Characterization of Paracompactness in T3 Space/Statement 5 implies Statement 6
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Theorem
Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ has an open $\sigma$-discrete refinement then:
- every open cover of $T$ has an open $\sigma$-locally finite refinement
Proof
Follows immediately from Sigma-Discrete Set of Subsets is Sigma-Locally Finite.
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology: Chapter $5$: Compact Spaces: $\S$Paracompactness: Theorem $28$