\begin{definition}[Definition:Generalized Ordered Space/Definition 3]
Let $\struct {S, \preceq}$ be a totally ordered set.
Let $\tau$ be a topology on $S$.
$\struct {S, \preceq, \tau}$ is a '''generalized ordered space''' {{iff}}:
:$(1): \quad \struct {S, \tau}$ is a Hausdorff space
:$(2): \quad$ there exists a sub-basis for $\struct {S, \tau}$ each of whose elements is an upper set or lower set in $S$.
\end{definition}