# nLab homeomorphism

topology

algebraic topology

## Examples

A homeomorphism (also spelt ‘homoeomorphism’ and ‘homœomorphism’ but not ‘homomorphism’) is an isomorphism in the category Top of topological spaces.

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.

Note that a continous bijection is not necessarily a homeomorphism; that is, $\mathrm{Top}$ is not a balanced category.

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

Revised on May 24, 2010 10:05:52 by Urs Schreiber (134.100.32.31)