Houston Journal of Mathematics, vol. 28, No. 3, 2002, 511-527 A NOTE ON p-BOUNDED AND QUASI-p-BOUNDED SUBSETS Manuel Sanchis and Angel Tamariz-Mascar´ ua Abstract. We discuss the relationship between p-boundedness and quasi-p-boun- dedness in the realm of GLOTS for p ∈ ω * . We show that p-pseudocompactness, p-compactness, quasi-p-pseudocompactness and quasi-p-compactness are equivalent properties for a GLOTS; that bounded subsets of a GLOTS are strongly-bounded; and C-compact subsets of a GLOTS are strongly-C-compact. We also show that a topologically orderable group is locally precompact if and only if it is metrizable. For bounded subsets of a GLOTS, a version of the classical Gilcksberg’s Theorem on pseudocompactness is obtained: if Aα is a bounded subset of a GLOTS Xα for each α ∈ Δ, then cl β(  α∈Δ Xα) (  α∈Δ Aα)=  α∈Δ cl β(Xα) Aα. Also we prove that there exists an ultrapseudocompact topological group which is not quasi-p-compact for any p ∈ ω * . To see this example, p-pseudocompactness and p-compactness are investigated in the field of Cπ-spaces, proving that ultracompactness, quasi-p- compactness for a p ∈ ω * and countable compactness (respectively, ultrapseudo- compactness, quasi-p-pseudocompactness for a p ∈ ω * and pseudocompactness) are equivalent properties in the class of spaces of the form Cπ (X, [0, 1]). 1. Introduction In this article we will assume that all spaces are Tychonoff unless otherwise stated. The set of natural numbers will be denoted by ω, and the Stone- ˘ Cech compactification of a space X will be denoted as β(X). The space β(ω) is identified with the set of ultrafilters on ω, and ω * = β(ω) \ ω is the set of free ultrafilters. For p ∈ ω * , Bernstein [B] introduced and investigated the concept of p-limit in connection with some problems in the theory of nonstandard analysis. Indepen- dently, Fr´ olik [F] and Kat˘ etov [K1], [K2] introduced this concept in a different form, and Ginsburg and Saks [GS] generalized this notion as follows: 1.1. Definition. Let p ∈ ω * and let (S n ) n<ω be a sequence of nonempty subsets of a space X. A point x ∈ X is a p-limit point of the sequence (S n ) n<ω , in symbols x = p - lim(S n ), if for every V ∈N (x), {n<ω : V ∩ S n = ∅} ∈ p. If x n ∈ X and S n = {x n } for each n<ω, then a p-limit point of (S n ) n<ω is a Bernstein’s p-limit point of the sequence (x n ) n<ω . Note that if there exists a p-limit point of a sequence (x n ) n<ω , this has to be unique, since X is Hausdorff; 1991 Mathematics Subject Classification. 54A20, 54A25, 54C35, 54G20. Key words and phrases. p-limit point; p-pseudocompact space; p-compact space; p-bounded set; quasi p-bounded set; strongly bounded set; C-compact set; spaces of continuous functions; Generalized Linearly Topological Spaces; P -space; α-b-discrete space; topological group. The first-listed author was partially supported by Ministerio de Educaci´ on y Ciencia of Spain under Grant PB95-0737. The second-listed author was supported by Proyecto de Cooperaci´ on Intercampus. This author is also pleased to thank the Department of Mathematics of Jaume I University for generous hospitality during February-March, 1997. Typeset by A M S-T E X 2