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 if and only if:

$(1): \quad \struct {S, \tau}$ is a Hausdorff space

$(2): \quad$ there exists a sub-basis for $\struct {S, \tau}$ each of whose elements is an upper section or lower section in $S$.

Also see

• Equivalence of Definitions of Generalized Ordered Space