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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protected hasBasis_uniformity_of_basis {ι : Sort*} {p : ι → Prop} {s : ι → Set (β × β)}
(h : (𝓤 β).HasBasis p s) : (𝓤 (α →ᵤ β)).HasBasis p (UniformFun.gen α β ∘ s) :=
(UniformFun.hasBasis_uniformity α β).to_hasBasis
(fun _ hU =>
let ⟨i, hi, hiU⟩ := h.mem_iff.mp hU
⟨i, hi, fun _ huv x => hiU (huv... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_uniformity_of_basis | The uniformity of `α →ᵤ β` admits the family `{(f, g) | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓑` as
a filter basis, for any basis `𝓑` of `𝓤 β` (in the case `𝓑 = (𝓤 β).as_basis` this is true by
definition). |
protected hasBasis_nhds_of_basis (f) {p : ι → Prop} {s : ι → Set (β × β)}
(h : HasBasis (𝓤 β) p s) :
(𝓝 f).HasBasis p fun i => { g | (f, g) ∈ UniformFun.gen α β (s i) } :=
nhds_basis_uniformity' (UniformFun.hasBasis_uniformity_of_basis α β h) | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_nhds_of_basis | For `f : α →ᵤ β`, `𝓝 f` admits the family `{g | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓑` as a filter
basis, for any basis `𝓑` of `𝓤 β`. |
protected hasBasis_nhds (f) :
(𝓝 f).HasBasis (fun V => V ∈ 𝓤 β) fun V => { g | (f, g) ∈ UniformFun.gen α β V } :=
UniformFun.hasBasis_nhds_of_basis α β f (Filter.basis_sets _)
variable {α} | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_nhds | For `f : α →ᵤ β`, `𝓝 f` admits the family `{g | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓤 β` as a
filter basis. |
uniformContinuous_eval (x : α) :
UniformContinuous (Function.eval x ∘ toFun : (α →ᵤ β) → β) := by
change _ ≤ _
rw [map_le_iff_le_comap,
(UniformFun.hasBasis_uniformity α β).le_basis_iff ((𝓤 _).basis_sets.comap _)]
exact fun U hU => ⟨U, hU, fun uv huv => huv x⟩
variable {β}
@[simp] | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_eval | Evaluation at a fixed point is uniformly continuous on `α →ᵤ β`. |
protected mem_gen {β} {f g : α →ᵤ β} {V : Set (β × β)} :
(f, g) ∈ UniformFun.gen α β V ↔ ∀ x, (toFun f x, toFun g x) ∈ V :=
.rfl | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | mem_gen | null |
protected mono : Monotone (@UniformFun.uniformSpace α γ) := fun _ _ hu =>
(UniformFun.gc α γ).monotone_u hu | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | mono | If `u₁` and `u₂` are two uniform structures on `γ` and `u₁ ≤ u₂`, then
`𝒰(α, γ, u₁) ≤ 𝒰(α, γ, u₂)`. |
protected iInf_eq {u : ι → UniformSpace γ} : 𝒰(α, γ, (⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i) := by
ext : 1
change UniformFun.filter α γ 𝓤[⨅ i, u i] = 𝓤[⨅ i, 𝒰(α, γ, u i)]
rw [iInf_uniformity, iInf_uniformity]
exact (UniformFun.gc α γ).u_iInf | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | iInf_eq | If `u` is a family of uniform structures on `γ`, then
`𝒰(α, γ, (⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i)`. |
protected inf_eq {u₁ u₂ : UniformSpace γ} :
𝒰(α, γ, u₁ ⊓ u₂) = 𝒰(α, γ, u₁) ⊓ 𝒰(α, γ, u₂) := by
rw [inf_eq_iInf, inf_eq_iInf, UniformFun.iInf_eq]
refine iInf_congr fun i => ?_
cases i <;> rfl | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | inf_eq | If `u₁` and `u₂` are two uniform structures on `γ`, then
`𝒰(α, γ, u₁ ⊓ u₂) = 𝒰(α, γ, u₁) ⊓ 𝒰(α, γ, u₂)`. |
postcomp_isUniformInducing [UniformSpace γ] {f : γ → β}
(hf : IsUniformInducing f) : IsUniformInducing (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) :=
⟨((UniformFun.hasBasis_uniformity _ _).comap _).eq_of_same_basis <|
UniformFun.hasBasis_uniformity_of_basis _ _ (hf.basis_uniformity (𝓤 β).basis_sets)⟩ | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | postcomp_isUniformInducing | Post-composition by a uniform inducing function is
a uniform inducing function for the uniform structures of uniform convergence.
More precisely, if `f : γ → β` is uniform inducing,
then `(f ∘ ·) : (α →ᵤ γ) → (α →ᵤ β)` is uniform inducing. |
protected postcomp_isUniformEmbedding [UniformSpace γ] {f : γ → β}
(hf : IsUniformEmbedding f) :
IsUniformEmbedding (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) where
toIsUniformInducing := UniformFun.postcomp_isUniformInducing hf.isUniformInducing
injective _ _ H := funext fun _ ↦ hf.injective (congrFun H... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | postcomp_isUniformEmbedding | Post-composition by a uniform embedding is
a uniform embedding for the uniform structures of uniform convergence.
More precisely, if `f : γ → β` is a uniform embedding,
then `(f ∘ ·) : (α →ᵤ γ) → (α →ᵤ β)` is a uniform embedding. |
protected comap_eq {f : γ → β} :
𝒰(α, γ, ‹UniformSpace β›.comap f) = 𝒰(α, β, _).comap (f ∘ ·) := by
letI : UniformSpace γ := .comap f ‹_›
exact (UniformFun.postcomp_isUniformInducing (f := f) ⟨rfl⟩).comap_uniformSpace.symm | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | comap_eq | If `u` is a uniform structures on `β` and `f : γ → β`, then
`𝒰(α, γ, comap f u) = comap (fun g ↦ f ∘ g) 𝒰(α, γ, u₁)`. |
protected postcomp_uniformContinuous [UniformSpace γ] {f : γ → β}
(hf : UniformContinuous f) :
UniformContinuous (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) := by
refine uniformContinuous_iff.mpr ?_
calc
𝒰(α, γ, _) ≤ 𝒰(α, γ, ‹UniformSpace β›.comap f) :=
UniformFun.mono (uniformContinuo... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | postcomp_uniformContinuous | Post-composition by a uniformly continuous function is uniformly continuous on `α →ᵤ β`.
More precisely, if `f : γ → β` is uniformly continuous, then `(fun g ↦ f ∘ g) : (α →ᵤ γ) → (α →ᵤ β)`
is uniformly continuous. |
protected congrRight [UniformSpace γ] (e : γ ≃ᵤ β) : (α →ᵤ γ) ≃ᵤ (α →ᵤ β) :=
{ Equiv.piCongrRight fun _ => e.toEquiv with
uniformContinuous_toFun := UniformFun.postcomp_uniformContinuous e.uniformContinuous
uniformContinuous_invFun := UniformFun.postcomp_uniformContinuous e.symm.uniformContinuous } | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | congrRight | Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ γ) ≃ᵤ (α →ᵤ β)` by
post-composing. |
protected precomp_uniformContinuous {f : γ → α} :
UniformContinuous fun g : α →ᵤ β => ofFun (toFun g ∘ f) := by
rw [UniformContinuous,
(UniformFun.hasBasis_uniformity α β).tendsto_iff (UniformFun.hasBasis_uniformity γ β)]
exact fun U hU => ⟨U, hU, fun uv huv x => huv (f x)⟩ | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | precomp_uniformContinuous | Pre-composition by any function is uniformly continuous for the uniform structures of
uniform convergence.
