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.

$\blacksquare$