# Countably Metacompact Lindelöf Space is Metacompact

## Theorem

Let $T = \struct {S, \tau}$ be a Lindelöf space which is also countably metacompact.

Then $T$ is metacompact.

## Proof

By the definitions:

If $T = \struct {S, \tau}$ is a Lindelöf space then every open cover of $S$ has a countable subcover.
If $T = \struct {S, \tau}$ is a countably metacompact space then every countable open cover of $S$ has an open refinement which is point finite.

It follows trivially that every open cover of $S$ has an open refinement which is point finite.

Hence the result by definition of metacompact.

$\blacksquare$