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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UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
(hA : UniformEquicontinuous (u ∘ (↑) : A → β → α)) (hu : Continuous u) :
UniformEquicontinuous (u ∘ (↑) : closure A → β → α) := by
rw [← uniformEquicontinuousOn_univ] at hA ⊢
exact hA.closure' (Pi.continuous_restrict _ |>.comp hu) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuous.closure' | If a set of functions is uniformly equicontinuous, the same is true for its closure in *any*
topology for which evaluation at any point is continuous. Since this will be applied to
`DFunLike` types, we state it for any topological space with a map to `β → α` satisfying the right
continuity conditions. See also `Set.UniformEquicontinuous.closure` for a more familiar statement. |
protected Set.UniformEquicontinuous.closure {A : Set <| β → α}
(hA : A.UniformEquicontinuous) : (closure A).UniformEquicontinuous :=
UniformEquicontinuous.closure' (u := id) hA continuous_id | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.UniformEquicontinuous.closure | If a set of functions is uniformly equicontinuous, its closure for the product topology is also
uniformly equicontinuous. |
protected Set.UniformEquicontinuousOn.closure {A : Set <| β → α} {S : Set β}
(hA : A.UniformEquicontinuousOn S) : (closure A).UniformEquicontinuousOn S :=
UniformEquicontinuousOn.closure' (u := id) hA (Pi.continuous_restrict _)
/-
Implementation note: The following lemma (as well as all the following variations) could
theoretically be deduced from the "closure" statements above. For example, we could do:
```lean | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.UniformEquicontinuousOn.closure | If a set of functions is uniformly equicontinuous on a set `S`, its closure for the product
topology is also uniformly equicontinuous. This would also be true for the coarser topology of
pointwise convergence on `S`, see `UniformEquicontinuousOn.closure'`. |
Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
{f : X → α} {x₀ : X} (h₁ : Tendsto F l (𝓝 f)) (h₂ : EquicontinuousAt F x₀) :
ContinuousAt f x₀ :=
(equicontinuousAt_iff_range.mp h₂).closure.continuousAt
⟨f, mem_closure_of_tendsto h₁ <| Eventually.of_forall mem_range_self⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.Tendsto.continuousAt_of_equicontinuousAt | null |
Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.NeBot]
{F : ι → β → α} {f : β → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : UniformEquicontinuous F) :
UniformContinuous f :=
(uniformEquicontinuous_iff_range.mp h₂).closure.uniformContinuous
⟨f, mem_closure_of_tendsto h₁ <| Eventually.of_forall mem_range_self⟩
```
Unfortunately, the proofs get painful when dealing with the relative case as one needs to change
the ambient topology. So it turns out to be easier to re-do the proof by hand.
-/ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous | null |
Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt {l : Filter ι} [l.NeBot]
{F : ι → X → α} {f : X → α} {S : Set X} {x₀ : X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
(h₂ : Tendsto (F · x₀) l (𝓝 (f x₀))) (h₃ : EquicontinuousWithinAt F S x₀) :
ContinuousWithinAt f S x₀ := by
intro U hU; rw [mem_map]
rcases UniformSpace.mem_nhds_iff.mp hU with ⟨V, hV, hVU⟩
rcases mem_uniformity_isClosed hV with ⟨W, hW, hWclosed, hWV⟩
filter_upwards [h₃ W hW, eventually_mem_nhdsWithin] with x hx hxS using
hVU <| ball_mono hWV (f x₀) <| hWclosed.mem_of_tendsto (h₂.prodMk_nhds (h₁ x hxS)) <|
Eventually.of_forall hx | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise on `S ∪ {x₀} : Set X`* along some nontrivial
filter, and if the family `𝓕` is equicontinuous at `x₀ : X` within `S`, then the limit is
continuous at `x₀` within `S`. |
Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
{f : X → α} {x₀ : X} (h₁ : Tendsto F l (𝓝 f)) (h₂ : EquicontinuousAt F x₀) :
ContinuousAt f x₀ := by
rw [← continuousWithinAt_univ, ← equicontinuousWithinAt_univ, tendsto_pi_nhds] at *
exact continuousWithinAt_of_equicontinuousWithinAt (fun x _ ↦ h₁ x) (h₁ x₀) h₂ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.Tendsto.continuousAt_of_equicontinuousAt | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is equicontinuous at some `x₀ : X`, then the limit is continuous at `x₀`. |
Filter.Tendsto.continuous_of_equicontinuous {l : Filter ι} [l.NeBot] {F : ι → X → α}
{f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f :=
continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.Tendsto.continuous_of_equicontinuous | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is equicontinuous, then the limit is continuous. |
Filter.Tendsto.continuousOn_of_equicontinuousOn {l : Filter ι} [l.NeBot] {F : ι → X → α}
{f : X → α} {S : Set X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
(h₂ : EquicontinuousOn F S) : ContinuousOn f S :=
fun x hx ↦ Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt h₁ (h₁ x hx) (h₂ x hx) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.Tendsto.continuousOn_of_equicontinuousOn | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise on `S : Set X`* along some nontrivial
filter, and if the family `𝓕` is equicontinuous, then the limit is continuous on `S`. |
Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn {l : Filter ι} [l.NeBot]
{F : ι → β → α} {f : β → α} {S : Set β} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
(h₂ : UniformEquicontinuousOn F S) :
UniformContinuousOn f S := by
intro U hU; rw [mem_map]
rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
filter_upwards [h₂ V hV, mem_inf_of_right (mem_principal_self _)]
rintro ⟨x, y⟩ hxy ⟨hxS, hyS⟩
exact hVU <| hVclosed.mem_of_tendsto ((h₁ x hxS).prodMk_nhds (h₁ y hyS)) <|
Eventually.of_forall hxy | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn | If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise on `S : Set β`* along some nontrivial
filter, and if the family `𝓕` is uniformly equicontinuous on `S`, then the limit is uniformly
continuous on `S`. |
Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.NeBot]
{F : ι → β → α} {f : β → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : UniformEquicontinuous F) :
UniformContinuous f := by
rw [← uniformContinuousOn_univ, ← uniformEquicontinuousOn_univ, tendsto_pi_nhds] at *
exact uniformContinuousOn_of_uniformEquicontinuousOn (fun x _ ↦ h₁ x) h₂ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous | If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. |
EquicontinuousAt.tendsto_of_mem_closure {l : Filter ι} {F : ι → X → α} {f : X → α}
{s : Set X} {x : X} {z : α} (hF : EquicontinuousAt F x) (hf : Tendsto f (𝓝[s] x) (𝓝 z))
(hs : ∀ y ∈ s, Tendsto (F · y) l (𝓝 (f y))) (hx : x ∈ closure s) :
Tendsto (F · x) l (𝓝 z) := by
rw [(nhds_basis_uniformity (𝓤 α).basis_sets).tendsto_right_iff] at hf ⊢
intro U hU
rcases comp_comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVs, hVU⟩
rw [mem_closure_iff_nhdsWithin_neBot] at hx
have : ∀ᶠ y in 𝓝[s] x, y ∈ s ∧ (∀ i, (F i x, F i y) ∈ V) ∧ (f y, z) ∈ V :=
eventually_mem_nhdsWithin.and <| ((hF V hV).filter_mono nhdsWithin_le_nhds).and (hf V hV)
rcases this.exists with ⟨y, hys, hFy, hfy⟩
filter_upwards [hs y hys (ball_mem_nhds _ hV)] with i hi
exact hVU ⟨_, ⟨_, hFy i, (mem_ball_symmetry hVs).2 hi⟩, hfy⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousAt.tendsto_of_mem_closure | If `F : ι → X → α` is a family of functions equicontinuous at `x`,
it tends to `f y` along a filter `l` for any `y ∈ s`,
the limit function `f` tends to `z` along `𝓝[s] x`, and `x ∈ closure s`,
then `(F · x)` tends to `z` along `l`.
In some sense, this is a converse of `EquicontinuousAt.closure`. |
Equicontinuous.isClosed_setOf_tendsto {l : Filter ι} {F : ι → X → α} {f : X → α}
(hF : Equicontinuous F) (hf : Continuous f) :
IsClosed {x | Tendsto (F · x) l (𝓝 (f x))} :=
closure_subset_iff_isClosed.mp fun x hx ↦
(hF x).tendsto_of_mem_closure (hf.continuousAt.mono_left inf_le_left) (fun _ ↦ id) hx | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Equicontinuous.isClosed_setOf_tendsto | If `F : ι → X → α` is an equicontinuous family of functions,
`f : X → α` is a continuous function, and `l` is a filter on `ι`,
then `{x | Filter.Tendsto (F · x) l (𝓝 (f x))}` is a closed set. |
UniformEquiv (α : Type*) (β : Type*) [UniformSpace α] [UniformSpace β] extends
α ≃ β where
/-- Uniform continuity of the function -/
uniformContinuous_toFun : UniformContinuous toFun
/-- Uniform continuity of the inverse -/
uniformContinuous_invFun : UniformContinuous invFun | structure | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | UniformEquiv | Uniform isomorphism between `α` and `β` |
protected symm (h : α ≃ᵤ β) : β ≃ᵤ α where
uniformContinuous_toFun := h.uniformContinuous_invFun
uniformContinuous_invFun := h.uniformContinuous_toFun
toEquiv := h.toEquiv.symm | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | symm | Uniform isomorphism between `α` and `β` -/
infixl:25 " ≃ᵤ " => UniformEquiv
namespace UniformEquiv
variable [UniformSpace α] [UniformSpace β] [UniformSpace γ] [UniformSpace δ]
theorem toEquiv_injective : Function.Injective (toEquiv : α ≃ᵤ β → α ≃ β)
| ⟨e, h₁, h₂⟩, ⟨e', h₁', h₂'⟩, h => by simpa only [mk.injEq]
instance : EquivLike (α ≃ᵤ β) α β where
coe h := h.toEquiv
inv h := h.toEquiv.symm
left_inv h := h.left_inv
right_inv h := h.right_inv
coe_injective' _ _ H _ := toEquiv_injective <| DFunLike.ext' H
@[simp]
theorem uniformEquiv_mk_coe (a : Equiv α β) (b c) : (UniformEquiv.mk a b c : α → β) = a :=
rfl
/-- Inverse of a uniform isomorphism. |
Simps.apply (h : α ≃ᵤ β) : α → β :=
h | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | Simps.apply | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. |
Simps.symm_apply (h : α ≃ᵤ β) : β → α :=
h.symm
initialize_simps_projections UniformEquiv (toFun → apply, invFun → symm_apply)
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | Simps.symm_apply | See Note [custom simps projection] |
coe_toEquiv (h : α ≃ᵤ β) : ⇑h.toEquiv = h :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | coe_toEquiv | null |
coe_symm_toEquiv (h : α ≃ᵤ β) : ⇑h.toEquiv.symm = h.symm :=
rfl
@[ext] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | coe_symm_toEquiv | null |
ext {h h' : α ≃ᵤ β} (H : ∀ x, h x = h' x) : h = h' :=
toEquiv_injective <| Equiv.ext H | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | ext | null |
@[simps! -fullyApplied apply]
protected refl (α : Type*) [UniformSpace α] : α ≃ᵤ α where
uniformContinuous_toFun := uniformContinuous_id
uniformContinuous_invFun := uniformContinuous_id
toEquiv := Equiv.refl α | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | refl | Identity map as a uniform isomorphism. |
protected trans (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) : α ≃ᵤ γ where
uniformContinuous_toFun := h₂.uniformContinuous_toFun.comp h₁.uniformContinuous_toFun
uniformContinuous_invFun := h₁.uniformContinuous_invFun.comp h₂.uniformContinuous_invFun
toEquiv := Equiv.trans h₁.toEquiv h₂.toEquiv
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | trans | Composition of two uniform isomorphisms. |
trans_apply (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) (a : α) : h₁.trans h₂ a = h₂ (h₁ a) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | trans_apply | null |
uniformEquiv_mk_coe_symm (a : Equiv α β) (b c) :
((UniformEquiv.mk a b c).symm : β → α) = a.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | uniformEquiv_mk_coe_symm | null |
refl_symm : (UniformEquiv.refl α).symm = UniformEquiv.refl α :=
rfl | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | refl_symm | null |
protected uniformContinuous (h : α ≃ᵤ β) : UniformContinuous h :=
h.uniformContinuous_toFun
@[continuity] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | uniformContinuous | null |
protected continuous (h : α ≃ᵤ β) : Continuous h :=
h.uniformContinuous.continuous | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | continuous | null |
protected uniformContinuous_symm (h : α ≃ᵤ β) : UniformContinuous h.symm :=
h.uniformContinuous_invFun
@[continuity] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | uniformContinuous_symm | null |
protected continuous_symm (h : α ≃ᵤ β) : Continuous h.symm :=
h.uniformContinuous_symm.continuous | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | continuous_symm | null |
protected toHomeomorph (e : α ≃ᵤ β) : α ≃ₜ β :=
{ e.toEquiv with
continuous_toFun := e.continuous
continuous_invFun := e.continuous_symm } | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | toHomeomorph | A uniform isomorphism as a homeomorphism. |
toHomeomorph_apply (e : α ≃ᵤ β) : (e.toHomeomorph : α → β) = e := rfl | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | toHomeomorph_apply | null |
toHomeomorph_symm_apply (e : α ≃ᵤ β) : (e.toHomeomorph.symm : β → α) = e.symm := rfl
@[simp] | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | toHomeomorph_symm_apply | null |
apply_symm_apply (h : α ≃ᵤ β) (x : β) : h (h.symm x) = x :=
h.toEquiv.apply_symm_apply x
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | apply_symm_apply | null |
symm_apply_apply (h : α ≃ᵤ β) (x : α) : h.symm (h x) = x :=
h.toEquiv.symm_apply_apply x | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | symm_apply_apply | null |
protected bijective (h : α ≃ᵤ β) : Function.Bijective h :=
h.toEquiv.bijective | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | bijective | null |
protected injective (h : α ≃ᵤ β) : Function.Injective h :=
h.toEquiv.injective | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | injective | null |
protected surjective (h : α ≃ᵤ β) : Function.Surjective h :=
h.toEquiv.surjective | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | surjective | null |
changeInv (f : α ≃ᵤ β) (g : β → α) (hg : Function.RightInverse g f) : α ≃ᵤ β :=
have : g = f.symm :=
funext fun x => calc
g x = f.symm (f (g x)) := (f.left_inv (g x)).symm
_ = f.symm x := by rw [hg x]
{ toFun := f
invFun := g
left_inv := by convert f.left_inv
right_inv := by convert f.right_inv using 1
uniformContinuous_toFun := f.uniformContinuous
uniformContinuous_invFun := by convert f.symm.uniformContinuous }
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | changeInv | Change the uniform equiv `f` to make the inverse function definitionally equal to `g`. |
symm_comp_self (h : α ≃ᵤ β) : (h.symm : β → α) ∘ h = id :=
funext h.symm_apply_apply
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | symm_comp_self | null |
self_comp_symm (h : α ≃ᵤ β) : (h : α → β) ∘ h.symm = id :=
funext h.apply_symm_apply | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | self_comp_symm | null |
range_coe (h : α ≃ᵤ β) : range h = univ := by simp | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | range_coe | null |
