AN EXISTENCE THEOREM CONCERNING ORDINARY SHAPE OF CARTESIAN PRODUCTS

Abstract

The paper is devoted to the question when is the Cartesian product \(X \times P\) of a compact metric space \(X\) and a polyhedron \(P\) a product in the shape category of topological spaces. The question consists of two parts. The existence part, which asks whether, for every topological space \(Z\), every shape morphism \(F : Z \to X\) and every homotopy class of mappings \([g] : Z \to P\), there exists a shape morphism \(H : Z \to X \times P\), whose compositions with the canonical projections of \(X \times P\) equal \(F\) and \([g]\), respectively. The uniqueness part asks whether \(H\) is unique. It is known that, in general, the uniqueness part does not hold even when \(Z\) is a polyhedron. The main result of the paper asserts that the existence part always holds. The proof is based on an analogous result for strong shape.