnion₂_subset_iff.1 hVU⟩ | theorem | lebesgue_number_of_compact_open | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsCompact",
"IsOpen",
"Set",
"UniformSpace.ball",
"uniformity_hasBasis_open"
] | A useful consequence of the Lebesgue number lemma: given any compact set `K` contained in an
open set `U`, we can find an (open) entourage `V` such that the ball of size `V` about any point of
`K` is contained in `U`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhdsSet_diagonal_eq_uniformity [CompactSpace α] : 𝓝ˢ (diagonal α) = 𝓤 α | by
refine nhdsSet_diagonal_le_uniformity.antisymm ?_
have :
(𝓤 (α × α)).HasBasis (fun U => U ∈ 𝓤 α) fun U =>
(fun p : (α × α) × α × α => ((p.1.1, p.2.1), p.1.2, p.2.2)) ⁻¹' U ×ˢ U := by
rw [uniformity_prod_eq_comap_prod]
exact (𝓤 α).basis_sets.prod_self.comap _
refine (isCompact_diagonal.nhds... | theorem | nhdsSet_diagonal_eq_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"CompactSpace",
"refl_mem_uniformity",
"uniformity_prod_eq_comap_prod"
] | On a compact uniform space, the topology determines the uniform structure, entourages are
exactly the neighborhoods of the diagonal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
compactSpace_uniformity [CompactSpace α] : 𝓤 α = ⨆ x, 𝓝 (x, x) | nhdsSet_diagonal_eq_uniformity.symm.trans (nhdsSet_diagonal _) | theorem | compactSpace_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"CompactSpace",
"nhdsSet_diagonal"
] | On a compact uniform space, the topology determines the uniform structure, entourages are
exactly the neighborhoods of the diagonal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
unique_uniformity_of_compact [t : TopologicalSpace γ] [CompactSpace γ]
{u u' : UniformSpace γ} (h : u.toTopologicalSpace = t) (h' : u'.toTopologicalSpace = t) :
u = u' | by
refine UniformSpace.ext ?_
have : @CompactSpace γ u.toTopologicalSpace := by rwa [h]
have : @CompactSpace γ u'.toTopologicalSpace := by rwa [h']
rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h'] | theorem | unique_uniformity_of_compact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"CompactSpace",
"TopologicalSpace",
"UniformSpace",
"UniformSpace.ext",
"compactSpace_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.relPreimage_of_isCompact [TopologicalSpace α] [TopologicalSpace β]
{s : SetRel α β} (hs : IsClosed s) {t : Set β} (ht : IsCompact t) :
IsClosed (s.preimage t) | by
rw [← isOpen_compl_iff, isOpen_iff_eventually] at hs ⊢
simp_rw [Set.mem_compl_iff, SetRel.mem_preimage, not_exists, not_and]
exact fun y hy => ht.eventually_forall_of_forall_eventually fun x hx => hs _ <| hy _ hx | theorem | IsClosed.relPreimage_of_isCompact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsClosed",
"IsCompact",
"Set",
"Set.mem_compl_iff",
"SetRel",
"SetRel.mem_preimage",
"TopologicalSpace",
"isOpen_compl_iff",
"isOpen_iff_eventually"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.relImage_of_isCompact [TopologicalSpace α] [TopologicalSpace β]
{s : SetRel α β} (hs : IsClosed s) {t : Set α} (ht : IsCompact t) :
IsClosed (s.image t) | hs.relInv.relPreimage_of_isCompact ht | theorem | IsClosed.relImage_of_isCompact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Compact.lean | [] | [
"IsClosed",
"IsCompact",
"Set",
"SetRel",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_iff_forall_isCompact_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
· -- 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 [← tendstoLocallyUnif... | theorem | ContinuousMap.tendsto_iff_forall_isCompact_tendstoUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"Filter",
"IsCompact",
"SetRel.symm",
"TendstoUniformlyOn",
"comp_open_symm_mem_uniformity_sets",
"inter_mem_nhdsWithin",
"lebesgue_number_of_compact_open",
"mem_uniformity_isClosed",
"tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toUniformOnFunIsCompact (f : C(α, β)) : α →ᵤ[{K | IsCompact K}] β | UniformOnFun.ofFun {K | IsCompact K} f | def | ContinuousMap.toUniformOnFunIsCompact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"IsCompact",
"UniformOnFun.ofFun"
