File: Mathlib/Topology/UniformSpace/Compact.lean

full_name: nhdsSet_diagonal_eq_uniformity
code: /-- On a compact uniform space, the topology determines the uniform structure, entourages are
exactly the neighborhoods of the diagonal. -/
theorem nhdsSet_diagonal_eq_uniformity [CompactSpace α] : 𝓝ˢ (diagonal α) = 𝓤 α 
goal: 𝓝ˢ (Set.diagonal α) = 𝓤 α

full_name: compactSpace_uniformity
code: /-- On a compact uniform space, the topology determines the uniform structure, entourages are
exactly the neighborhoods of the diagonal. -/
theorem compactSpace_uniformity [CompactSpace α] : 𝓤 α = ⨆ x, 𝓝 (x, x) 
goal: 𝓤 α = ⨆ x, 𝓝 (x, x)

full_name: unique_uniformity_of_compact
code: theorem unique_uniformity_of_compact [t : TopologicalSpace γ] [CompactSpace γ]
    {u u' : UniformSpace γ} (h : u.toTopologicalSpace = t) (h' : u'.toTopologicalSpace = t) :
    u = u' 
goal: u = u'

full_name: uniformSpaceOfCompactT2
code: /-- The unique uniform structure inducing a given compact topological structure. -/
def uniformSpaceOfCompactT2 [TopologicalSpace γ] [CompactSpace γ] [T2Space γ] : UniformSpace γ 
goal: UniformSpace γ

full_name: CompactSpace.uniformContinuous_of_continuous
code: /-- Heine-Cantor: a continuous function on a compact uniform space is uniformly
continuous. -/
theorem CompactSpace.uniformContinuous_of_continuous [CompactSpace α] {f : α → β}
    (h : Continuous f) : UniformContinuous f 
goal: UniformContinuous f

full_name: IsCompact.uniformContinuousOn_of_continuous
code: /-- Heine-Cantor: a continuous function on a compact set of a uniform space is uniformly
continuous. -/
theorem IsCompact.uniformContinuousOn_of_continuous {s : Set α} {f : α → β} (hs : IsCompact s)
    (hf : ContinuousOn f s) : UniformContinuousOn f s 
goal: UniformContinuousOn f s

full_name: IsCompact.uniformContinuousAt_of_continuousAt
code: /-- 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`). -/
theorem 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 
goal: {x | x.1 ∈ s → (f x.1, f x.2) ∈ r} ∈ 𝓤 α

full_name: Continuous.uniformContinuous_of_tendsto_cocompact
code: theorem Continuous.uniformContinuous_of_tendsto_cocompact {f : α → β} {x : β}
    (h_cont : Continuous f) (hx : Tendsto f (cocompact α) (𝓝 x)) : UniformContinuous f 
goal: UniformContinuous f

full_name: HasCompactSupport.uniformContinuous_of_continuous
code: @[to_additive]
theorem HasCompactMulSupport.uniformContinuous_of_continuous {f : α → β} [One β]
    (h1 : HasCompactMulSupport f) (h2 : Continuous f) : UniformContinuous f 
goal: UniformContinuous f

full_name: HasCompactMulSupport.uniformContinuous_of_continuous
code: @[to_additive]
theorem HasCompactMulSupport.uniformContinuous_of_continuous {f : α → β} [One β]
    (h1 : HasCompactMulSupport f) (h2 : Continuous f) : UniformContinuous f 
goal: UniformContinuous f

full_name: ContinuousOn.tendstoUniformly
code: /-- 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`. -/
theorem 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) 
goal: TendstoUniformly f (f x) (𝓝 x)

full_name: Continuous.tendstoUniformly
code: /-- 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. -/
theorem Continuous.tendstoUniformly [WeaklyLocallyCompactSpace α] [CompactSpace β] [UniformSpace γ]
    (f : α → β → γ) (h : Continuous ↿f) (x : α) : TendstoUniformly f (f x) (𝓝 x) 
goal: TendstoUniformly f (f x) (𝓝 x)

full_name: IsCompact.mem_uniformity_of_prod
code: /-- 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`. -/
lemma 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 
goal: ∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ k, (f p x, f q x) ∈ u

full_name: CompactSpace.uniformEquicontinuous_of_equicontinuous
code: /-- An equicontinuous family of functions defined on a compact uniform space is automatically
uniformly equicontinuous. -/
theorem CompactSpace.uniformEquicontinuous_of_equicontinuous {ι : Type*} {F : ι → β → α}
    [CompactSpace β] (h : Equicontinuous F) : UniformEquicontinuous F 
goal: UniformEquicontinuous F