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 :
 * $(1): \quad \left({X, \tau}\right)$ is a Hausdorff space.
 * $(2): \quad$ there exists a sub-basis for $\left({X, \tau}\right)$ each of whose elements is an upper set or lower set in $X$.

Also see

 * Equivalence of Definitions of Generalized Ordered Space