Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 6

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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 an open $\sigma$-locally finite refinement

Proof

Let $T$ be paracompact.

By definition of paracompact:

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

From Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets

every open cover of $T$ has an open $\sigma$-locally finite refinement

$\blacksquare$

Sources

• 1970: Stephen Willard: General Topology: Chapter $6$: Compactness: $\S20$: Paracompactness: Theorem $20.7$