Definition:Order Topology

Definition

Definition 1

Let $\struct {S, \preceq}$ be a totally ordered set.

Let $\XX$ be the set of open rays in $S$.

Let $\tau$ be the topology on $S$ generated by $\XX$.

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

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

Also known as

The order topology is also known as the interval topology.

Also see

• Results about order topologies can be found here.