Characterization of Paracompactness in T3 Space/Statement 6 implies Statement 2
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Theorem
Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ has an open $\sigma$-locally finite refinement then:
- every open cover of $T$ has a locally finite refinement
Proof
Let every open cover of $T$ have an open $\sigma$-locally finite refinement.
Let $\UU$ be an open cover of $T$.
Let $\VV$ be an open $\sigma$-locally finite refinement of $\UU$ by hypothesis.
From Sigma-Locally Finite Cover has Locally Finite Refinement:
- there exists a locally finite refinement $\AA$ of $\VV$
From Refinement of a Refinement is Refinement of Cover:
- $\AA$ is a locally finite refinement of $\UU$
Since $\UU$ was arbitrary, it follows that:
- every open cover of $T$ has a locally finite refinement
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology: Chapter $5$: Compact Spaces: $\S$Paracompactness: Lemma $34$
- 1970: Stephen Willard: General Topology: Chapter $6$: Compactness: $\S20$: Paracompactness: Theorem $20.7$