Products of Quasi-p-Pseudocompact Spaces

Abstract

Given p ∈β (ω) ω, we determine when a product of quasi-p-pseudocompact spaces preserves this property. In particular, we analyze the product of quasi-p-pseudocompact subspaces of β(ω) containing ω. We give examples of spaces X, Y, X _s , Ys which are quasi-p-pseudocompact for every p ∈ω*, but X Y is not pseudocompact, and X _s Y _s is pseudocompact and it is not quasi-s-pseudocompact for each s ∈*. Besides, we prove that every pseudocompact space X of βω with ω ⊂ X, is quasi-p-pseudocompact for some p ∈ω*. Finally, we introduce, for each p ∈ ω*, the class P _p of all spaces X such that X × Y is quasi-p-pseudocompact when so is Y; and we prove: (1) the intersection of classes P _p ( _p ∈ω*) coincides with the Frol"ik class; (2) every class P _p is closed under arbitrary products; (3) the partial ordered set ( P _p p∈ ,⊃) is isomorphic to the set of equivalence classes of free ultrafilters on ω with the Rudin–Keisler order. A topological characterization of RK-minimal ultrafilters is also given.