image_symm (h : α ≃ᵤ β) : image h.symm = preimage h :=
funext h.symm.toEquiv.image_eq_preimage | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | image_symm | null |
preimage_symm (h : α ≃ᵤ β) : preimage h.symm = image h :=
(funext h.toEquiv.image_eq_preimage).symm
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | preimage_symm | null |
image_preimage (h : α ≃ᵤ β) (s : Set β) : h '' (h ⁻¹' s) = s :=
h.toEquiv.image_preimage s
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | image_preimage | null |
preimage_image (h : α ≃ᵤ β) (s : Set α) : h ⁻¹' (h '' s) = s :=
h.toEquiv.preimage_image s | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | preimage_image | null |
isUniformInducing (h : α ≃ᵤ β) : IsUniformInducing h :=
IsUniformInducing.of_comp h.uniformContinuous h.symm.uniformContinuous <| by
simp only [symm_comp_self, IsUniformInducing.id] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | isUniformInducing | null |
comap_eq (h : α ≃ᵤ β) : UniformSpace.comap h ‹_› = ‹_› :=
h.isUniformInducing.comap_uniformSpace | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | comap_eq | null |
isUniformEmbedding (h : α ≃ᵤ β) : IsUniformEmbedding h := ⟨h.isUniformInducing, h.injective⟩ | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | isUniformEmbedding | null |
completeSpace_iff (h : α ≃ᵤ β) : CompleteSpace α ↔ CompleteSpace β :=
completeSpace_congr h.isUniformEmbedding | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | completeSpace_iff | null |
noncomputable ofIsUniformEmbedding (f : α → β) (hf : IsUniformEmbedding f) :
α ≃ᵤ Set.range f where
uniformContinuous_toFun := hf.isUniformInducing.uniformContinuous.subtype_mk _
uniformContinuous_invFun := by
rw [hf.isUniformInducing.uniformContinuous_iff, Equiv.invFun_as_coe,
Equiv.self_comp_ofInjective_symm]
exact uniformContinuous_subtype_val
toEquiv := Equiv.ofInjective f hf.injective | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | ofIsUniformEmbedding | Uniform equiv given a uniform embedding. |
setCongr {s t : Set α} (h : s = t) : s ≃ᵤ t where
uniformContinuous_toFun := uniformContinuous_subtype_val.subtype_mk _
uniformContinuous_invFun := uniformContinuous_subtype_val.subtype_mk _
toEquiv := Equiv.setCongr h | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | setCongr | If two sets are equal, then they are uniformly equivalent. |
prodCongr (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : α × γ ≃ᵤ β × δ where
uniformContinuous_toFun :=
(h₁.uniformContinuous.comp uniformContinuous_fst).prodMk
(h₂.uniformContinuous.comp uniformContinuous_snd)
uniformContinuous_invFun :=
(h₁.symm.uniformContinuous.comp uniformContinuous_fst).prodMk
(h₂.symm.uniformContinuous.comp uniformContinuous_snd)
toEquiv := h₁.toEquiv.prodCongr h₂.toEquiv
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | prodCongr | Product of two uniform isomorphisms. |
prodCongr_symm (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) :
(h₁.prodCongr h₂).symm = h₁.symm.prodCongr h₂.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | prodCongr_symm | null |
coe_prodCongr (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : ⇑(h₁.prodCongr h₂) = Prod.map h₁ h₂ :=
rfl | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | coe_prodCongr | null |
prodComm : α × β ≃ᵤ β × α where
uniformContinuous_toFun := uniformContinuous_snd.prodMk uniformContinuous_fst
uniformContinuous_invFun := uniformContinuous_snd.prodMk uniformContinuous_fst
toEquiv := Equiv.prodComm α β
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | prodComm | `α × β` is uniformly isomorphic to `β × α`. |
prodComm_symm : (prodComm α β).symm = prodComm β α :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | prodComm_symm | null |
coe_prodComm : ⇑(prodComm α β) = Prod.swap :=
rfl | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | coe_prodComm | null |
prodAssoc : (α × β) × γ ≃ᵤ α × β × γ where
uniformContinuous_toFun :=
(uniformContinuous_fst.comp uniformContinuous_fst).prodMk
((uniformContinuous_snd.comp uniformContinuous_fst).prodMk uniformContinuous_snd)
uniformContinuous_invFun :=
(uniformContinuous_fst.prodMk (uniformContinuous_fst.comp
uniformContinuous_snd)).prodMk (uniformContinuous_snd.comp uniformContinuous_snd)
toEquiv := Equiv.prodAssoc α β γ | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | prodAssoc | `(α × β) × γ` is uniformly isomorphic to `α × (β × γ)`. |
@[simps! -fullyApplied apply]
prodPunit : α × PUnit ≃ᵤ α where
toEquiv := Equiv.prodPUnit α
uniformContinuous_toFun := uniformContinuous_fst
uniformContinuous_invFun := uniformContinuous_id.prodMk uniformContinuous_const | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | prodPunit | `α × {*}` is uniformly isomorphic to `α`. |
punitProd : PUnit × α ≃ᵤ α :=
(prodComm _ _).trans (prodPunit _)
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | punitProd | `{*} × α` is uniformly isomorphic to `α`. |
coe_punitProd : ⇑(punitProd α) = Prod.snd :=
rfl | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | coe_punitProd | null |
@[simps toEquiv, simps! -isSimp apply]
piCongrLeft {ι ι' : Type*} {β : ι' → Type*} [∀ j, UniformSpace (β j)]
(e : ι ≃ ι') : (∀ i, β (e i)) ≃ᵤ ∀ j, β j where
uniformContinuous_toFun := uniformContinuous_pi.mpr <| e.forall_congr_right.mp fun i ↦ by
simpa only [Equiv.toFun_as_coe, Equiv.piCongrLeft_apply_apply] using
Pi.uniformContinuous_proj _ i
uniformContinuous_invFun := Pi.uniformContinuous_precomp' _ e
toEquiv := Equiv.piCongrLeft _ e
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piCongrLeft | `Equiv.piCongrLeft` as a uniform isomorphism: this is the natural isomorphism
`Π i, β (e i) ≃ᵤ Π j, β j` obtained from a bijection `ι ≃ ι'`. |
piCongrLeft_refl {ι : Type*} {X : ι → Type*} [∀ i, UniformSpace (X i)] :
piCongrLeft (.refl ι) = .refl (∀ i, X i) :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piCongrLeft_refl | null |
piCongrLeft_symm_apply {ι ι' : Type*} {X : ι' → Type*} [∀ j, UniformSpace (X j)]
(e : ι ≃ ι') : ⇑(piCongrLeft (β := X) e).symm = (· <| e ·) :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piCongrLeft_symm_apply | null |
piCongrLeft_apply_apply {ι ι' : Type*} {X : ι' → Type*} [∀ j, UniformSpace (X j)]
(e : ι ≃ ι') (x : ∀ i, X (e i)) i : piCongrLeft e x (e i) = x i :=
Equiv.piCongrLeft_apply_apply .. | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piCongrLeft_apply_apply | null |
@[simps! apply toEquiv]
piCongrRight {ι : Type*} {β₁ β₂ : ι → Type*} [∀ i, UniformSpace (β₁ i)]
[∀ i, UniformSpace (β₂ i)] (F : ∀ i, β₁ i ≃ᵤ β₂ i) : (∀ i, β₁ i) ≃ᵤ ∀ i, β₂ i where
uniformContinuous_toFun := Pi.uniformContinuous_postcomp' _ fun i ↦ (F i).uniformContinuous
uniformContinuous_invFun := Pi.uniformContinuous_postcomp' _ fun i ↦ (F i).symm.uniformContinuous
toEquiv := Equiv.piCongrRight fun i => (F i).toEquiv
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piCongrRight | `Equiv.piCongrRight` as a uniform isomorphism: this is the natural isomorphism
`Π i, β₁ i ≃ᵤ Π j, β₂ i` obtained from uniform isomorphisms `β₁ i ≃ᵤ β₂ i` for each `i`. |
piCongrRight_symm {ι : Type*} {β₁ β₂ : ι → Type*} [∀ i, UniformSpace (β₁ i)]
[∀ i, UniformSpace (β₂ i)] (F : ∀ i, β₁ i ≃ᵤ β₂ i) :
(piCongrRight F).symm = piCongrRight fun i => (F i).symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piCongrRight_symm | null |
piCongrRight_refl {ι : Type*} {X : ι → Type*} [∀ i, UniformSpace (X i)] :
piCongrRight (fun i ↦ .refl (X i)) = .refl (∀ i, X i) :=
rfl | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piCongrRight_refl | null |
@[simps! apply toEquiv]
piCongr {ι₁ ι₂ : Type*} {β₁ : ι₁ → Type*} {β₂ : ι₂ → Type*}
[∀ i₁, UniformSpace (β₁ i₁)] [∀ i₂, UniformSpace (β₂ i₂)]
(e : ι₁ ≃ ι₂) (F : ∀ i₁, β₁ i₁ ≃ᵤ β₂ (e i₁)) : (∀ i₁, β₁ i₁) ≃ᵤ ∀ i₂, β₂ i₂ :=
(UniformEquiv.piCongrRight F).trans (UniformEquiv.piCongrLeft e) | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piCongr | `Equiv.piCongr` as a uniform isomorphism: this is the natural isomorphism
`Π i₁, β₁ i ≃ᵤ Π i₂, β₂ i₂` obtained from a bijection `ι₁ ≃ ι₂` and isomorphisms
`β₁ i₁ ≃ᵤ β₂ (e i₁)` for each `i₁ : ι₁`. |
ulift : ULift.{v, u} α ≃ᵤ α :=
{ Equiv.ulift with
uniformContinuous_toFun := uniformContinuous_comap
uniformContinuous_invFun := by
have hf : IsUniformInducing (@Equiv.ulift.{v, u} α).toFun := ⟨rfl⟩
simp_rw [hf.uniformContinuous_iff]
exact uniformContinuous_id } | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | ulift | Uniform equivalence between `ULift α` and `α`. |
@[simps! -fullyApplied]
funUnique (ι α : Type*) [Unique ι] [UniformSpace α] : (ι → α) ≃ᵤ α where
toEquiv := Equiv.funUnique ι α
uniformContinuous_toFun := Pi.uniformContinuous_proj _ _
uniformContinuous_invFun := uniformContinuous_pi.mpr fun _ => uniformContinuous_id | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | funUnique | If `ι` has a unique element, then `ι → α` is uniformly isomorphic to `α`. |
@[simps! -fullyApplied]
piFinTwo (α : Fin 2 → Type u) [∀ i, UniformSpace (α i)] : (∀ i, α i) ≃ᵤ α 0 × α 1 where
toEquiv := piFinTwoEquiv α
uniformContinuous_toFun := (Pi.uniformContinuous_proj _ 0).prodMk (Pi.uniformContinuous_proj _ 1)
uniformContinuous_invFun :=
uniformContinuous_pi.mpr <| Fin.forall_fin_two.2 ⟨uniformContinuous_fst, uniformContinuous_snd⟩ | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | piFinTwo | Uniform isomorphism between dependent functions `Π i : Fin 2, α i` and `α 0 × α 1`. |
@[simps! -fullyApplied]
finTwoArrow (α : Type*) [UniformSpace α] : (Fin 2 → α) ≃ᵤ α × α :=
{ piFinTwo fun _ => α with toEquiv := finTwoArrowEquiv α } | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | finTwoArrow | Uniform isomorphism between `α² = Fin 2 → α` and `α × α`. |
image (e : α ≃ᵤ β) (s : Set α) : s ≃ᵤ e '' s where
uniformContinuous_toFun := (e.uniformContinuous.comp uniformContinuous_subtype_val).subtype_mk _
uniformContinuous_invFun :=
