A resolution for the product of a compactum with a polyhedron

In shape theory one associates with compact Hausdorff spaces X limits p :X→X of inverse systems of compact polyhedra (or compact ANRs). However, in the case of noncompact spaces, polyhedral limits have to be replaced by polyhedral resolutions p :X→X. If Y is an arbitrary space and p :X→X is an inverse limit, then p×1 :X×Y→X×Y is also an inverse limit. However, the analogue of this statement is false for resolutions, even in the case when p is the limit of an inverse sequence of compact polyhedra and Y is an infinite discrete space. In the present paper, with the limit p :X→X of an arbitrary inverse system of compact Hausdorff spaces and with an arbitrary simplicial complex K one associates a resolution q :X×P→Y, where P=|K| is the geometric realization of K, this resolution extends the limit p×1 :X×P→X×P and it consists of paracompact spaces. Moreover, if the members of X are compact polyhedra, the members of Y have the homotopy type of polyhedra. The resolution does not depend on the choice of the triangulation K of P (up to isomorphism in pro-Top).