Is the following statement true? A continuous bijection is a homeomorphism

No. id : (ℝ, discrete) → (ℝ, usual), or t ↦ (cos t, sin t) from [0, 2π) to S¹ Needs compact domain and Hausdorff codomain.

Counterexample

Is this true?

A continuous bijection is a homeomorphism

No — it is false.

The counterexample

id : $(\mathbb{R}$ , discrete $) \to (\mathbb{R}$ , usual), or t ↦ (cos t, sin t) from $[0, 2\pi)$ to $S^{1}$

Why it fails

Needs compact domain and Hausdorff codomain.

Keep practicing

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.