Multivalue and shape for paracompacta

Abstract

We give a new description of the shape category of paracompact spaces in terms of multivalued maps that allows us to find new characterizations, in some way intrinsic, of calmness, movability and maps inducing shape equivalences, and to extend the concepts of internal movability and internal shape for such spaces. By using the approach to shape described here we introduce a topology on the set Sh(X, Y ). These spaces are homeomorphic to the spaces introduced in [the authors, Topologizing the set of shape morphisms of topological spaces (preprint)]. We use some properties of these spaces to obtain results concerning the above generalizations. We prove that for every paracompact spaces X, Y such that Sh(X, Y ) is discrete every c refinable map between them is a shape equivalence. As a corollary we generalize to this context a result of Kato. We give a short proof of a known theorem of Kozlowski- Segal about the shape of zero dimensional paracompacta.