\begin{definition}[Definition:Order Topology/Definition 2]
Let $\struct {S, \preceq}$ be a totally ordered set.
Define:
:$\map {\Uparrow} S = \set {s^\succ: s \in S}$
:$\map {\Downarrow} S = \set {s^\prec: s \in S}$
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 $\map {\Uparrow} S \cup \map {\Downarrow} S$.
\end{definition}