Indiscrete Space is Separable

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 1

By definition, $T$ is separable if and only if 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.

$\blacksquare$

Proof 2

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

The result follows from Second-Countable Space is Separable.

$\blacksquare$