Countably Metacompact Lindelöf Space is Metacompact

Theorem
Let $T = \left({X, \vartheta}\right)$ be a Lindelöf space which is also countably metacompact.

Then $T$ is metacompact.

Proof
By the definitions:
 * If $T = \left({X, \vartheta}\right)$ is a Lindelöf space then every open cover of $X$ has a countable subcover.


 * If $T = \left({X, \vartheta}\right)$ is a countably metacompact space then every countable open cover of $X$ has an open refinement which is point finite.

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

Hence the result by definition of metacompact.