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