Characterization of Paracompactness in T3 Space/Statement 2 implies Statement 3

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Theorem

Let $T = \struct{X, \tau}$ be a $T_3$ space.

If every open cover of $T$ has a locally finite refinement then:

every open cover of $T$ has a closed locally finite refinement

Proof

Let every open cover of $T$ have a locally finite refinement.

Let $\UU$ be an open cover of $T$.

Let $\VV = \set{V \in \tau : \exists U \in \UU : V^- \subseteq U}$ where $V^-$ denotes the closure of $V$ in $T$.

Lemma 1

$\VV$ is an open cover of $T$

$\Box$

By assumption:

there exists a locally finite refinement $\AA$ of $\VV$.

Let:

$\BB = \set{A^- : A \in \AA}$

From Closures of Elements of Locally Finite Set is Locally Finite:

$\BB$ is locally finite

Lemma 2

$\BB$ is a cover of $T$ consisting of closed sets

$\Box$

Lemma 3

$\BB$ is a refinement of $\UU$

$\Box$

Since $\UU$ was an arbitrary open cover of $T$ it follows that:

every open cover of $T$ has a closed locally finite refinement.

$\blacksquare$

Sources

• 1955: John L. Kelley: General Topology: Chapter $5$: Compact Spaces: $\S$Paracompactness: Lemma $29$

• 1970: Stephen Willard: General Topology: Chapter $6$: Compactness: $\S20$: Paracompactness: Theorem $20.7$