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.

$\blacksquare$