# Definition:Homeomorphism/Topological Spaces/Definition 3

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## Contents

## Definition

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 $f$ is both an open mapping and a continuous mapping.

## Terminology

Let a **homeomorphism** exist between $T_\alpha$ and $T_\beta$.

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 see

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

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions