Definition:Order Topology/Definition 2

Definition
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 $\map {\Uparrow} S \cup \map {\Downarrow} S$.

Also see

 * Equivalence of Definitions of Order Topology