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GO-spaces with $\sigma$-closed discrete dense subsets

Authors: Harold R. Bennett, Robert W. Heath and David J. Lutzer
Journal: Proc. Amer. Math. Soc. 129 (2001), 931-939
MSC (1991): Primary 54F05, 54E35; Secondary 54D15
Published electronically: October 16, 2000
MathSciNet review: 1707135
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In this paper we study the question ``When does a perfect generalized ordered space have a $\sigma$-closed-discrete dense subset?'' and we characterize such spaces in terms of their subspace structure, $s$-mappings to metric spaces, and special open covers. We also give a metrization theorem for generalized ordered spaces that have a $\sigma$-closed-discrete dense set and a weak monotone ortho-base. That metrization theorem cannot be proved in ZFC for perfect GO-spaces because if there is a Souslin line, then there is a non-metrizable, perfect, linearly ordered topological space that has a weak monotone ortho-base.

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Additional Information

Harold R. Bennett
Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409

Robert W. Heath
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15213

David J. Lutzer
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187

Keywords: Generalized ordered space, linearly ordered space, perfect, $\sigma$-discrete dense subset, $G_\delta$-diagonal, metrization, weak monotone ortho-base, Souslin line
Received by editor(s): September 9, 1998
Received by editor(s) in revised form: June 6, 1999
Published electronically: October 16, 2000
Dedicated: Dedicated to the memory of F. Burton Jones
Communicated by: Alan Dow
Article copyright: © Copyright 2000 American Mathematical Society