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 ⌀ |
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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.compRel monotone_id).comap _).mem_iff.1 hi⟩
choose ind W hW hWU using this
rcases hK.elim_nhds_subcover' (fun x hx ↦ ball x (W x hx)) (fun x hx ↦ ball_mem_nhds _ (hW x hx))
with ⟨t, ht⟩
refine ⟨⋂ x ∈ t, W x x.2, (biInter_finset_mem _).2 fun x _ ↦ hW x x.2, fun x hx ↦ ?_⟩
rcases mem_iUnion₂.1 (ht hx) with ⟨y, hyt, hxy⟩
exact ⟨ind y y.2, fun z hz ↦ hWU _ _ ⟨x, hxy, mem_iInter₂.1 hz _ hyt⟩⟩ | 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_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 | 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_uni, hV⟩
exact ⟨V, V_uni, fun x hx => (hV x hx).imp fun _ hy => hy.trans interior_subset⟩ | 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 hx).imp fun _ hy => (inter_subset _ _ _).2 hy⟩ | 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_subset _ _ _).2 hy⟩ | 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 (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 | 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 (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 | 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 hK hU)
exact fun s t hst ht x hx ↦ (ht x hx).imp fun y hy ↦ Subset.trans (ball_mono hst _) hy | 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.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 | 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_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 | 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, 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 _ ↦ isOpen_interior) HKU
exact ⟨i, hpi, hi.trans interior_subset⟩
· rintro ⟨i, hpi, hi⟩
refine mem_of_superset (bUnion_mem_nhdsSet fun x _ ↦ ?_) hi
exact ball_mem_nhds _ <| hbasis.mem_of_mem hpi | 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 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_iUnion₂]
rintro ⟨a, ha, hxa⟩ ⟨b, hb, hxb⟩
rw [mem_ball_symmetry hVsymm] at hxa hxb
exact hUAB (mem_iUnion₂_of_mem ha <| hVU <| mem_comp_of_mem_ball hVsymm hxa hxb) hb | 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, 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 | 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 hVU⟩ | 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]
exact (𝓤 α).basis_sets.prod_self.comap _
refine (isCompact_diagonal.nhdsSet_basis_uniformity this).ge_iff.2 fun U hU => ?_
exact mem_of_superset hU fun ⟨x, y⟩ hxy => mem_iUnion₂.2
⟨(x, x), rfl, refl_mem_uniformity hU, hxy⟩ | 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 := by rwa [h']
rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h'] | 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 h K hK
rw [← tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK]
intro U hU x _
rcases comp_open_symm_mem_uniformity_sets hU with ⟨V, hV, hVo, hVsymm, hVU⟩
rcases mem_uniformity_isClosed hV with ⟨W, hW, hWc, hWU⟩
set s := K ∩ f ⁻¹' ball (f x) W
have hnhds : s ∈ 𝓝[K] x := inter_mem_nhdsWithin _ <| f.continuousAt _ (ball_mem_nhds _ hW)
have hcomp : IsCompact s := hK.inter_right <| (isClosed_ball _ hWc).preimage f.continuous
have hmaps : MapsTo f s (ball (f x) V) := fun x hx ↦ hWU hx.2
use s, hnhds
refine (h s hcomp _ (isOpen_ball _ hVo) hmaps).mono fun g hg y hy ↦ ?_
exact hVU ⟨f x, hVsymm.mk_mem_comm.2 <| hmaps hy, hg hy⟩
· -- Now we prove that uniform convergence on compacts
intro h K hK U hU hf
rcases lebesgue_number_of_compact_open (hK.image (map_continuous f)) hU hf.image_subset
with ⟨V, hV, -, hVf⟩
filter_upwards [h K hK V hV] with g hg x hx using hVf _ (mem_image_of_mem f hx) (hg x hx) | 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, UniformOnFun.tendsto_iff_tendstoUniformlyOn]
rfl | 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 the compact-open topology
and replace the topology with `compactOpen` to avoid non-defeq diamonds,
see Note [forgetful inheritance]. |
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_toUniformOnFunIsCompact.comap_uniformity]
exact .comap _ <| UniformOnFun.hasBasis_uniformity_of_basis _ _ {K | IsCompact K}
⟨∅, isCompact_empty⟩ (directedOn_of_sup_mem fun _ _ ↦ IsCompact.union) 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 | 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_of_covering_of_basis {K | IsCompact K} K.isCompact
(Monotone.directed_le K.subset) (fun _ ↦ K.exists_superset_of_isCompact) hb).comap _ | 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
K.subset (fun _ ↦ K.exists_superset_of_isCompact) hb).comap _ | 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)] :
IsCountablyGenerated (𝓤 (C(α, β))) :=
let ⟨_V, hV⟩ := exists_antitone_basis (𝓤 β)
((CompactExhaustion.choice α).hasAntitoneBasis_compactConvergenceUniformity
hV).isCountablyGenerated
variable {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f}
/-- Locally uniform convergence implies convergence in the compact-open topology. |
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 | 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.uniformContinuous | 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.isUniformInducing | 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.uniformContinuous | 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.right_inv (f _) |>.trans <| congrArg f <| φ.right_inv _
uniformContinuous_toFun := uniformContinuous_comp _ ψ.uniformContinuous |>.comp <|
uniformContinuous_comp_left _
uniformContinuous_invFun := uniformContinuous_comp _ ψ.symm.uniformContinuous |>.comp <|
uniformContinuous_comp_left _ | 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⟩, ⟨isCompact_univ, hV⟩, fun _fg hfg x => hfg x (mem_univ x)⟩ | 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.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⟩
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 | _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 f p s := by
rw [ContinuousMap.tendsto_iff_tendstoUniformly, tendstoUniformlyOn_iff_tendstoUniformly_comp_coe]
congr!
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 | _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.continuous_iff_continuous_uniformFun, UniformOnFun.continuous_rng_iff]
simp [Function.comp_def] | 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) inferInstance ⊓
.comap (comp · φ₂) inferInstance := by
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.image φ₁.continuous
have h_image₂ : MapsTo (φ₂ '' ·) 𝔗₂ 𝔖 := fun K hK ↦ hK.image φ₂.continuous
have h_preimage₁ : MapsTo (φ₁ ⁻¹' ·) 𝔖 𝔗₁ := fun K ↦ h_proper₁.isCompact_preimage
have h_preimage₂ : MapsTo (φ₂ ⁻¹' ·) 𝔖 𝔗₂ := fun K ↦ h_proper₂.isCompact_preimage
have h_cover' : ∀ S ∈ 𝔖, S ⊆ range φ₁ ∪ range φ₂ := fun S _ ↦ h_cover ▸ subset_univ _
simp_rw +zetaDelta [compactConvergenceUniformSpace, replaceTopology_eq,
UniformOnFun.uniformSpace_eq_inf_precomp_of_cover _ _ _ _ _
h_image₁ h_image₂ h_preimage₁ h_preimage₂ h_cover',
UniformSpace.comap_inf, ← UniformSpace.comap_comap]
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 | 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 := by
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
have h_preimage : ∀ i, MapsTo (φ i ⁻¹' ·) 𝔖 (𝔗 i) := fun i K ↦ (h_proper i).isCompact_preimage
have h_cover' : ∀ S ∈ 𝔖, ∃ I : Set ι, I.Finite ∧ S ⊆ ⋃ i ∈ I, range (φ i) := fun S hS ↦ by
refine ⟨{i | (range (φ i) ∩ S).Nonempty}, h_lf.finite_nonempty_inter_compact hS,
inter_eq_right.mp ?_⟩
simp_rw [iUnion₂_inter, mem_setOf, iUnion_nonempty_self, ← iUnion_inter, h_cover, univ_inter]
simp_rw +zetaDelta [compactConvergenceUniformSpace, replaceTopology_eq,
UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover _ _ _ h_image h_preimage h_cover',
UniformSpace.comap_iInf, ← UniformSpace.comap_comap]
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 | 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_continuous
CompactlyCoherentSpace.isCoherentWith).completeSpace_coe
@[deprecated (since := "2025-06-03")]
alias completeSpace_of_isCoherentWith := instCompleteSpaceOfCompactlyCoherentSpace
@[deprecated (since := "2025-04-08")]
alias completeSpace_of_restrictGenTopology := instCompleteSpaceOfCompactlyCoherentSpace | 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_inseparable ?_ ((continuous_eval_const x).continuousAt.mono_left hf')
refine tendsto_const_nhds.congr' <| .filter_mono ?_ hlf
exact fun _ h ↦ (h hx).symm
have H₂ (x) : Inseparable (s.piecewise f f' x) (f' x) := by
by_cases hx : x ∈ s <;> simp [hx, H₁, Inseparable.refl]
set g : C(α, β) :=
⟨s.piecewise f f', (continuous_congr_of_inseparable H₂).mpr <| map_continuous f'⟩
refine ⟨g, Set.piecewise_eqOn _ _ _, hf'.trans_eq ?_⟩
rwa [eq_comm, ← Inseparable, ← inseparable_coe, inseparable_pi] | 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 between two points
and the space of homotopies between two continuous maps are complete spaces,
without assuming that the codomain is a Hausdorff space. |
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,
abs_sub_comm] | 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.uniformSpace_eq]
exact Rat.isUniformEmbedding_coe_real.isUniformInducing)