More precisely, for any `f : γ → α`, the function `(· ∘ f) : (α →ᵤ β) → (γ →ᵤ β)` is uniformly
continuous. |
protected congrLeft (e : γ ≃ α) : (γ →ᵤ β) ≃ᵤ (α →ᵤ β) where
toEquiv := e.arrowCongr (.refl _)
uniformContinuous_toFun := UniformFun.precomp_uniformContinuous
uniformContinuous_invFun := UniformFun.precomp_uniformContinuous | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | congrLeft | Turn a bijection `γ ≃ α` into a uniform isomorphism
`(γ →ᵤ β) ≃ᵤ (α →ᵤ β)` by pre-composing. |
protected uniformContinuous_toFun : UniformContinuous (toFun : (α →ᵤ β) → α → β) := by
rw [uniformContinuous_pi]
intro x
exact uniformContinuous_eval β x | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_toFun | The natural map `UniformFun.toFun` from `α →ᵤ β` to `α → β` is uniformly continuous.
In other words, the uniform structure of uniform convergence is finer than that of pointwise
convergence, aka the product uniform structure. |
protected tendsto_iff_tendstoUniformly {F : ι → α →ᵤ β} {f : α →ᵤ β} :
Tendsto F p (𝓝 f) ↔ TendstoUniformly (toFun ∘ F) (toFun f) p := by
rw [(UniformFun.hasBasis_nhds α β f).tendsto_right_iff, TendstoUniformly]
simp only [mem_setOf, UniformFun.gen, Function.comp_def] | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | tendsto_iff_tendstoUniformly | The topology of uniform convergence is T₂. -/
instance [T2Space β] : T2Space (α →ᵤ β) :=
.of_injective_continuous toFun.injective UniformFun.uniformContinuous_toFun.continuous
/-- The topology of uniform convergence indeed gives the same notion of convergence as
`TendstoUniformly`. |
protected uniformEquivProdArrow [UniformSpace γ] : (α →ᵤ β × γ) ≃ᵤ (α →ᵤ β) × (α →ᵤ γ) :=
Equiv.toUniformEquivOfIsUniformInducing (Equiv.arrowProdEquivProdArrow _ _ _) <| by
constructor
change
comap (Prod.map (Equiv.arrowProdEquivProdArrow _ _ _) (Equiv.arrowProdEquivProdArrow _ _ _))
_ = _
... | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformEquivProdArrow | The natural bijection between `α → β × γ` and `(α → β) × (α → γ)`, upgraded to a uniform
isomorphism between `α →ᵤ β × γ` and `(α →ᵤ β) × (α →ᵤ γ)`. |
protected uniformEquivPiComm : UniformEquiv (α →ᵤ ∀ i, δ i) (∀ i, α →ᵤ δ i) :=
@Equiv.toUniformEquivOfIsUniformInducing
_ _ 𝒰(α, ∀ i, δ i, Pi.uniformSpace δ)
(@Pi.uniformSpace ι (fun i => α → δ i) fun i => 𝒰(α, δ i, _)) (Equiv.piComm _) <| by
refine @IsUniformInducing.mk ?_ ?_ ?_ ?_ ?_ ?_
chan... | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformEquivPiComm | The natural bijection between `α → Π i, δ i` and `Π i, α → δ i`, upgraded to a uniform
isomorphism between `α →ᵤ (Π i, δ i)` and `Π i, α →ᵤ δ i`. |
isClosed_setOf_continuous [TopologicalSpace α] :
IsClosed {f : α →ᵤ β | Continuous (toFun f)} := by
refine isClosed_iff_forall_filter.2 fun f u _ hu huf ↦ ?_
rw [← tendsto_id', UniformFun.tendsto_iff_tendstoUniformly] at huf
exact huf.continuous (le_principal_iff.mp hu)
variable {α} (β) in | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | isClosed_setOf_continuous | The set of continuous functions is closed in the uniform convergence topology.
This is a simple wrapper over `TendstoUniformly.continuous`. |
uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} (φ₁ : δ₁ → α) (φ₂ : δ₂ → α)
(h_cover : range φ₁ ∪ range φ₂ = univ) :
𝒰(α, β, _) =
.comap (ofFun ∘ (· ∘ φ₁) ∘ toFun) 𝒰(δ₁, β, _) ⊓
.comap (ofFun ∘ (· ∘ φ₂) ∘ toFun) 𝒰(δ₂, β, _) := by
ext : 1
refine le_antisymm (le_inf ?_ ?_) ?_
· exact ten... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformSpace_eq_inf_precomp_of_cover | null |
uniformSpace_eq_iInf_precomp_of_cover {δ : ι → Type*} (φ : Π i, δ i → α)
(h_cover : ∃ I : Set ι, I.Finite ∧ ⋃ i ∈ I, range (φ i) = univ) :
𝒰(α, β, _) = ⨅ i, .comap (ofFun ∘ (· ∘ φ i) ∘ toFun) 𝒰(δ i, β, _) := by
ext : 1
simp_rw [iInf_uniformity, uniformity_comap]
refine le_antisymm (le_iInf fun i ↦ tends... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformSpace_eq_iInf_precomp_of_cover | null |
protected gen (𝔖) (S : Set α) (V : Set (β × β)) : Set ((α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)) :=
{ uv : (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β) | ∀ x ∈ S, (toFun 𝔖 uv.1 x, toFun 𝔖 uv.2 x) ∈ V } | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | gen | Basis sets for the uniformity of `𝔖`-convergence: for `S : Set α` and `V : Set (β × β)`,
`gen 𝔖 S V` is the set of pairs `(f, g)` of functions `α →ᵤ[𝔖] β` such that
`∀ x ∈ S, (f x, g x) ∈ V`. Note that the family `𝔖 : Set (Set α)` is only used to specify which
type alias of `α → β` to use here. |
protected gen_eq_preimage_restrict {𝔖} (S : Set α) (V : Set (β × β)) :
UniformOnFun.gen 𝔖 S V =
Prod.map (S.restrict ∘ UniformFun.toFun) (S.restrict ∘ UniformFun.toFun) ⁻¹'
UniformFun.gen S β V := by
ext uv
exact ⟨fun h ⟨x, hx⟩ => h x hx, fun h x hx => h ⟨x, hx⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | gen_eq_preimage_restrict | For `S : Set α` and `V : Set (β × β)`, we have
`UniformOnFun.gen 𝔖 S V = (S.restrict × S.restrict) ⁻¹' (UniformFun.gen S β V)`.
