Indiscrete Space is Separable
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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.
$\blacksquare$