Compact Space is Paracompact

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