(e.symm.uniformContinuous.comp uniformContinuous_subtype_val).subtype_mk _
toEquiv := e.toEquiv.image s | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | image | A subset of a uniform space is uniformly isomorphic to its image under a uniform isomorphism. |
Equiv.toUniformEquivOfIsUniformInducing [UniformSpace α] [UniformSpace β] (f : α ≃ β)
(hf : IsUniformInducing f) : α ≃ᵤ β :=
{ f with
uniformContinuous_toFun := hf.uniformContinuous
uniformContinuous_invFun := hf.uniformContinuous_iff.2 <| by simpa using uniformContinuous_id } | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.Pi"
] | Mathlib/Topology/UniformSpace/Equiv.lean | Equiv.toUniformEquivOfIsUniformInducing | A uniform inducing equiv between uniform spaces is a uniform isomorphism. |
CompactSpace.uniformContinuous_of_continuous [CompactSpace α] {f : α → β}
(h : Continuous f) : UniformContinuous f :=
calc map (Prod.map f f) (𝓤 α)
= map (Prod.map f f) (𝓝ˢ (diagonal α)) := by rw [nhdsSet_diagonal_eq_uniformity]
_ ≤ 𝓝ˢ (diagonal β) := (h.prodMap h).tendsto_nhdsSet mapsTo_prodMap_diagonal
_ ≤ 𝓤 β := nhdsSet_diagonal_le_uniformity | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | CompactSpace.uniformContinuous_of_continuous | Heine-Cantor: a continuous function on a compact uniform space is uniformly
continuous. |
IsCompact.uniformContinuousOn_of_continuous {s : Set α} {f : α → β} (hs : IsCompact s)
(hf : ContinuousOn f s) : UniformContinuousOn f s := by
rw [uniformContinuousOn_iff_restrict]
rw [isCompact_iff_compactSpace] at hs
rw [continuousOn_iff_continuous_restrict] at hf
exact CompactSpace.uniformContinuous_of_continuous hf | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | IsCompact.uniformContinuousOn_of_continuous | Heine-Cantor: a continuous function on a compact set of a uniform space is uniformly
continuous. |
IsCompact.uniformContinuousAt_of_continuousAt {r : Set (β × β)} {s : Set α}
(hs : IsCompact s) (f : α → β) (hf : ∀ a ∈ s, ContinuousAt f a) (hr : r ∈ 𝓤 β) :
{ x : α × α | x.1 ∈ s → (f x.1, f x.2) ∈ r } ∈ 𝓤 α := by
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr
choose U hU T hT hb using fun a ha =>
exists_mem_nhds_ball_subset_of_mem_nhds ((hf a ha).preimage_mem_nhds <| mem_nhds_left _ ht)
obtain ⟨fs, hsU⟩ := hs.elim_nhds_subcover' U hU
apply mem_of_superset ((biInter_finset_mem fs).2 fun a _ => hT a a.2)
rintro ⟨a₁, a₂⟩ h h₁
obtain ⟨a, ha, haU⟩ := Set.mem_iUnion₂.1 (hsU h₁)
apply htr
refine ⟨f a, htsymm.mk_mem_comm.1 (hb _ _ _ haU ?_), hb _ _ _ haU ?_⟩
exacts [mem_ball_self _ (hT a a.2), mem_iInter₂.1 h a ha] | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | IsCompact.uniformContinuousAt_of_continuousAt | If `s` is compact and `f` is continuous at all points of `s`, then `f` is
"uniformly continuous at the set `s`", i.e. `f x` is close to `f y` whenever `x ∈ s` and `y` is
close to `x` (even if `y` is not itself in `s`, so this is a stronger assertion than
`UniformContinuousOn s`). |
Continuous.uniformContinuous_of_tendsto_cocompact {f : α → β} {x : β}
(h_cont : Continuous f) (hx : Tendsto f (cocompact α) (𝓝 x)) : UniformContinuous f :=
uniformContinuous_def.2 fun r hr => by
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr
obtain ⟨s, hs, hst⟩ := mem_cocompact.1 (hx <| mem_nhds_left _ ht)
apply
mem_of_superset
(symmetrize_mem_uniformity <|
(hs.uniformContinuousAt_of_continuousAt f fun _ _ => h_cont.continuousAt) <|
symmetrize_mem_uniformity hr)
rintro ⟨b₁, b₂⟩ h
by_cases h₁ : b₁ ∈ s; · exact (h.1 h₁).1
by_cases h₂ : b₂ ∈ s; · exact (h.2 h₂).2
apply htr
exact ⟨x, htsymm.mk_mem_comm.1 (hst h₁), hst h₂⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | Continuous.uniformContinuous_of_tendsto_cocompact | null |
HasCompactMulSupport.uniformContinuous_of_continuous {f : α → β} [One β]
(h1 : HasCompactMulSupport f) (h2 : Continuous f) : UniformContinuous f :=
h2.uniformContinuous_of_tendsto_cocompact h1.is_one_at_infty | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | HasCompactMulSupport.uniformContinuous_of_continuous | null |
ContinuousOn.tendstoUniformly [LocallyCompactSpace α] [CompactSpace β] [UniformSpace γ]
{f : α → β → γ} {x : α} {U : Set α} (hxU : U ∈ 𝓝 x) (h : ContinuousOn ↿f (U ×ˢ univ)) :
TendstoUniformly f (f x) (𝓝 x) := by
rcases LocallyCompactSpace.local_compact_nhds _ _ hxU with ⟨K, hxK, hKU, hK⟩
have : UniformContinuousOn ↿f (K ×ˢ univ) :=
IsCompact.uniformContinuousOn_of_continuous (hK.prod isCompact_univ)
(h.mono <| prod_mono hKU Subset.rfl)
exact this.tendstoUniformly hxK | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | ContinuousOn.tendstoUniformly | A family of functions `α → β → γ` tends uniformly to its value at `x` if `α` is locally compact,
`β` is compact and `f` is continuous on `U × (univ : Set β)` for some neighborhood `U` of `x`. |
Continuous.tendstoUniformly [WeaklyLocallyCompactSpace α] [CompactSpace β] [UniformSpace γ]
(f : α → β → γ) (h : Continuous ↿f) (x : α) : TendstoUniformly f (f x) (𝓝 x) :=
