Discrete Space is Separable iff Countable

Theorem
Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Then:
 * $T$ is separable $S$ is countable.

Sufficient Condition
Let $T$ be separable.

By Space is Separable iff Density not greater than Aleph Zero:
 * $ d \left({T}\right) \aleph_0$

where
 * $d \left({T}\right)$ denotes the density of $T$,
 * $\aleph$ denotes the aleph mapping.

By definition of {[Definition:Density of Topological Space|density]]:
 * $\exists A \subseteq S: A$ is dense $\land d \left({T}\right) = \left\vert{A}\right\vert$

where $ \left\vert{A}\right\vert$ denotes the cardinality of $A$.

By definition of dense set:
 * $A^- = S$

where $A^-$ denotes the closure of $A$.

By Set in Discrete Topology is Clopen:
 * $A$ is closed

Then by Closed Set equals its Topological Closure:
 * $A^- = A$

Thus by Set is Countable iff Cardinality not greater Aleph Zero
 * $S$ is [Definition:Countable Set|countable]].

{{qed|lemma}

Necessary Condition
Let $S$ be countable.

By Underlying Set of Topological Space is Everywhere Dense
 * $S$ is dense

Thus by definition
 * $T$ is separable.