Characterization of Paracompactness in T3 Space/Lemma 3

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Theorem

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

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

Let:

$\VV = \set{V \in \tau : \exists U \in \UU : V^- \subseteq U}$ be cover of $T$

where $V^-$ denotes the closure of $V$ in $T$.

Let $\AA$ be a locally finite refinement of $\VV$.

Let $\BB = \set{A^- : A \in \AA}$ be locally finite.

Then:

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

Proof

Lemma 2

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

$\Box$

Let $B \in \BB$.

By definition of $\BB$:

$\exists A \in \AA : A^- = B$

By definition of refinement:

$\exists V \in \VV : A \subseteq V$

From Set Closure Preserves Set Inclusion:

$B = A^- \subseteq V^-$

By definition of $\VV$:

$\exists U \in \UU : V^- \subseteq U$

From Subset Relation is Transitive:

$B \subseteq U$

It follows that $\BB$ is a refinement of $\UU$ by definition.

$\blacksquare$