] | 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)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toUniformOnFun_toFun (f : C(α, β)) :
UniformOnFun.toFun _ f.toUniformOnFunIsCompact = f | rfl | theorem | ContinuousMap.toUniformOnFun_toFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"UniformOnFun.toFun"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_toUniformOnFunIsCompact :
range (toUniformOnFunIsCompact) = {f : UniformOnFun α β {K | IsCompact K} | Continuous f} | Set.ext fun f ↦ ⟨fun g ↦ g.choose_spec ▸ g.choose.2, fun hf ↦ ⟨⟨f, hf⟩, rfl⟩⟩ | theorem | ContinuousMap.range_toUniformOnFunIsCompact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"Continuous",
"IsCompact",
"Set.ext",
"UniformOnFun"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compactConvergenceUniformSpace : UniformSpace C(α, β) | .replaceTopology (.comap toUniformOnFunIsCompact inferInstance) <| by
refine TopologicalSpace.ext_nhds fun f ↦ eq_of_forall_le_iff fun l ↦ ?_
simp_rw [← tendsto_id', tendsto_iff_forall_isCompact_tendstoUniformlyOn,
nhds_induced, tendsto_comap_iff, UniformOnFun.tendsto_iff_tendstoUniformlyOn]
rfl | instance | ContinuousMap.compactConvergenceUniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"UniformOnFun.tendsto_iff_tendstoUniformlyOn",
"UniformSpace",
"comap",
"eq_of_forall_le_iff",
"nhds_induced"
] | 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_isCompact_tendstoUniformlyOn`
to show that the induced topology agrees with th... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isUniformEmbedding_toUniformOnFunIsCompact :
IsUniformEmbedding (toUniformOnFunIsCompact : C(α, β) → α →ᵤ[{K | IsCompact K}] β) | where
comap_uniformity := rfl
injective := DFunLike.coe_injective | theorem | ContinuousMap.isUniformEmbedding_toUniformOnFunIsCompact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"DFunLike.coe_injective",
"IsCompact",
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_iff_continuous_uniformOnFun {X : Type*} [TopologicalSpace X] (f : X → C(α, β)) :
Continuous f ↔ Continuous (fun x ↦ ofFun {K | IsCompact K} (f x)) | isUniformEmbedding_toUniformOnFunIsCompact.isInducing.continuous_iff | theorem | ContinuousMap.continuous_iff_continuous_uniformOnFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"Continuous",
"IsCompact",
"TopologicalSpace"
] | `f : X → C(α, β)` is continuous if any only if it is continuous when reinterpreted as a
map `f : X → α →ᵤ[{K | IsCompact K}] β`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_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
rw [← isUniformEmbedding_toUniformOnFunIsCompact.comap_uniformity]
exact .comap _ <| UniformOnFun.hasBasis_uniformity_of_basis _ _ {K | IsCompact K}
⟨∅, isCompact_empty⟩ (directedOn_of_sup_mem fun _ _ ↦ IsCompact.union) h | theorem | Filter.HasBasis.compactConvergenceUniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"IsCompact",
"IsCompact.union",
"Set",
"UniformOnFun.hasBasis_uniformity_of_basis",
"comap",
"directedOn_of_sup_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | ContinuousMap.hasBasis_compactConvergenceUniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | theorem | ContinuousMap.mem_compactConvergence_entourage_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"IsCompact",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | CompactExhaustion.hasBasis_compactConvergenceUniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"CompactExhaustion",
"IsCompact",
"Monotone.directed_le",
"Set",
"UniformOnFun.hasBasis_uniformity_of_covering_of_basis",
"comap"
] | 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)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_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 _ | theorem | CompactExhaustion.hasAntitoneBasis_compactConvergenceUniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"CompactExhaustion",
"IsCompact",
"Set",
"UniformOnFun.hasAntitoneBasis_uniformity",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a => F i a) f p) :
Tendsto F p (𝓝 f) | by
rw [tendsto_iff_forall_isCompact_tendstoUniformlyOn]
