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 $20.7$