Definition:Generalized Ordered Space/Definition 1

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 basis for $\left({X,\tau}\right)$ whose elements are convex in $X$.