statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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postcomp_uniformContinuous [UniformSpace γ] {f : γ → β}
(hf : UniformContinuous f) : UniformContinuous (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) | by
-- This is a direct consequence of `UniformOnFun.comap_eq`
rw [uniformContinuous_iff_le_comap]
exact (UniformOnFun.mono (uniformContinuous_iff_le_comap.mp hf)
subset_rfl).trans_eq UniformOnFun.comap_eq | theorem | UniformOnFun.postcomp_uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"UniformContinuous",
"UniformOnFun.comap_eq",
"UniformOnFun.mono",
"UniformSpace",
"subset_rfl",
"uniformContinuous_iff_le_comap"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
postcomp_isUniformInducing [UniformSpace γ] {f : γ → β}
(hf : IsUniformInducing f) : IsUniformInducing (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) | by
-- This is a direct consequence of `UniformOnFun.comap_eq`
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)) _ = _
rw [← uniformity_comap] at hf ⊢
congr
rw [← UniformSpace.ext hf, Unifo... | lemma | UniformOnFun.postcomp_isUniformInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"IsUniformInducing",
"UniformOnFun.comap_eq",
"UniformSpace",
"UniformSpace.ext",
"comap",
"uniformity_comap"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformOnFun.postcomp_isUniformEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"IsUniformEmbedding",
"UniformOnFun.postcomp_isUniformInducing",
"UniformSpace"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformOnFun.congrRight | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Equiv.piCongrRight",
"UniformOnFun.postcomp_uniformContinuous",
"UniformSpace"
] | Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β)`
by post-composing. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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
-- This follows from the fact that `(· ∘ f) × (· ∘ f)` maps `gen (f '' t) V` to `gen t V`.
simp_rw [UniformContinuous, UniformOnFun.uniformity_eq, tendsto_iInf]
refine fun t ht V hV ↦ tendsto_iInf' (f '' t) <| tendsto_iInf' (hf ht) <|
tendsto_iInf' V <| tendsto_iInf' hV ?_
simpa only [tendsto_principal_p... | theorem | UniformOnFun.precomp_uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Set",
"UniformContinuous",
"UniformOnFun.gen",
"UniformOnFun.uniformity_eq"
] | 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... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 [Equiv.image_symm_eq_preimage]
exact he' hs
uniformContinuous_invFun := UniformOnFun.precomp_uniformContinuous he } | def | UniformOnFun.congrLeft | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Equiv.arrowCongr",
"Equiv.image_symm_eq_preimage",
"Equiv.refl",
"Set",
"UniformOnFun.precomp_uniformContinuous"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 hx | theorem | UniformOnFun.t2Space_of_covering | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"T2Space",
"separated_by_continuous"
] | If `𝔖` covers `α`, then the topology of `𝔖`-convergence is T₂. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformOnFun.uniformContinuous_restrict_toFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Set",
"UniformContinuous",
"uniformContinuous_pi"
] | The restriction map from `α →ᵤ[𝔖] β` to `⋃₀ 𝔖 → β` is uniformly continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformOnFun.isUniformInducing_pi_restrict | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"IsUniformInducing",
"Pi.uniformSpace_eq",
"Set",
"UniformFun.ofFun",
"UniformSpace.comap_comap",
"UniformSpace.comap_iInf",
"isUniformInducing_iff_uniformSpace"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_toFun (h : ⋃₀ 𝔖 = univ) :
UniformContinuous (toFun 𝔖 : (α →ᵤ[𝔖] β) → α → β) | by
rw [uniformContinuous_pi]
exact uniformContinuous_eval h | theorem | UniformOnFun.uniformContinuous_toFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"UniformContinuous",
"uniformContinuous_pi"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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,
tendsto_lift']
intro U hU
rcases h𝔖 with ⟨V, hV, hVx⟩
filter_upwards [prod_mem_nhds (UniformOnFun.gen_mem_nhds _ _ _ hV hU)
(inter_mem hVx <| hc <| UniformSpace.ball_mem_nhds _ hU)]
with ⟨g, y⟩ ⟨hg, hyV, hy⟩ ... | theorem | UniformOnFun.continuousAt_eval₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"ContinuousAt",
"TopologicalSpace",
"UniformOnFun.gen_mem_nhds",
"UniformSpace.ball_mem_nhds",
"lift'_comp_uniformity",
"nhds_eq_comap_uniformity",
"prod_mem_nhds"
] | 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)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformOnFun.continuousOn_eval₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"ContinuousAt",
"ContinuousOn",
"TopologicalSpace",
"UniformOnFun.continuousAt_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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformOnFun.tendsto_iff_tendstoUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"TendstoUniformlyOn",
"UniformOnFun.nhds_eq"
] | Convergence in the topology of `𝔖`-convergence means uniform convergence on `S` (in the sense
of `TendstoUniformlyOn`) for all `S ∈ 𝔖`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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.tendsto_iff_tendstoUniformly,
tendstoUniformlyOn_iff_tendstoUniformly_comp_coe, @forall_comm X,
Function.comp_def, restrict_eq, UniformFun.toFun_ofFun] | lemma | UniformOnFun.continuous_rng_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Continuous",
"ContinuousAt",
"TopologicalSpace",
"UniformFun.ofFun",
"UniformFun.tendsto_iff_tendstoUniformly",
"UniformFun.toFun_ofFun",
"UniformOnFun.tendsto_iff_tendstoUniformlyOn",
"UniformOnFun.toFun",
"continuous_iff_continuousAt",
"tendstoUniformlyOn_iff_tendstoUniformly_comp_coe"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquivProdArrow [UniformSpace γ] :
(α →ᵤ[𝔖] β × γ) ≃ᵤ (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] γ) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (𝒱(α, β, 𝔖, uβ) × 𝒱(α, γ, 𝔖, uγ)) = 𝒱(α, β × γ, 𝔖, uβ × uγ)`.
