Metacompact Space is Countably Metacompact

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \tau}\right)$ 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.

$\blacksquare$


Sources