# Countable Complement Space is not Countably Metacompact

## Theorem

Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.

Then $T$ is not countably metacompact.

## Proof

So for any open cover $\CC$ of $T$, every point is in an infinite number of open sets of $T$.

So no refinement of any open cover of $T$ can be point finite.

Hence the result by definition of countably metacompact.

$\blacksquare$