Product of quasi-p-pseudocompact spaces
by
Angel Tamariz-Mascarua
Universidad Nacional Autonoma de Mexico
Coauthors: Manuel Sanchis (Universitat Jaume I)

\omega is the set of natural numbers, \beta(\omega) is its Stone-Cech compactification and \omega^* = \beta(\omega)\\omega. The Rudin-Keisler pre-order <= _RK on \beta(\omega) is defined by p <= _RK q if there exists a function g :\omega --> \omega such that g^\beta(q) = p, where g^\beta is the continuous extension to \beta(\omega) of g. For p in \omega^*, we set P_RK(p) = {r in \omega^* : r <= _RK p}.

Definitions 1.- For p in \omega ^* , a point x in X is said to be a p-limit point of a sequence (U_n)_{n < \omega} of nonempty subsets of X (in symbols: x = p-lim(U_n)_{n < \omega}) if for each neighborhood V of x, the set {n < \omega: U_n \cap V =/= \emptyset} belongs to p.

2.- A space X is quasi-p-pseudocompact if for every sequence (U_n)_{n < \omega} of nonempty open sets in X, there are p in P_RK(p) and x in X such that x = p-lim(U_n)_{n < \omega}.

Given p in \omega^*, we determine when a product of quasi-p-pseudocompact spaces preserve this property. In particular, we analyze the product of quasi-p-pseudocompact subspaces of \beta(\omega) containing \omega. We give examples of spaces X, Y, X_s, Y_s which are quasi-p-pseudocompact for every p in \omega^*, but X ×Y is not pseudocompact, and X_s ×Y_s is pseudocompact and it is not quasi-s-pseudocompact for each s in \omega^*. Besides, we prove that every pseudocompact space X of \beta(\omega) with \omega subset X, is quasi-p-pseudocompact for some p in \omega^*. Finally, we introduce, for each p in \omega^*, the class \Cal P_p of all the spaces X such that X ×Y is quasi-p-pseudocompact when Y so is; and we prove: (1) the intersection of classes \Cal P_p (p in \omega^*) coincide with the Frolík class; (2) every class \Cal P_p is closed under arbitrary products; (3) the partial order set ({\Cal P_p : p in \omega^*}, contains ) is isomorphic to the set of equivalence classes of free ultrafilters on \omega with the Rudin-Keisler order. A topological characterization of RK-minimal ultrafilters is also given.

Date received: June 20, 2000