A metric space of A. H. Stone and an example concerning 𝜎-minimal bases

Bennett, Harold; Lutzer, David

doi:10.1090/S0002-9939-98-04785-6

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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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