# Definition:Homeomorphism

*This page is about homeomorphisms, the isomorphisms in topology. For other uses, see Definition:Isomorphism.*

*Not to be confused with Definition:Homomorphism.*

## Contents

## Definition

### Topological Spaces

Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces.

Let $f: T_\alpha \to T_\beta$ be a bijection.

$f$ is a **homeomorphism** if and only if both $f$ and $f^{-1}$ are continuous.

### Metric Spaces

The same definition applies to metric spaces:

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $f: A_1 \to A_2$ be a bijection such that:

- $f$ is continuous from $M_1$ to $M_2$
- $f^{-1}$ is continuous from $M_2$ to $M_1$.

Then:

- $f$ is a
**homeomorphism** - $M_1$ and $M_2$ are
**homeomorphic**.

### Manifolds

The same definition applies to manifolds:

Let $X$ and $Y$ be manifolds.

A **homeomorphism** of $X$ to $Y$ is a continuous bijection such that the inverse is also continuous.

## Also known as

Also known as:

- a
**topological equivalence**, usually used when the spaces in question are metric spaces - an
**isomorphism**, usually used when the spaces in question are manifolds.

## Caution

Not to be confused with homomorphism.

## Also see

- Results about
**Homeomorphisms**can be found here.