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

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

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

Let $G$ be open in $T$.

Then:
 * $G$ is an $F_\sigma$ set

Proof
Let:
 * $\CC = \set{B \in \BB : B^- \subseteq G}$

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

Lemma 2
Let $\BB = \ds \bigcup_{n \mathop \in \N} \BB_n$ be a $\sigma$-locally finite basis where $\BB_n$ is a locally finite set of subsets for each $n \in \N$.

For each $n \in \N$, let:
 * $\CC_n = \CC \cap \BB_n$

From Subset of Locally Finite Set of Subsets is Locally Finite:
 * $\CC_n$ is locally finite

For each $n \in \N$, let:
 * $C_n = \cup \set{C^- : C \in \CC_n}$

From Union of Closures of Elements of Locally Finite Set is Closed:
 * For each $n \in \N$, $C_n$ is closed in $T$

From Union of Subsets is Subset:
 * $\forall n \in \N: C_n \subseteq G$

and
 * $\ds \bigcup_{n \mathop \in \N} C_n \subseteq G$

By definition of cover:
 * $\forall x \in G : \exists n \in \N : \exists C \in \CC_n : x \in C^-$

From Set is Subset of Union:
 * $\forall n \in \N, C \in \CC_n : C^- \subseteq C_n \subseteq \ds \bigcup_{n \mathop \in \N} C_n$

Hence:
 * $\forall x \in G : x \in \ds \bigcup_{n \mathop \in \N} C_n$

By definition of subset:
 * $G \subseteq \ds \bigcup_{n \mathop \in \N} C_n$

By definition of set equality:
 * $G = \ds \bigcup_{n \mathop \in \N} C_n$

Hence $G$ is an $F_\sigma$ set by definition.