Definition:Metrizable Topology

Definition
Let $\struct {S, d}$ be a metric space.

Let $\struct {S, \tau}$ be the topological space induced by $d$.

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

Also see

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


 * Definition:Completely Metrizable Topology