intro K hK
rw [← tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK]
exact h.tendstoLocallyUniformlyOn | theorem | ContinuousMap.tendsto_of_tendstoLocallyUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"TendstoLocallyUniformly",
"tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact"
] | Locally uniform convergence implies convergence in the compact-open topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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_isCompact_tendstoUniformlyOn] at h
obtain ⟨n, hn₁, hn₂⟩ := exists_compact_mem_nhds x
exact ⟨n, hn₂, h n hn₁ V hV⟩ | theorem | ContinuousMap.tendsto_iff_tendstoLocallyUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"TendstoLocallyUniformly",
"WeaklyLocallyCompactSpace"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_comp (g : C(β, δ)) (hg : UniformContinuous g) :
UniformContinuous (ContinuousMap.comp g : C(α, β) → C(α, δ)) | isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous_iff.mpr <|
UniformOnFun.postcomp_uniformContinuous hg |>.comp
isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous | theorem | ContinuousMap.uniformContinuous_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"ContinuousMap.comp",
"UniformContinuous",
"UniformOnFun.postcomp_uniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformInducing_comp (g : C(β, δ)) (hg : IsUniformInducing g) :
IsUniformInducing (ContinuousMap.comp g : C(α, β) → C(α, δ)) | isUniformEmbedding_toUniformOnFunIsCompact.isUniformInducing.of_comp_iff.mp <|
UniformOnFun.postcomp_isUniformInducing hg |>.comp
isUniformEmbedding_toUniformOnFunIsCompact.isUniformInducing | theorem | ContinuousMap.isUniformInducing_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"ContinuousMap.comp",
"IsUniformInducing",
"UniformOnFun.postcomp_isUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_comp (g : C(β, δ)) (hg : IsUniformEmbedding g) :
IsUniformEmbedding (ContinuousMap.comp g : C(α, β) → C(α, δ)) | isUniformEmbedding_toUniformOnFunIsCompact.of_comp_iff.mp <|
UniformOnFun.postcomp_isUniformEmbedding hg |>.comp
isUniformEmbedding_toUniformOnFunIsCompact | theorem | ContinuousMap.isUniformEmbedding_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"ContinuousMap.comp",
"IsUniformEmbedding",
"UniformOnFun.postcomp_isUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_comp_left (g : C(α, γ)) :
UniformContinuous (fun f ↦ f.comp g : C(γ, β) → C(α, β)) | isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous_iff.mpr <|
UniformOnFun.precomp_uniformContinuous (fun _ hK ↦ hK.image g.continuous) |>.comp
isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous | theorem | ContinuousMap.uniformContinuous_comp_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"UniformContinuous",
"UniformOnFun.precomp_uniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.UniformEquiv.arrowCongr (φ : α ≃ₜ γ) (ψ : β ≃ᵤ δ) :
C(α, β) ≃ᵤ C(γ, δ) | where
toFun f := .comp ψ.toHomeomorph <| f.comp φ.symm
invFun f := .comp ψ.symm.toHomeomorph <| f.comp φ
left_inv f := ext fun _ ↦ ψ.left_inv (f _) |>.trans <| congrArg f <| φ.left_inv _
right_inv f := ext fun _ ↦ ψ.right_inv (f _) |>.trans <| congrArg f <| φ.right_inv _
uniformContinuous_toFun := uniformCont... | def | UniformEquiv.arrowCongr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"trans"
] | Any pair of a homeomorphism `X ≃ₜ Z` and an isomorphism `Y ≃ᵤ T` of uniform spaces gives rise
to an isomorphism `C(X, Y) ≃ᵤ C(Z, T)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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} | 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)⟩ | theorem | ContinuousMap.hasBasis_compactConvergenceUniformity_of_compact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Filter.HasBasis.compactConvergenceUniformity_of_compact
{ι : Sort*} {p : ι → Prop} {V : ι → Set (β × β)} (h : (𝓤 β).HasBasis p V) :
HasBasis (𝓤 C(α, β)) p fun i ↦ {fg : C(α, β) × C(α, β) | ∀ x, (fg.1 x, fg.2 x) ∈ V i} | hasBasis_compactConvergenceUniformity_of_compact.to_hasBasis
