# Countable Complement Space is not Sigma-Compact

## Theorem

Let $T = \struct {S, \tau}$ 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.

$\blacksquare$