The hyperspace HSmn(X) for a finite graph X is unique

For a metric continuum X and a positive integer n, we consider the hyperspaces C-n(X) (respectively, F-n(X)) of all nonempty closed subsets of X having at most n components (respectively, n points). Given positive integers n and m such that n >= m, we define HSmn (X) as the quotient space C-n(X)/F-m(X) which is obtained from Cn(X) by shrinking F-m(X) to a point. In this paper we prove that if X is a finite graph and Y is a continuum such that HSmn(X) is homeomorphic to HSmn(Y), then X is homeomorphic to Y. (C) 2017 Elsevier B.V. All rights reserved.