Hyperspaces and Frolík sums
by
Jerry E. Vaughan
University of North Carolina at Greensboro
Coauthors: István Juhász

Let H(X) denote the set of all non-empty closed subsets of a topological space X with the Vietoris topology. A subbase for the Vietoris topology consists of sets of the form U⁺, U^- where U is open in X and U⁺={C ∈ H(X):C ⊂ U} and U^-={C ∈ H(X): C∩U ≠ ∅}. We prove that for any family of spaces {X_a:a < k}, the product P_{a < k}H(X_a) can be embedded as a closed subset of H(F(X_a:a < k)), the hyperspace of the Frolík sum of the family of spaces (the Frolík sum is also called the "one-point countable-compactification" of the family). Among many corollaries we obtain the result of Cao, Nogura and Tomita that if the hyperspace of the Frolík sum of a family of T₂-space H(F(X_a:a < k)) is countably compact, then the product P_{a < k}X_a is countably compact.

Date received: June 8, 2006