Indiscrete Space is Separable/Proof 2

Theorem

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space such that $S$ has more than one element.

Then $T$ is separable.

Proof

By Indiscrete Space is Second-Countable, $T$ is second-countable.

The result follows from Second-Countable Space is Separable.

$\blacksquare$