ON p-SPACES AND ¿¿-SPACES

Abstract

In this paper it is proved that when .¥/ is a kR-space then pX (the smallest subspace of ßX containing X with the property that each of its bounded closed subsets is compact) also is a /cÄ-space; an example is given of a kR -space X such that its Hewitt realcompactification, vX, is not a /cÄ-space. We show with an example that there is a non-/cÄ-space X such that vX and pX are kR-spaces. Also we answer negatively a question posed by Buchwalter: Is pX the union of the closures in vX of the bounded subsets of A"? Finally, without using the continuum hypothesis, we give an example of a locally compact space X of cardinality n, such that vX is not a /c-space. Introduction. The topological spaces used here will always be completely regular Hausdorff spaces. If X is a topological space we write C(X) for the ring of the continuous real-valued functions on X, and ßX (resp. vX) for the Stone-Cech compactification (resp. Hewitt realcompactification) of X. A subset M of X is said to be bounded if g\M is bounded for all g E C(X). A space is said to be a p-space if every closed bounded subset is compact. Realcompact spaces (closed subspaces of a product of real lines) and Pspaces (spaces in which every Gs is open) are p-spaces. Write pX for the smallest subspace of ßX that contains X and is a p-space. A real-valued function g on I is called kR-continuous if g\K is continuous in K for all compact subsets K of X. A space such that every kR-continuous function is continuous is called a /c^-space. The associated /<Ä-space of a space X, denoted by kRX, will be X provided with the coarsest topology for which every kR-continuous function on X is continuous. It is easy to see that kRX is a completely regular Hausdorff space. Our work provides the solutions to the following questions: (1) If A' is a kR -space, is pX a kR -space? (2) If ® is the family of all closed bounded subsets of X, does the relation pX = \J{BvX: B E <ft}hold? (3) If vX or pX is a kR-spact, is A" a kR-spa.cel (4) If X is a kR-space, is vX a kR-space? (5) If X is a realcompact space, is kRX realcompact?1 Received by the editors April 13, 1975 and, in revised form, June 29, 1975, October 13, 1975, and February 2, 1976. AMS (MOS) subject classifications (1970). Primary 54A05, 54C35, 54D50, 54D60, 54G20.