Sorgenfrey Line is Topology

Theorem
The Sorgenfrey Line is a topological space.

Proof
We have to check that $\BB = \set {\hointr a b: a, b \in \R}$ fulfills the axioms of being a basis for a topology.

By definition of synthetic basis we only have to check that:
 * $(1): \quad \bigcup \BB = \R$
 * $(2): \quad \forall B_1, B_2 \in \BB: \exists V \in \BB: V \subseteq B_1 \cap B_2$

We have that:
 * $\forall n \in \Z: \hointr n {n + 1} \in \BB$
 * $\R = \ds \bigcup_{n \mathop \in \Z} \hointr n {n + 1} \subseteq \bigcup \BB$

Hence $\R = \bigcup \BB$ and condition $(1)$ is fulfilled.

Now take $ B_1, B_2 \in \BB$ where:
 * $B_1 = \hointr {a_1} {b_1}$
 * $B_2 = \hointr {a_2} {b_2}$

Let $B_3$ be constructed as:
 * $B_3 := \hointr {\max \set {a_1, a_2} } {\min \set {b_1, b_2} } \in \BB$

From the method of construction, it is clear that $B_3 = B_1 \cap B_2$.

Thus taking $V = B_3$, condition $(2)$ is fulfilled.