Definition:Metrizable Topology

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

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

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

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

Linguistic Note
The UK English spelling of this is metrisable, but it is rarely found.