Countable Complement Space is not Sigma-Compact

Theorem
Let $T = \left({S, \tau}\right)$ be a countable complement topology on an uncountable set $S$.

Then $T$ is not a $\sigma$-compact space.

Proof
From Compact Sets in Countable Complement Space, the only compact sets in $T$ are finite.

A countable union of finite sets can not be an uncountable set.

Hence the result by definition of $\sigma$-compact space.