Topology/Order Topology

Let X be a totally ordered set with ordering $\leq$.

Consider all subsets of X of the form

${x\in X|x<a}$

and

${x\in X|x>a}$

where a is any element of X. We call these the open rays of X. Since the union of all open rays is X, this is a semibase of some topology in this set.

We define the order topology of this ordered set to be the topology $\tau$ that is generated by this semibase.

We define the open intervals in this set to be all sets of the form

${x\in X|a<x<b}$.

A base of this topology is the set of all open rays and open intervals. This is because the set of all open rays and open intervals together is the set of all finite intersections of the semibase of open rays.