Countable Discrete Space is Separable

Theorem

Let $T = \struct {S, \tau}$ be a countable discrete topological space.

Then $T$ is separable.

Proof 1

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

From Countable Discrete Space is Second-Countable:

$T$ is second-countable.

From Second-Countable Space is Separable:

$T$ is separable.

$\blacksquare$

Proof 2

Follows immediately from Countable Space is Separable.

$\blacksquare$

Also see

• Uncountable Discrete Space is not Separable

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $2$. Countable Discrete Topology: $8$