On the hyperespace Cn(X)/CnK(X)

Abstract

Let X be a continuum and n a positive integer. Let Cn(X) be the hyperspace of all nonempty closed subsets of X with at most n components endowed with the Hausdorff metric. For K compact subset of X, define the hyperspace CnK(X) = {A ∈ C (X) : K ⊂ A}. In this paper, we consider the quotient space Cn(X)/ CnK(X) which can be a tool to study the space Cn(X). We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility.