Let be an ordinal. The set ordered by is a well-ordered set. becomes a topological space if we equip with the interval topology. An ordinal space is a topological space such that (with the interval topology) for some ordinal . In this entry, we will always assume that , or .
Before examining some basic topological structures of , let us look at some of its order structures.
First, it is easy to see that , for any . Here, is the upper set of .
In any ordinal space where , a typical open interval may be written , where . If is not a limit ordinal, we can also write where . This means that is a clopen set if is not a limit ordinal. In particular, if is not a limit ordinal, then is clopen, where , so that is an isolated point. For example, any finite ordinal is an isolated point in .
Conversely, an isolated point can not be a limit ordinal. If is isolated, then is open. Write as the union of open intervals . So . Since covers , each must be or would contain more than a point. If is a limit ordinal, then so that, again, would contain more than just . Therefore, can not be a limit ordinal and all must be the same. Therefore , where is the predecessor of : .
Several basic properties of an ordinal space are:
Some interesting ordinal spaces are
- 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
|Date of creation||2013-03-22 17:10:56|
|Last modified on||2013-03-22 17:10:56|
|Last modified by||CWoo (3771)|