Pseudocompactness of hyperspaces

We consider the following question of Ginsburg: Is there an), relationship between the pseudocompactness of X-omega and that of the hyperspace 2(X) ? We do that first in the context of Mrowka-Isbell spaces psi (A) associated with a maximal almost disjoint (MAD) family A on omega answenng a question of J. Cao and T. Nogura. The space psi (A)(omega) is pseudocompact for every MAD family A. We show that (1) (p = c) 2(Psi) ((A)) is pseudocompact for every MAD family A. (2) (h < c) There is a MAD family A such that 2(Psi(A)) is not pseudocompact. We also construct a ZFC example of a space X such that X-omega is pseudocompact, yet 2(X) is not. (C) 2007 Elsevier B.V. All rights reserved