Definition:Order Topology/Definition 2

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

Define:


 * ${\Uparrow} \left({S}\right) = \left\{{ {\dot\uparrow} \left({s}\right): s \in S}\right\}$
 * ${\Downarrow} \left({S}\right) = \left\{{ {\dot\downarrow} \left({s}\right): s \in S}\right\}$

where ${\dot\uparrow} \left({s}\right)$ and ${\dot \downarrow} \left({s}\right)$ denote strict up-set and strict down-set of $s$, respectively.

Let $\tau$ be the topology on $S$ generated by ${\Uparrow} \left({S}\right) \cup {\Downarrow} \left({S}\right)$.

Then $\tau$ is called the order topology on $S$.