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

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 ( X ), consisting of all nonempty closed subsets of X and of all nonempty closed subsets of X having at most n components, respectively. If 7t ( X ) E { 2 X , C n ( X ) } , 7t ( X ; K ) denotes the hyperspace of all elements in 7t ( X ) intersecting K . In this paper we present some topological properties of the quotient space 7t ( X ) /7t ( 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 ( X ) and C n ( X ) /C n ( X ; K ) are homeomorphic, obtaining in this direction some characterizations. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.