---
tag: RCWHDMWP1T
kind: concept
name: Order topology
msc: [54F05, 06F30]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Order_topology
same_as:
  wikidata: Q1321469
  wikipedia: Order topology
---

## Summary

The order topology is the natural topology on a [[totally ordered set|0FRXBVVR03]], generated by the open rays $\set{x : x > a}$ and $\set{x : x < b}$.
The usual topology of the real line is the order topology of its usual order.
Ordinals, the long line and many classical counterexamples of general topology are built as order topologies.
Every order topology is Hausdorff, and in fact normal.

## Explanations

### L1 human

On a line of ordered points, call a set open when, around each of its points, it contains a whole open interval.
This is exactly how the open sets of the real line are described, but it uses only the order and not the distance.
So any totally ordered set, such as the rationals, the integers or a set of ordinals, gets a topology in the same way.
On the integers every point $n$ is the open interval $(n - 1, n + 1)$, so the order topology is discrete.