Dendrites and symmetric products

For a given continuum X and a natural number n, we consider the hyperspace Fn(X) of all nonempty subsets of X with at most n points, metrized by the Hausdorff metric. In this paper we show that if X is a dendrite whose set of end points is closed, n N and Y is a continuum such that the hyperspaces Fn(X) and Fn(Y) are homeomorphic, then Y is a dendrite whose set of end points is closed.