Definition:Metrizable Topology

Definition
Let $\left({S, d}\right)$ be a metric space.

Let $\left({S, \tau}\right)$ be the topological space induced by $d$.

Then for any topological space which is homeomorphic to such a $\left({S, \tau}\right)$, it and its topology are defined as metrizable.

Also see

 * Indiscrete Topology is not Metrizable: thus, not all topological spaces are metrizable