# Upper Set with no Smallest Element is Open in GO-Space

## Theorem

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

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

Then $U$ is open in $\left({S, \preceq, \tau}\right)$.

## Proof

$U = \bigcup \left\{{u^\succ: u \in U}\right\}$

where $u^\succ$ is the strict upper closure of $u$.

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

$\blacksquare$