-- But `uβ × uγ` is defined as `comap fst uβ ⊓ comap snd uγ`, so we just have to apply
-- `UniformOnFun.inf_eq` and `UniformOnFun.comap_eq`,
-- which leaves us to check that some sq... | def | UniformOnFun.uniformEquivProdArrow | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Equiv.arrowProdEquivProdArrow",
"UniformOnFun.comap_eq",
"UniformOnFun.inf_eq",
"UniformOnFun.ofFun",
"UniformSpace",
"UniformSpace.comap",
"inf_uniformity",
"trans",
"uniformity",
"uniformity_comap",
"uniformity_prod"
] | 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)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformEquivPiComm : (α →ᵤ[𝔖] ((i : ι) → δ i)) ≃ᵤ ((i : ι) → α →ᵤ[𝔖] δ i) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (Π i, 𝒱(α, δ i, 𝔖, uδ i)) = 𝒱(α, (Π i, δ i), 𝔖, (Π i, uδ i))`.
-- But `Π i, uδ i` is defined as `⨅ i, comap (eval i) (uδ i)`, so we just have to apply
-- `UniformOnFun.iInf_eq` and `UniformOnFun.comap_eq`,
-- which leaves us to check that some s... | def | UniformOnFun.uniformEquivPiComm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Equiv.piComm",
"Equiv.toUniformEquivOfIsUniformInducing",
"Function.swap",
"Pi.uniformSpace",
"UniformOnFun.comap_eq",
"UniformOnFun.iInf_eq",
"UniformSpace.comap_comap",
"UniformSpace.comap_iInf",
"UniformSpace.ofCoreEq_toCore",
"comap",
"uniformity_comap"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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).continuousOn <| Eventually.frequently <| hu fun _ ↦ Continuous.continuousOn | theorem | UniformOnFun.isClosed_setOf_continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Continuous",
"Continuous.continuousOn",
"IsClosed",
"TopologicalSpace",
"UniformOnFun.tendsto_iff_tendstoUniformlyOn"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 ⊆ ... | by
set ψ₁ : Π S : Set α, φ₁ ⁻¹' S → S := fun S ↦ S.restrictPreimage φ₁
set ψ₂ : Π S : Set α, φ₂ ⁻¹' S → S := fun S ↦ S.restrictPreimage φ₂
have : ∀ S ∈ 𝔖, 𝒰(S, β, _) = .comap (· ∘ ψ₁ S) 𝒰(_, β, _) ⊓ .comap (· ∘ ψ₂ S) 𝒰(_, β, _) := by
refine fun S hS ↦ UniformFun.uniformSpace_eq_inf_precomp_of_cover β _ _ ... | theorem | UniformOnFun.uniformSpace_eq_inf_precomp_of_cover | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Set",
"Subtype.range_val",
"UniformFun.uniformSpace_eq_inf_precomp_of_cover",
"UniformOnFun.precomp_uniformContinuous",
"UniformSpace.comap_comap",
"UniformSpace.comap_iInf",
"UniformSpace.comap_inf",
"comap",
"le_antisymm",
"le_rfl",
"uniformContinuous_iff_le_comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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... | by
set ψ : Π S : Set α, Π i : ι, (φ i) ⁻¹' S → S := fun S i ↦ S.restrictPreimage (φ i)
have : ∀ S ∈ 𝔖, 𝒰(S, β, _) = ⨅ i, .comap (· ∘ ψ S i) 𝒰(_, β, _) := fun S hS ↦ by
rcases h_cover S hS with ⟨I, I_finite, I_cover⟩
refine UniformFun.uniformSpace_eq_iInf_precomp_of_cover β _ ⟨I, I_finite, ?_⟩
simpa o... | theorem | UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Set",
"Subtype.range_val",
"UniformFun.uniformSpace_eq_iInf_precomp_of_cover",
"UniformOnFun.precomp_uniformContinuous",
"UniformSpace.comap_comap",
"UniformSpace.comap_iInf",
"comap",
"le_antisymm",
"le_rfl",
"uniformContinuous_iff_le_comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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'
lift f to α → s using hf with f' hf'
rw [tendstoUniformly_iff_tendsto] at h
have : Tendsto (fun q ↦ (f' q.2, F' q.1 q.2)) (p ×ˢ ⊤) (𝓤 s) :=
h.of_tendsto_comp isUniformEmbedding_subtype_val.comap_uniformity.le
apply ... | theorem | UniformContinuousOn.comp_tendstoUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"TendstoUniformly",
"UniformContinuous.comp_tendstoUniformly",
"UniformContinuousOn",
"tendstoUniformly_iff_tendsto",
"uniformContinuousOn_iff_restrict"
] | Composing on the left by a uniformly continuous function preserves uniform convergence | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 F' : ι → α → β := fun i x => if i ∈ s' then F i x else f x
have hF : F =ᶠ[p] F' := by
rw [eventuallyEq_iff_exists_mem]
refine ⟨s', hs', fun y hy => by grind⟩
have h' : TendstoUniformly F' f p := by
rwa [tendstoUniformly_congr... | theorem | UniformContinuousOn.comp_tendstoUniformly_eventually | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"TendstoUniformly",
"UniformContinuousOn",
"UniformContinuousOn.comp_tendstoUniformly",
"tendstoUniformly_congr"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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_restrict]
apply UniformContinuousOn.comp_tendstoUniformly_eventually (by simpa using hF)
(by simpa using hf) hg (tendstoUniformlyOn_iff_restrict.mp h) | theorem | UniformContinuousOn.comp_tendstoUniformlyOn_eventually | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | [] | [
"Set",
"TendstoUniformlyOn",
"UniformContinuousOn",
"UniformContinuousOn.comp_tendstoUniformly_eventually",
"tendstoUniformlyOn_iff_restrict"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | isUniformInducing_iff_uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"UniformSpace.ext_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | isUniformInducing_iff' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"UniformContinuous",
"comap",
"le_antisymm_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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' i → (x, y... | by
