Definition:Homeomorphism/Metric Spaces/Definition 3

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 $V \subseteq A_1$, $V$ is a closed set of $M_1$ $f \left[{V}\right]$ is a closed set of $M_2$.

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

 * Equivalence of Definitions of Homeomorphic Metric Spaces