T3 Space with Sigma-Locally Finite Basis is T4 Space/Lemma 2

Theorem

Let $T = \struct {S, \tau}$ be a $T_3$ topological space.

Let $\BB$ be a $\sigma$-locally finite basis.

Let $F$ be a closed subset of $T$.

Let $x \in S \setminus F$.

Then:

$\exists U \in \BB : x \in U : U^- \cap F = \O$

Proof

From Characterization of T3 Space:

$\exists W \in \tau: x \in W : W^- \cap F = \O$

By definition of basis:

$\exists U \in \BB : x \in U \subseteq W$

From Topological Closure of Subset is Subset of Topological Closure:

$U^- \subseteq W^-$

From Set Intersection Preserves Subsets:

$U^- \cap F = \O$

$\blacksquare$