simp [isUniformInducing_iff', h.uniformContinuous_iff h', (h'.comap _).le_basis_iff h, subset_def] | lemma | Filter.HasBasis.isUniformInducing_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"isUniformInducing_iff'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsUniformInducing.mk' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.id : IsUniformInducing (@id α) | ⟨by rw [← Prod.map_def, Prod.map_id, comap_id]⟩ | theorem | IsUniformInducing.id | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"Prod.map_def",
"comap_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsUniformInducing.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsUniformInducing.of_comp_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsUniformInducing.basis_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsUniformInducing.cauchy_map_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Cauchy",
"Filter",
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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.le_comap | theorem | IsUniformInducing.of_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"Prod.map_def",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.uniformContinuous {f : α → β} (hf : IsUniformInducing f) :
UniformContinuous f | (isUniformInducing_iff'.1 hf).1 | theorem | IsUniformInducing.uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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]
@[deprecated (since := "2026-03-17")]
alias IsUniformInducing.isUniformInducing_comp_iff := IsUniformInducing.of_comp_iff | theorem | IsUniformInducing.uniformContinuous_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Filter.map_map",
"IsUniformInducing",
"IsUniformInducing.of_comp_iff",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsUniformInducing.uniformContinuousOn_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Filter.map_map",
"IsUniformInducing",
"Set",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.isInducing {f : α → β} (h : IsUniformInducing f) : IsInducing f | by
obtain rfl := h.comap_uniformSpace
exact .induced f | theorem | IsUniformInducing.isInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsUniformInducing.prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"UniformSpace",
"uniformity_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.isDenseInducing (h : IsUniformInducing f) (hd : DenseRange f) :
IsDenseInducing f | where
toIsInducing := h.isInducing
dense := hd | lemma | IsUniformInducing.isDenseInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"DenseRange",
"IsDenseInducing",
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparationQuotient.isUniformInducing_mk :
IsUniformInducing (mk : α → SeparationQuotient α) | ⟨comap_mk_uniformity⟩ | lemma | SeparationQuotient.isUniformInducing_mk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"SeparationQuotient"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.injective [T0Space α] {f : α → β} (h : IsUniformInducing f) :
Injective f | h.isInducing.injective | theorem | IsUniformInducing.injective | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | isUniformEmbedding_iff' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"UniformContinuous",
"comap",
"isUniformInducing_iff'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.isUniformEmbedding_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} :
IsUniformEmbedding f ↔ Injective 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 ... | by
rw [isUniformEmbedding_iff, and_comm, h.isUniformInducing_iff h'] | theorem | Filter.HasBasis.isUniformEmbedding_iff' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.isUniformEmbedding_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} :
IsUniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) | by
simp only [h.isUniformEmbedding_iff' h', h.uniformContinuous_iff h'] | theorem | Filter.HasBasis.isUniformEmbedding_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_subtype_val {p : α → Prop} :
IsUniformEmbedding (Subtype.val : Subtype p → α) | { comap_uniformity := rfl
injective := Subtype.val_injective } | theorem | isUniformEmbedding_subtype_val | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"Subtype.val_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_set_inclusion {s t : Set α} (hst : s ⊆ t) :
IsUniformEmbedding (inclusion hst) | where
comap_uniformity := by rw [uniformity_subtype, uniformity_subtype, comap_comap]; rfl
injective := inclusion_injective hst | theorem | isUniformEmbedding_set_inclusion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"Set",
"uniformity_subtype"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformEmbedding.comp {g : β → γ} (hg : IsUniformEmbedding g) {f : α → β}
(hf : IsUniformEmbedding f) : IsUniformEmbedding (g ∘ f) | where
toIsUniformInducing := hg.isUniformInducing.comp hf.isUniformInducing
injective := hg.injective.comp hf.injective | theorem | IsUniformEmbedding.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformEmbedding.of_comp_iff {g : β → γ} (hg : IsUniformEmbedding g) {f : α → β} :
IsUniformEmbedding (g ∘ f) ↔ IsUniformEmbedding f | by
simp_rw [isUniformEmbedding_iff, hg.isUniformInducing.of_comp_iff, hg.injective.of_comp_iff f] | theorem | IsUniformEmbedding.of_comp_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformEmbedding.of_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f)
