A result concerning the Lipschitz realcompactification of the product of two metric spaces

Abstract

For a metric space (X, d), we consider the so-called Lipschitz realcompactification of X, denoted by H(Lipd(X)). In this note we give a result concerning the equality H(Lipd+ρ(X × Y )) = H(Lipd(X)) × H(Lipρ(Y )) for the product of the two metric spaces (X, d) and (Y, ρ). More precisely, we prove that such equality holds if and only if H(Lipd(X)) = X or H(Lipρ(Y )) = Y , where X and Y denote the completion of X and Y respectively, or equivalently, if and only if the Lipschitz realcompactification of one of the factors X or Y is as simple as possible. We also point out that our result is, in fact, a true generalization of a known theorem by Woods about the Samuel compactification of the product of two metric spaces.