nLab
homeomorphism

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Equality and Equivalence

Contents

Definition

A homeomorphism (also spelt ‘homoeomorphism’ and ‘homœomorphism’ but nothomomorphism’) 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.

But see prop. 1.

Properties

Proposition

(homeomorphisms are the continuous and open bijections)

Let f:(X,τ X)(Y,τ Y)f \;\colon\; (X, \tau_X) \longrightarrow (Y,\tau_Y) be a continuous function between topological spaces. Then the following are equivalence:

  1. ff is a homeomorphism;

  2. ff is a bijection and an open map;

  3. ff is a bijection and a closed map.

Proof

It is clear from the definition that a homeomorphism in particular has to be a bijection. The condition that the inverse function YX:gY \leftarrow X \colon g be continuous means that the pre-image function of gg sends open subsets to open subsets. But by gg being the inverse to ff, that pre-image function is equal to ff, regarded as a function on subsets:

g 1=f:P(X)P(Y). g^{-1} = f \;\colon\; P(X) \to P(Y) \,.

Hence g 1g^{-1} sends opens to opens precisely if ff does, which is the case precisely if ff is an open map, by definition. This shows the equivalence of the first two items. The equivalence between the first and the third follows similarly via prop. \ref{ClosedSubsetContinuity}.

Revised on June 11, 2017 06:38:14 by Urs Schreiber (88.77.226.246)