Let X be a continuum. Let CðX Þ be the hyperspace of all closed, connected and nonempty subsets of X , with the Hausdor¤ metric. For a mapping f : X ! Y between continua, let Cð f Þ : CðX Þ ! CðY Þ be the induced mapping by f , given by Cð f ÞðAÞ ¼ f ðAÞ. In this paper we study the hyperspace CðX Þ ¼ fCðAÞ : A A CðX Þg as a subspace of CðCðX ÞÞ, and define an induced function Cð f Þ between CðX Þ and CðY Þ. We prove some relationships between the functions f , Cð f Þ and Cð f Þ for the following classes of mapping: confluent, light, monotone and weakly confluent.

In this paper we study the hyperspace of all nonempty closed totally disconnected subsets of a space, equipped with the Vietoris topology. We show results of compactness, connectedness and local connectedness for this hyperspace. We also include a study of path connectedness, particularly we prove that for a smooth dendroid this hyperspace is pathwise connected, and we present a general result which implies that for an Euclidean space this hyperspace has uncountably many arc components.