Characterization of Paracompactness in T3 Space/Statement 3 implies Statement 4

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Theorem

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

If every open cover of $T$ has a closed locally finite refinement then:

every open cover of $T$ is even

Proof

This follows immediately from Open Cover with Closed Locally Finite Refinement is Even Cover.

$\blacksquare$

Sources

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