Discrete Space is Separable iff Countable

Theorem

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

Then:

$T$ is separable if and only if $S$ is countable.

Proof

Sufficient Condition

Immediate from Separable Discrete Space is Countable.

$\Box$

Necessary Condition

Immediate from Countable Space is Separable.

$\blacksquare$

Sources

• 1989: Ryszard Engelking: General Topology (revised and completed ed.)

• Mizar article TOPGEN_4:11