Countably compact hyperspaces and Frolík sums

Let H 0(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 {Zα : α < κ} of closed subsets of X separated by pairwise disjoint open sets and any family {kα : α < κ} of natural numbers, the product ∏α<κ Zα α 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 X n 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”) F(Xα : α < κ) of a family {Xα : α < κ}. 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<κ H (Xα) embeds into H(F(Xα : α < κ)). © 2007 Elsevier B.V. All rights reserved. MSC: 54A25; 54B10; 54B20; 54C25; 54D10; 54D20