Upper and Lower Closures of Open Set in GO-Space are Open

Theorem
Let $(X, \preceq, \tau)$ be a Generalized Ordered Space/Definition 1.

Let $A$ be open in $X$.

Then the upper and lower closures of $A$ are open.

Proof
We will show that the upper closure, $U$, of $A$ is open. The lower closure can be proven open by the same method.

By the definition of upper closure, $U = \{ u \in X: \exists a \in A: a \preceq u \}$.

But then

By Open Ray is Open in GO-Space/Definition 1, each ${\uparrow}a$ is open.

Thus $U$ is a union of open sets, and thus open by the definition of a topology.