Local dendrites with unique hyperspace

For a continuum X we denote by C ( X ) the hyperspace of subcontinua of X, metrized by the Hausdorff metric. Let D be the class of dendrites whose set of end points is closed and let LD be the class of local dendrites X such that every point of X has a neighborhood which is in D . In this paper we study the structure of the classes D and LD . As an application, we show that if X ∈ LD is different from an arc and a simple closed curve, and Y is a continuum such that the hyperspaces C ( X ) and C ( Y ) are homeomorphic, then X is homeomorphic to Y.