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 $T$.

Proof
From Closures of Elements of Locally Finite Set is Locally Finite:
 * $\set{A^- : A \in \AA}$ is also locally finite

From Union of Closed Locally Finite Set of Subsets is Closed:
 * $\bigcup \set{A^- : A \in \AA}$ is closed in $T$

We have:

From Closure of Union contains Union of Closures
 * $\ds \bigcup \set{A^- : A \in \AA} \subseteq \paren {\bigcup \AA}^-$

By definition of set equality:
 * $\ds \paren {\bigcup \AA}^- = \bigcup \set{A^- : A \in \AA}$