(hg : UniformContinuous g) (hgf : IsUniformEmbedding (g ∘ f)) : IsUniformEmbedding f | ⟨.of_comp hf hg hgf.isUniformInducing, .of_comp hgf.injective⟩ | theorem | IsUniformEmbedding.of_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Equiv.isUniformEmbedding {α β : Type*} [UniformSpace α] [UniformSpace β] (f : α ≃ β)
(h₁ : UniformContinuous f) (h₂ : UniformContinuous f.symm) : IsUniformEmbedding f | isUniformEmbedding_iff'.2 ⟨f.injective, h₁, by rwa [← Equiv.prodCongr_apply, ← map_equiv_symm]⟩ | theorem | Equiv.isUniformEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_inl : IsUniformEmbedding (Sum.inl : α → α ⊕ β) | isUniformEmbedding_iff'.2 ⟨Sum.inl_injective, uniformContinuous_inl, fun s hs =>
⟨Prod.map Sum.inl Sum.inl '' s ∪ range (Prod.map Sum.inr Sum.inr),
union_mem_sup (image_mem_map hs) range_mem_map,
fun x h => by simpa [Prod.map_apply'] using h⟩⟩ | theorem | isUniformEmbedding_inl | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"Prod.map_apply'",
"uniformContinuous_inl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_inr : IsUniformEmbedding (Sum.inr : β → α ⊕ β) | isUniformEmbedding_iff'.2 ⟨Sum.inr_injective, uniformContinuous_inr, fun s hs =>
⟨range (Prod.map Sum.inl Sum.inl) ∪ Prod.map Sum.inr Sum.inr '' s,
union_mem_sup range_mem_map (image_mem_map hs),
fun x h => by simpa [Prod.map_apply'] using h⟩⟩ | theorem | isUniformEmbedding_inr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"Prod.map_apply'",
"uniformContinuous_inr"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.isUniformEmbedding [T0Space α] {f : α → β}
(hf : IsUniformInducing f) : IsUniformEmbedding f | ⟨hf, hf.isInducing.injective⟩ | theorem | IsUniformInducing.isUniformEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"IsUniformInducing",
"T0Space"
] | If the domain of a `IsUniformInducing` map `f` is a T₀ space, then `f` is injective,
hence it is a `IsUniformEmbedding`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isUniformEmbedding_iff_isUniformInducing [T0Space α] {f : α → β} :
IsUniformEmbedding f ↔ IsUniformInducing f | ⟨IsUniformEmbedding.isUniformInducing, IsUniformInducing.isUniformEmbedding⟩ | theorem | isUniformEmbedding_iff_isUniformInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"IsUniformInducing",
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_uniformity_of_spaced_out {α} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
(hf : Pairwise fun x y => (f x, f y) ∉ s) : comap (Prod.map f f) (𝓤 β) = 𝓟 SetRel.id | by
refine le_antisymm ?_ (@refl_le_uniformity α (UniformSpace.comap f _))
calc
comap (Prod.map f f) (𝓤 β) ≤ comap (Prod.map f f) (𝓟 s) := comap_mono (le_principal_iff.2 hs)
_ = 𝓟 (Prod.map f f ⁻¹' s) := comap_principal
_ ≤ 𝓟 SetRel.id := principal_mono.2 ?_
rintro ⟨x, y⟩; simpa [not_imp_not] using... | theorem | comap_uniformity_of_spaced_out | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Pairwise",
"Set",
"SetRel.id",
"UniformSpace.comap",
"comap",
"le_antisymm",
"not_imp_not",
"refl_le_uniformity"
] | If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
`s ∈ 𝓤 β`, then `f` is uniform inducing with respect to the discrete uniformity on `α`:
the preimage of `𝓤 β` under `Prod.map f f` is the principal filter generated by the diagonal in
`α × α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isUniformEmbedding_of_spaced_out {α} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
(hf : Pairwise fun x y => (f x, f y) ∉ s) : @IsUniformEmbedding α β ⊥ ‹_› f | by
let _ : UniformSpace α := ⊥; have := discreteTopology_bot α
exact IsUniformInducing.isUniformEmbedding ⟨comap_uniformity_of_spaced_out hs hf⟩ | theorem | isUniformEmbedding_of_spaced_out | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"IsUniformInducing.isUniformEmbedding",
"Pairwise",
"Set",
"UniformSpace",
"discreteTopology_bot"
] | If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
`s ∈ 𝓤 β`, then `f` is a uniform embedding with respect to the discrete uniformity on `α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformEmbedding.isEmbedding {f : α → β} (h : IsUniformEmbedding f) :
IsEmbedding f | where
toIsInducing := h.toIsUniformInducing.isInducing
injective := h.injective | lemma | IsUniformEmbedding.isEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformEmbedding.isDenseEmbedding {f : α → β} (h : IsUniformEmbedding f)
(hd : DenseRange f) : IsDenseEmbedding f | { h.isEmbedding with dense := hd } | theorem | IsUniformEmbedding.isDenseEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"DenseRange",
"IsDenseEmbedding",
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedEmbedding_of_spaced_out {α} [TopologicalSpace α] [DiscreteTopology α]
[T0Space β] {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
(hf : Pairwise fun x y => (f x, f y) ∉ s) : IsClosedEmbedding f | by
rcases @DiscreteTopology.eq_bot α _ _ with rfl; let _ : UniformSpace α := ⊥
exact
{ (isUniformEmbedding_of_spaced_out hs hf).isEmbedding with
isClosed_range := isClosed_range_of_spaced_out hs hf } | theorem | isClosedEmbedding_of_spaced_out | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"DiscreteTopology",
"Pairwise",
"Set",
"T0Space",
"TopologicalSpace",
"UniformSpace",
"isClosed_range_of_spaced_out",
"isUniformEmbedding_of_spaced_out"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_image_mem_nhds_of_isUniformInducing {s : Set (α × α)} {e : α → β} (b : β)
