§ 14. The Order Topology

Definition 14.1 : Open Interval / Closed Interval / Half-Open Interval

Suppose that \(X\) is a set having a simple order relation \(<\). Given elements \(a\) and \(b\) of \(X\) such that \(a<b\), there are four subsets of \(X\) that are called the intervals determined by \(a\) and \(b\). They are the following :

\[ \begin{aligned} & (a, b)=\{x \mid a<x<b\}, \\ & (a, b]=\{x \mid a<x \leq b\}, \\ & {[a, b)=\{x \mid a \leq x<b\},} \\ & {[a, b]=\{x \mid a \leq x \leq b\} .} \end{aligned} \]

A set of the first type is called an open interval in \(X\), a set of the last type is called a closed interval in \(X\), and sets of the second and third types are called half-open intervals.

Definition 14.2 : Order Topology

Let \(X\) be a set with a simple order relation; assume \(X\) has more than one element. Let \(\mathcal{B}\) be the collection of all sets of the following types:

1. All open intervals \((a, b)\) in \(X\).
2. All intervals of the form \(\left[a_{0}, b\right)\), where \(a_{0}\) is the smallest element (if any) of \(X\).
3. All intervals of the form \(\left(a, b_{0}\right]\), where \(b_{0}\) is the largest element (if any) of \(X\).

The collection \(\mathcal{B}\) is a basis for a topology on \(X\), which is called the order topology.

Definition 14.3 : Open Rays / Closed Rays

If \(X\) is an ordered set, and \(a\) is an element of \(X\), there are four subsets of \(X\) that are called the rays determined by \(a\). They are the following:

\[ \begin{aligned} & (a,+\infty)=\{x \mid x>a\}, \\ & (-\infty, a)=\{x \mid x<a\}, \\ & {[a,+\infty)=\{x \mid x \geq a\},} \\ & (-\infty, a]=\{x \mid x \leq a\} . \end{aligned} \]

Sets of the first two types are called open rays, and sets of the last two types are called closed rays.

Theorem 14.1

The use of the term "open" suggests that open rays in \(X\) are open sets in the order topology. And so they are.

Proof

Consider, for example, the ray \((a,+\infty)\). If \(X\) has a largest element \(b_{0}\), then \((a,+\infty)\) equals the basis element \(\left(a, b_{0}\right]\). If \(X\) has no largest element, then \((a,+\infty)\) equals the union of all basis elements of the form \((a, x)\), for \(x>a\). In either case, \((a,+\infty)\) is open. A similar argument applies to the ray \((-\infty, a)\).

Theorem 14.2

The open rays, form a subbasis for the order topology on \(X\).

Proof

Because the open rays are open in the order topology, the topology they generate is contained in the order topology.
On the other hand, every basis element for the order topology equals a finite intersection of open rays; the interval \((a, b)\) equals the intersection of \((-\infty, b)\) and \((a,+\infty)\), while \(\left[a_{0}, b\right)\) and \(\left(a, b_{0}\right]\), if they exist, are themselves open rays. Hence the topology generated by the open rays contains the order topology.