Continua with almost unique hyperspace

For a metric continuum X, let C(X) denote the hyperspace of subcontinua of X. The continuum X is said to have unique hyperspace provided that if Y is a continuum and C(X) is homeomorphic to C(Y), then X is homeomorphic to Y. Among other results, we show in this paper the following: (1) indecomposable continua such that all their proper and nondegenerate subcontinua are arcs, have unique hyperspace, (2) there are metric compactifications of the space (−∞,∞), with nondegenerate and connected remainder, that do not have unique hyperspace, (3) if X and Y are arcwise connected circle-like continua such that C(X) is homeomorphic to C(Y), then X is homeomorphic to Y. This last result is a partial answer to a question by S.B. Nadler Jr.