(he₁ : IsUniformInducing e) (he₂ : IsDenseInducing e) (hs : s ∈ 𝓤 α) :
∃ a, closure (e '' { a' | (a, a') ∈ s }) ∈ 𝓝 b | by
obtain ⟨U, ⟨hU, hUo, hsymm⟩, hs⟩ :
∃ U, (U ∈ 𝓤 β ∧ IsOpen U ∧ SetRel.IsSymm U) ∧ Prod.map e e ⁻¹' U ⊆ s := by
rwa [← he₁.comap_uniformity, (uniformity_hasBasis_open_symmetric.comap _).mem_iff] at hs
rcases he₂.dense.mem_nhds (UniformSpace.ball_mem_nhds b hU) with ⟨a, ha⟩
refine ⟨a, mem_of_superset ?... | theorem | closure_image_mem_nhds_of_isUniformInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsDenseInducing",
"IsOpen",
"IsUniformInducing",
"Set",
"SetRel.IsSymm",
"UniformSpace.ball",
"UniformSpace.ball_mem_nhds",
"UniformSpace.ball_mono",
"UniformSpace.isOpen_ball",
"closure",
"closure_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_subtypeEmb (p : α → Prop) {e : α → β} (ue : IsUniformEmbedding e)
(de : IsDenseEmbedding e) : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e) | { comap_uniformity := by
simp [comap_comap, Function.comp_def, IsDenseEmbedding.subtypeEmb, uniformity_subtype,
ue.comap_uniformity.symm]
injective := (de.subtype p).injective } | theorem | isUniformEmbedding_subtypeEmb | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsDenseEmbedding",
"IsDenseEmbedding.subtypeEmb",
"IsUniformEmbedding",
"uniformity_subtype"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformEmbedding.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : IsUniformEmbedding e₁) (h₂ : IsUniformEmbedding e₂) :
IsUniformEmbedding fun p : α × β => (e₁ p.1, e₂ p.2) | where
toIsUniformInducing := h₁.isUniformInducing.prod h₂.isUniformInducing
injective := h₁.injective.prodMap h₂.injective | theorem | IsUniformEmbedding.prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isComplete_image_iff {m : α → β} {s : Set α} (hm : IsUniformInducing m) :
IsComplete (m '' s) ↔ IsComplete s | by
have fact1 : SurjOn (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 <| m '' s) := surjOn_image .. |>.filter_map_Iic
have fact2 : MapsTo (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 <| m '' s) := mapsTo_image .. |>.filter_map_Iic
simp_rw [IsComplete, imp.swap (a := Cauchy _), ← mem_Iic (b := 𝓟 _), fact1.forall fact2,
hm.cauchy_map_i... | theorem | isComplete_image_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Cauchy",
"IsComplete",
"IsUniformInducing",
"Set"
] | A set is complete iff its image under a uniform inducing map is complete. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.isComplete_iff {f : α → β} {s : Set α} (hf : IsUniformInducing f) :
IsComplete (f '' s) ↔ IsComplete s | isComplete_image_iff hf | theorem | IsUniformInducing.isComplete_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsComplete",
"IsUniformInducing",
"Set",
"isComplete_image_iff"
] | If `f : X → Y` is an `IsUniformInducing` map, the image `f '' s` of a set `s` is complete
if and only if `s` is complete. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformEmbedding.isComplete_iff {f : α → β} {s : Set α} (hf : IsUniformEmbedding f) :
IsComplete (f '' s) ↔ IsComplete s | hf.isUniformInducing.isComplete_iff | theorem | IsUniformEmbedding.isComplete_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsComplete",
"IsUniformEmbedding",
"Set"
] | If `f : X → Y` is an `IsUniformEmbedding`, the image `f '' s` of a set `s` is complete
if and only if `s` is complete. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Subtype.isComplete_iff {p : α → Prop} {s : Set { x // p x }} :
IsComplete s ↔ IsComplete ((↑) '' s : Set α) | isUniformEmbedding_subtype_val.isComplete_iff.symm
alias ⟨isComplete_of_complete_image, _⟩ := isComplete_image_iff | theorem | Subtype.isComplete_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsComplete",
"Set",
"isComplete_image_iff"
] | Sets of a subtype are complete iff their image under the coercion is complete. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
completeSpace_iff_isComplete_range {f : α → β} (hf : IsUniformInducing f) :
CompleteSpace α ↔ IsComplete (range f) | by
rw [completeSpace_iff_isComplete_univ, ← isComplete_image_iff hf, image_univ]
alias ⟨_, IsUniformInducing.completeSpace⟩ := completeSpace_iff_isComplete_range | theorem | completeSpace_iff_isComplete_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsComplete",
"IsUniformInducing",
"completeSpace_iff_isComplete_univ",
"isComplete_image_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.isComplete_range [CompleteSpace α] (hf : IsUniformInducing f) :
IsComplete (range f) | (completeSpace_iff_isComplete_range hf).1 ‹_› | lemma | IsUniformInducing.isComplete_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsComplete",
"IsUniformInducing",
"completeSpace_iff_isComplete_range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.completeSpace_congr {f : α → β} (hf : IsUniformInducing f)
(hsurj : f.Surjective) : CompleteSpace α ↔ CompleteSpace β | by
rw [completeSpace_iff_isComplete_range hf, hsurj.range_eq, completeSpace_iff_isComplete_univ] | theorem | IsUniformInducing.completeSpace_congr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsUniformInducing",
"completeSpace_iff_isComplete_range",
"completeSpace_iff_isComplete_univ"
] | If `f` is a surjective uniform inducing map,
then its domain is a complete space iff its codomain is a complete space.
