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

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Theorem

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

$\struct {S, \preceq, \tau}$ 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 $S$.

Then $\struct {S, \preceq, \tau}$ is a generalized ordered space by Definition 1:

$\struct {S, \preceq, \tau}$ is a generalized ordered space if and only if:

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

$(2): \quad$ there exists a basis for $\struct {S, \tau}$ whose elements are convex in $S$.

Proof

Let $\SS$ be a sub-basis for $\tau$ consisting of upper sections and lower sections.

Let $\BB$ be the set of intersections of finite subsets of $\SS$.

By Upper Section is Convex, Lower Section is Convex and Intersection of Convex Sets is Convex Set (Order Theory) :

the elements of $\BB$ are convex.

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But $\BB$ is a basis for $\tau$.

Therefore $\tau$ has a basis consisting of convex sets.

$\blacksquare$