On the hyperspace $C_n(X)/{C_n}_K(X)$

Abstract

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 ${C_n}_K(X)=\{A\in C_n(X)\colon K\subset A\}$. In this paper, we consider the hyperspace $C_K^n(X)=C_n(X)/{C_n}_K(X)$, 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.