Page 44 - DMTH503_TOPOLOGY

Topology

Notes
Example 4: The set X = {1, 2} × Z in the dictionary order is another example of an ordered
+
set with a smallest element. Denoting 1 × n by a and 2 × n by b , we can represent X by
n n
a , a , …. ; b , b , ….
1 2 1 2
The order topology on X is not the discrete topology. Most one-point sets are open, but there is
an exception the one-point set {b }. Any open set containing b must contain a basis element
1 1
about b (by definition), and any basis element containing b contains points of the a sequence.
1 1 i
3.1.3 Rays

Definition: 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:
(a, + ) = {x|x > a}
(– , a) = {x|x < a},

[a, + ) = {x|x  a},
(– , a] = {x|x  a}.
sets of first two types are called open rays; and sets of the last two types are called closed rays.

Notes
(1) The use of the term “open” suggests that open rays in X are open sets in the order
topology. And so they are (consider, for example, the ray (a, + ). If X has a largest
element b , then (a, + ) equals the basis element (a, b ]. If X has no largest element,
o o
then (a, + ) equals the union of all basis elements of the form (a, x), for x > a. In
either case, (a, + ) is open. A similar argument applies to the ray (–, a).
(2) The open rays, in fact, form a sub-basis for the order topology on X, as we now show.
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 (–, b) and (a, + ), while [a , b) and (a, b ], if they exist, are themselves
o o
open rays. Hence the topology generated by the open rays contains the order
topology.

3.1.4 Order Topology on the Linearly Ordered Set

The order topology T on the linearly ordered set X is the topology generated by all open rays.
<
A linearly ordered space is a linearly ordered set with the order topology.

3.1.5 Lemma (Basis for the Order Topology)

Let (X, <) be a linearly ordered set.
(1) The union of all open rays and all open intervals is a basis for the order topology T .
<
(2) If X has no smallest and no largest element, then the set {(a, b)|a, b  X, a < b} of all open
intervals is a basis for the order topology.

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