Definition:Homeomorphism/Metric Spaces/Definition 4

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.