p-pseudocompactness and related topics in topological spaces

Abstract

We prove some basic properties of p-bounded subsets (p epsilon omega) in terms of z-ultrafilters and families of continuous functions, We analyze the relations between p-pseudocompactness with other pseudocompact like-properties as p-compactness and alpha-pseudocompactness where alpha is a cardinal number We give an example of a sequentially compact ultrapseudocompact alpha-pseudocompact space which is not ultracompact, and we also give an example of an ultrapseudocompact totally countably compact alpha-pseudocompact space which is not q-compact for any q epsilon omega, answering affirmatively to a question posed by S. García-Ferreira and Kocinac (1996). We show the distribution law cl(gamma(XxY)) (A x B) = cl(gamma X) A x cl(gamma Y)B, where gamma Z denotes the Dieudonne completion of Z, for p-bounded subsets and we generalize the classical Glisckberg Theorem on pseudocompactness in the realm of p-boundedness, These results are applied to study the degree of pseudocompactness in the product of p-bounded subsets. (C) 1999 Published by Elsevier Science B.V. All rights reserved.