Uniqueness of the (n,m)-fold hyperspace suspension for continua

Abstract

Let n,m∈N with m≤n and X be a metric continuum. We consider the hyperspaces Cn(X) (respectively, Fn(X)) of all nonempty closed subsets of X with at most n components (respectively, n points). The (n,m)-fold hyperspace suspension on X was introduced in 2018 by Anaya, Maya, and Vázquez-Juárez, to be the quotient space Cn(X)/Fm(X) which is obtained from Cn(X) by identifying Fm(X) into a one-point set. In this paper: we present some properties of this hyperspace. In particular, if X is a meshed continuum, Y is a continuum and Cn(X)/Fm(X) is homeomorphic to Cn(Y)/Fm(Y), then X is homeomorphic to Y. Furthermore, we prove that if X,Y are almost meshed locally connected continua such that Cn(X)/Fm(X) is homeomorphic to Cn(Y)/Fm(Y), then X is homeomorphic to Y, for each n>2 and m>1. We also provide examples of almost meshed locally connected continua X which do not have unique hyperspace Cn(X)/Fm(X).