On the uniqueness of the n-fold pseudo-hyperspace suspension for locally connected continua

Abstract

Let X be a metric continuum. Let n be a positive integer, we consider the hyperspace Cn(X) of all nonempty closed subsets of X with at most n components and F1(X)={{x}:x∈X}. The n-fold pseudo-hyperspace suspension of X is the quotient space Cn(X)/F1(X) and it is denoted by PHSn(X). In this paper we prove that: (1) if X is a meshed continuum and Y is a continuum such that PHSn(X) is homeomorphic to PHSn(Y), then X is homeomorphic to Y, for each n>1. (2) There are locally connected continua without unique hyperspace PHSn(X).