T3 Space with Sigma-Locally Finite Basis is T4 Space/Proof 1

Proof
Let $A$ and $B$ be disjoint closed subsets of $T$.

Lemma 1
From Lemma 1, there exists a countable open cover $\UU = \set{U_n : n \in \N}$ of $A$:
 * $\forall n \in \N : U_n^- \cap B = \O$

and there exists a countable open cover $\VV = \set{V_n : n \in \N}$ of $B$:
 * $\forall n \in \N : V_n^- \cap A = \O$

From Countable Open Covers Condition for Separated Sets:
 * $A$ and $B$ can be separated in $T$

By definition of separated:
 * there exists $U, V \in \tau$ such that $A \subseteq U, B \subseteq V$ and $U \cap V = \O$

Since $A$ and $B$ were arbitrary disjoint closed subsets of $T$, by definition, $T$ is a $T_4$ space.