Definition:Order Topology

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Definition

Definition 1

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

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

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


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


Definition 2

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 ${\Uparrow} \left({S}\right) \cup {\Downarrow} \left({S}\right)$.


Also known as

The order topology is also known as the interval topology.


Also see

  • Results about order topologies can be found here.