See also `_root_.completeSpace_congr` for a version that assumes `f` to be an equivalence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
SeparationQuotient.completeSpace_iff :
CompleteSpace (SeparationQuotient α) ↔ CompleteSpace α | .symm <| isUniformInducing_mk.completeSpace_congr surjective_mk | theorem | SeparationQuotient.completeSpace_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"SeparationQuotient",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
SeparationQuotient.instCompleteSpace [CompleteSpace α] :
CompleteSpace (SeparationQuotient α) | completeSpace_iff.2 ‹_› | instance | SeparationQuotient.instCompleteSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"SeparationQuotient"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completeSpace_congr {e : α ≃ β} (he : IsUniformEmbedding e) :
CompleteSpace α ↔ CompleteSpace β | he.completeSpace_congr e.surjective | theorem | completeSpace_congr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsUniformEmbedding"
] | See also `IsUniformInducing.completeSpace_congr`
for a version that works for non-injective maps. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
completeSpace_coe_iff_isComplete {s : Set α} : CompleteSpace s ↔ IsComplete s | by
rw [completeSpace_iff_isComplete_range isUniformEmbedding_subtype_val.isUniformInducing,
Subtype.range_coe]
alias ⟨_, IsComplete.completeSpace_coe⟩ := completeSpace_coe_iff_isComplete | theorem | completeSpace_coe_iff_isComplete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsComplete",
"Set",
"Subtype.range_coe",
"completeSpace_iff_isComplete_range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.completeSpace_coe [CompleteSpace α] {s : Set α} [hs : IsClosed s] :
CompleteSpace s | hs.isComplete.completeSpace_coe | instance | IsClosed.completeSpace_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsClosed",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completeSpace_ulift_iff : CompleteSpace (ULift α) ↔ CompleteSpace α | IsUniformInducing.completeSpace_congr ⟨rfl⟩ ULift.down_surjective | theorem | completeSpace_ulift_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsUniformInducing.completeSpace_congr",
"ULift.down_surjective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ULift.instCompleteSpace [CompleteSpace α] : CompleteSpace (ULift α) | completeSpace_ulift_iff.2 ‹_› | instance | ULift.instCompleteSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace"
] | The lift of a complete space to another universe is still complete. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
completeSpace_extension {m : β → α} (hm : IsUniformInducing m) (dense : DenseRange m)
(h : ∀ f : Filter β, Cauchy f → ∃ x : α, map m f ≤ 𝓝 x) : CompleteSpace α | ⟨fun {f : Filter α} (hf : Cauchy f) =>
let p : Set (α × α) → Set α → Set α := fun s t => { y : α | ∃ x : α, x ∈ t ∧ (x, y) ∈ s }
let g := (𝓤 α).lift fun s => f.lift' (p s)
have mp₀ : Monotone p := fun _ _ h _ _ ⟨x, xs, xa⟩ => ⟨x, xs, h xa⟩
have mp₁ : ∀ {s}, Monotone (p s) := fun h _ ⟨y, ya, yxs⟩ => ⟨y,... | theorem | completeSpace_extension | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Cauchy",
"ClusterPt",
"CompleteSpace",
"DenseRange",
"Filter",
"Filter.comap",
"IsUniformInducing",
"Monotone",
"Set",
"SetRel.prodMk_mem_comp",
"comap",
"comp_mem_uniformity_sets",
"le_nhds_iff_adhp_of_cauchy",
"le_nhds_of_cauchy_adhp",
"le_of_eq",
"mem_nhds_left",
"monotone_const"... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.totallyBounded_map_iff {f : α → β} {F : Filter α} (hf : IsUniformInducing f) :
(F.map f).TotallyBounded ↔ F.TotallyBounded | by
refine ⟨fun hs ↦ ?_, fun h ↦ h.map hf.uniformContinuous⟩
simp_rw [(hf.basis_uniformity (basis_sets _)).filter_totallyBounded_iff]
intro t ht
rcases exists_subset_image_finite_and.1 (hs.exists_subset_of_mem (F.image_mem_map F.univ_mem) ht)
with ⟨u, -, hfin, h⟩
use u, hfin
simp_rw [SetRel.preimage, exi... | lemma | Filter.totallyBounded_map_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Filter",
"IsUniformInducing",
"SetRel.preimage",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_image_iff {f : α → β} {s : Set α} (hf : IsUniformInducing f) :
TotallyBounded (f '' s) ↔ TotallyBounded s | by
simp_rw [← totallyBounded_principal_iff, ← map_principal, totallyBounded_map_iff hf] | lemma | totallyBounded_image_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_preimage {f : α → β} {s : Set β} (hf : IsUniformInducing f)
(hs : TotallyBounded s) : TotallyBounded (f ⁻¹' s) | (totallyBounded_image_iff hf).1 <| hs.subset <| image_preimage_subset .. | theorem | totallyBounded_preimage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"Set",
"TotallyBounded",
"totallyBounded_image_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.totallyBounded_comap {f : α → β} {F : Filter β} (hf : IsUniformInducing f)
(hF : F.TotallyBounded) : (F.comap f).TotallyBounded | (totallyBounded_map_iff hf).1 <| hF.mono map_comap_le | theorem | Filter.totallyBounded_comap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Filter",
"IsUniformInducing",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompleteSpace.sum [CompleteSpace α] [CompleteSpace β] : CompleteSpace (α ⊕ β) | by
rw [completeSpace_iff_isComplete_univ, ← range_inl_union_range_inr]
exact isUniformEmbedding_inl.isUniformInducing.isComplete_range.union
isUniformEmbedding_inr.isUniformInducing.isComplete_range | instance | CompleteSpace.sum | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"completeSpace_iff_isComplete_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformEmbedding.discreteUniformity [DiscreteUniformity β] {f : α → β}
(hf : IsUniformEmbedding f) : DiscreteUniformity α | by
simp_rw [discreteUniformity_iff_eq_principal_setRelId, ← hf.comap_uniformity,
DiscreteUniformity.eq_principal_setRelId, comap_principal, SetRel.id, preimage_setOf_eq,
hf.injective.eq_iff] | theorem | IsUniformEmbedding.discreteUniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"DiscreteUniformity",
"DiscreteUniformity.eq_principal_setRelId",
"IsUniformEmbedding",
"SetRel.id",
"discreteUniformity_iff_eq_principal_setRelId"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_comap {α : Type*} {β : Type*} {f : α → β} [u : UniformSpace β]
(hf : Function.Injective f) : @IsUniformEmbedding α β (UniformSpace.comap f u) u f | @IsUniformEmbedding.mk _ _ (UniformSpace.comap f u) _ _
(@IsUniformInducing.mk _ _ (UniformSpace.comap f u) _ _ rfl) hf | theorem | isUniformEmbedding_comap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"UniformSpace",
"UniformSpace.comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Topology.IsEmbedding.comapUniformSpace {α β} [TopologicalSpace α] [u : UniformSpace β]
(f : α → β) (h : IsEmbedding f) : UniformSpace α | (u.comap f).replaceTopology h.eq_induced | def | Topology.IsEmbedding.comapUniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"TopologicalSpace",
"UniformSpace"
] | Pull back a uniform space structure by an embedding, adjusting the new uniform structure to
make sure that its topology is defeq to the original one. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Embedding.to_isUniformEmbedding {α β} [TopologicalSpace α] [u : UniformSpace β] (f : α → β)
(h : IsEmbedding f) : @IsUniformEmbedding α β (h.comapUniformSpace f) u f | let _ := h.comapUniformSpace f
{ comap_uniformity := rfl
injective := h.injective } | theorem | Embedding.to_isUniformEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformEmbedding",
"TopologicalSpace",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformly_extend_exists [CompleteSpace γ] (a : α) : ∃ c, Tendsto f (comap e (𝓝 a)) (𝓝 c) | let de := h_e.isDenseInducing h_dense
have : Cauchy (𝓝 a) := cauchy_nhds
have : Cauchy (comap e (𝓝 a)) :=
this.comap' (le_of_eq h_e.comap_uniformity) (de.comap_nhds_neBot _)
have : Cauchy (map f (comap e (𝓝 a))) := this.map h_f
CompleteSpace.complete this | theorem | uniformly_extend_exists | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Cauchy",
"CompleteSpace",
"cauchy_nhds",
"comap",
"le_of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniform_extend_subtype [CompleteSpace γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β}
{s : Set α} (hf : UniformContinuous fun x : Subtype p => f x.val) (he : IsUniformEmbedding e)
(hd : ∀ x : β, x ∈ closure (range e)) (hb : closure (e '' s) ∈ 𝓝 b) (hs : IsClosed s)
(hp : ∀ x ∈ s, p x) : ∃ c, Tendsto f (c... | by
have de : IsDenseEmbedding e := he.isDenseEmbedding hd
have de' : IsDenseEmbedding (IsDenseEmbedding.subtypeEmb p e) := de.subtype p
have ue' : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e) :=
isUniformEmbedding_subtypeEmb _ he de
have : b ∈ closure (e '' { x | p x }) :=
(closure_mono <| monot... | theorem | uniform_extend_subtype | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsClosed",
"IsDenseEmbedding",
"IsDenseEmbedding.subtypeEmb",
"IsUniformEmbedding",
"Set",
"Subtype.range_coe_subtype",
"UniformContinuous",
"closure",
"closure_mono",
"comap",
"isUniformEmbedding_subtypeEmb",
"mem_of_mem_nhds",
"nhds_subtype_eq_comap",
"uniformly_exten... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformly_extend_spec [CompleteSpace γ] (a : α) : Tendsto f (comap e (𝓝 a)) (𝓝 (ψ a)) | by
simpa only [IsDenseInducing.extend] using
