Union of Closures of Elements of Locally Finite Set is Closed

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

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

Then:
 * $\ds \paren {\bigcup \AA}^- = \bigcup \set{A^- : A \in \AA}$

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