Topological properties on n-fold pseudo-hyperspace suspension of a continuum

Abstract

Given a continuum X we denote by Cn(X) the hyperspace of nonempty closed subsets of X having at most n components and by F1(X) the space of singletons of X, topologized with the Hausdorff metric [11]. In 2008, J. Macías, defined the n-fold pseudo-hyperspace suspension of a continuum X[16], PHSn(X), as the quotient space Cn(X)/F1(X), topologized with the quotient topology. Given a map f:X→Y, we consider the induced maps Cn(f):Cn(X)→Cn(Y) and PHSn(f):PHSn(X)→PHSn(Y). In this paper we present some topological properties of the hyperspace PHSn(X), we investigate some relationships between the induced maps, Cn(f), PHSn(f) and the function f. Moreover, we partially answered a question posed by J. Macias and S. Macias in [17, Question 8.2].