nLab
homeomorphism

Contents

Definition

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.

See also at topology – Introduction.

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.

Counter-examples

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

For example exp(2πi())[0,1)S 1\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.

Revised on January 23, 2017 14:48:16 by Urs Schreiber (178.6.113.169)