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 $20.7$