Retractions in hyperspaces

Let C(X) and 2X denote the hyperspaces of subcontinua and of closed subsets of a metric continuum X, respectively. For p∈X, define the hyperspace 2Xp={A∈2X: p∈A}. Moreover, consider the mapping A↦A∪{p}. In this paper we find necessary and sufficient conditions under which this mapping is a deformation retraction or a strong deformation retraction from 2X onto 2Xp. We obtain similar results for the case in which the domain of the mapping is C(X). Finally, we characterize local connectedness of a continuum in terms of the behavior of these mappings.