# 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.