About the hyperspace H(X)/H(X;K)

Abstract

Let X be a continuum, K a nonempty closed subset of X, and let n be a positive integer. In this paper, we consider the hyperspaces $2^X$ and $C_n\left(X\right)$, consisting of all nonempty closed subsets of X and of all nonempty closed subsets of X having at most n components, respectively. If $H\left(X\right)\in \{2^X,C_n(X\left)\right\}$, $H(X;K)$ denotes the hyperspace of all elements in $H\left(X\right)$ intersecting K. In this paper we present some topological properties of the quotient space $H\left(X\right)/H(X;K)$, going forward in its study in the available literature. In the class of finite graphs, we study the problem of determining conditions on X and K such that $C_n\left(X\right)$ and $C_n\left(X\right)/C_n(X;K)$ are homeomorphic, obtaining in this direction some characterizations.