Open Ray is Open in GO-Space

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.