Compact Sets in Countable Complement Space

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

Then the only compact sets of $T$ are finite.

Proof
Let $H \subseteq S$.

Consider the open cover of $H$ defined as:


 * $\mathcal C = \left\{{S \setminus \left\{{x}\right\}: x \in H}\right\}$

If $S$ is finite then $H$ is a finite subcover of itself.

But if $H$ is countably infinite, there is no such finite subcover.

Hence the result.