Order topology

Summary

The order topology is the natural topology on a totally ordered set, 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.