Countable Space is Separable
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Theorem
Let $T = \struct {S, \tau}$ be a topological space where $S$ is a countable set.
Then $T$ is a separable space.
Proof
By definition, a topological space $T = \struct {S, \tau}$ 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.
$\blacksquare$