Compactifications of [0,infinity) with unique hyperspace Fn(X)

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 ANR, then X is homeomorphic to Y. The question if the result in (a) is valid for n = 3 remains open