Metrization of Regular Second Countable Space

Theorem
Let $T = \struct {S, \tau}$ be a $T_1$ space.


 * $(1): \quad T$ is regular and second-countable
 * $(2): \quad T$ is homeomorphic to a metric subspace of the Hilbert cube $I^\omega$
 * $(3): \quad T$ is metrizable and separable