Characterization of Paracompactness in T3 Space/Lemma 1
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Theorem
Let $T = \struct{X, \tau}$ be a $T_3$ space.
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 $T$.
Then:
- $\VV$ is an open cover of $T$
Proof
Let $x \in S$.
By definition of open cover:
- $\exists U \in \UU : x \in U$
From Characterization of T3 Space:
- $\exists V \in \tau : x \in V : V^- \subseteq U$
Hence:
- $V \in \VV$
Since $x$ was arbitrary, $\VV$ is an open cover by definition.
$\blacksquare$