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Polyadic spaces of arbitrary compactness numbers. (English) Zbl 0587.54039
The author and J. van Mill [Fundam. Math. 106, 163-173 (1980; Zbl 0431.54010)] extended the notion of supercompact introduced by J. de Groot [Contrib. Extens. Theory Topol. Struct., Proc. Sympos. Berlin 1967, 89-90 (1969; Zbl 0191.212)]. They defined the compactness number of a compact Hausdorff space to be the least integer k (or infinity if no such k exists) for which there is an open subbase with the property that every cover by subbase elements contains a subcover with at most k members (k is 2 for supercompact). Using de Groot’s notion of superextension they give a sequence of spaces exhibiting all possible compactness numbers. Another such sequence using Stone spaces was later described by the author [Can. J. Math. 35, 824-838 (1983; Zbl 0533.06007)]. The paper under review gives a third using subsets of \(2^{\omega_ 1}\) whose members have finite support, which is more easily described and answers other questions as well. E.g., it yields examples of Eberlein spaces (homeomorphs of weak compacta in Banach spaces) which are not supercompact, answering problem 3 of M. Hušek [Categorical Topology, Proc. Int. Conf. Berlin 1978, Lect. Notes Math. 719, 167-195 (1979; Zbl 0433.54013)]. Also the Alexandroff one-point compactification of the disjoint union of such a sequence has infinite compactness number but is a continuous image of a space of finite compactness number, answering question 4.3 of the first paper quoted above.
Reviewer: D.E.Sanderson

54D30 Compactness
03E05 Other combinatorial set theory
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