Equivalence of Definitions of Generalized Ordered Space

Theorem
Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $\tau$ be a topology for $S$.

The following definitions for $\left({S, \preceq, \tau}\right)$ to be a generalized ordered space are equivalent:

Definition $(3)$ implies Definition $(2)$
This follows from GO-Space Embeds Densely into Linearly Ordered Space.