On Dense Subspaces of Generalized Ordered Spaces
by
Steven D. Purisch
Coauthors: Harold R. Bennett, David J. Lutzer

The work of Souslin and Kurepa made it clear how important it is to know whether a given linearly ordered or generalized ordered space has a dense subspace with special metric related propertied. We study the relationship amoung the following four properties on generalized ordered spaces:

1. X has a sigma-discrete dense subset;
2. X has a dense metrizable subspace;
3. X has property III, i.e., there are open sets U(n) and relatively closed (in U(n)) discrete subsets D(n) contained in U(n) such that if G is open and p is an element of G, then for some positive integer n we have p is in U(n) and G intersect D(n) is not empty;
4. X has a dense set that is the countable union of discrete (but not necessarily closed) subspaces.

We study which generalized ordered spaces have these properties, investigate to what extent the propertied are hereditary, and which generalized ordered spaces embed in a generalized ordered space with one of these four properties.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caai-89.