# Definition:Homeomorphism/Metric Spaces/Definition 4

Jump to navigation
Jump to search

## Contents

## Definition

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:

- for all $a \in A_1$ and $N \subseteq A_1$, $N$ is a neighborhood of $a$ if and only if $f \left[{N}\right]$ is a neighborhood of $f \left({a}\right)$.

Then:

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

## Also known as

A **homeomorphism** is also known as a **topological equivalence**.

Two **homeomorphic** metric spaces can be described as **topologically equivalent**.

## Also see

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.7$: Subspaces and Equivalence of Metric Spaces: Theorem $7.10$