Definition:Order Topology/Definition 2

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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 known as

The order topology is also known as the interval topology.


Also see