Theorems · Definition · general topology

Equiv.toHomeomorphOfContinuousClosed

{X : Type u_1} →
  {Y : Type u_2} →
    [inst : TopologicalSpace X] → [inst_1 : TopologicalSpace Y] → (e : X ≃ Y) → Continuous ⇑e → IsClosedMap ⇑e → X ≃ₜ Y

If a bijective map e : X ≃ Y is continuous and open, then it is a homeomorphism.

Defined in

Mathlib.Topology.Homeomorph.Defs

Cited by

4 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace