engleski

An existence theorem concerning ordinary shape of Cartesian products

The paper is devoted to the question when is the Cartesian product X × 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 → X and every homotopy class of mappings [g] : Z → P, there exists a shape morphism H : Z → X × P, whose compositions with the canonical projections of X × 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.

shape ; ordinary shape ; strong shape ; direct product ; Cartesian product ; resolution ; homotopy mapping of systems ; inverse limit ; coherent homotopy

nije evidentirano