Theorems · Theorem · general topology

ContinuousOn.mono_dom

∀ {α : Type u_5} {β : Type u_6} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β},
  t₂ ≤ t₁ → ∀ {s : Set α} {f : α → β}, ContinuousOn f s → ContinuousOn f s

If a function is continuous on a set for some topologies, then it is continuous on the same set with respect to any finer topology on the source space.

Defined in

Mathlib.Topology.ContinuousOn

Cited by

1 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound