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

Theorem
Let $\struct {X, \preceq, \tau}$ 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:
 * $U = \set {u \in X: \exists a \in A: a \preceq u}$

But then:

where $a^\preceq$ denotes the upper closure of $a$.

By Open Ray is Open in GO-Space/Definition 1, each $a^\succeq$ is open.

Thus $U$ is a union of open sets.

Thus $U$ is open by the definition of a topology.