# Definition:Homeomorphism/Topological Spaces

< Definition:Homeomorphism(Redirected from Definition:Homeomorphism (Topological Spaces))

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## 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.

### 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:

- $\forall U \subseteq S_\alpha: U \in \tau_\alpha \iff f \sqbrk U \in \tau_\beta$

### 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.

## 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 known as

A **homeomorphism** is 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.

## Also see

- Results about
**homeomorphisms**in the context of**topological spaces**can be found**here**.