This is the crucial fact for proving that the family `UniformOnFun.gen S V` for `S ∈ 𝔖` and
`V ∈ 𝓤 β` is indeed a basis for the uniformity `α →ᵤ[𝔖] β` endowed with `𝒱(α, β, 𝔖, uβ)`
the u... |
protected gen_mono {𝔖} {S S' : Set α} {V V' : Set (β × β)} (hS : S' ⊆ S) (hV : V ⊆ V') :
UniformOnFun.gen 𝔖 S V ⊆ UniformOnFun.gen 𝔖 S' V' := fun _uv h x hx => hV (h x <| hS hx) | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | gen_mono | `UniformOnFun.gen` is antitone in the first argument and monotone in the second. |
protected isBasis_gen (𝔖 : Set (Set α)) (h : 𝔖.Nonempty) (h' : DirectedOn (· ⊆ ·) 𝔖)
(𝓑 : FilterBasis <| β × β) :
IsBasis (fun SV : Set α × Set (β × β) => SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓑) fun SV =>
UniformOnFun.gen 𝔖 SV.1 SV.2 :=
⟨h.prod 𝓑.nonempty, fun {U₁V₁ U₂V₂} h₁ h₂ =>
let ⟨U₃, hU₃, hU₁₃, hU₂₃⟩ := ... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | isBasis_gen | If `𝔖 : Set (Set α)` is nonempty and directed and `𝓑` is a filter basis on `β × β`, then the
family `UniformOnFun.gen 𝔖 S V` for `S ∈ 𝔖` and `V ∈ 𝓑` is a filter basis on
`(α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)`.
We will show in `has_basis_uniformity_of_basis` that, if `𝓑` is a basis for `𝓤 β`, then the
corresponding filter... |
uniformSpace : UniformSpace (α →ᵤ[𝔖] β) :=
⨅ (s : Set α) (_ : s ∈ 𝔖),
.comap (UniformFun.ofFun ∘ s.restrict ∘ UniformOnFun.toFun 𝔖) 𝒰(s, β, _)
local notation "𝒱(" α ", " β ", " 𝔖 ", " u ")" => @UniformOnFun.uniformSpace α β u 𝔖 | instance | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformSpace | Uniform structure of `𝔖`-convergence, i.e uniform convergence on the elements of `𝔖`,
declared as an instance on `α →ᵤ[𝔖] β`. It is defined as the infimum, for `S ∈ 𝔖`, of the pullback
by `S.restrict`, the map of restriction to `S`, of the uniform structure `𝒰(s, β, uβ)` on
`↥S →ᵤ β`. We will denote it `𝒱(α, β, �... |
topologicalSpace : TopologicalSpace (α →ᵤ[𝔖] β) :=
𝒱(α, β, 𝔖, _).toTopologicalSpace | instance | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | topologicalSpace | Topology of `𝔖`-convergence, i.e uniform convergence on the elements of `𝔖`, declared as an
instance on `α →ᵤ[𝔖] β`. |
protected topologicalSpace_eq :
UniformOnFun.topologicalSpace α β 𝔖 =
⨅ (s : Set α) (_ : s ∈ 𝔖), TopologicalSpace.induced
(UniformFun.ofFun ∘ s.restrict ∘ toFun 𝔖) (UniformFun.topologicalSpace s β) := by
simp only [UniformOnFun.topologicalSpace, UniformSpace.toTopologicalSpace_iInf]
rfl | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | topologicalSpace_eq | The topology of `𝔖`-convergence is the infimum, for `S ∈ 𝔖`, of topology induced by the map
of `S.restrict : (α →ᵤ[𝔖] β) → (↥S →ᵤ β)` of restriction to `S`, where `↥S →ᵤ β` is endowed with
the topology of uniform convergence. |
protected hasBasis_uniformity_of_basis_aux₁ {p : ι → Prop} {s : ι → Set (β × β)}
(hb : HasBasis (𝓤 β) p s) (S : Set α) :
(@uniformity (α →ᵤ[𝔖] β) ((UniformFun.uniformSpace S β).comap S.restrict)).HasBasis p fun i =>
UniformOnFun.gen 𝔖 S (s i) := by
simp_rw [UniformOnFun.gen_eq_preimage_restrict, unif... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_uniformity_of_basis_aux₁ | null |
protected hasBasis_uniformity_of_basis_aux₂ (h : DirectedOn (· ⊆ ·) 𝔖) {p : ι → Prop}
{s : ι → Set (β × β)} (hb : HasBasis (𝓤 β) p s) :
DirectedOn
((fun s : Set α => (UniformFun.uniformSpace s β).comap (s.restrict : (α →ᵤ β) → s →ᵤ β)) ⁻¹'o
GE.ge)
𝔖 :=
h.mono fun _ _ hst =>
((Unifor... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_uniformity_of_basis_aux₂ | null |
protected hasBasis_uniformity_of_basis (h : 𝔖.Nonempty) (h' : DirectedOn (· ⊆ ·) 𝔖)
{p : ι → Prop} {s : ι → Set (β × β)} (hb : HasBasis (𝓤 β) p s) :
(𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun Si : Set α × ι => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si =>
UniformOnFun.gen 𝔖 Si.1 (s Si.2) := by
simp only [iInf_uniformity]
ex... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_uniformity_of_basis | If `𝔖 : Set (Set α)` is nonempty and directed and `𝓑` is a filter basis of `𝓤 β`, then the
uniformity of `α →ᵤ[𝔖] β` admits the family `{(f, g) | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and
`V ∈ 𝓑` as a filter basis. |
protected hasBasis_uniformity (h : 𝔖.Nonempty) (h' : DirectedOn (· ⊆ ·) 𝔖) :
(𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun SV : Set α × Set (β × β) => SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β) fun SV =>
UniformOnFun.gen 𝔖 SV.1 SV.2 :=
UniformOnFun.hasBasis_uniformity_of_basis α β 𝔖 h h' (𝓤 β).basis_sets
variable {α β} | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_uniformity | If `𝔖 : Set (Set α)` is nonempty and directed, then the uniformity of `α →ᵤ[𝔖] β` admits the
family `{(f, g) | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓤 β` as a filter basis. |
protected hasBasis_uniformity_of_covering_of_basis {ι ι' : Type*} [Nonempty ι]
{t : ι → Set α} {p : ι' → Prop} {V : ι' → Set (β × β)} (ht : ∀ i, t i ∈ 𝔖)
(hdir : Directed (· ⊆ ·) t) (hex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i) (hb : HasBasis (𝓤 β) p V) :
(𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i : ι × ι' ↦ p i.2) fun i ↦
... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_uniformity_of_covering_of_basis | Let `t i` be a nonempty directed subfamily of `𝔖`
such that every `s ∈ 𝔖` is included in some `t i`.
Let `V` bounded by `p` be a basis of entourages of `β`.
Then `UniformOnFun.gen 𝔖 (t i) (V j)` bounded by `p j` is a basis of entourages of `α →ᵤ[𝔖] β`. |
protected hasAntitoneBasis_uniformity {ι : Type*} [Preorder ι] [IsDirected ι (· ≤ ·)]
{t : ι → Set α} {V : ι → Set (β × β)}
(ht : ∀ n, t n ∈ 𝔖) (hmono : Monotone t) (hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n)
(hb : HasAntitoneBasis (𝓤 β) V) :
(𝓤 (α →ᵤ[𝔖] β)).HasAntitoneBasis fun n ↦ UniformOnFun.gen 𝔖 (t n) (V ... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasAntitoneBasis_uniformity | If `t n` is a monotone sequence of sets in `𝔖`
such that each `s ∈ 𝔖` is included in some `t n`
and `V n` is an antitone basis of entourages of `β`,
then `UniformOnFun.gen 𝔖 (t n) (V n)` is an antitone basis of entourages of `α →ᵤ[𝔖] β`. |
protected isCountablyGenerated_uniformity [IsCountablyGenerated (𝓤 β)] {t : ℕ → Set α}
(ht : ∀ n, t n ∈ 𝔖) (hmono : Monotone t) (hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n) :
IsCountablyGenerated (𝓤 (α →ᵤ[𝔖] β)) :=
let ⟨_V, hV⟩ := exists_antitone_basis (𝓤 β)
(UniformOnFun.hasAntitoneBasis_uniformity 𝔖 ht hmono hex ... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | isCountablyGenerated_uniformity | null |
protected hasBasis_nhds_of_basis (f : α →ᵤ[𝔖] β) (h : 𝔖.Nonempty)
(h' : DirectedOn (· ⊆ ·) 𝔖) {p : ι → Prop} {s : ι → Set (β × β)} (hb : HasBasis (𝓤 β) p s) :
(𝓝 f).HasBasis (fun Si : Set α × ι => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si =>
{ g | (g, f) ∈ UniformOnFun.gen 𝔖 Si.1 (s Si.2) } :=
letI : UniformSpac... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_nhds_of_basis | For `f : α →ᵤ[𝔖] β`, where `𝔖 : Set (Set α)` is nonempty and directed, `𝓝 f` admits the
family `{g | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓑` as a filter basis, for any basis
`𝓑` of `𝓤 β`. |
protected hasBasis_nhds (f : α →ᵤ[𝔖] β) (h : 𝔖.Nonempty) (h' : DirectedOn (· ⊆ ·) 𝔖) :
(𝓝 f).HasBasis (fun SV : Set α × Set (β × β) => SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β) fun SV =>
{ g | (g, f) ∈ UniformOnFun.gen 𝔖 SV.1 SV.2 } :=
UniformOnFun.hasBasis_nhds_of_basis α β 𝔖 f h h' (Filter.basis_sets _) | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | hasBasis_nhds | For `f : α →ᵤ[𝔖] β`, where `𝔖 : Set (Set α)` is nonempty and directed, `𝓝 f` admits the
family `{g | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓤 β` as a filter basis. |
protected uniformContinuous_restrict (h : s ∈ 𝔖) :
UniformContinuous (UniformFun.ofFun ∘ (s.restrict : (α → β) → s → β) ∘ toFun 𝔖) := by
change _ ≤ _
simp only [UniformOnFun.uniformSpace, map_le_iff_le_comap, iInf_uniformity]
exact iInf₂_le s h
variable {α} | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_restrict | If `S ∈ 𝔖`, then the restriction to `S` is a uniformly continuous map from `α →ᵤ[𝔖] β` to
`↥S →ᵤ β`. |
protected uniformity_eq_of_basis {ι : Sort*} {p : ι → Prop} {V : ι → Set (β × β)}
(h : (𝓤 β).HasBasis p V) :
𝓤 (α →ᵤ[𝔖] β) = ⨅ s ∈ 𝔖, ⨅ (i) (_ : p i), 𝓟 (UniformOnFun.gen 𝔖 s (V i)) := by
simp_rw [iInf_uniformity, uniformity_comap,
(UniformFun.hasBasis_uniformity_of_basis _ _ h).eq_biInf, comap_iInf... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformity_eq_of_basis | A version of `UniformOnFun.hasBasis_uniformity_of_basis`
with weaker conclusion and weaker assumptions.
We make no assumptions about the set `𝔖`
but conclude only that the uniformity is equal to some indexed infimum. |
protected uniformity_eq : 𝓤 (α →ᵤ[𝔖] β) = ⨅ s ∈ 𝔖, ⨅ V ∈ 𝓤 β, 𝓟 (UniformOnFun.gen 𝔖 s V) :=
UniformOnFun.uniformity_eq_of_basis _ _ (𝓤 β).basis_sets | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformity_eq | null |
protected gen_mem_uniformity (hs : s ∈ 𝔖) {V : Set (β × β)} (hV : V ∈ 𝓤 β) :
UniformOnFun.gen 𝔖 s V ∈ 𝓤 (α →ᵤ[𝔖] β) := by
rw [UniformOnFun.uniformity_eq]
apply_rules [mem_iInf_of_mem, mem_principal_self] | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | gen_mem_uniformity | null |
protected nhds_eq_of_basis {ι : Sort*} {p : ι → Prop} {V : ι → Set (β × β)}
(h : (𝓤 β).HasBasis p V) (f : α →ᵤ[𝔖] β) :
𝓝 f = ⨅ s ∈ 𝔖, ⨅ (i) (_ : p i), 𝓟 {g | ∀ x ∈ s, (toFun 𝔖 f x, toFun 𝔖 g x) ∈ V i} := by
simp_rw [nhds_eq_comap_uniformity, UniformOnFun.uniformity_eq_of_basis _ _ h, comap_iInf,
co... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | nhds_eq_of_basis | A version of `UniformOnFun.hasBasis_nhds_of_basis`
with weaker conclusion and weaker assumptions.
We make no assumptions about the set `𝔖`
but conclude only that the neighbourhoods filter is equal to some indexed infimum. |
protected nhds_eq (f : α →ᵤ[𝔖] β) :
𝓝 f = ⨅ s ∈ 𝔖, ⨅ V ∈ 𝓤 β, 𝓟 {g | ∀ x ∈ s, (toFun 𝔖 f x, toFun 𝔖 g x) ∈ V} :=
UniformOnFun.nhds_eq_of_basis _ _ (𝓤 β).basis_sets f | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | nhds_eq | null |
protected gen_mem_nhds (f : α →ᵤ[𝔖] β) (hs : s ∈ 𝔖) {V : Set (β × β)} (hV : V ∈ 𝓤 β) :
{g | ∀ x ∈ s, (toFun 𝔖 f x, toFun 𝔖 g x) ∈ V} ∈ 𝓝 f := by
rw [UniformOnFun.nhds_eq]
apply_rules [mem_iInf_of_mem, mem_principal_self] | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | gen_mem_nhds | null |
uniformContinuous_ofUniformFun :
UniformContinuous fun f : α →ᵤ β ↦ ofFun 𝔖 (UniformFun.toFun f) := by
simp only [UniformContinuous, UniformOnFun.uniformity_eq, tendsto_iInf, tendsto_principal,
(UniformFun.hasBasis_uniformity _ _).eventually_iff]
exact fun _ _ U hU ↦ ⟨U, hU, fun f hf x _ ↦ hf x⟩ | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_ofUniformFun | null |
uniformEquivUniformFun (h : univ ∈ 𝔖) : (α →ᵤ[𝔖] β) ≃ᵤ (α →ᵤ β) where
toFun f := UniformFun.ofFun <| toFun _ f
invFun f := ofFun _ <| UniformFun.toFun f
uniformContinuous_toFun := by
simp only [UniformContinuous, (UniformFun.hasBasis_uniformity _ _).tendsto_right_iff]
intro U hU
filter_upwards [Unif... | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformEquivUniformFun | The uniformity on `α →ᵤ[𝔖] β` is the same as the uniformity on `α →ᵤ β`,
provided that `Set.univ ∈ 𝔖`.
