A metric space of A. H. Stone

and an example concerning -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 due to A. H. Stone and one of its completions to construct a linearly ordered topological space that is \v{C}ech complete, has a -closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a -minimal base for its relative topology. However, is not metrizable and is not quasi-developable. The construction of 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

DOI:
http://dx.doi.org/10.1090/S0002-9939-98-04785-6

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