A metric space of A. H. Stone and an example concerning $\sigma$-minimal bases
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- by Harold R. Bennett and David J. Lutzer
- Proc. Amer. Math. Soc. 126 (1998), 2191-2196
- DOI: https://doi.org/10.1090/S0002-9939-98-04785-6
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Abstract:
In this paper we use a metric space $Y$ due to A. H. Stone and one of its completions $X$ to construct a linearly ordered topological space $E = E(Y,X)$ that is Čech complete, has a $\sigma$-closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a $\sigma$-minimal base for its relative topology. However, $E$ is not metrizable and is not quasi-developable. The construction of $E(Y,X)$ is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone’s metric space and our construction.References
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Bibliographic Information
- Harold R. Bennett
- Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409
- David J. Lutzer
- Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187
- Received by editor(s): April 25, 1996
- Received by editor(s) in revised form: January 1, 1997
- Communicated by: Franklin D. Tall
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2191-2196
- MSC (1991): Primary 54F05, 54D18, 54D30, 54E35
- DOI: https://doi.org/10.1090/S0002-9939-98-04785-6
- MathSciNet review: 1487358