Definition:Metrizable Topology

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

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

Then any topological space which is homeomorphic to such a $\left({S, \vartheta_{\left({S, d}\right)}}\right)$ is defined as metrizable.

Also see
Not all topological spaces are metrizable - see Indiscrete Topology Not Metrizable.