Here we formulate it as a `UniformEquiv`. |
uniformContinuous_ofFun_toFun (𝔗 : Set (Set α)) (h : ∀ s ∈ 𝔖, ∃ T ⊆ 𝔗, T.Finite ∧ s ⊆ ⋃₀ T) :
UniformContinuous (ofFun 𝔗 ∘ toFun 𝔖 : (α →ᵤ[𝔗] β) → α →ᵤ[𝔖] β) := by
simp only [UniformContinuous, UniformOnFun.uniformity_eq, iInf₂_comm (ι₂ := Set (β × β))]
refine tendsto_iInf_iInf fun V ↦ tendsto_iInf_iInf ... | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_ofFun_toFun | If `𝔖` and `𝔗` are families of sets in `α`, then the identity map
`(α →ᵤ[𝔗] β) → (α →ᵤ[𝔖] β)` is uniformly continuous if every `s ∈ 𝔖` is contained in a finite
union of elements of `𝔗`.
With more API around `Order.Ideal`, this could be phrased in that language instead. |
protected mono ⦃u₁ u₂ : UniformSpace γ⦄ (hu : u₁ ≤ u₂) ⦃𝔖₁ 𝔖₂ : Set (Set α)⦄
(h𝔖 : 𝔖₂ ⊆ 𝔖₁) : 𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₂) :=
calc
𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₁) := iInf_le_iInf_of_subset h𝔖
_ ≤ 𝒱(α, γ, 𝔖₂, u₂) := iInf₂_mono fun _i _hi => UniformSpace.comap_mono <| UniformFun.mono ... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | mono | Let `u₁`, `u₂` be two uniform structures on `γ` and `𝔖₁ 𝔖₂ : Set (Set α)`. If `u₁ ≤ u₂` and
`𝔖₂ ⊆ 𝔖₁` then `𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₂)`. |
uniformContinuous_eval_of_mem {x : α} (hxs : x ∈ s) (hs : s ∈ 𝔖) :
UniformContinuous ((Function.eval x : (α → β) → β) ∘ toFun 𝔖) :=
(UniformFun.uniformContinuous_eval β (⟨x, hxs⟩ : s)).comp
(UniformOnFun.uniformContinuous_restrict α β 𝔖 hs) | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_eval_of_mem | If `x : α` is in some `S ∈ 𝔖`, then evaluation at `x` is uniformly continuous on
`α →ᵤ[𝔖] β`. |
uniformContinuous_eval_of_mem_sUnion {x : α} (hx : x ∈ ⋃₀ 𝔖) :
UniformContinuous ((Function.eval x : (α → β) → β) ∘ toFun 𝔖) :=
let ⟨_s, hs, hxs⟩ := hx
uniformContinuous_eval_of_mem _ _ hxs hs
variable {β} {𝔖} | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_eval_of_mem_sUnion | null |
uniformContinuous_eval (h : ⋃₀ 𝔖 = univ) (x : α) :
UniformContinuous ((Function.eval x : (α → β) → β) ∘ toFun 𝔖) :=
uniformContinuous_eval_of_mem_sUnion _ _ <| h.symm ▸ mem_univ _ | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_eval | null |
protected iInf_eq {u : ι → UniformSpace γ} :
𝒱(α, γ, 𝔖, ⨅ i, u i) = ⨅ i, 𝒱(α, γ, 𝔖, u i) := by
simp_rw [UniformOnFun.uniformSpace, UniformFun.iInf_eq, UniformSpace.comap_iInf]
rw [iInf_comm]
exact iInf_congr fun s => iInf_comm | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | iInf_eq | If `u` is a family of uniform structures on `γ`, then
`𝒱(α, γ, 𝔖, (⨅ i, u i)) = ⨅ i, 𝒱(α, γ, 𝔖, u i)`. |
protected inf_eq {u₁ u₂ : UniformSpace γ} :
𝒱(α, γ, 𝔖, u₁ ⊓ u₂) = 𝒱(α, γ, 𝔖, u₁) ⊓ 𝒱(α, γ, 𝔖, u₂) := by
rw [inf_eq_iInf, inf_eq_iInf, UniformOnFun.iInf_eq]
refine iInf_congr fun i => ?_
cases i <;> rfl | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | inf_eq | If `u₁` and `u₂` are two uniform structures on `γ`, then
`𝒱(α, γ, 𝔖, u₁ ⊓ u₂) = 𝒱(α, γ, 𝔖, u₁) ⊓ 𝒱(α, γ, 𝔖, u₂)`. |
protected comap_eq {f : γ → β} :
𝒱(α, γ, 𝔖, ‹UniformSpace β›.comap f) = 𝒱(α, β, 𝔖, _).comap (f ∘ ·) := by
simp_rw [UniformOnFun.uniformSpace, UniformSpace.comap_iInf, UniformFun.comap_eq, ←
UniformSpace.comap_comap]
rfl | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | comap_eq | If `u` is a uniform structure on `β` and `f : γ → β`, then
`𝒱(α, γ, 𝔖, comap f u) = comap (fun g ↦ f ∘ g) 𝒱(α, γ, 𝔖, u₁)`. |
protected postcomp_uniformContinuous [UniformSpace γ] {f : γ → β}
(hf : UniformContinuous f) : UniformContinuous (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) := by
rw [uniformContinuous_iff]
exact (UniformOnFun.mono (uniformContinuous_iff.mp hf) subset_rfl).trans_eq UniformOnFun.comap_eq | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | postcomp_uniformContinuous | Post-composition by a uniformly continuous function is uniformly continuous for the
uniform structures of `𝔖`-convergence.
More precisely, if `f : γ → β` is uniformly continuous, then
`(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is uniformly continuous. |
postcomp_isUniformInducing [UniformSpace γ] {f : γ → β}
(hf : IsUniformInducing f) : IsUniformInducing (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) := by
constructor
replace hf : (𝓤 β).comap (Prod.map f f) = _ := hf.comap_uniformity
change comap (Prod.map (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖)) _... | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | postcomp_isUniformInducing | Post-composition by a uniform inducing is a uniform inducing for the
uniform structures of `𝔖`-convergence.
