Characterization of Paracompactness in T3 Space/Lemma 2

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Theorem

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

Let $\AA$ be a locally finite cover of $T$.

Let $\BB = \set{A^- : A \in \AA}$ be locally finite, where $A^-$ denotes the closure of $A$ in $T$.

Then:

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

Proof

Let $x \in X$.

By definition of refinement:

$\AA$ is a cover of $X$

By definition of cover of set:

$\exists A \in \AA : x \in A$

From Set is Subset of its Topological Closure:

$A \subseteq A^-$

By definition of subset:

$x \in A^-$

By definition of $\BB$:

$A^- \in \BB$

Since $x$ was arbitrary, $\BB$ is a cover of $T$ by definition.

From Topological Closure is Closed, every element of $\BB$ is closed in $T$

$\blacksquare$