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

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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

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

By Upper Set with no Minimal Element:

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

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.

$\blacksquare$


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