Hilbert Cube is Second-Countable

Theorem
Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.

Then $M$ is a second-countable space.

Proof
From Hilbert Cube is Separable, $M$ is a separable space.

We also have that Hilbert Cube is Metric Space.

The result follows from Separable Metric Space is Second-Countable.