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 Cn(X) (respectively, Fn(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 Cn(X)/Fm(X) which is obtained from Cn(X) by shrinking Fm(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.