(fun _U hU ↦ (h.mem_iff.mp hU).imp fun _i ⟨hpi, hi⟩ ↦ ⟨hpi, fun _ h a ↦ hi <| h a⟩)
fun i hi ↦ ⟨V i, h.mem_of_mem hi, .rfl⟩ | theorem | Filter.HasBasis.compactConvergenceUniformity_of_compact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_uniformFunOfFun :
IsUniformEmbedding ((ofFun ·) : C(α, β) → α →ᵤ β) | where
comap_uniformity := UniformOnFun.uniformEquivUniformFun β _ isCompact_univ
|>.isUniformEmbedding.comp isUniformEmbedding_toUniformOnFunIsCompact
|>.comap_uniformity
injective := DFunLike.coe_injective | theorem | ContinuousMap.isUniformEmbedding_uniformFunOfFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"DFunLike.coe_injective",
"IsUniformEmbedding",
"UniformOnFun.uniformEquivUniformFun",
"isCompact_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_iff_tendstoUniformly :
Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a => F i a) f p | by
simp [isUniformEmbedding_uniformFunOfFun.isInducing.tendsto_nhds_iff,
UniformFun.tendsto_iff_tendstoUniformly, Function.comp_def] | theorem | ContinuousMap.tendsto_iff_tendstoUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"TendstoUniformly",
"UniformFun.tendsto_iff_tendstoUniformly"
] | Convergence in the compact-open topology is the same as uniform convergence for sequences of
continuous functions on a compact space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuous_iff_continuous_uniformFun {X : Type*} [TopologicalSpace X] (f : X → C(α, β)) :
Continuous f ↔ Continuous (fun x ↦ ofFun (f x)) | isUniformEmbedding_uniformFunOfFun.isInducing.continuous_iff | theorem | ContinuousMap.continuous_iff_continuous_uniformFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"Continuous",
"TopologicalSpace"
] | When `α` is compact, `f : X → C(α, β)` is continuous if any only if it is continuous when
reinterpreted as a map `f : X → α →ᵤ β`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.ContinuousOn.tendsto_restrict_iff_tendstoUniformlyOn {s : Set α} [CompactSpace s]
{f : α → β} (hf : ContinuousOn f s) {ι : Type*} {p : Filter ι}
{F : ι → α → β} (hF : ∀ i, ContinuousOn (F i) s) :
Tendsto (fun i ↦ ⟨_, (hF i).restrict⟩ : ι → C(s, β)) p (𝓝 ⟨_, hf.restrict⟩) ↔
TendstoUniformlyOn F... | by
rw [ContinuousMap.tendsto_iff_tendstoUniformly, tendstoUniformlyOn_iff_tendstoUniformly_comp_coe]
congr! | theorem | ContinuousOn.tendsto_restrict_iff_tendstoUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"CompactSpace",
"ContinuousMap.tendsto_iff_tendstoUniformly",
"ContinuousOn",
"Filter",
"Set",
"TendstoUniformlyOn",
"tendstoUniformlyOn_iff_tendstoUniformly_comp_coe"
] | Given functions `F i, f` which are continuous on a compact set `s`, `F` tends to `f`
uniformly on `s` if and only if the restrictions (as elements of `C(s, β)`) converge. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.ContinuousOn.continuous_restrict_iff_continuous_uniformOnFun
{X : Type*} [TopologicalSpace X] {f : X → α → β} {s : Set α}
(hf : ∀ x, ContinuousOn (f x) s) [CompactSpace s] :
Continuous (fun x ↦ ⟨_, (hf x).restrict⟩ : X → C(s, β)) ↔
Continuous (fun x ↦ ofFun {s} (f x)) | by
rw [ContinuousMap.continuous_iff_continuous_uniformFun, UniformOnFun.continuous_rng_iff]
simp [Function.comp_def] | theorem | ContinuousOn.continuous_restrict_iff_continuous_uniformOnFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"CompactSpace",
"Continuous",
"ContinuousMap.continuous_iff_continuous_uniformFun",
"ContinuousOn",
"Set",
"TopologicalSpace",
"UniformOnFun.continuous_rng_iff"
] | A family `f : X → α → β`, each of which is continuous on a compact set `s : Set α` is
continuous in the topology `X → α →ᵤ[{s}] β` if and only if the family of continuous restrictions
`X → C(s, β)` is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} [TopologicalSpace δ₁]
[TopologicalSpace δ₂] (φ₁ : C(δ₁, α)) (φ₂ : C(δ₂, α)) (h_proper₁ : IsProperMap φ₁)
(h_proper₂ : IsProperMap φ₂) (h_cover : range φ₁ ∪ range φ₂ = univ) :
((inferInstance : UniformSpace C(α, β))) =
.comap (comp · φ₁) inferInstanc... | by
-- We check the analogous result for `UniformOnFun` using
-- `UniformOnFun.uniformSpace_eq_inf_precomp_of_cover`...
