Open Ray is Open in GO-Space/Definition 1

Theorem
Let $\left({X, \preceq, \tau}\right)$ be a generalized ordered space.

Let $p \in X$.

Then ${\dot\downarrow} p$ and ${\dot\uparrow} p$ are $\tau$-open.

Here, ${\dot\downarrow}p$ and ${\dot\uparrow}p$ are the strict down-set and strict up-set of $p$, respectively.

Proof
We will prove that $U = {\dot\uparrow} p$ is open.

That ${\dot\downarrow}p$ is open will follow by duality.

Let $u \in U$.

Since $p \notin U$, $p \ne u$.

By the definition of GO-space, $\tau$ is Hausdorff, and therefore $\mathrm T_1$.

Thus by the definition of GO-space, there is an open, convex set $M$ such that $u \in M$ and $p \notin M$.

Next we will show that $M \subseteq U$:

Let $x \in X \setminus U$.

Then $x \preceq p \preceq u$.

Suppose for the sake of contradiction that $x \in M$.

Since $x, u \in M$, $p \in M$ because $M$ is convex, contradicting the choice of $M$.

Thus $x \notin M$.

Since this hold for all $x \in X \setminus U$, $M \subseteq U$.

Thus $U$ contains a neighborhood of each of its points, so it is open.