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A metric space of A. H. Stone
and an example concerning $\sigma $-minimal bases

Authors: Harold R. Bennett and David J. Lutzer
Journal: Proc. Amer. Math. Soc. 126 (1998), 2191-2196
MSC (1991): Primary 54F05, 54D18, 54D30, 54E35
MathSciNet review: 1487358
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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 \v{C}ech 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.

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Additional 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

Keywords: Linearly ordered space, generalized ordered space, \v Cech complete, paracompact, perfect space, $\sigma $-minimal base, metrization theory
Received by editor(s): April 25, 1996
Received by editor(s) in revised form: January 1, 1997
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1998 American Mathematical Society