GO-spaces with $\sigma$-closed discrete dense subsets
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- by Harold R. Bennett, Robert W. Heath and David J. Lutzer PDF
- Proc. Amer. Math. Soc. 129 (2001), 931-939 Request permission
Abstract:
In this paper we study the question βWhen does a perfect generalized ordered space have a $\sigma$-closed-discrete dense subset?β 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.References
- K. Alster, Subparacompactness in Cartesian products of generalized ordered topological spaces, Fund. Math. 87 (1975), 7β28. MR 451210, DOI 10.4064/fm-87-1-7-28
- Bennett, H., On quasi-developable spaces, Ph.D. Thesis, Arizona State University, 1968.
- H. R. Bennett and D. J. Lutzer, Generalized ordered spaces with capacities, Pacific J. Math. 112 (1984), no.Β 1, 11β19. MR 739137
- Harold R. Bennett and David J. Lutzer, A note on property III in generalized ordered spaces, Topology Proc. 21 (1996), 15β24. MR 1489188
- Bennett, H., Lutzer, D., and Purisch, S., On dense subspaces of generalized ordered spaces, Topology & Appl. 93 (1999), 191-205.
- M. J. Faber, Metrizability in generalized ordered spaces, Mathematical Centre Tracts, No. 53, Mathematisch Centrum, Amsterdam, 1974. MR 0418053
- D. J. Lutzer, On generalized ordered spaces, Dissertationes Math. (Rozprawy Mat.) 89 (1971), 32. MR 324668
- Peter J. Nyikos, Order-theoretic base axioms, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp.Β 367β398. MR 564107
- Peter J. Nyikos, Some surprising base properties in topology. II, Set-theoretic topology (Papers, Inst. Medicine and Math., Ohio Univ., Athens, Ohio, 1975β1976) Academic Press, New York, 1977, pp.Β 277β305. MR 0442889
- Ponomarev, V., Metrizability of a finally compact p-space with a point-countable base, Sov. Math. Dokl. 8 (1967), 765-768.
- Qiao, Y-Q. and Tall, F., Perfectly normal non-metrizable non-archimedean spaces are generalized Souslin lines, Proc. Amer. Math. Soc., to appear.
- Wei-Xue Shi, Extensions of perfect GO-spaces and $\sigma$-discrete dense sets, Proc. Amer. Math. Soc. 127 (1999), no.Β 2, 615β618. MR 1468203, DOI 10.1090/S0002-9939-99-04554-2
- H. E. White Jr., First countable spaces that have special pseudo-bases, Canad. Math. Bull. 21 (1978), no.Β 1, 103β112. MR 482615, DOI 10.4153/CMB-1978-016-5
- J. M. van Wouwe, GO-spaces and generalizations of metrizability, Mathematical Centre Tracts, vol. 104, Mathematisch Centrum, Amsterdam, 1979. MR 541832
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
- Received by editor(s): September 9, 1998
- Received by editor(s) in revised form: June 6, 1999
- Published electronically: October 16, 2000
- Communicated by: Alan Dow
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 931-939
- MSC (1991): Primary 54F05, 54E35; Secondary 54D15
- DOI: https://doi.org/10.1090/S0002-9939-00-05582-9
- MathSciNet review: 1707135
Dedicated: Dedicated to the memory of F. Burton Jones