Abstract
Let X be a metric continuum. Let n be a positive integer, and let Cn(X) be the space of all nonempty closed subsets of X with at most n components topologized with the Hausdorff metric. We consider the quotient space C1n(X)=Cn(X)/C1(X), with the quotient topology. We prove that given a graph X and a continuum Y, we have that C1n(X) is homeomorphic to C1n(Y) if and only if X is homeomorphic to Y; in the case when X has no ramification points, this answers a question by J. Camargo and S. Macías.
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