Countable Space is Separable

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $S$ is a countable set.

Then $T$ is a separable space.

Proof
By definition, a topological space $T = \left({S, \vartheta}\right)$ is separable if there exists a countable subset of $S$ which is everywhere dense in $T$.

The closure of $S$ in $S$ is trivially $S$.

So, by definition, $S$ is everywhere dense in $S$.

As $S$ is countable by definition, the result follows.