nLab
homeomorphism

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

Equality and Equivalence

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.

For more see at Introduction to Topology -- 1 the section Homeomorphisms- 1#Homeomorphisms).

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 April 20, 2017 10:25:11 by Urs Schreiber (46.183.103.8)