# Compact Space is Paracompact

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## Theorem

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

Then $T$ is paracompact.

## Proof

From the definition, $T$ is compact if and only if every open cover of $S$ has a finite subcover.

From Subcover is Refinement of Cover, it follows that every open cover of $S$ has an open refinement which is locally finite.

This is precisely the definition of paracompact.

$\blacksquare$

## 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