Framed continua have unique n-fold hyperspace suspension

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 with at most n components (respectively, n points). Let HSn(X) be the quotient space Cn(X)/Fn(X) which is obtained from Cn(X) by identifying Fn(X) into a one-point set. In this paper we prove that if X is a framed continuum and Y is a continuum such that HSn(X) is homeomorphic to HSn(Y), then X is homeomorphic to Y. This result generalizes the work of David Herrera-Carrasco, Alejandro Illanes, Fernando Macías-Romero and Francisco Vázquez-Juárez who previously proved the corresponding result for finite graphs. This answers a question by Alejandro Illanes.