Retractions and contractibility in hyperspaces

Abstract

Given a metric continuum X, let 2(X) denote the hyperspace of all nonempty closed subsets of X. For each positive integer k let C-k(X) stand for the hyperspace of members of 2(X) having at most k components. Consider mappings phi(B) : C-k (X) --> Ck+m (X) (where B is an element of C-m (X)) and psi(B) : 2(X) --> 2(X) both defined by A bar right arrow A boolean OR B. We give necessary and sufficient conditions under which these mappings are deformation retractions (under a special convention for phi(B)). The conditions are related to the contractibility of the corresponding hyperspaces. (C) 2006 Elsevier B.V. All rights reserved.