# Paracompact Space is Countably Paracompact

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## Theorem

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

Then $T$ is a countably paracompact space.

## Proof

From the definition, $T$ is paracompact space if and only if every open cover of $T$ has an open refinement which is locally finite.

This also applies to all countable open covers.

So every countable open cover of $T$ has an open refinement which is locally finite.

This is precisely the definition for a countably paracompact space.

$\blacksquare$

## 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