tendsto_nhds_limUnder (uniformly_extend_exists h_e ‹_› h_f _)
include h_f in | theorem | uniformly_extend_spec | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"IsDenseInducing.extend",
"comap",
"tendsto_nhds_limUnder",
"uniformly_extend_exists"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_uniformly_extend [CompleteSpace γ] : UniformContinuous ψ | fun d hd =>
let ⟨s, hs, hs_comp⟩ := comp3_mem_uniformity hd
have h_pnt : ∀ {a m}, m ∈ 𝓝 a → ∃ c ∈ f '' e ⁻¹' m, (c, ψ a) ∈ s ∧ (ψ a, c) ∈ s :=
fun {a m} hm =>
have nb : NeBot (map f (comap e (𝓝 a))) :=
((h_e.isDenseInducing h_dense).comap_nhds_neBot _).map _
have :
f '' (e ⁻¹' m) ∩ ({ c | ... | theorem | uniformContinuous_uniformly_extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"UniformContinuous",
"comap",
"comp3_mem_uniformity",
"interior",
"interior_mem_uniformity",
"interior_subset",
"mem_nhds_left",
"mem_nhds_right",
"uniformly_extend_spec"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformly_extend_of_ind (b : β) : ψ (e b) = f b | IsDenseInducing.extend_eq_at _ h_f.continuous.continuousAt | theorem | uniformly_extend_of_ind | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsDenseInducing.extend_eq_at"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformly_extend_unique {g : α → γ} (hg : ∀ b, g (e b) = f b) (hc : Continuous g) : ψ = g | IsDenseInducing.extend_unique _ hg hc | theorem | uniformly_extend_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Continuous",
"IsDenseInducing.extend_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformInducing_val (s : Set α) :
IsUniformInducing ((↑) : s → α) | ⟨uniformity_setCoe⟩ | theorem | isUniformInducing_val | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_rangeFactorization_iff {f : α → β} :
UniformContinuous (rangeFactorization f) ↔ UniformContinuous f | (isUniformInducing_val _).uniformContinuous_iff | theorem | uniformContinuous_rangeFactorization_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"UniformContinuous",
"isUniformInducing_val"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.rangeFactorization {f : α → β} (hf : UniformContinuous f) :
UniformContinuous (rangeFactorization f) | uniformContinuous_rangeFactorization_iff.mpr hf | theorem | UniformContinuous.rangeFactorization | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformInducing_rangeFactorization_iff {f : α → β} :
IsUniformInducing (rangeFactorization f) ↔ IsUniformInducing f | (isUniformInducing_val (range f)).of_comp_iff.symm | theorem | isUniformInducing_rangeFactorization_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing",
"isUniformInducing_val"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.rangeFactorization {f : α → β} (hf : IsUniformInducing f) :
IsUniformInducing (rangeFactorization f) | isUniformInducing_rangeFactorization_iff.2 hf | theorem | IsUniformInducing.rangeFactorization | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend_exists [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) (a : α) :
∃ b, Tendsto f (comap (↑) (𝓝 a)) (𝓝 b) | uniformly_extend_exists (isUniformInducing_val s) hs.denseRange_val hf a | theorem | Dense.extend_exists | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"Dense",
"UniformContinuous",
"comap",
"isUniformInducing_val",
"uniformly_extend_exists"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend_spec [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) (a : α) :
Tendsto f (comap (↑) (𝓝 a)) (𝓝 (hs.extend f a)) | uniformly_extend_spec (isUniformInducing_val s) hs.denseRange_val hf a | theorem | Dense.extend_spec | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"Dense",
"UniformContinuous",
"comap",
"isUniformInducing_val",
"uniformly_extend_spec"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_extend [CompleteSpace β] (hs : Dense s) (hf : UniformContinuous f) :
UniformContinuous (hs.extend f) | uniformContinuous_uniformly_extend (isUniformInducing_val s) hs.denseRange_val hf | theorem | Dense.uniformContinuous_extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"CompleteSpace",
"Dense",
"UniformContinuous",
"isUniformInducing_val",
"uniformContinuous_uniformly_extend"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
extend_of_ind (hs : Dense s) (hf : UniformContinuous f) (x : s) :
hs.extend f x = f x | IsDenseInducing.extend_eq_at _ hf.continuous.continuousAt | theorem | Dense.extend_of_ind | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | [] | [
"Dense",
"IsDenseInducing.extend_eq_at",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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