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 $20.7$