Countable Complement Space is not Countably Metacompact

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Theorem

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


Then $T$ is not countably metacompact.


Proof

From Uncountable Subset of Countable Complement Space Intersects Open Sets, the intersection of any open sets is uncountable.

So for any open cover $\mathcal C$ 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.

$\blacksquare$


Sources