fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
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).me... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | lebesgue_number_lemma | 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`. |
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_... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | lebesgue_number_lemma_nhds' | null |
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_u... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | lebesgue_number_lemma_nhds | null |
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 h... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | lebesgue_number_lemma_nhdsWithin' | null |
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_... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | lebesgue_number_lemma_nhdsWithin | null |
protected 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 (l... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | Filter.HasBasis.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. |
protected 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... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | Filter.HasBasis.lebesgue_number_lemma_nhds' | null |
protected 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 ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | Filter.HasBasis.lebesgue_number_lemma_nhds | null |
protected 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.exi... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | Filter.HasBasis.lebesgue_number_lemma_nhdsWithin' | null |
protected 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_numbe... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | Filter.HasBasis.lebesgue_number_lemma_nhdsWithin | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | lebesgue_number_lemma_sUnion | 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`. |
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,... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | IsCompact.nhdsSet_basis_uniformity | 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`. |
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 ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | Disjoint.exists_uniform_thickening | null |
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, hV... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | Disjoint.exists_uniform_thickening_of_basis | null |
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 h... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | lebesgue_number_of_compact_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`. |
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]
... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | nhdsSet_diagonal_eq_uniformity | On a compact uniform space, the topology determines the uniform structure, entourages are
exactly the neighborhoods of the diagonal. |
compactSpace_uniformity [CompactSpace α] : 𝓤 α = ⨆ x, 𝓝 (x, x) :=
nhdsSet_diagonal_eq_uniformity.symm.trans (nhdsSet_diagonal _) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | compactSpace_uniformity | On a compact uniform space, the topology determines the uniform structure, entourages are
exactly the neighborhoods of the diagonal. |
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 :=... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/UniformSpace/Compact.lean | unique_uniformity_of_compact | null |
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
intro... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | tendsto_iff_forall_isCompact_tendstoUniformlyOn | 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. |
toUniformOnFunIsCompact (f : C(α, β)) : α →ᵤ[{K | IsCompact K}] β :=
UniformOnFun.ofFun {K | IsCompact K} f
@[simp] | def | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | toUniformOnFunIsCompact | 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)`. |
toUniformOnFun_toFun (f : C(α, β)) :
UniformOnFun.toFun _ f.toUniformOnFunIsCompact = f := rfl | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | toUniformOnFun_toFun | null |
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⟩⟩
open UniformSpace in | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | range_toUniformOnFunIsCompact | null |
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,... | instance | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | compactConvergenceUniformSpace | 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... |
isUniformEmbedding_toUniformOnFunIsCompact :
IsUniformEmbedding (toUniformOnFunIsCompact : C(α, β) → α →ᵤ[{K | IsCompact K}] β) where
comap_uniformity := rfl
injective := DFunLike.coe_injective
open UniformOnFun in | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | isUniformEmbedding_toUniformOnFunIsCompact | null |
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 | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | continuous_iff_continuous_uniformOnFun | `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`. |
_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_... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | _root_.Filter.HasBasis.compactConvergenceUniformity | null |
