Rationals are Everywhere Dense in Sorgenfrey Line

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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:

$U \cap \Q \ne \O$

$\blacksquare$


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