On the Hewitt Realcompactification of a Product Space

Abstract

One of the themes of Edwin Hewitt's fundamental and stimulating work [16] is that the g-spaces (now called realcompact spaces) introduced there, although they are not in general compact, enjoy many attributes similar to those possessed by compact spaces; and that the canonical realcompactification vX associated with a given completely regular Hausdorff space X bears much the same relation to the ring C(X) of real-valued continuous functions on X as does the Stone-Cech compactification ß^to the ring C*(X) of bounded elements of C(X). Much of the Gillman-Jerison textbook [12] may be considered to be an amplification of this point. Over and over again the reader is treated to a "/?" theorem and then, some pages later, to its "v" analogue.One of the most elegant "jS" theorems whose "v" analogue remains unproved and even unstated is the following, given by Irving Glicksberg in [13] and reproved later by another method in [11] by Zdenëk Frolik: For infinite spaces X and Y, the relation ß(Xx F)=JSXx^Fholds if and only if Zx Fis pseudocompact. The present paper is an outgrowth of the author's unsuccessful attempt to characterize those pairs of spaces (X, Y) for which v(Xx Y) = vXxvY. It is shown (Theorem 2.4) that, barring the existence of measurable cardinals, the relation holds whenever y is a ¿-space and uXis locally compact; and more generally (Theorem 4.5) that the relation holds whenever the ¿-space Y and the locally compact space vX admit no compact subsets of measurable cardinal. The question arises naturally as to when it will occur that vX is locally compact. Our best result in this direction, a part of Theorem 4.8, is, again, not definitive: X is locally pseudocompact if and only if there is a locally compact space Y for which X<^ F<= VX.Some of the questions treated here are susceptible to attack when thrown into the uniform space context. See Onuchic [18] and, more intensively, Isbell [17, especially Chapter 8].I am indebted to Tony Hager for several references, and for allowing me to announce here that, in addition to the results of [15], he has achieved certain new theorems on the relation v(Xx Y) = vXxvY, which being "uniform" and not "topological" have negligible overlap with those of this paper.

On the product of twob f-spaces

“…If X is a space in S, from Hu~ek's result quoted in Example 12, X is locally bounded and therefore locally pseudocompact ( [5], Proposition 4.2). The sufficiency follows from part (a) of Corollary 10.…”

On the product of twob R-spaces and the class 45-145-145-1of Frolík

“…Necessity. From ( [3], Proposition 4.2) a bounded regular closed subset (of a completely regular space) is pseudocompact. Then, from the former observation, every space of the class ~ is locally pseudocompact.…”

Local compactness and Hewitt realcompactifications of products

“…More precisely, we can prove the following theorems. The implication (a) -> (b) of Theorem 1 was proved by Comfort [1]. Theorem 1.…”

“…For the notions of locally pseudocompact spaces and A>spaces see [1]. For an ordinal a, we denote by Wia) the set of all ordinals less than a topologized with order topology, and by co0 the first infinite ordinal.…”

“…Remarks. (1) If vX is locally compact, then X is locally pseudocompact, but the converse is false (see [1]). …”