Hilbert Cube is Second-Countable

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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.

$\blacksquare$


Sources