Indiscrete Space is Separable/Proof 1

Proof
By definition, $T$ is separable there exists a countable subset of $S$ which is everywhere dense in $T$.

Let $x \in T$.

Then $\set x \subseteq T$ and $\set x$ is (trivially) countable.

From Subset of Indiscrete Space is Everywhere Dense we have that $\set x$ is everywhere dense.

Hence the result by definition of separable space.