Definition:Order Topology/Definition 2

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

Define:


 * ${\Uparrow} \left({S}\right) = \left\{{s^\succ: s \in S}\right\}$
 * ${\Downarrow} \left({S}\right) = \left\{{s^\prec: s \in S}\right\}$

where $s^\succ$ and $s^\prec$ denote the strict upper closure and strict lower closure of $s$, respectively.

The order topology $\tau$ on $S$ is the topology on $S$ generated by ${\Uparrow} \left({S}\right) \cup {\Downarrow} \left({S}\right)$.