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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A non-metrizable compact
linearly ordered topological space,
every subspace of which
has a \begin{math}\sigma\end{math}-minimal base


Author: Wei-Xue Shi
Journal: Proc. Amer. Math. Soc. 127 (1999), 2783-2791
MSC (1991): Primary 54F05, 54G20, 54E35
Published electronically: April 15, 1999
MathSciNet review: 1600133
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Abstract: A collection \begin{math}\mathcal{D}\end{math} of subsets of a space is minimal if each element of \begin{math}\mathcal{D}\end{math} contains a point which is not contained in any other element of \begin{math}\mathcal{D}\end{math}. A base of a topological space is \begin{math}\sigma\end{math}-minimal if it can be written as a union of countably many minimal collections. We will construct a compact linearly ordered space \begin{math}X\end{math} satisfying that \begin{math}X\end{math} is not metrizable and every subspace of \begin{math}X\end{math} has a \begin{math}\sigma\end{math}-minimal base for its relative topology. This answers a problem of Bennett and Lutzer in the negative.


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Additional Information

Wei-Xue Shi
Affiliation: Department of Mathematics, Changchun Teachers College, Changchun 130032, China
Address at time of publication: Institute of Mathematics, University of Tsukuba, Tsukuba-shi, Ibaraki 305, Japan
Email: shi@abel.math.tsukuba.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04853-4
PII: S 0002-9939(99)04853-4
Keywords: \(\sigma\)-minimal base, metrizable, linearly ordered topological space, special Aronszajn tree, quasi-developable
Received by editor(s): October 25, 1996
Received by editor(s) in revised form: November 15, 1997
Published electronically: April 15, 1999
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society