let ⟨K, hK, hxK⟩ := exists_compact_mem_nhds x
have : UniformContinuousOn ↿f (K ×ˢ univ) :=
IsCompact.uniformContinuousOn_of_continuous (hK.prod isCompact_univ) h.continuousOn
this.tendstoUniformly hxK | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | Continuous.tendstoUniformly | A continuous family of functions `α → β → γ` tends uniformly to its value at `x`
if `α` is weakly locally compact and `β` is compact. |
IsCompact.mem_uniformity_of_prod
{α β E : Type*} [TopologicalSpace α] [TopologicalSpace β] [UniformSpace E]
{f : α → β → E} {s : Set α} {k : Set β} {q : α} {u : Set (E × E)}
(hk : IsCompact k) (hf : ContinuousOn f.uncurry (s ×ˢ k)) (hq : q ∈ s) (hu : u ∈ 𝓤 E) :
∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ k, (f p x, f q x) ∈ u := by
apply hk.induction_on (p := fun t ↦ ∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ t, (f p x, f q x) ∈ u)
· exact ⟨univ, univ_mem, by simp⟩
· intro t' t ht't ⟨v, v_mem, hv⟩
exact ⟨v, v_mem, fun p hp x hx ↦ hv p hp x (ht't hx)⟩
· intro t t' ⟨v, v_mem, hv⟩ ⟨v', v'_mem, hv'⟩
refine ⟨v ∩ v', inter_mem v_mem v'_mem, fun p hp x hx ↦ ?_⟩
rcases hx with h'x|h'x
· exact hv p hp.1 x h'x
· exact hv' p hp.2 x h'x
· rcases comp_symm_of_uniformity hu with ⟨u', u'_mem, u'_symm, hu'⟩
intro x hx
obtain ⟨v, hv, w, hw, hvw⟩ :
∃ v ∈ 𝓝[s] q, ∃ w ∈ 𝓝[k] x, v ×ˢ w ⊆ f.uncurry ⁻¹' {z | (f q x, z) ∈ u'} :=
mem_nhdsWithin_prod_iff.1 (hf (q, x) ⟨hq, hx⟩ (mem_nhds_left (f q x) u'_mem))
refine ⟨w, hw, v, hv, fun p hp y hy ↦ ?_⟩
have A : (f q x, f p y) ∈ u' := hvw (⟨hp, hy⟩ : (p, y) ∈ v ×ˢ w)
have B : (f q x, f q y) ∈ u' := hvw (⟨mem_of_mem_nhdsWithin hq hv, hy⟩ : (q, y) ∈ v ×ˢ w)
exact hu' (prodMk_mem_compRel (u'_symm A) B) | lemma | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | IsCompact.mem_uniformity_of_prod | In a product space `α × β`, assume that a function `f` is continuous on `s × k` where `k` is
compact. Then, along the fiber above any `q ∈ s`, `f` is transversely uniformly continuous, i.e.,
if `p ∈ s` is close enough to `q`, then `f p x` is uniformly close to `f q x` for all `x ∈ k`. |
CompactSpace.uniformEquicontinuous_of_equicontinuous {ι : Type*} {F : ι → β → α}
[CompactSpace β] (h : Equicontinuous F) : UniformEquicontinuous F := by
rw [equicontinuous_iff_continuous] at h
rw [uniformEquicontinuous_iff_uniformContinuous]
exact CompactSpace.uniformContinuous_of_continuous h | theorem | Topology | [
"Mathlib.Topology.Algebra.Support",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.Topology.UniformSpace.Equicontinuity"
] | Mathlib/Topology/UniformSpace/HeineCantor.lean | CompactSpace.uniformEquicontinuous_of_equicontinuous | An equicontinuous family of functions defined on a compact uniform space is automatically
uniformly equicontinuous. |
more commonly encountered in the literature. The reason is that in our definition the
neighborhood `v` of `x` can depend on the entourage `u`; so our condition is *a priori* weaker than
the usual one, although the two conditions are equivalent if the domain is locally compact. See
`tendstoLocallyUniformlyOn_of_forall_exists_nhds` for the one-way implication; the equivalence
assuming local compactness is part of `tendstoLocallyUniformlyOn_TFAE`.
We adopt this weaker condition because it is more general but appears to be sufficient for
the standard applications of locally-uniform convergence (in particular, for proving that a
locally-uniform limit of continuous functions is continuous).
We also define variants for locally uniform convergence on a subset, called
`TendstoLocallyUniformlyOn F f p s`. | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | more | null |
TendstoLocallyUniformlyOn (F : ι → α → β) (f : α → β) (p : Filter ι) (s : Set α) :=
∀ u ∈ 𝓤 β, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | TendstoLocallyUniformlyOn | A sequence of functions `Fₙ` converges locally uniformly on a set `s` to a limiting function
`f` with respect to a filter `p` if, for any entourage of the diagonal `u`, for any `x ∈ s`, one
has `p`-eventually `(f y, Fₙ y) ∈ u` for all `y` in a neighborhood of `x` in `s`. |
TendstoLocallyUniformly (F : ι → α → β) (f : α → β) (p : Filter ι) :=
∀ u ∈ 𝓤 β, ∀ x : α, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | TendstoLocallyUniformly | A sequence of functions `Fₙ` converges locally uniformly to a limiting function `f` with respect
to a filter `p` if, for any entourage of the diagonal `u`, for any `x`, one has `p`-eventually
`(f y, Fₙ y) ∈ u` for all `y` in a neighborhood of `x`. |
tendstoLocallyUniformlyOn_univ :
TendstoLocallyUniformlyOn F f p univ ↔ TendstoLocallyUniformly F f p := by
simp [TendstoLocallyUniformlyOn, TendstoLocallyUniformly, nhdsWithin_univ] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | tendstoLocallyUniformlyOn_univ | null |
tendstoLocallyUniformlyOn_iff_forall_tendsto :
TendstoLocallyUniformlyOn F f p s ↔
∀ x ∈ s, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β) :=
