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

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+rho(X x Y)) = H(Lipd(X)) x H(Lip rho(Y)) for the product of the two metric spaces (X, d) and (Y, rho). More precisely, we prove that such equality holds if and only if H(Lipd(X)) = X ⠂ or H(Lip rho(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.(c) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).