---
tag: E4TEZEDHY8
kind: definition
name: Order topology
msc: [54F05, 06F30]
defines: RCWHDMWP1T
uses: [0FRXBVVR03, 0CQHFPGJ45, 09S8DV34Y0]
sources:
  - url: https://en.wikipedia.org/wiki/Order_topology
---

## Definition

Let $(X, <)$ be a [[totally ordered set|0FRXBVVR03]].
The *order topology* on $X$ is the [[topology|0CQHFPGJ45]] on $X$ with [[subbasis|09S8DV34Y0]] the open rays
$$(a, \to) = \set{x \in X : a < x} \quad\text{and}\quad (\gets, b) = \set{x \in X : x < b}, \qquad a, b \in X.$$

## Remarks

When $X$ has more than one point, the open intervals $(a, b)$ together with the open rays form a basis of the order topology.

Example: on $\R$ with its usual order, the open rays generate the open intervals, so the order topology is the usual topology of $\R$.

Non-example: let $Y = \set{-1} \cup \set{1/n : n \ge 1}$ as a subset of $\R$.
With the subspace topology from $\R$, the point $-1$ is isolated.
In the order topology of $Y$, every open set containing $-1$ contains a ray $(\gets, 1/n)$, which contains $1/m$ for all $m > n$.
So the subspace topology of $Y$ is not its order topology, although the order of $Y$ is the restriction of the order of $\R$.

Some authors also use the name for the topology generated by the same rays on an arbitrary partially ordered set.