nLab homeomorphism

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 $f : X \to Y$ 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.

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 $Top$, 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, $Top$ is not a balanced category.

For example $\exp(2\pi i(-)) [0,1) \to S^1$ 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. .

Properties

Proposition

(homeomorphisms are the continuous and open bijections)

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

1. $f$ is a homeomorphism;

2. $f$ is a bijection and an open map;

3. $f$ 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 $Y \leftarrow X \colon g$ be continuous means that the pre-image function of $g$ sends open subsets to open subsets. But by $g$ being the inverse to $f$, that pre-image function is equal to $f$, regarded as a function on subsets:

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

Hence $g^{-1}$ sends opens to opens precisely if $f$ does, which is the case precisely if $f$ 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. .

Related concepts

• homeomorphism invariant

• homotopy equivalence

• weak homotopy equivalence

References

The terminology “homeomorphism” is due to:

• Henri Poincaré, Analysis Situs, Journal de l’École Polytechnique. (2). 1: 1–123 (1895) (gallica:12148/bpt6k4337198/f7, Engl: pdf, pdf)