Countable Discrete Space is Separable/Proof 1

Theorem

Let $T = \struct {S, \tau}$ be a countable discrete topological space.

Then $T$ is separable.

Proof

Let $T = \left({S, \tau}\right)$ be a countable discrete topological space.

From Countable Discrete Space is Second-Countable:

$T$ is second-countable.

From Second-Countable Space is Separable:

$T$ is separable.

$\blacksquare$