Products of quasi-p-pseudocompact spaces

Abstract

(2) every class P-P is closed under arbitrary products

Given p is an element of beta(w) \ w, 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 beta(w) containing w. We give examples of spaces X, Y, X-s, Y-s which are quasi-p-pseudocompact for every p is an element of w, but X x Y is not pseudocompact, and X-s x Y-s is pseudocompact and it is not quasi-s-pseudocompact for each s is an element of w. Besides, we prove that every pseudocompact space X of beta(w) with w subset of X, is quasi-p-pseudocompact for some p is an element of w. Finally, we introduce, for each p is an element of w, the class P-P of all spaces X such that X x Y is quasi-p-pseudocompact when so is Y

and we prove: (1) the intersection of classes P-p (p is an element of w) coincides with the Frolik class

(3) the partial ordered set ({P-p : p is an element of w}, superset of) is isomorphic to the set of equivalence classes of free ultrafilters on w with the Rudin-Keisler order. A topological characterization of RK-minimal ultrafilters is also given.