LOCAL CONNECTEDNESS AND RETRACTIONS IN HYPERSPACES

Given a metric continuum X, let 2(X) denote the hyperspace of all nonempty closed subsets of X; further, let C(X) (and C-2(X)) be the hyperspace of all members of 2(X) which are connected (composed of at most two components, respectively). For a point p is an element of X we consider mappings phi(p) : C(X) -> C-2(X) and psi(p) : 2(X) -> 2(X) both defined by A bar right arrow A boolean OR {p}. Relations between local connectedness of X at p and conditions under which these mappings are deformation retractions or strong deformation retractions (under a special convention for phi(p)) are studied. The conditions are related to contractibility of the corresponding hyperspaces and to the existence of a special local base of the point p.