nLab
homeomorphism

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

Note that a continous bijection is not necessarily a homeomorphism; that is, TopTop is not a balanced category.

The term ‘homeomorphism’ is also applied to isomorphisms in categories that generalise TopTop, 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)