The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P = vertical bar K vertical bar a resolution R(X, K) of X x P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X, K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X, K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X, K) is a bifunctor. Using this result, it is proved that the Cartesian product X x Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt) x Sh(Top) -> Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X x Z need not be a direct product in Sh(Top). (c) 2009 Elsevier B.V. All rights reserved.