Definition:Generalized Ordered Space/Definition 3

Definition
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 :


 * $(1): \quad \struct {S, \tau}$ 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 $S$.

Also see

 * Equivalence of Definitions of Generalized Ordered Space