forall₂_swap.trans <| forall₄_congr fun _ _ _ _ => by
simp_rw [mem_map, mem_prod_iff_right, mem_preimage]
nonrec theorem IsOpen.tendstoLocallyUniformlyOn_iff_forall_tendsto (hs : IsOpen s) :
TendstoLocallyUniformlyOn F f p s ↔
∀ x ∈ s, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β) :=
tendstoLocallyUniformlyOn_iff_forall_tendsto.trans <| forall₂_congr fun x hx => by
rw [hs.nhdsWithin_eq hx] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | tendstoLocallyUniformlyOn_iff_forall_tendsto | null |
tendstoLocallyUniformly_iff_forall_tendsto :
TendstoLocallyUniformly F f p ↔
∀ x, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β) := by
simp [← tendstoLocallyUniformlyOn_univ, isOpen_univ.tendstoLocallyUniformlyOn_iff_forall_tendsto] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | tendstoLocallyUniformly_iff_forall_tendsto | null |
tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe :
TendstoLocallyUniformlyOn F f p s ↔
TendstoLocallyUniformly (fun i (x : s) => F i x) (f ∘ (↑)) p := by
simp only [tendstoLocallyUniformly_iff_forall_tendsto, Subtype.forall', tendsto_map'_iff,
tendstoLocallyUniformlyOn_iff_forall_tendsto, ← map_nhds_subtype_val, prod_map_right]; rfl | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe | null |
protected TendstoUniformlyOn.tendstoLocallyUniformlyOn (h : TendstoUniformlyOn F f p s) :
TendstoLocallyUniformlyOn F f p s := fun u hu _ _ =>
⟨s, self_mem_nhdsWithin, by simpa using h u hu⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | TendstoUniformlyOn.tendstoLocallyUniformlyOn | null |
protected TendstoUniformly.tendstoLocallyUniformly (h : TendstoUniformly F f p) :
TendstoLocallyUniformly F f p := fun u hu _ => ⟨univ, univ_mem, by simpa using h u hu⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | TendstoUniformly.tendstoLocallyUniformly | null |
TendstoLocallyUniformlyOn.mono (h : TendstoLocallyUniformlyOn F f p s) (h' : s' ⊆ s) :
TendstoLocallyUniformlyOn F f p s' := by
intro u hu x hx
rcases h u hu x (h' hx) with ⟨t, ht, H⟩
exact ⟨t, nhdsWithin_mono x h' ht, H.mono fun n => id⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | TendstoLocallyUniformlyOn.mono | null |
tendstoLocallyUniformlyOn_iUnion {ι' : Sort*} {S : ι' → Set α} (hS : ∀ i, IsOpen (S i))
(h : ∀ i, TendstoLocallyUniformlyOn F f p (S i)) :
TendstoLocallyUniformlyOn F f p (⋃ i, S i) :=
(isOpen_iUnion hS).tendstoLocallyUniformlyOn_iff_forall_tendsto.2 fun _x hx =>
let ⟨i, hi⟩ := mem_iUnion.1 hx
(hS i).tendstoLocallyUniformlyOn_iff_forall_tendsto.1 (h i) _ hi | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | tendstoLocallyUniformlyOn_iUnion | null |
tendstoLocallyUniformlyOn_biUnion {s : Set γ} {S : γ → Set α} (hS : ∀ i ∈ s, IsOpen (S i))
(h : ∀ i ∈ s, TendstoLocallyUniformlyOn F f p (S i)) :
TendstoLocallyUniformlyOn F f p (⋃ i ∈ s, S i) :=
tendstoLocallyUniformlyOn_iUnion (fun i => isOpen_iUnion (hS i))
fun i ↦ tendstoLocallyUniformlyOn_iUnion (hS i) (h i) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | tendstoLocallyUniformlyOn_biUnion | null |
tendstoLocallyUniformlyOn_sUnion (S : Set (Set α)) (hS : ∀ s ∈ S, IsOpen s)
(h : ∀ s ∈ S, TendstoLocallyUniformlyOn F f p s) : TendstoLocallyUniformlyOn F f p (⋃₀ S) := by
rw [sUnion_eq_biUnion]
exact tendstoLocallyUniformlyOn_biUnion hS h | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | tendstoLocallyUniformlyOn_sUnion | null |
TendstoLocallyUniformlyOn.union (hs₁ : IsOpen s) (hs₂ : IsOpen s')
(h₁ : TendstoLocallyUniformlyOn F f p s) (h₂ : TendstoLocallyUniformlyOn F f p s') :
TendstoLocallyUniformlyOn F f p (s ∪ s') := by
rw [← sUnion_pair]
refine tendstoLocallyUniformlyOn_sUnion _ ?_ ?_ <;> simp [*] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | TendstoLocallyUniformlyOn.union | null |
protected TendstoLocallyUniformly.tendstoLocallyUniformlyOn
(h : TendstoLocallyUniformly F f p) : TendstoLocallyUniformlyOn F f p s :=
(tendstoLocallyUniformlyOn_univ.mpr h).mono (subset_univ _) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | TendstoLocallyUniformly.tendstoLocallyUniformlyOn | null |
tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace [CompactSpace α] :
TendstoLocallyUniformly F f p ↔ TendstoUniformly F f p := by
refine ⟨fun h V hV => ?_, TendstoUniformly.tendstoLocallyUniformly⟩
choose U hU using h V hV
obtain ⟨t, ht⟩ := isCompact_univ.elim_nhds_subcover' (fun k _ => U k) fun k _ => (hU k).1
replace hU := fun x : t => (hU x).2
rw [← eventually_all] at hU
refine hU.mono fun i hi x => ?_
specialize ht (mem_univ x)
simp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right] at ht
obtain ⟨y, ⟨hy₁, hy₂⟩, hy₃⟩ := ht
exact hi ⟨⟨y, hy₁⟩, hy₂⟩ x hy₃ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergence"
] | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace | On a compact space, locally uniform convergence is just uniform convergence. |
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