Definition:Generalized Ordered Space/Definition 3

Definition
Let $\left({X, \preceq}\right)$ be a totally ordered set.

Let $\tau$ be a topology for $X$.

Then $\left({X, \preceq, \tau}\right)$ is a generalized ordered space iff:
 * $\left({X, \tau}\right)$ is a Hausdorff space.
 * There exists a sub-basis for $\left({X, \tau}\right)$ each of whose elements is an upper set or lower set in $X$.