Corollary

Tag 0B97, permanent. Cite it as [Bourbaki, Tag 0B97].

General Topology, chapter I, § 9. Compact spaces and locally compact spaces

Let $X$ be a locally compact paracompact space. Then every open covering $\mathfrak{R}$ of $X$ has a locally finite open refinement $\mathfrak{R}'$ formed of relatively compact sets. If $X$ is $\sigma$ -compact then $\mathfrak{R}'$ can be taken to be countable.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

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