The Hewitt realcompactification of products

Abstract

For a completely regular Hausdorff space X X , υ X \upsilon X denotes the Hewitt realcompactification of X X . Given a topological property P \mathcal {P} of spaces, our interest is in characterizing the class R ( P ) \mathcal {R}(\mathcal {P}) of all spaces X X such that υ ( X × Y ) = υ X × υ Y \upsilon (X \times Y) = \upsilon X \times \upsilon Y holds for each P \mathcal {P} -space Y Y . In the present paper, we obtain such characterizations in the case that P \mathcal {P} is locally compact and in the case that P \mathcal {P} is metrizable.