set 𝔖 : Set (Set α) := {K | IsCompact K}
set 𝔗₁ : Set (Set δ₁) := {K | IsCompact K}
set 𝔗₂ : Set (Set δ₂) := {K | IsCompact K}
have h_image₁ : MapsTo (φ₁ '' ·) 𝔗₁ 𝔖 := fun K hK ↦ hK.im... | theorem | ContinuousMap.uniformSpace_eq_inf_precomp_of_cover | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"IsCompact",
"IsProperMap",
"Set",
"TopologicalSpace",
"UniformOnFun.uniformSpace_eq_inf_precomp_of_cover",
"UniformSpace",
"UniformSpace.comap_comap",
"UniformSpace.comap_inf",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformSpace_eq_iInf_precomp_of_cover {δ : ι → Type*} [∀ i, TopologicalSpace (δ i)]
(φ : Π i, C(δ i, α)) (h_proper : ∀ i, IsProperMap (φ i))
(h_lf : LocallyFinite fun i ↦ range (φ i)) (h_cover : ⋃ i, range (φ i) = univ) :
((inferInstance : UniformSpace C(α, β))) = ⨅ i, .comap (comp · (φ i)) inferInstance | by
-- We check the analogous result for `UniformOnFun` using
-- `UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover`...
set 𝔖 : Set (Set α) := {K | IsCompact K}
set 𝔗 : Π i, Set (Set (δ i)) := fun i ↦ {K | IsCompact K}
have h_image : ∀ i, MapsTo (φ i '' ·) (𝔗 i) 𝔖 := fun i K hK ↦ hK.image (φ i).continuous... | theorem | ContinuousMap.uniformSpace_eq_iInf_precomp_of_cover | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"IsCompact",
"IsProperMap",
"LocallyFinite",
"Set",
"TopologicalSpace",
"UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover",
"UniformSpace",
"UniformSpace.comap_comap",
"UniformSpace.comap_iInf",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instCompleteSpaceOfCompactlyCoherentSpace [CompactlyCoherentSpace α] :
CompleteSpace C(α, β) | by
rw [completeSpace_iff_isComplete_range
isUniformEmbedding_toUniformOnFunIsCompact.isUniformInducing,
range_toUniformOnFunIsCompact, ← completeSpace_coe_iff_isComplete]
exact (UniformOnFun.isClosed_setOf_continuous
CompactlyCoherentSpace.isCoherentWith).completeSpace_coe | instance | ContinuousMap.instCompleteSpaceOfCompactlyCoherentSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"CompactlyCoherentSpace",
"CompleteSpace",
"UniformOnFun.isClosed_setOf_continuous",
"completeSpace_coe_iff_isComplete",
"completeSpace_iff_isComplete_range"
] | If the topology on `α` is generated by its restrictions to compact sets, then the space of
continuous maps `C(α, β)` is complete (w.r.t. the compact convergence uniformity).
Sufficient conditions on `α` to satisfy this condition are (weak) local compactness and sequential
compactness. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isComplete_setOf_eqOn [CompleteSpace C(α, β)] (f : α → β) (s : Set α) :
IsComplete {g : C(α, β) | EqOn g f s} | by
classical
intro l hlc hlf
rcases CompleteSpace.complete hlc with ⟨f', hf'⟩
have := hlc.1
have H₁ : ∀ x ∈ s, Inseparable (f x) (f' x) := fun x hx ↦ by
refine tendsto_nhds_unique_inseparable ?_ ((continuous_eval_const x).continuousAt.mono_left hf')
refine tendsto_const_nhds.congr' <| .filter_mono ?_ ... | theorem | ContinuousMap.isComplete_setOf_eqOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompactConvergence.lean | [] | [
"CompleteSpace",
"Inseparable",
"Inseparable.refl",
"IsComplete",
"Set",
"Set.piecewise_eqOn",
"by_cases",
"continuous_congr_of_inseparable",
"inseparable_pi",
"symm",
"tendsto_nhds_unique_inseparable"
] | If `C(α, β)` is a complete space, then for any (possibly, discontinuous) function `f`
and any set `s`, the set of functions `g : C(α, β)` that are equal to `f` on `s`
is a complete set.
Note that this set does not have to be a closed set when `β` is not T0.
This lemma is useful to prove that, e.g., the space of paths ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Rat.uniformSpace_eq :
(AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace | by
ext s
rw [(AbsoluteValue.hasBasis_uniformity _).mem_iff, Metric.uniformity_basis_dist_rat.mem_iff]
simp only [Rat.dist_eq, AbsoluteValue.abs_apply, ← Rat.cast_sub, ← Rat.cast_abs, Rat.cast_lt,
_root_.abs_sub_comm] | theorem | Rat.uniformSpace_eq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [
"AbsoluteValue",
"AbsoluteValue.abs",
"AbsoluteValue.hasBasis_uniformity",
"Rat.cast_abs",
"Rat.cast_lt",
"Rat.cast_sub",
"Rat.dist_eq"
] | The metric space uniform structure on ℚ (which presupposes the existence
of real numbers) agrees with the one coming directly from (abs : ℚ → ℚ). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
rationalCauSeqPkg : @AbstractCompletion ℚ <| (@AbsoluteValue.abs ℚ _).uniformSpace | @AbstractCompletion.mk
(space := ℝ)
(coe := ((↑) : ℚ → ℝ))
(uniformStruct := by infer_instance)
(complete := by infer_instance)
(separation := by infer_instance)
(isUniformInducing := by
rw [Rat.uniformSpace_eq]
exact Rat.isUniformEmbedding_coe_real.isUniformInducing)
(dense := R... | def | rationalCauSeqPkg | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [
"AbsoluteValue.abs",
"Rat.uniformSpace_eq"
] | Cauchy reals packaged as a completion of ℚ using the absolute value route. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Q | ℚ deriving CommRing, Inhabited | def | CompareReals.Q | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [
"CommRing"
] | Type wrapper around ℚ to make sure the absolute value uniform space instance is picked up
instead of the metric space one. We proved in `Rat.uniformSpace_eq` that they are equal,
but they are not definitionaly equal, so it would confuse the type class system (and probably
also human readers). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformSpace : UniformSpace Q | fast_instance% (@AbsoluteValue.abs ℚ _).uniformSpace | instance | CompareReals.uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [
"AbsoluteValue.abs",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Bourbakiℝ : Type | Completion Q deriving Inhabited | def | CompareReals.Bourbakiℝ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [] | Real numbers constructed as in Bourbaki. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Bourbaki.uniformSpace : UniformSpace Bourbakiℝ | fast_instance% Completion.uniformSpace Q | instance | CompareReals.Bourbaki.uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
bourbakiPkg : AbstractCompletion Q | Completion.cPkg | def | CompareReals.bourbakiPkg | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [] | Bourbaki reals packaged as a completion of Q using the general theory. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
compareEquiv : Bourbakiℝ ≃ᵤ ℝ | bourbakiPkg.compareEquiv rationalCauSeqPkg | def | CompareReals.compareEquiv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [
"rationalCauSeqPkg"
] | The uniform bijection between Bourbaki and Cauchy reals. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
compare_uc : UniformContinuous compareEquiv | bourbakiPkg.uniformContinuous_compareEquiv rationalCauSeqPkg | theorem | CompareReals.compare_uc | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [
"UniformContinuous",
"rationalCauSeqPkg"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compare_uc_symm : UniformContinuous compareEquiv.symm | bourbakiPkg.uniformContinuous_compareEquiv_symm rationalCauSeqPkg | theorem | CompareReals.compare_uc_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompareReals.lean | [] | [
"UniformContinuous",
"rationalCauSeqPkg"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsComplete.isClosed [UniformSpace α] [T0Space α] {s : Set α} (h : IsComplete s) :
IsClosed s | isClosed_iff_clusterPt.2 fun a ha => by
let f := 𝓝[s] a
have : Cauchy f := cauchy_nhds.mono' ha inf_le_left
rcases h f this inf_le_right with ⟨y, ys, fy⟩
rwa [(tendsto_nhds_unique' ha inf_le_left fy : a = y)] | theorem | IsComplete.isClosed | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompleteSeparated.lean | [] | [
"Cauchy",
"IsClosed",
"IsComplete",
"Set",
"T0Space",
"UniformSpace",
"tendsto_nhds_unique'"
] | In a separated space, a complete set is closed. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformEmbedding.isClosedEmbedding [UniformSpace α] [UniformSpace β] [CompleteSpace α]
[T0Space β] {f : α → β} (hf : IsUniformEmbedding f) :
IsClosedEmbedding f | ⟨hf.isEmbedding, hf.isUniformInducing.isComplete_range.isClosed⟩ | theorem | IsUniformEmbedding.isClosedEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompleteSeparated.lean | [] | [
"CompleteSpace",
"IsUniformEmbedding",
"T0Space",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_extend_of_cauchy {e : α → β} {f : α → γ} (de : IsDenseInducing e)
(h : ∀ b : β, Cauchy (map f (comap e <| 𝓝 b))) : Continuous (de.extend f) | de.continuous_extend fun b => CompleteSpace.complete (h b) | theorem | IsDenseInducing.continuous_extend_of_cauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/CompleteSeparated.lean | [] | [
"Cauchy",
"Continuous",
"IsDenseInducing",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CauchyFilter (α : Type u) [UniformSpace α] : Type u | { f : Filter α // Cauchy f } | def | CauchyFilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"Cauchy",
"Filter",
"UniformSpace"
] | Space of Cauchy filters
This is essentially the completion of a uniform space. The embeddings are the neighbourhood filters.