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 | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | hasBasis_compactConvergenceUniformity | null |
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 | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | mem_compactConvergence_entourage_iff | null |
_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_o... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | _root_.CompactExhaustion.hasBasis_compactConvergenceUniformity | 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)`. |
_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
... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | _root_.CompactExhaustion.hasAntitoneBasis_compactConvergenceUniformity | null |
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 | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | tendsto_of_tendstoLocallyUniformly | 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)] :
IsCounta... |
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... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | tendsto_iff_tendstoLocallyUniformly | 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`. |
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.uniformC... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | uniformContinuous_comp | null |
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.i... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | isUniformInducing_comp | null |
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 | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | isUniformEmbedding_comp | null |
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.uni... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | uniformContinuous_comp_left | null |
protected _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.righ... | def | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | _root_.UniformEquiv.arrowCongr | 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)`. |
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⟩, ⟨i... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | hasBasis_compactConvergenceUniformity_of_compact | null |
_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... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | _root_.Filter.HasBasis.compactConvergenceUniformity_of_compact | null |
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 | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | isUniformEmbedding_uniformFunOfFun | null |
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]
open UniformFun in | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | tendsto_iff_tendstoUniformly | Convergence in the compact-open topology is the same as uniform convergence for sequences of
continuous functions on a compact space. |
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 | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | continuous_iff_continuous_uniformFun | 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`. |
_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... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | _root_.ContinuousOn.tendsto_restrict_iff_tendstoUniformlyOn | 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. |
_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.... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | _root_.ContinuousOn.continuous_restrict_iff_continuous_uniformOnFun | 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. |
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) :
(inferInstanceAs <| UniformSpace C(α, β)) =
.comap (comp · φ₁) inferInstan... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | uniformSpace_eq_inf_precomp_of_cover | null |
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) :
(inferInstanceAs <| UniformSpace C(α, β)) = ⨅ i, .comap (comp · (φ i)) inferInstance :... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | uniformSpace_eq_iInf_precomp_of_cover | null |
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_co... | instance | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | instCompleteSpaceOfCompactlyCoherentSpace | 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. |
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_inseparab... | theorem | Topology | [
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.Compactness.CompactlyCoherentSpace",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/CompactConvergence.lean | isComplete_setOf_eqOn | 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 ... |
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,
... | theorem | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | Rat.uniformSpace_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). |
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.uniform... | def | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | rationalCauSeqPkg | Cauchy reals packaged as a completion of ℚ using the absolute value route. |
Q :=
ℚ deriving CommRing, Inhabited | def | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | Q | 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). |
uniformSpace : UniformSpace Q :=
(@AbsoluteValue.abs ℚ _).uniformSpace | instance | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | uniformSpace | null |
Bourbakiℝ : Type :=
