# Paracompact Space is Countably Paracompact

## 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.

$\blacksquare$