Definition:Order Topology

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

Define:


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

where $\mathop{\uparrow} \left({s}\right)$ and $\mathop{\downarrow} \left({s}\right)$ denote strict upper closure and strict lower closure of $s$, respectively.

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

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