Countable Discrete Space is Lindelöf

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Let $S$ be a countable set, thereby making $\vartheta$ the countable discrete topology on $S$..

Then $T$ is a Lindelöf space.

If $S$ is uncountable, then $T$ is not a Lindelöf space.

Proof
We have:
 * A countable discrete space is $\sigma$-compact.
 * A $\sigma$-compact space is Lindelöf.

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