To find more examples, let us look at the singletons of a door space . For each , either is open or closed. Call a point in open or closed according to whether is open or closed. Let be the collection of open points in . If , then is discrete. So suppose now that . We look at the special case when . It is now easy to see that the topology generated by all the open singletons makes a door space:
When has more than one element, the situation is a little more complicated. We know that if is door, then its topology is strictly finer then the topology generated by all the open singletons. McCartan has shown that for some ultrafilter in . In fact, McCartan showed , as well as the previous two examples, are the only types of possible topologies on a set making it a door space.
- 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
- 2 S.D. McCartan, Door Spaces are identifiable, Proc. Roy. Irish Acad. Sect. A, 87 (1) 1987, pp. 13-16.
|Date of creation||2013-03-22 18:46:11|
|Last modified on||2013-03-22 18:46:11|
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