Equivalence of Definitions of Generalized Ordered Space/Definition 1 implies Definition 3

Theorem
Let $\struct {S, \preceq, \tau}$ be a generalized ordered space by Definition 1:

Then $\struct {S, \preceq, \tau}$ is a generalized ordered space by Definition 3:

Proof
 Let $\BB$ be a basis for $\tau$ consisting of convex sets.

Let:
 * $\SS = \set {U^\succeq: U \in \BB} \cup \set {U^\preceq: U \in \BB}$

where $U^\succeq$ and $U^\preceq$ denote the upper closure and lower closure respectively of $U$.

By Upper Closure is Upper Set and Lower Closure is Lower Set, the elements of $\SS$ are upper and lower sets.

It is to be shown that $\SS$ is a sub-basis for $\tau$.

By Upper and Lower Closures of Open Set in GO-Space are Open:
 * $\SS \subseteq \tau$

By Convex Set Characterization (Order Theory), each element of $\BB$ is the intersection of its upper closure with its lower closure.

Thus each element of $\BB$ is generated by $\SS$.

Thus $\SS$ is a sub-basis for $\tau$.