# Countably Metacompact Lindelöf Space is Metacompact

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## 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.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness