Sorgenfrey Line is Perfectly Normal

Theorem

Let $T=(\R,\tau)$ 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 $T1$ space and that any closed set is $G_\delta$.

From T2 Space is T1 Space and Sorgenfrey Line is Hausdorff we get that the Sorgenfrey Line is a $T1$ space.

From Complement of F-Sigma Set is G-Delta Set it's the same to prove that any closed set is $G_\delta$ and that any open set is $F_\sigma$.

Let $W$ be any open set in $T$, and $O\subseteq W$ the interior of $W$ with respect to the metric space $\R=(\R,d)$ which is defined in Real Number Line is Metric Space.

We claim that $W\setminus O$ is countable:

From the definition of $T$, $W=\bigcup_{i\in I} [a_i,b_i)$ and from the definition of $\R$, $O=\bigcup_{i\in I}(a_i,b_i)$.

Note that for any two distinct points $x,y\in W\setminus O$, $\exists [x,h(x))\subseteq W$ and $\exists [y,h(y))\subseteq W$.

It is also true that $[x,h(x))\cap [y,h(y))=\emptyset$ because otherwise $x<y<h(x)$ or $y<x<h(y)$ which implies $y\in (x,h(x))\subseteq O$ or $x\in (y,h(y))\subseteq O$; eitherway a contradiction by definition of $x$ and $y$.

From the definition of real numbers there is a rational number $q(x)$ such that $x<q(x)<h(x)$, hence there is an injective mapping from $W\setminus O$ to a countable set. From Injection from Infinite to Countably Infinite Set it follows that $W\setminus O$ is countable.

From Metric Space is Perfectly T4 $O$ is a $F_\sigma$ set in $\R$ and thus in the Sorgenfrey Line from Sorgenfrey Line is Expansion of Real Line.

Since $W\setminus O$ is countable is the union of the singletons; which are closed because the space is $T1$, thus $W=O\cup (W\setminus O)$ is also $F_\sigma$ set.

$\blacksquare$