Definition:Metrizable Topology

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.

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