More precisely, if `f : γ → β` is a uniform inducing, then
`(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform inducing. |
protected postcomp_isUniformEmbedding [UniformSpace γ] {f : γ → β}
(hf : IsUniformEmbedding f) : IsUniformEmbedding (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) where
toIsUniformInducing := UniformOnFun.postcomp_isUniformInducing hf.isUniformInducing
injective _ _ H := funext fun _ ↦ hf.injective (congrFun H _) | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | postcomp_isUniformEmbedding | Post-composition by a uniform embedding is a uniform embedding for the
uniform structures of `𝔖`-convergence.
More precisely, if `f : γ → β` is a uniform embedding, then
`(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform embedding. |
protected congrRight [UniformSpace γ] (e : γ ≃ᵤ β) : (α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β) :=
{ Equiv.piCongrRight fun _a => e.toEquiv with
uniformContinuous_toFun := UniformOnFun.postcomp_uniformContinuous e.uniformContinuous
uniformContinuous_invFun := UniformOnFun.postcomp_uniformContinuous e.symm.uniformContinuous... | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | congrRight | Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β)`
by post-composing. |
protected precomp_uniformContinuous {𝔗 : Set (Set γ)} {f : γ → α}
(hf : MapsTo (f '' ·) 𝔗 𝔖) :
UniformContinuous fun g : α →ᵤ[𝔖] β => ofFun 𝔗 (toFun 𝔖 g ∘ f) := by
simp_rw [UniformContinuous, UniformOnFun.uniformity_eq, tendsto_iInf]
refine fun t ht V hV ↦ tendsto_iInf' (f '' t) <| tendsto_iInf' (hf h... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | precomp_uniformContinuous | Let `f : γ → α`, `𝔖 : Set (Set α)`, `𝔗 : Set (Set γ)`, and assume that `∀ T ∈ 𝔗, f '' T ∈ 𝔖`.
Then, the function `(fun g ↦ g ∘ f) : (α →ᵤ[𝔖] β) → (γ →ᵤ[𝔗] β)` is uniformly continuous.
Note that one can easily see that assuming `∀ T ∈ 𝔗, ∃ S ∈ 𝔖, f '' T ⊆ S` would work too, but
we will get this for free when we... |
protected congrLeft {𝔗 : Set (Set γ)} (e : γ ≃ α) (he : 𝔗 ⊆ image e ⁻¹' 𝔖)
(he' : 𝔖 ⊆ preimage e ⁻¹' 𝔗) : (γ →ᵤ[𝔗] β) ≃ᵤ (α →ᵤ[𝔖] β) :=
{ Equiv.arrowCongr e (Equiv.refl _) with
uniformContinuous_toFun := UniformOnFun.precomp_uniformContinuous fun s hs ↦ by
change e.symm '' s ∈ 𝔗
rw [← prei... | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | congrLeft | Turn a bijection `e : γ ≃ α` such that we have both `∀ T ∈ 𝔗, e '' T ∈ 𝔖` and
`∀ S ∈ 𝔖, e ⁻¹' S ∈ 𝔗` into a uniform isomorphism `(γ →ᵤ[𝔗] β) ≃ᵤ (α →ᵤ[𝔖] β)` by pre-composing. |
t2Space_of_covering [T2Space β] (h : ⋃₀ 𝔖 = univ) : T2Space (α →ᵤ[𝔖] β) where
t2 f g hfg := by
obtain ⟨x, hx⟩ := not_forall.mp (mt funext hfg)
obtain ⟨s, hs, hxs⟩ : ∃ s ∈ 𝔖, x ∈ s := mem_sUnion.mp (h.symm ▸ True.intro)
exact separated_by_continuous (uniformContinuous_eval_of_mem β 𝔖 hxs hs).continuous... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | t2Space_of_covering | If `𝔖` covers `α`, then the topology of `𝔖`-convergence is T₂. |
uniformContinuous_restrict_toFun :
UniformContinuous ((⋃₀ 𝔖).restrict ∘ toFun 𝔖 : (α →ᵤ[𝔖] β) → ⋃₀ 𝔖 → β) := by
rw [uniformContinuous_pi]
intro ⟨x, hx⟩
obtain ⟨s : Set α, hs : s ∈ 𝔖, hxs : x ∈ s⟩ := mem_sUnion.mpr hx
exact uniformContinuous_eval_of_mem β 𝔖 hxs hs | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_restrict_toFun | The restriction map from `α →ᵤ[𝔖] β` to `⋃₀ 𝔖 → β` is uniformly continuous. |
isUniformInducing_pi_restrict :
IsUniformInducing
(fun f : α →ᵤ[𝔖] β ↦ fun s : 𝔖 ↦ UniformFun.ofFun ((s : Set α).restrict (toFun 𝔖 f))) := by
simp_rw [isUniformInducing_iff_uniformSpace, Pi.uniformSpace_eq, UniformSpace.comap_iInf,
← UniformSpace.comap_comap, iInf_subtype]
rfl | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | isUniformInducing_pi_restrict | The map sending a function `f : α →ᵤ[𝔖] β` to the family of restrictions of `f` to each `s ∈ 𝔖`
(each coordinate equipped with its respective uniform structure `s →ᵤ β`) induces the uniformity on
`α →ᵤ[𝔖] β`. |
protected uniformContinuous_toFun (h : ⋃₀ 𝔖 = univ) :
UniformContinuous (toFun 𝔖 : (α →ᵤ[𝔖] β) → α → β) := by
rw [uniformContinuous_pi]
exact uniformContinuous_eval h | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformContinuous_toFun | If `𝔖` covers `α`, the natural map `UniformOnFun.toFun` from `α →ᵤ[𝔖] β` to `α → β` is
uniformly continuous.
