# Definition:Homeomorphism/Topological Spaces

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## Contents

## Definition

Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces.

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

### Definition 1

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

### Definition 2

$f$ is a **homeomorphism** if and only if:

- $\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \left[{U}\right] \in \tau_\beta$

### Definition 3

$f$ is a **homeomorphism** if and only if $f$ is both an open mapping and a continuous mapping.

### Definition 4

$f$ is a **homeomorphism** if and only if $f$ is both a closed mapping and a continuous mapping.

If such a **homeomorphism** exists, then $T_\alpha$ and $T_\beta$ are said to be **homeomorphic**.

The symbolism $T_\alpha \sim T_\beta$ is often seen to denote that $T_\alpha$ is **homeomorphic** to $T_\beta$.

## Also known as

Also known as:

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

## Also see

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