Closures of Elements of Locally Finite Set is Locally Finite

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

Let $\AA$ be a locally finite set of subsets of $T$.

Then:
 * $\set{A^- : A \in \AA}$ is locally finite in $T$

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