Pseudocompactness of hyperspaces

摘要

We consider the following question of Ginsburg: Is there any relationship between the pseudocompactness of X~ω and that of the hyperspace 2~X? We do that first in the context of Mrσwka-Isbell spaces ψ(A) associated with a maximal almost disjoint (MAD) family A on ω answering a question of J. Cao and T. Nogura. The space ψ (A)~ω is pseudocompact for every MAD family A. We show thatrn(1) (ρ = c) 2~(ψ(A)) is pseudocompact for every MAD family A.rn(2) (η ＜ c) There is a MAD family A such that 2(ψ(A)) is not pseudocompact.rnWe also construct a ZFC example of a space X such that X~ω is pseudocompact, yet 2~X is not.