Countable Space is Sigma-Compact

Theorem
Let $T = \left({X, \tau}\right)$ be a topological space, where $X$ is a countable set.

Then $T$ is $\sigma$-compact.

Proof
From Finite Space Satisfies All Compactness Properties, for every $p \in X$, $\left\{{p}\right\}$ is compact.

Then $\displaystyle X = \bigcup_{p \in X} \left\{{p}\right\}$ is a countable union of compact sets.

Thus by definition $T$ is $\sigma$-compact.