Rationals are Everywhere Dense in Sorgenfrey Line

Theorem
$\Q$ is everywhere dense in the Sorgenfrey line.

Proof
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line.

Define:
 * $\BB := \set {\hointr x y: x, y \in \R}$

where $\hointr x y$ denotes the right half-open real interval between $x$ and $y$.

By definition of Sorgenfrey line:
 * $\BB$ is basis of $T$.

By definition of subset:
 * $\Q^- \subseteq \R$

where $\Q^-$ denotes the topological closure of $\Q$ in $T$.

By definition of set equality to prove the equality: $\Q^- = \R$, it is necessary to show:
 * $\R \subseteq \Q^-$

Let $x \in \R$.

By Characterization of Closure by Basis it suffices to prove that:
 * $\forall U \in \BB: x \in U \implies U \cap \Q \ne \O$

Let $U \in \BB$.

By definition of $\BB$:
 * $\exists y, z \in \R: U = \hointr y z$

Assume:
 * $x \in U$

By definition of half-open real interval:
 * $y \le x < z$

By Between two Real Numbers exists Rational Number:
 * $\exists q \in \Q: y < q < z$

By definition of half-open real interval:
 * $q \in U$

Thus by definitions of intersection and empty set:
 * $U \cap \Q \ne \O$