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Unit 3: The Order Topology

The notation used here is familiar to you already in the case where X is the real line, but these are Notes
intervals in an arbitrary ordered set.

 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.
 Sets of the second and third types are called half-open intervals.

Note The use of the term “open” in this connection suggests that open intervals in X
should turn out to be open sets when we put a topology on X and so they will.

3.1.2 Order Topology

Definition: Let X be a set with a simple order relation; assume X has more than one element. Let
 be the collection of all sets of the following types:
(1) All open intervals (a, b) in X.
(2) All intervals of the form [a , b), where a is the smallest element (if any) of X.
o o
(3) All intervals of the form (a , b ], where b is the largest element (if any) of X.
o o o
The collection  is a basis for a topology on X, which is called the order topology. If X has no
smallest element, there are no sets of type (2), and if X has no largest element, there are no sets
of type (3).

Notes One has to check that  satisfies the requirements for a basis.
(A) First, note that every element x of X lies in at least one element of  : The smallest
element (if any) lies in all sets of type (2), the largest element (if any) lies in all sets
of type (3), and every other element lies in a set of type (1).

(B) Second, note that the intersection of any two sets of the preceding types is again a set
of one of these types, or is empty.

Example 1: The standard topology on  is just the order topology derived from the
usual order on .

Example 2: Consider the set  ×  in the dictionary order; we shall denote the general
element of  ×  by x × y, to avoid difficulty with notation. The set  ×  has neither a largest
nor a smallest element, so the order topology on  ×  has as basis the collection of all open
intervals of the form (a × b, c × d) for a < c, and for a = c and b < d. The subcollection consisting
of only intervals of the second type is also a basis for the order topology on  × , as you can
check.

Example 3: The positive integers Z form an ordered set with a smallest element. The
+
order topology on Z is the discrete topology, for every one-point set is open : If n > 1, then the
+
one-point set {n} = {n–1, n+1} is a basis element; and if n=1, the one-point set {1} = [1, 2) is a basis
element.

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