Section 14: The Order Topology

Let $X$ be ordered by $<$ .

An interval is a subset of $X$ of the form $(a,b)$ , $(a,b]$ , $[a,b)$ or $[a,b]$ . The first one is an open interval, the second and third intervals are half-open intervals, and the fourth one is the closed interval.

An ray is a set of the form $(a,+\infty)$ , $(-\infty,b)$ , $[a,+\infty)$ or $(-\infty,b]$ . The first two are open rays, and the last two are closed rays.

The order topology on $X$ is the one generated by the basis consisting of the following three types of subsets:

open intervals,
half-open intervals $[a_{0},b)$ where $a_{0}$ is the smallest element of $X$ (if there is one), and
half-open intervals $(a,b_{0}]$ where $b_{0}$ is the largest element of $X$ (if there is one).

The order topology is also the one generated by the set of all open rays as a subbasis.

Examples

The order topology on the real line is the standard topology.
The order topology on the set of positive integers is the discrete topology.
The order topology on the product $\{1,2\}\times\mathbb{Z}_{+}$ in the dictionary order has a basis consisting of all singletons except $(2,1)$ and all intervals $((1,n),(2,1)]$ .