Theorems · Theorem · general topology

ContinuousOn.tendstoUniformly

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : UniformSpace α] [inst_1 : UniformSpace β] [LocallyCompactSpace α]
  [CompactSpace β] [inst_4 : UniformSpace γ] {f : α → β → γ} {x : α} {U : Set α},
  U ∈ nhds x → ContinuousOn (↿f) (U ×ˢ Set.univ) → TendstoUniformly f (f x) (nhds x)

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.

Defined in

Mathlib.Topology.UniformSpace.HeineCantor

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

UniformSpaceUniformSpaceLocallyCompactSpaceCompactSpaceUniformSpace