Theorems · Theorem · general topology

ContinuousMap.continuous_iff_continuous_uniformFun

∀ {α : Type u₁} {β : Type u₂} [inst : TopologicalSpace α] [inst_1 : UniformSpace β] [CompactSpace α] {X : Type u_1}
  [inst_3 : TopologicalSpace X] (f : X → C(α, β)), Continuous f ↔ Continuous fun x => UniformFun.ofFun ⇑(f x)

When α is compact, f : X → C(α, β) is continuous if any only if it is continuous when reinterpreted as a map f : X → α →ᵤ β.

Defined in

Mathlib.Topology.UniformSpace.CompactConvergence

Cited by

1 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceUniformSpaceCompactSpaceTopologicalSpace