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

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

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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $20$. Countable Complement Topology: $5$