\begin{definition}[Definition:Homeomorphism/Metric Spaces/Definition 3]
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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 $M_1$ {{iff}} $f \sqbrk V$ is a closed set of $M_2$.
Then:
:$f$ is a '''homeomorphism'''
:$M_1$ and $M_2$ are '''homeomorphic'''.
\end{definition}