This space is not minimal, the separated uniform space (i.e. quotiented on the intersection of all
entourages) is necessary for this. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
gen (s : SetRel α α) : SetRel (CauchyFilter α) (CauchyFilter α) | { p | s ∈ p.1.val ×ˢ p.2.val } | def | CauchyFilter.gen | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"SetRel"
] | The pairs of Cauchy filters generated by a set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
monotone_gen : Monotone (gen : SetRel α α → _) | monotone_setOf fun p => @Filter.monotone_mem _ (p.1.val ×ˢ p.2.val) | theorem | CauchyFilter.monotone_gen | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"Filter.monotone_mem",
"Monotone",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_gen : map Prod.swap ((𝓤 α).lift' gen) ≤ (𝓤 α).lift' gen | by
let f := fun s : SetRel α α =>
{ p : CauchyFilter α × CauchyFilter α | s ∈ (p.2.val ×ˢ p.1.val : Filter (α × α)) }
have h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f := by
delta gen
simp [f, map_lift'_eq, monotone_setOf, Filter.monotone_mem, Function.comp_def,
image_swap_eq_preimag... | theorem | CauchyFilter.symm_gen | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"Filter",
"Filter.monotone_mem",
"Filter.prod_comm",
"SetRel",
"le_rfl",
"uniformity_lift_le_swap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
subset_gen_relComp {s t : SetRel α α} : gen s ○ gen t ⊆ gen (s ○ t) | fun ⟨f, g⟩ ⟨h, h₁, h₂⟩ =>
let ⟨t₁, (ht₁ : t₁ ∈ f.val), t₂, (ht₂ : t₂ ∈ h.val), (h₁ : t₁ ×ˢ t₂ ⊆ s)⟩ := mem_prod_iff.mp h₁
let ⟨t₃, (ht₃ : t₃ ∈ h.val), t₄, (ht₄ : t₄ ∈ g.val), (h₂ : t₃ ×ˢ t₄ ⊆ t)⟩ := mem_prod_iff.mp h₂
have : t₂ ∩ t₃ ∈ h.val := inter_mem ht₂ ht₃
let ⟨x, xt₂, xt₃⟩ := h.property.left.nonempty_of_m... | theorem | CauchyFilter.subset_gen_relComp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_gen : ((𝓤 α).lift' gen).lift' (fun s ↦ s ○ s) ≤ (𝓤 α).lift' gen | calc
((𝓤 α).lift' gen).lift' (fun s ↦ s ○ s)
_ = (𝓤 α).lift' fun s ↦ gen s ○ gen s := by
rw [lift'_lift'_assoc]
· exact monotone_gen
· exact monotone_id.relComp monotone_id
_ ≤ (𝓤 α).lift' fun s ↦ gen <| s ○ s := lift'_mono' fun _ _hs => subset_gen_relComp
_ = ((𝓤 α).lift' fun ... | theorem | CauchyFilter.comp_gen | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"SetRel",
"comp_le_uniformity",
"le_rfl",
"monotone_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_uniformity {s : Set (CauchyFilter α × CauchyFilter α)} :
s ∈ 𝓤 (CauchyFilter α) ↔ ∃ t ∈ 𝓤 α, gen t ⊆ s | mem_lift'_sets monotone_gen | theorem | CauchyFilter.mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
basis_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → SetRel α α} (h : (𝓤 α).HasBasis p s) :
(𝓤 (CauchyFilter α)).HasBasis p (gen ∘ s) | h.lift' monotone_gen | theorem | CauchyFilter.basis_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_uniformity' {s : Set (CauchyFilter α × CauchyFilter α)} :
s ∈ 𝓤 (CauchyFilter α) ↔ ∃ t ∈ 𝓤 α, ∀ f g : CauchyFilter α, t ∈ f.1 ×ˢ g.1 → (f, g) ∈ s | by
refine mem_uniformity.trans (exists_congr (fun t => and_congr_right_iff.mpr (fun _h => ?_)))
exact ⟨fun h _f _g ht => h ht, fun h _p hp => h _ _ hp⟩ | theorem | CauchyFilter.mem_uniformity' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
pureCauchy (a : α) : CauchyFilter α | ⟨pure a, cauchy_pure⟩ | def | CauchyFilter.pureCauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter"
] | Embedding of `α` into its completion `CauchyFilter α` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isUniformInducing_pureCauchy : IsUniformInducing (pureCauchy : α → CauchyFilter α) | ⟨have : (preimage fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen = id :=
funext fun s =>
Set.ext fun ⟨a₁, a₂⟩ => by simp [preimage, gen, pureCauchy]
calc
comap (fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ((𝓤 α).lift' gen) =
(𝓤 α).lift' ((preimage fun x : α ×... | theorem | CauchyFilter.isUniformInducing_pureCauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"IsUniformInducing",
