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Communicated by Milos Kurilic Abstract. Let X be a Hausdorff continuum (a compact connected Hausdorff space). Let 2 X (respectively, Cn(X)) denote the hyperspace of nonempty closed subsets of X (respectively, nonempty closed subsets of X with at most n components), with the Vietoris topology. We prove that if X is hereditarily indecomposable, Y is a Hausdorff continuum and 2X (respectively Cn(X)) is homeomorphic to 2Y (respectively, Cn(Y )), then X is homeomorphic to Y.

Hereditarily indecomposable Hausdorff continua have unique hyperspaces 2X and Cn(X)