Characterization of Paracompactness in T3 Space/Lemma 2

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 $T$