Theorems · Theorem · general topology

ContinuousMap.continuous_of_continuous_uncurry

∀ {X : Type u_2} {Y : Type u_3} {Z : Type u_4} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
  [inst_2 : TopologicalSpace Z] (f : X → C(Y, Z)), Continuous (Function.uncurry fun x y => (f x) y) → Continuous f

To show continuity of a map α → C(β, γ), it suffices to show that its uncurried form α × β → γ is continuous.

Defined in

Mathlib.Topology.CompactOpen

Cited by

11 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceTopologicalSpace