Countable Discrete Space is Lindelöf

Theorem

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

Then $T$ is a Lindelöf space.

Proof

We have:

Countable Discrete Space is $\sigma$-Compact

$\sigma$-Compact Space is Lindelöf

So if $S$ is countable, $T$ is a Lindelöf space.

$\blacksquare$

Also see

• Uncountable Discrete Space is not Lindelöf

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: $8$