Completion Q deriving Inhabited | def | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | Bourbakiℝ | Real numbers constructed as in Bourbaki. |
Bourbaki.uniformSpace : UniformSpace Bourbakiℝ :=
Completion.uniformSpace Q | instance | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | Bourbaki.uniformSpace | null |
bourbakiPkg : AbstractCompletion Q :=
Completion.cPkg | def | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | bourbakiPkg | Bourbaki reals packaged as a completion of Q using the general theory. |
noncomputable compareEquiv : Bourbakiℝ ≃ᵤ ℝ :=
bourbakiPkg.compareEquiv rationalCauSeqPkg | def | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | compareEquiv | The uniform bijection between Bourbaki and Cauchy reals. |
compare_uc : UniformContinuous compareEquiv :=
bourbakiPkg.uniformContinuous_compareEquiv rationalCauSeqPkg | theorem | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | compare_uc | null |
compare_uc_symm : UniformContinuous compareEquiv.symm :=
bourbakiPkg.uniformContinuous_compareEquiv_symm rationalCauSeqPkg | theorem | Topology | [
"Mathlib.Topology.Instances.Rat",
"Mathlib.Topology.UniformSpace.AbsoluteValue",
"Mathlib.Topology.UniformSpace.Completion"
] | Mathlib/Topology/UniformSpace/CompareReals.lean | compare_uc_symm | null |
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 : ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] | Mathlib/Topology/UniformSpace/CompleteSeparated.lean | IsComplete.isClosed | In a separated space, a complete set is closed. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] | Mathlib/Topology/UniformSpace/CompleteSeparated.lean | IsUniformEmbedding.isClosedEmbedding | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformEmbedding"
] | Mathlib/Topology/UniformSpace/CompleteSeparated.lean | continuous_extend_of_cauchy | null |
CauchyFilter (α : Type u) [UniformSpace α] : Type u :=
{ f : Filter α // Cauchy f } | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | CauchyFilter | 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. |
gen (s : Set (α × α)) : Set (CauchyFilter α × CauchyFilter α) :=
{ p | s ∈ p.1.val ×ˢ p.2.val } | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | gen | The pairs of Cauchy filters generated by a set. |
monotone_gen : Monotone (gen : Set (α × α) → _) :=
monotone_setOf fun p => @Filter.monotone_mem _ (p.1.val ×ˢ p.2.val) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | monotone_gen | null |
private symm_gen : map Prod.swap ((𝓤 α).lift' gen) ≤ (𝓤 α).lift' gen := by
let f := fun s : Set (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... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | symm_gen | null |
private compRel_gen_gen_subset_gen_compRel {s t : Set (α × α)} :
compRel (gen s) (gen t) ⊆ (gen (compRel s t) : Set (CauchyFilter α × CauchyFilter α)) :=
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, (... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | compRel_gen_gen_subset_gen_compRel | null |
private comp_gen : (((𝓤 α).lift' gen).lift' fun s => compRel s s) ≤ (𝓤 α).lift' gen :=
calc
(((𝓤 α).lift' gen).lift' fun s => compRel s s) =
(𝓤 α).lift' fun s => compRel (gen s) (gen s) := by
rw [lift'_lift'_assoc]
· exact monotone_gen
· exact monotone_id.compRel monotone_id
_ ≤ ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | comp_gen | null |
mem_uniformity {s : Set (CauchyFilter α × CauchyFilter α)} :
s ∈ 𝓤 (CauchyFilter α) ↔ ∃ t ∈ 𝓤 α, gen t ⊆ s :=
mem_lift'_sets monotone_gen | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | mem_uniformity | null |
basis_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
(𝓤 (CauchyFilter α)).HasBasis p (gen ∘ s) :=
h.lift' monotone_gen | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | basis_uniformity | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | mem_uniformity' | null |
pureCauchy (a : α) : CauchyFilter α :=
⟨pure a, cauchy_pure⟩ | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | pureCauchy | Embedding of `α` into its completion `CauchyFilter α` |
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.fs... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | isUniformInducing_pureCauchy | null |
isUniformEmbedding_pureCauchy : IsUniformEmbedding (pureCauchy : α → CauchyFilter α) where
__ := isUniformInducing_pureCauchy
injective _a₁ _a₂ h := pure_injective <| Subtype.ext_iff.1 h | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | isUniformEmbedding_pureCauchy | null |
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
hav... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | denseRange_pureCauchy | null |
isDenseInducing_pureCauchy : IsDenseInducing (pureCauchy : α → CauchyFilter α) :=
isUniformInducing_pureCauchy.isDenseInducing denseRange_pureCauchy | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | isDenseInducing_pureCauchy | null |
isDenseEmbedding_pureCauchy : IsDenseEmbedding (pureCauchy : α → CauchyFilter α) :=
isUniformEmbedding_pureCauchy.isDenseEmbedding denseRange_pureCauchy | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | isDenseEmbedding_pureCauchy | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | nonempty_cauchyFilter_iff | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extend | Extend a uniformly continuous function `α → β` to a function `CauchyFilter α → β`.
