A metric space of A. H. Stone, \sigma-minimal bases, and small diagonals
by
David J. Lutzer
College of William and Mary

This talk reports on recent joint work with Hal Bennett. In his 1965 paper [On \sigma-discreteness and Borel isomorphism, Amer J.Math., 85 (1963), 655-666], A.H. Stone described a metric space X with cardinality \omega₁ such that X is not \sigma-discrete and every separable subset of X is countable. The space X is a special subset of D^\omega, where D is an uncountable discrete space with cardinality \omega₁. Let Y be the closure of X in D^\omega. Then Y is completely metrizable and, according to a 1965 theorem of Herrlich, there is some linear ordering of Y whose open-interval topology coincides with the topology that Y inherits from D^\omega. That allows us to construct a linearly ordered space E that is non-metrizable, first-countable, paracompact, perfect, Cech-complete, and has a dense subspace that is the union of countably many closed discrete subspaces of E. The purpose of this talk is to discuss E and its relationships to two older problems concerning \sigma-minimal bases and small diagonals.

1) An open problem asks whether a compact linearly ordered topological space Z must be metrizable if every subspace of Z has a \sigma-minimal base (in the sense of Aull) for its relative topology. Such a space Z must be first-countable and hereditarily paracompact, and D. Burke pointed out every subspace of Z must contain a dense metrizable subspace. The space E above is not compact, of course, but it is Cech complete, and one can show that every subspace of E has a \sigma-minimal base. Therefore, while the question for compact ordered spaces remains unresolved, the space E places a limit on how far a positive result could be generalized. Even if it turns out to be true that a compact LOTS having \sigma-minimal bases hereditarily must be metrizable, that result could not be generalized beyond the class of locally compact ordered spaces.

2) Recently, Juhasz and Szentmiklossy [Proc. Amer. Math. Soc., 116 (1992), 1153-60] proved that (*) under CH, a compact Hausdorff space with a small diagonal (in the sense of Husek) must be metrizable. One would expect that their result could be proved for the successively larger classes of paracompact, locally compact spaces, then paracompact Cech-complete spaces, and finally for paracompact p- spaces. The space E shows that is not the case: the generalization must stop with paracompact, locally compact spaces. In addition, some years ago, van Douwen and I proved that (**) a compact (or even Lindelöf) linearly ordered topological space with a small diagonal must be metrizable, and the space E shows that (**) also resists generalization. Indeed, the space E has an even stronger property called an "H-diagonal": for any regular uncountable cardinal \kappa < |E|, if T is a subsetof E^2 - (E) has cardinality \kappa, then there is an open neighborhood W of \Delta(E) in E² such that |T - W| = \kappa.

Date received: January 26, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaa-12.