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.