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

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 2 X (respectively Cn(X)) is homeomorphic to 2 Y (respectively, Cn(Y)), then X is homeomorphic to Y .