Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2

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Theorem

Let $T = \struct{X, \tau}$ be a topological space.

If $T$ is paracompact then:

every open cover of $T$ has a locally finite refinement

Proof

Let $T$ be paracompact.

By definition of paracompact:

every open cover of $S$ has an open refinement which is locally finite.

By definition of open refinement:

every open refinement of a cover is a refinement of the cover.

It follows that:

every open cover of $T$ has a locally finite refinement.

$\blacksquare$

Sources

• 1955: John L. Kelley: General Topology: Chapter $5$: Compact Spaces: $\S$ Paracompactness: Theorem $28$