GOspaces with closed discrete dense subsets
Authors:
Harold R. Bennett, Robert W. Heath and David J. Lutzer
Journal:
Proc. Amer. Math. Soc. 129 (2001), 931939
MSC (1991):
Primary 54F05, 54E35; Secondary 54D15
Published electronically:
October 16, 2000
MathSciNet review:
1707135
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: In this paper we study the question ``When does a perfect generalized ordered space have a closeddiscrete dense subset?'' and we characterize such spaces in terms of their subspace structure, mappings to metric spaces, and special open covers. We also give a metrization theorem for generalized ordered spaces that have a closeddiscrete dense set and a weak monotone orthobase. That metrization theorem cannot be proved in ZFC for perfect GOspaces because if there is a Souslin line, then there is a nonmetrizable, perfect, linearly ordered topological space that has a weak monotone orthobase.
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 [BL1]
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 [BL2]
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 [BLP]
 Bennett, H., Lutzer, D., and Purisch, S., On dense subspaces of generalized ordered spaces, Topology & Appl. 93 (1999), 191205. CMP 99:12
 [F]
 Faber, M., Metrizability in generalized ordered spaces, Math. Centre Tract 53, Mathematical Centre, Amsterdam, 1974. MR 54:6097
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 Lutzer, D., On generalized ordered spaces, Dissertationes Math., vol. 89, 1971. MR 48:3018
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 Qiao, YQ. and Tall, F., Perfectly normal nonmetrizable nonarchimedean spaces are generalized Souslin lines, Proc. Amer. Math. Soc., to appear.
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 Shi, WX., Extensions of perfect GOspaces and discrete dense sets, Proc. Amer. Math. Soc. 127 (1999), 615618. MR 99c:54045
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 White, H., First countable spaces that have special pseudobases, Canadian Math. Bull. 21 (1978), 103112. MR 58:2675
 [vW]
 van Wouwe, J., GOspaces and generalizations of metrizability, MC Tract 104, Mathematical Centre, Amsterdam, 1979. MR 80m:54046
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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
Email:
lutzer@math.wm.edu
DOI:
http://dx.doi.org/10.1090/S0002993900055829
PII:
S 00029939(00)055829
Keywords:
Generalized ordered space,
linearly ordered space,
perfect,
$\sigma$discrete dense subset,
$G_\delta$diagonal,
metrization,
weak monotone orthobase,
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
