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:
 * a closed set of $T$ is $G_\delta$
 * 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 $\R$.

It is to be shown that that $W \setminus O$ is countable, as follows:

From the definition of $T$:
 * $\displaystyle W = \bigcup_{i \mathop \in I} \left[{a_i \,.\,.\, b_i}\right)$

and from the definition of $\R$:
 * $\displaystyle O = \bigcup_{i \mathop \in I} \left({a_i \,.\,.\, b_i}\right)$

Note that for any two distinct points $x, y \in W \setminus O$:
 * $\exists \left[{x \,.\,.\, h \left({x}\right)}\right) \subseteq W$

and:
 * $\exists \left[{y \,.\,.\, h \left({y}\right)}\right) \subseteq W$

It is also true that:
 * $\left[{x \,.\,.\, h \left({x}\right)}\right) \cap \left[{y \,.\,.\, h \left({y}\right)}\right) = \varnothing$

because otherwise $x < y < h \left({x}\right)$ or $y < x < h\left({y}\right)$ which implies $y \in \left({x \,.\,.\, h \left({x}\right)}\right) \subseteq O$ or $x \in \left({y \,.\,.\, h \left({y}\right)}\right) \subseteq O$; eitherway a contradiction by definition of $x$ and $y$.

From the definition of real numbers there is a rational number $q \left({x}\right)$ such that:
 * $x < q \left({x}\right) < h \left({x}\right)$

Hence there is an injective mapping from $W \setminus O$ to a countable set.

From Domain of Injection to Countable Set is Countable it follows that $W \setminus O$ is countable.

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

Thus from Sorgenfrey Line is Expansion of Real Line:
 * $O$ is in the Sorgenfrey line.

Since $W \setminus O$ is countable is the union of the singletons.

These are closed because $T$ is a $T_1$ space.

Thus $W = O \cup \left({W \setminus O}\right)$ is also an $F_\sigma$ set.