Hewitt Realcompactifications of Products

Abstract

The Hewitt realcompactification vX of a completely regular Hausdorff space X has been widely investigated since its introduction by Hewitt [17]. An important open question in the theory concerns when the equality v(X × Y) = vX × vY is valid. Glicksberg [10] settled the analogous question in the parallel theory of Stone-Čech compactifications: for infinite spaces X and Y, β(X × Y) = βX × β Y if and only if the product X × Y is pseudocompact. Work of others, notably Comfort [3; 4] and Hager [13], makes it seem likely that Glicksberg's theorem has no equally specific analogue for v(X × Y) = vX × vY. In the absence of such a general result, particular instances may tend to be attacked by ad hoc techniques resulting in much duplication of effort.

Local Compactness and Hewitt Realcompactifications of Products

“…In this note we prove McArthur's conjecture [6]: If card X is nonmeasurable and if v(X X Y) = vX XvY holds for each space Y, then X is locally compact. Consequently, we can completely characterize the class of all spaces X such that for each space Y, v(X X Y) = vX XvY holds.…”

“…For a space X, vX denotes the Hewitt realcompactification of X, and the symbolism v(X X Y) = vX X vY means that X X Y is C-embedded in vX XvY. Following [6], let 91 denote the class of all spaces X such that for each space Y, v(X x Y) =vX XvY holds. It is known that a locally compact realcompact space of nonmeasurable cardinal is a member of 91 and that every member of 91 is realcompact (Comfort [1, Corollary 2.2], McArthur [6,Theorem 5.2]).…”

“…It is known that a locally compact realcompact space of nonmeasurable cardinal is a member of 91 and that every member of 91 is realcompact (Comfort [1, Corollary 2.2], McArthur [6,Theorem 5.2]). In [6], McArthur conjectured that if card X is nonmeasurable and A" is a member of 91, then X is locally compact. The main purpose of this note is to establish his conjecture positively.…”

“…(1) If vX is locally compact, then X is locally pseudocompact, but the converse is false (see [1]). [6] to show that every member of 91 is realcompact.…”

“…Following [6], let 91 denote the class of all spaces X such that for each space Y, v(X x Y) =vX XvY holds. It is known that a locally compact realcompact space of nonmeasurable cardinal is a member of 91 and that every member of 91 is realcompact (Comfort [1, Corollary 2.2], McArthur [6,Theorem 5.2]). In [6], McArthur conjectured that if card X is nonmeasurable and A" is a member of 91, then X is locally compact.…”