Theorems · Theorem · general topology

OpenPartialHomeomorph.continuousOn_iff_continuousOn_comp_left

∀ {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
  [inst_2 : TopologicalSpace Z] (e : OpenPartialHomeomorph X Y) {f : Z → X} {s : Set Z},
  s ⊆ f ⁻¹' e.source → (ContinuousOn f s ↔ ContinuousOn (↑e ∘ f) s)

A function is continuous on a set if and only if its composition with an open partial homeomorphism on the left is continuous on the corresponding set.

Defined in

Mathlib.Topology.OpenPartialHomeomorph.Continuity

Cited by

1 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceTopologicalSpace