Definition:Order Topology
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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.