"Set.ext",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_pureCauchy : IsUniformEmbedding (pureCauchy : α → CauchyFilter α) | where
__ := isUniformInducing_pureCauchy
injective _a₁ _a₂ h := pure_injective <| Subtype.ext_iff.1 h | theorem | CauchyFilter.isUniformEmbedding_pureCauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) | fun f => by
have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
let ⟨t', ht'₁, ht'₂⟩ := comp_mem_uniformity_sets ht''₁
have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
let ⟨t, ht, (h : t ... | theorem | CauchyFilter.denseRange_pureCauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"ClusterPt",
"DenseRange",
"Set.inter_comm",
"SetRel.prodMk_mem_comp",
"closure_eq_cluster_pts",
"comp_mem_uniformity_sets",
"nhds_eq_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isDenseInducing_pureCauchy : IsDenseInducing (pureCauchy : α → CauchyFilter α) | isUniformInducing_pureCauchy.isDenseInducing denseRange_pureCauchy | theorem | CauchyFilter.isDenseInducing_pureCauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"IsDenseInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isDenseEmbedding_pureCauchy : IsDenseEmbedding (pureCauchy : α → CauchyFilter α) | isUniformEmbedding_pureCauchy.isDenseEmbedding denseRange_pureCauchy | theorem | CauchyFilter.isDenseEmbedding_pureCauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"IsDenseEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α | by
constructor <;> rintro ⟨c⟩
· have := eq_univ_iff_forall.1 isDenseEmbedding_pureCauchy.isDenseInducing.closure_range c
obtain ⟨_, ⟨_, a, _⟩⟩ := mem_closure_iff.1 this _ isOpen_univ trivial
exact ⟨a⟩
· exact ⟨pureCauchy c⟩ | theorem | CauchyFilter.nonempty_cauchyFilter_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"isOpen_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend (f : α → β) : CauchyFilter α → β | if UniformContinuous f then isDenseInducing_pureCauchy.extend f
else fun x => f (nonempty_cauchyFilter_iff.1 ⟨x⟩).some | def | CauchyFilter.extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"UniformContinuous"
] | Extend a uniformly continuous function `α → β` to a function `CauchyFilter α → β`.
Outputs junk when `f` is not uniformly continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
extend_pureCauchy {f : α → β} (hf : UniformContinuous f) (a : α) :
extend f (pureCauchy a) = f a | by
rw [extend, if_pos hf]
exact uniformly_extend_of_ind isUniformInducing_pureCauchy denseRange_pureCauchy hf _ | theorem | CauchyFilter.extend_pureCauchy | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"UniformContinuous",
"uniformly_extend_of_ind"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_extend {f : α → β} : UniformContinuous (extend f) | by
by_cases hf : UniformContinuous f
· rw [extend, if_pos hf]
exact uniformContinuous_uniformly_extend isUniformInducing_pureCauchy denseRange_pureCauchy hf
· rw [extend, if_neg hf]
exact uniformContinuous_of_const fun a _b => by congr | theorem | CauchyFilter.uniformContinuous_extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"UniformContinuous",
"by_cases",
"uniformContinuous_of_const",
"uniformContinuous_uniformly_extend"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inseparable_iff {f g : CauchyFilter α} : Inseparable f g ↔ f.1 ×ˢ g.1 ≤ 𝓤 α | (basis_uniformity (basis_sets _)).inseparable_iff_uniformity | theorem | CauchyFilter.inseparable_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"Inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inseparable_iff_of_le_nhds {f g : CauchyFilter α} {a b : α}
(ha : f.1 ≤ 𝓝 a) (hb : g.1 ≤ 𝓝 b) : Inseparable a b ↔ Inseparable f g | by
rw [← tendsto_id'] at ha hb
rw [inseparable_iff, (ha.comp tendsto_fst).inseparable_iff_uniformity (hb.comp tendsto_snd)]
simp only [Function.comp_apply, id_eq, Prod.mk.eta, ← Function.id_def, tendsto_id'] | theorem | CauchyFilter.inseparable_iff_of_le_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Completion.lean | [] | [
"CauchyFilter",
"Inseparable",
"Prod.mk.eta"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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