Outputs junk when `f` is not uniformly continuous. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extend_pureCauchy | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformContinuous_extend | null |
inseparable_iff {f g : CauchyFilter α} : Inseparable f g ↔ f.1 ×ˢ g.1 ≤ 𝓤 α :=
(basis_uniformity (basis_sets _)).inseparable_iff_uniformity | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | inseparable_iff | null |
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,... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | inseparable_iff_of_le_nhds | null |
inseparable_lim_iff [CompleteSpace α] {f g : CauchyFilter α} :
haveI := f.2.1.nonempty; Inseparable (lim f.1) (lim g.1) ↔ Inseparable f g :=
inseparable_iff_of_le_nhds f.2.le_nhds_lim g.2.le_nhds_lim | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | inseparable_lim_iff | null |
cauchyFilter_eq {α : Type*} [UniformSpace α] [CompleteSpace α] [T0Space α]
{f g : CauchyFilter α} :
haveI := f.2.1.nonempty; lim f.1 = lim g.1 ↔ Inseparable f g := by
rw [← inseparable_iff_eq, inseparable_lim_iff] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | cauchyFilter_eq | null |
separated_pureCauchy_injective {α : Type*} [UniformSpace α] [T0Space α] :
Function.Injective fun a : α => SeparationQuotient.mk (pureCauchy a) := fun a b h ↦
Inseparable.eq <| (inseparable_iff_of_le_nhds (pure_le_nhds a) (pure_le_nhds b)).2 <|
SeparationQuotient.mk_eq_mk.1 h | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | separated_pureCauchy_injective | null |
Completion := SeparationQuotient (CauchyFilter α) | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | Completion | Hausdorff completion of `α` |
inhabited [Inhabited α] : Inhabited (Completion α) :=
inferInstanceAs <| Inhabited (Quotient _) | instance | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | inhabited | null |
uniformSpace : UniformSpace (Completion α) :=
SeparationQuotient.instUniformSpace | instance | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformSpace | null |
completeSpace : CompleteSpace (Completion α) :=
SeparationQuotient.instCompleteSpace | instance | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | completeSpace | null |
t0Space : T0Space (Completion α) := SeparationQuotient.instT0Space
variable {α} in | instance | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | t0Space | null |
@[coe] coe' : α → Completion α := SeparationQuotient.mk ∘ pureCauchy | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | coe' | The map from a uniform space to its completion. |
cPkg {α : Type*} [UniformSpace α] : AbstractCompletion α where
space := Completion α
coe := (↑)
uniformStruct := by infer_instance
complete := by infer_instance
separation := by infer_instance
isUniformInducing := Completion.isUniformInducing_coe α
dense := Completion.denseRange_coe | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | cPkg | Automatic coercion from `α` to its completion. Not always injective. -/
instance : Coe α (Completion α) :=
⟨coe'⟩
-- note [use has_coe_t]
protected theorem coe_eq : ((↑) : α → Completion α) = SeparationQuotient.mk ∘ pureCauchy := rfl
theorem isUniformInducing_coe : IsUniformInducing ((↑) : α → Completion α) :=
Se... |
AbstractCompletion.inhabited : Inhabited (AbstractCompletion α) :=
⟨cPkg⟩
attribute [local instance]
AbstractCompletion.uniformStruct AbstractCompletion.complete AbstractCompletion.separation | instance | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | AbstractCompletion.inhabited | null |
nonempty_completion_iff : Nonempty (Completion α) ↔ Nonempty α :=
cPkg.dense.nonempty_iff.symm | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | nonempty_completion_iff | null |
uniformContinuous_coe : UniformContinuous ((↑) : α → Completion α) :=
cPkg.uniformContinuous_coe | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformContinuous_coe | null |
continuous_coe : Continuous ((↑) : α → Completion α) :=
cPkg.continuous_coe | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | continuous_coe | null |
isUniformEmbedding_coe [T0Space α] : IsUniformEmbedding ((↑) : α → Completion α) :=
{ comap_uniformity := comap_coe_eq_uniformity α
injective := separated_pureCauchy_injective } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | isUniformEmbedding_coe | null |
coe_injective [T0Space α] : Function.Injective ((↑) : α → Completion α) :=
IsUniformEmbedding.injective (isUniformEmbedding_coe _)
variable {α}
@[simp] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | coe_injective | null |
coe_inj [T0Space α] {a b : α} : (a : Completion α) = b ↔ a = b :=
(coe_injective _).eq_iff | lemma | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | coe_inj | null |
isDenseInducing_coe : IsDenseInducing ((↑) : α → Completion α) :=
{ (isUniformInducing_coe α).isInducing with dense := denseRange_coe } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | isDenseInducing_coe | null |
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