Local dendrites with unique n-fold hyperspace

Let Z be a metric continuum and n be a positive integer. Let C n ( Z ) be the hyperspace of the nonempty closed subsets of Z with at most n components. In this paper we prove the following result: Let X be a local dendrite such that every point of X has a neighborhood which is a dendrite whose set of end points is closed and Z is any continuum such that C n ( X ) is homeomorphic to C m ( Z ) for some n , m ⩾ 3 , then X is homeomorphic to Z.