Urysohn's Metrization Theorem

Theorem
Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable.

Then $T$ is metrizable.

Also see

 * Metrizable Space is not necessarily Second-Countable, indicating that the converse does not hold.


 * Metrization of Regular Second Countable Space, for necessary and sufficient conditions for the metrization of regular second-countable spaces.