Framed continua have unique n-fold hyperspace suspension

摘要

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 with at most n components (respectively, n points). Let HSn (X) be the quotient space C-n(X)/F-n(X) which is obtained from C-n (X) by identifying F-n (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 Macias-Romero and Francisco Vazquez-Juarez who previously proved the corresponding result for finite graphs. This answers a question by Alejandro Illanes. (C) 2015 Published by Elsevier B.V.