Countably compact hyperspaces and Frolيk 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. ine the property R ( κ ) : for every family { Z α : α < κ } of closed subsets of X separated by pairwise disjoint open sets and any family { k α : α < κ } of natural numbers, the product ∏ α < κ Z α k α is countably compact, and prove that if H ( X ) is countably compact for a T 2 -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 T 2 and H ( X ) is countably compact, then so is X n for all n < ω . We also prove that, for κ < t , if the T 3 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 X κ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita: if X is T 3 , homogeneous, and H ( X ) is countably compact, then so is X ω . e study the Frolík sum (also called “one-point countable-compactification”) F ( X α : α < κ ) of a family { X α : 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 0 ( X α ) embeds into H ( F ( X α : α < κ ) ) .