Countable Discrete Space is Separable

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Let $S$ be a countable set, thereby making $\vartheta$ the countable discrete topology on $S$..

Then $T$ is separable.

If $S$ is uncountable, then $T$ is not separable.

Proof
We have that a Countable Discrete Space is Second-Countable.

We also have that a second-countable space is separable.

So if $S$ is countable, $T$ is separable.

Let $S$ be uncountable.