Theorems · Theorem · general topology

UniformOnFun.postcomp_uniformContinuous

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : UniformSpace β] {𝔖 : Set (Set α)} [inst_1 : UniformSpace γ]
  {f : γ → β},
  UniformContinuous f → UniformContinuous (⇑(UniformOnFun.ofFun 𝔖) ∘ (fun x => f ∘ x) ∘ ⇑(UniformOnFun.toFun 𝔖))

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.

Defined in

Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Cited by

1 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

UniformSpaceUniformSpace