Upper Section with no Smallest Element is Open in GO-Space

Theorem
Let $(S, \preceq, \tau)$ be a generalized ordered set.

Let $U$ be an upper set in $S$ with no minimal element.

Then $U$ is open in $(S, \preceq, \tau)$.

Proof
By Upper Set with no Minimal Elements:
 * $U = \bigcup \left\{{ {\dot\uparrow}u: u \in U }\right\}$

where ${\dot\uparrow}u$ is the strict up-set of $u$.

By Open Ray is Open in GO-Space and the fact that a union of open sets is open, $U$ is open.