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Given a metric continuum X, we consider the hyperspace Cn(X) of all nonempty closed subsets of X with at most n components. In this paper we prove that if n≠ 2, X is an indecomposable continuum such that all its proper nondegenerate subcontinua are arcs and Y is a continuum such that Cn(X) is homeomorphic to Cn(Y), then X is homeomorphic to Y (that is, X has unique hyperspace Cn(X)).

Given a metric continuum X, we consider the hyperspace Cn(X) of all nonempty closed subsets of X with at most n components. In this paper we prove that if n≠ 2, X is an indecomposable continuum such that all its proper nondegenerate subcontinua are arcs and Y is a continuum such that Cn(X) is homeomorphic to Cn(Y), then X is homeomorphic to Y (that is, X has unique hyperspace Cn(X)).

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.

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.