Definition:Order Topology/Definition 2

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

Let:
 * $U = \left\{{ {\dot\uparrow} \left({s}\right): s \in S}\right\}$
 * $L = \left\{{ {\dot\downarrow} \left({s}\right): s \in S}\right\}$

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

Let $\tau$ be the topology on $S$ generated by $U \cup L$.

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