(dense := Rat.isDenseEmbedding_coe_real.dense) | 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 : a = y)] | 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_setOf, Filter.monotone_mem, Function.comp_def,
image_swap_eq_preimage_swap]
have h₂ : (𝓤 α).lift' f ≤ (𝓤 α).lift' gen :=
uniformity_lift_le_swap
(monotone_principal.comp
(monotone_setOf fun p => @Filter.monotone_mem _ (p.2.val ×ˢ p.1.val)))
(by
have h := fun p : CauchyFilter α × CauchyFilter α => @Filter.prod_comm _ _ p.2.val p.1.val
simp only [Function.comp, h, mem_map, f]
exact le_rfl)
exact h₁.trans_le h₂ | 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, (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_mem this
(f.val ×ˢ g.val).sets_of_superset (prod_mem_prod ht₁ ht₄)
fun ⟨a, b⟩ ⟨(ha : a ∈ t₁), (hb : b ∈ t₄)⟩ =>
⟨x, h₁ (show (a, x) ∈ t₁ ×ˢ t₂ from ⟨ha, xt₂⟩), h₂ (show (x, b) ∈ t₃ ×ˢ t₄ from ⟨xt₃, hb⟩)⟩ | 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
_ ≤ (𝓤 α).lift' fun s => gen <| compRel s s :=
lift'_mono' fun _ _hs => compRel_gen_gen_subset_gen_compRel
_ = ((𝓤 α).lift' fun s : Set (α × α) => compRel s s).lift' gen := by
rw [lift'_lift'_assoc]
· exact monotone_id.compRel monotone_id
· exact monotone_gen
_ ≤ (𝓤 α).lift' gen := lift'_mono comp_le_uniformity le_rfl | 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.fst, pureCauchy x.snd)) ((𝓤 α).lift' gen) =
(𝓤 α).lift' ((preimage fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen) :=
comap_lift'_eq
_ = 𝓤 α := by simp [this]
⟩ | 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
have : t' ∈ f.val ×ˢ f.val := f.property.right ht'₁
let ⟨t, ht, (h : t ×ˢ t ⊆ t')⟩ := mem_prod_same_iff.mp this
let ⟨x, (hx : x ∈ t)⟩ := f.property.left.nonempty_of_mem ht
have : t'' ∈ f.val ×ˢ pure x :=
mem_prod_iff.mpr
⟨t, ht, { y : α | (x, y) ∈ t' }, h <| mk_mem_prod hx hx,
fun ⟨a, b⟩ ⟨(h₁ : a ∈ t), (h₂ : (x, b) ∈ t')⟩ =>
ht'₂ <| prodMk_mem_compRel (@h (a, x) ⟨h₁, hx⟩) h₂⟩
⟨x, ht''₂ <| by dsimp [gen]; exact this⟩
simp only [closure_eq_cluster_pts, ClusterPt, nhds_eq_uniformity, lift'_inf_principal_eq,
Set.inter_comm _ (range pureCauchy), mem_setOf_eq]
refine (lift'_neBot_iff ?_).mpr (fun s hs => ?_)
· exact monotone_const.inter monotone_preimage
· let ⟨y, hy⟩ := h_ex s hs
have : pureCauchy y ∈ range pureCauchy ∩ { y : CauchyFilter α | (f, y) ∈ s } :=
⟨mem_range_self y, hy⟩
exact ⟨_, this⟩ | 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, ← Function.id_def, tendsto_id'] | 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) :=
SeparationQuotient.isUniformInducing_mk.comp isUniformInducing_pureCauchy
theorem comap_coe_eq_uniformity :
((𝓤 _).comap fun p : α × α => ((p.1 : Completion α), (p.2 : Completion α))) = 𝓤 α :=
(isUniformInducing_coe _).1
variable {α} in
theorem denseRange_coe : DenseRange ((↑) : α → Completion α) :=
SeparationQuotient.surjective_mk.denseRange.comp denseRange_pureCauchy
SeparationQuotient.continuous_mk
/-- The Hausdorff completion as an abstract completion. |
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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