That is, a homeomorphism $f : X \to Y$ is a continuous map of topological spaces such that there is an inverse$f^{-1}: Y \to X$ that is also a continuous map of topological spaces. Equivalently, $f$ is a bijection between the underlying sets such that both $f$ and its inverse are continuous.

The term ‘homeomorphism’ is also applied to isomorphisms in categories that generalise $Top$, such as the category of convergence spaces and the category of locales.

Counter-examples

Beware that a continous bijection is not necessarily a homeomorphism; that is, $Top$ is not a balanced category.

For example $\exp(2\pi i(-)) [0,1) \to S^1$ is continuous and the underlying function of sets is a bijection, but the inverse function at the level of sets is not continuous.