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

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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

$\CC$ is a is a cover of $G$

$\Box$

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.

$\blacksquare$