In other words, if `𝔖` covers `α`, then the uniform structure of `𝔖`-convergence is finer than
that of pointwise convergence. |
protected continuousAt_eval₂ [TopologicalSpace α] {f : α →ᵤ[𝔖] β} {x : α}
(h𝔖 : ∃ V ∈ 𝔖, V ∈ 𝓝 x) (hc : ContinuousAt (toFun 𝔖 f) x) :
ContinuousAt (fun fx : (α →ᵤ[𝔖] β) × α ↦ toFun 𝔖 fx.1 fx.2) (f, x) := by
rw [ContinuousAt, nhds_eq_comap_uniformity, tendsto_comap_iff, ← lift'_comp_uniformity,
tend... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | continuousAt_eval₂ | If `f : α →ᵤ[𝔖] β` is continuous at `x` and `x` admits a neighbourhood `V ∈ 𝔖`,
then evaluation of `g : α →ᵤ[𝔖] β` at `y : α` is continuous in `(g, y)` at `(f, x)`. |
protected continuousOn_eval₂ [TopologicalSpace α] (h𝔖 : ∀ x, ∃ V ∈ 𝔖, V ∈ 𝓝 x) :
ContinuousOn (fun fx : (α →ᵤ[𝔖] β) × α ↦ toFun 𝔖 fx.1 fx.2)
{fx | ContinuousAt (toFun 𝔖 fx.1) fx.2} := fun (_f, x) hc ↦
(UniformOnFun.continuousAt_eval₂ (h𝔖 x) hc).continuousWithinAt | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | continuousOn_eval₂ | If each point of `α` admits a neighbourhood `V ∈ 𝔖`,
then the evaluation of `f : α →ᵤ[𝔖] β` at `x : α` is continuous in `(f, x)`
on the set of `(f, x)` such that `f` is continuous at `x`. |
protected tendsto_iff_tendstoUniformlyOn {F : ι → α →ᵤ[𝔖] β} {f : α →ᵤ[𝔖] β} :
Tendsto F p (𝓝 f) ↔ ∀ s ∈ 𝔖, TendstoUniformlyOn (toFun 𝔖 ∘ F) (toFun 𝔖 f) p s := by
simp only [UniformOnFun.nhds_eq, tendsto_iInf, tendsto_principal, TendstoUniformlyOn,
Function.comp_apply, mem_setOf] | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | tendsto_iff_tendstoUniformlyOn | Convergence in the topology of `𝔖`-convergence means uniform convergence on `S` (in the sense
of `TendstoUniformlyOn`) for all `S ∈ 𝔖`. |
protected continuous_rng_iff {X : Type*} [TopologicalSpace X] {f : X → (α →ᵤ[𝔖] β)} :
Continuous f ↔ ∀ s ∈ 𝔖,
Continuous (UniformFun.ofFun ∘ s.restrict ∘ UniformOnFun.toFun 𝔖 ∘ f) := by
simp only [continuous_iff_continuousAt, ContinuousAt,
UniformOnFun.tendsto_iff_tendstoUniformlyOn, UniformFun.tends... | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | continuous_rng_iff | null |
protected uniformEquivProdArrow [UniformSpace γ] :
(α →ᵤ[𝔖] β × γ) ≃ᵤ (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] γ) :=
((UniformOnFun.ofFun 𝔖).symm.trans <| (Equiv.arrowProdEquivProdArrow _ _ _).trans <|
(UniformOnFun.ofFun 𝔖).prodCongr (UniformOnFun.ofFun 𝔖)).toUniformEquivOfIsUniformInducing <| by
constructor
... | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformEquivProdArrow | The natural bijection between `α → β × γ` and `(α → β) × (α → γ)`, upgraded to a uniform
isomorphism between `α →ᵤ[𝔖] β × γ` and `(α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] γ)`. |
protected uniformEquivPiComm : (α →ᵤ[𝔖] ((i : ι) → δ i)) ≃ᵤ ((i : ι) → α →ᵤ[𝔖] δ i) :=
@Equiv.toUniformEquivOfIsUniformInducing (α →ᵤ[𝔖] ((i : ι) → δ i)) ((i : ι) → α →ᵤ[𝔖] δ i)
_ _ (Equiv.piComm _) <| by
constructor
change comap (Prod.map Function.swap Function.swap) _ = _
erw [← uniformity_com... | def | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformEquivPiComm | The natural bijection between `α → Π i, δ i` and `Π i, α → δ i`, upgraded to a uniform
isomorphism between `α →ᵤ[𝔖] (Π i, δ i)` and `Π i, α →ᵤ[𝔖] δ i`. |
isClosed_setOf_continuous [TopologicalSpace α] (h : IsCoherentWith 𝔖) :
IsClosed {f : α →ᵤ[𝔖] β | Continuous (toFun 𝔖 f)} := by
refine isClosed_iff_forall_filter.2 fun f u _ hu huf ↦ h.continuous_iff.2 fun s hs ↦ ?_
rw [← tendsto_id', UniformOnFun.tendsto_iff_tendstoUniformlyOn] at huf
exact (huf s hs).con... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | isClosed_setOf_continuous | Suppose that the topology on `α` is defined by its restrictions to the sets of `𝔖`.
Then the set of continuous functions is closed
in the topology of uniform convergence on the sets of `𝔖`. |
uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} (φ₁ : δ₁ → α) (φ₂ : δ₂ → α)
(𝔗₁ : Set (Set δ₁)) (𝔗₂ : Set (Set δ₂))
(h_image₁ : MapsTo (φ₁ '' ·) 𝔗₁ 𝔖) (h_image₂ : MapsTo (φ₂ '' ·) 𝔗₂ 𝔖)
(h_preimage₁ : MapsTo (φ₁ ⁻¹' ·) 𝔖 𝔗₁) (h_preimage₂ : MapsTo (φ₂ ⁻¹' ·) 𝔖 𝔗₂)
(h_cover : ∀ S ∈ 𝔖, S ⊆ ... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformSpace_eq_inf_precomp_of_cover | null |
uniformSpace_eq_iInf_precomp_of_cover {δ : ι → Type*} (φ : Π i, δ i → α)
(𝔗 : ∀ i, Set (Set (δ i))) (h_image : ∀ i, MapsTo (φ i '' ·) (𝔗 i) 𝔖)
(h_preimage : ∀ i, MapsTo (φ i ⁻¹' ·) 𝔖 (𝔗 i))
(h_cover : ∀ S ∈ 𝔖, ∃ I : Set ι, I.Finite ∧ S ⊆ ⋃ i ∈ I, range (φ i)) :
𝒱(α, β, 𝔖, _) = ⨅ i, .comap (ofFun... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | uniformSpace_eq_iInf_precomp_of_cover | null |
UniformContinuousOn.comp_tendstoUniformly
(hF : ∀ i x, F i x ∈ s) (hf : ∀ x, f x ∈ s) (hg : UniformContinuousOn g s)
(h : TendstoUniformly F f p) :
TendstoUniformly (fun i x => g (F i x)) (fun x => g (f x)) p := by
rw [uniformContinuousOn_iff_restrict] at hg
lift F to ι → α → s using hF with F' hF'
li... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | UniformContinuousOn.comp_tendstoUniformly | Composing on the left by a uniformly continuous function preserves uniform convergence |
UniformContinuousOn.comp_tendstoUniformly_eventually
(hF : ∀ᶠ i in p, ∀ x, F i x ∈ s) (hf : ∀ x, f x ∈ s) (hg : UniformContinuousOn g s)
(h : TendstoUniformly F f p) :
TendstoUniformly (fun i x ↦ g (F i x)) (fun x ↦ g (f x)) p := by
classical
obtain ⟨s', hs', hs⟩ := eventually_iff_exists_mem.mp hF
let... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | UniformContinuousOn.comp_tendstoUniformly_eventually | null |
UniformContinuousOn.comp_tendstoUniformlyOn_eventually {t : Set α}
(hF : ∀ᶠ i in p, ∀ x ∈ t, F i x ∈ s) (hf : ∀ x ∈ t, f x ∈ s)
{g : β → γ} (hg : UniformContinuousOn g s) (h : TendstoUniformlyOn F f p t) :
TendstoUniformlyOn (fun i x ↦ g (F i x)) (fun x => g (f x)) p t := by
rw [tendstoUniformlyOn_iff_res... | theorem | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.UniformApproximation"
] | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | UniformContinuousOn.comp_tendstoUniformlyOn_eventually | null |
@[mk_iff]
IsUniformInducing (f : α → β) : Prop where
/-- The uniformity filter on the domain is the pullback of the uniformity filter on the codomain
under `Prod.map f f`. -/
comap_uniformity : comap (fun x : α × α => (f x.1, f x.2)) (𝓤 β) = 𝓤 α | structure | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing | A map `f : α → β` between uniform spaces is called *uniform inducing* if the uniformity filter
on `α` is the pullback of the uniformity filter on `β` under `Prod.map f f`. If `α` is a separated
