homeomorphism

**topology**
algebraic topology
## Basic concepts
* space
* locale
* topological space
* continuous map
* homeomorphism
* Top
* nice topological space
* nice category of spaces
* convenient category of topological spaces
* **homotopy theory**
* homotopy group
* covering space
## Theorems
* Whitehead's theorem
* Freudenthal suspension theorem
* nerve theorem
## Extra stuff, structure, properties
* CW-complex, Hausdorff space, second-countable space, sober space
* compact space, paracompact space
* connected space, locally connected space, contractible space, locally contractible space
* topological vector space, Banach space, Hilbert space
* manifold
## Examples
* point, real line, plane
* sphere, ball, annulus
* polytope, polyhedron
* loop space, path space
* Cantor space, Sierpinski space
* long line, Warsaw circle

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.

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

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.

Revised on May 24, 2010 10:05:52
by Urs Schreiber
(134.100.32.31)