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

Theorem

Let $\struct {S, \preceq, \tau}$ be a generalized ordered space.

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

Then $U$ is open in $\struct {S, \preceq, \tau}$.

Proof

By Minimal Element in Toset is Unique and Smallest, $U$ has no minimal element.

By Upper Section with no Minimal Element:

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

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.

$\blacksquare$

Also see

• Lower Section with no Greatest Element is Open in GO-Space