Pseudocompactness of hyperspaces

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 Mrówka–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 that(1) c ) 2 Ψ ( A ) is pseudocompact for every MAD family A . t; c ) There is a MAD family A such that 2 Ψ ( A ) is not pseudocompact. so construct a ZFC example of a space X such that X ω is pseudocompact, yet 2 X is not.