Countably compact hyperspaces and Frolík sums |
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Authors: | Istvá n Juhá sz,Jerry E. Vaughan |
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Affiliation: | a Alfréd Rényi Institute of Mathematics, P.O. Box 127, 1364 Budapest, Hungary b Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27402, USA |
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Abstract: | Let H0(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u∈ω∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction.We define the property R(κ): for every family of closed subsets of X separated by pairwise disjoint open sets and any family of natural numbers, the product is countably compact, and prove that if H(X) is countably compact for a T2-space X then X satisfies R(κ) for all κ. A space has R(1) iff all its finite powers are countably compact, so this generalizes a theorem of J. Ginsburg: if X is T2 and H(X) is countably compact, then so is Xn for all n<ω. We also prove that, for κ<t, if the T3 space X satisfies a weak form of R(κ), the orbit of every point in X is dense, and X contains κ pairwise disjoint open sets, then Xκ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita: if X is T3, homogeneous, and H(X) is countably compact, then so is Xω.Then we study the Frolík sum (also called “one-point countable-compactification”) of a family . We use the Frolík sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact. We also prove that any product ∏α<κH0(Xα) embeds into . |
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Keywords: | 54A25 54B10 54B20 54C25 54D10 54D20 |
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