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 $28$