space, then this implies that `f` is injective, hence it is a `IsUniformEmbedding`. |
isUniformInducing_iff_uniformSpace {f : α → β} :
IsUniformInducing f ↔ ‹UniformSpace β›.comap f = ‹UniformSpace α› := by
rw [isUniformInducing_iff, UniformSpace.ext_iff, Filter.ext_iff]
rfl
protected alias ⟨IsUniformInducing.comap_uniformSpace, _⟩ := isUniformInducing_iff_uniformSpace | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformInducing_iff_uniformSpace | null |
isUniformInducing_iff' {f : α → β} :
IsUniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α := by
rw [isUniformInducing_iff, UniformContinuous, tendsto_iff_comap, le_antisymm_iff, and_comm]; rfl | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformInducing_iff' | null |
protected Filter.HasBasis.isUniformInducing_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
IsUniformInducing f ↔
(∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s'... | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | Filter.HasBasis.isUniformInducing_iff | null |
IsUniformInducing.mk' {f : α → β}
(h : ∀ s, s ∈ 𝓤 α ↔ ∃ t ∈ 𝓤 β, ∀ x y : α, (f x, f y) ∈ t → (x, y) ∈ s) : IsUniformInducing f :=
⟨by simp [eq_comm, Filter.ext_iff, subset_def, h]⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.mk' | null |
IsUniformInducing.id : IsUniformInducing (@id α) :=
⟨by rw [← Prod.map_def, Prod.map_id, comap_id]⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.id | null |
IsUniformInducing.comp {g : β → γ} (hg : IsUniformInducing g) {f : α → β}
(hf : IsUniformInducing f) : IsUniformInducing (g ∘ f) :=
⟨by rw [← hf.1, ← hg.1, comap_comap]; rfl⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.comp | null |
IsUniformInducing.of_comp_iff {g : β → γ} (hg : IsUniformInducing g) {f : α → β} :
IsUniformInducing (g ∘ f) ↔ IsUniformInducing f := by
refine ⟨fun h ↦ ?_, hg.comp⟩
rw [isUniformInducing_iff, ← hg.comap_uniformity, comap_comap, ← h.comap_uniformity,
Function.comp_def, Function.comp_def] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.of_comp_iff | null |
IsUniformInducing.basis_uniformity {f : α → β} (hf : IsUniformInducing f) {ι : Sort*}
{p : ι → Prop} {s : ι → Set (β × β)} (H : (𝓤 β).HasBasis p s) :
(𝓤 α).HasBasis p fun i => Prod.map f f ⁻¹' s i :=
hf.1 ▸ H.comap _ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.basis_uniformity | null |
IsUniformInducing.cauchy_map_iff {f : α → β} (hf : IsUniformInducing f) {F : Filter α} :
Cauchy (map f F) ↔ Cauchy F := by
simp only [Cauchy, map_neBot_iff, prod_map_map_eq, map_le_iff_le_comap, ← hf.comap_uniformity] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.cauchy_map_iff | null |
IsUniformInducing.of_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f)
(hg : UniformContinuous g) (hgf : IsUniformInducing (g ∘ f)) : IsUniformInducing f := by
refine ⟨le_antisymm ?_ hf.le_comap⟩
rw [← hgf.1, ← Prod.map_def, ← Prod.map_def, ← Prod.map_comp_map f f g g, ← comap_comap]
exact comap_mono hg... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.of_comp | null |
IsUniformInducing.uniformContinuous {f : α → β} (hf : IsUniformInducing f) :
UniformContinuous f := (isUniformInducing_iff'.1 hf).1 | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.uniformContinuous | null |
IsUniformInducing.uniformContinuous_iff {f : α → β} {g : β → γ} (hg : IsUniformInducing g) :
UniformContinuous f ↔ UniformContinuous (g ∘ f) := by
dsimp only [UniformContinuous, Tendsto]
simp only [← hg.comap_uniformity, ← map_le_iff_le_comap, Filter.map_map, Function.comp_def] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.uniformContinuous_iff | null |
protected IsUniformInducing.isUniformInducing_comp_iff {f : α → β} {g : β → γ}
(hg : IsUniformInducing g) : IsUniformInducing (g ∘ f) ↔ IsUniformInducing f := by
simp only [isUniformInducing_iff, ← hg.comap_uniformity, comap_comap, Function.comp_def] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.isUniformInducing_comp_iff | null |
IsUniformInducing.uniformContinuousOn_iff {f : α → β} {g : β → γ} {S : Set α}
(hg : IsUniformInducing g) :
UniformContinuousOn f S ↔ UniformContinuousOn (g ∘ f) S := by
dsimp only [UniformContinuousOn, Tendsto]
rw [← hg.comap_uniformity, ← map_le_iff_le_comap, Filter.map_map, comp_def, comp_def] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.uniformContinuousOn_iff | null |
IsUniformInducing.isInducing {f : α → β} (h : IsUniformInducing f) : IsInducing f := by
obtain rfl := h.comap_uniformSpace
exact .induced f | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.isInducing | null |
IsUniformInducing.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : IsUniformInducing e₁) (h₂ : IsUniformInducing e₂) :
IsUniformInducing fun p : α × β => (e₁ p.1, e₂ p.2) :=
⟨by simp [Function.comp_def, uniformity_prod, ← h₁.1, ← h₂.1, comap_inf, comap_comap... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.prod | null |
IsUniformInducing.isDenseInducing (h : IsUniformInducing f) (hd : DenseRange f) :
IsDenseInducing f where
toIsInducing := h.isInducing
dense := hd | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.isDenseInducing | null |
SeparationQuotient.isUniformInducing_mk :
IsUniformInducing (mk : α → SeparationQuotient α) :=
⟨comap_mk_uniformity⟩ | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | SeparationQuotient.isUniformInducing_mk | null |
protected IsUniformInducing.injective [T0Space α] {f : α → β} (h : IsUniformInducing f) :
Injective f :=
h.isInducing.injective
/-! | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformInducing.injective | null |
@[mk_iff]
IsUniformEmbedding (f : α → β) : Prop extends IsUniformInducing f where
/-- A uniform embedding is injective. -/
injective : Function.Injective f | structure | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformEmbedding | A map `f : α → β` between uniform spaces is a *uniform embedding* if it is uniform inducing and
injective. If `α` is a separated space, then the latter assumption follows from the former. |
IsUniformEmbedding.isUniformInducing (hf : IsUniformEmbedding f) : IsUniformInducing f :=
hf.toIsUniformInducing | lemma | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | IsUniformEmbedding.isUniformInducing | null |
isUniformEmbedding_iff' {f : α → β} :
IsUniformEmbedding f ↔
Injective f ∧ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α := by
rw [isUniformEmbedding_iff, and_comm, isUniformInducing_iff'] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.DenseEmbedding"
] | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | isUniformEmbedding_iff' | null |
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