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Given a metric continuum X, Fn(X) denotes the hyperspace of nonempty subsets of X with at most n elements. In this paper we show the following result. Suppose that X is a metric compactification of [0, ∞), Y is a continuum and Fn(X) is homemorphic to Fn(Y ). Then: (a) if n = 3, then X is homeomorphic to Y , (b) if n = 3 and the remainder of X is an AN R, then X is homeomorphic to Y . The question if the result in (a) is valid for n = 3 remains open. 2000 Mathematics Subject Classification.doi:10.3336/gm.44.2.12 fatcat:vgx5wdknzjhzjcww2oya45zosi