Proceedings of the American Mathematical Society

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

Author(s): Wei-Xue Shi
Journal: Proc. Amer. Math. Soc. 127 (1999), 2783-2791.
MSC (1991): Primary 54F05, 54G20, 54E35
Posted: April 15, 1999
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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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