Sorgenfrey Line is Perfectly Normal

Theorem
Let $T = \left({\R, \tau}\right)$ be the Sorgenfrey line.

Then $T$ is perfectly normal.

Proof
From the definition of perfectly normal space, it is necessary to prove that $T$ is a $T_1$ space and that any closed set is $G_\delta$.

From $T_2$ Space is $T_1$ Space and Sorgenfrey Line is Hausdorff:
 * the Sorgenfrey line is a $T_1$ space.

From Complement of $F_\sigma$ Set is $G_\delta$ Set it is sufficient to prove that an open set of $T$ is $F_\sigma$.

Let $W$ be any open set in $T$.

Let $O \subseteq W$ be the interior of $W$ with respect to the metric space:
 * $\R = \left({\R, d}\right)$

where $d$ is the usual metric on $\R$.

From the definition of $T$, for each $x \in W \setminus O$, we can choose $h_x \in W$ such that $\left[{x \,.\,.\, h_x}\right) \subseteq W$.

Suppose $\left[{x \,.\,.\, h_x}\right) \cap \left[{y \,.\,.\, h_y}\right) \neq \varnothing$ for some distinct points $x, y \in W \setminus O$.

Then either $x < y < h_x$ or $y < x < h_y$. If $x < y < h_x$, then $y \in \left({x \,.\,.\, h_x}\right) \subseteq O$, contradicting $y \in W \setminus O$.

Similarly, if $y < x < h_y$, then $x \in \left({y \,.\,.\, h_y}\right) \subseteq O$, contradicting $x \in W \setminus O$.

Thus $\left \langle{\left[{x \,.\,.\, h_x}\right)}\right \rangle_{x \mathop \in W \setminus O}$ is a pairwise disjoint indexed family of open sets of $T$.

From Sorgenfrey Line is Separable and Separable Space satisfies Countable Chain Condition:
 * $\left\{{\left[{x \,.\,.\, h_x}\right) : x \in W \setminus O}\right\}$ is countable

and thus:
 * $W \setminus O$ is countable.

From Metric Space is Perfectly T4:
 * $O$ is an $F_\sigma$ set in $\R$.

Thus from Sorgenfrey Line is Expansion of Real Line:
 * $O$ is an $F_\sigma$ set in the Sorgenfrey line.

Since $W \setminus O$ is a countable union of singletons and $T$ is a $T_1$ space:
 * $W \setminus O$ is an $F_\sigma$ set in $T$.

Since $W = O \cup \left({W \setminus O}\right)$ and $F_\sigma$ sets are closed under unions:
 * $W$ is an $F_\sigma$ set in $T$.