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

Let X be a continuum and n a positive integer. Let C-n(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 C-nK(X) = {A is an element of C-n (X): K subset of A}. In this paper, we consider the hyperspace C-K(n) (X) = C-n(X)/C-nK (X), which can be a tool to study the space C-n (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.