www.elsevier.com/locate/topol Retractions and contractibility in hyperspaces

Given a metric continuum X, let 2X denote the hyperspace of all nonempty closed subsets of X. For each positive integer k let Ck(X) stand for the hyperspace of members of 2X having at most k components. Consider mappings ϕB:Ck(X) → Ck+m(X) (where B ∈ Cm(X)) and ψB: 2X → 2X both defined by A → A ∪ B. We give necessary and sufficient conditions under which these mappings are deformation retractions (under a special convention for ϕB). The conditions are related to the contractibility of the corresponding hyperspaces