# Metacompact Space is Countably Metacompact

## Theorem

Let $T = \struct {S, \tau}$ be a metacompact space.

Then $T$ is a countably metacompact space.

## Proof

From the definition, $T$ is metacompact space iff every open cover of $S$ has an open refinement which is point finite.

This also applies to all countable open covers.

So every countable open cover of $S$ has an open refinement which is point finite.

This is precisely the definition for a countably metacompact space.

$\blacksquare$