About the hyperspace \(\mathcal{H}(X) / \mathcal{H}(X; K)\) (Q6559426)

For a metric continuum \(X\) and the natural number \(n\) let the symbol \(2^{X}\) denote the hyperspace of nonempty closed subsets of \(X\), the symbol \(F_{n}(X)\) -- the \(n\)-fold symmetric product of \(X\), the symbol \(C_{n}(X)\) -- the \(n\)-fold hyperspace of \(X\). Let \(K \in 2^{X} \setminus \{X\}\). For a given hyperspace \(\mathcal{H}(X) \in \{2^{X}, F_{n}(X), C_{n}(X) \}\) the \textit{intersection hyperspace in \(\mathcal{H}(X)\)} is defined \(\mathcal{H}(X;K) = \{A \in\mathcal{H}(X): K \cap A \neq \emptyset \}\). The hyperspace \(\mathcal{H}(X;K)\) is a subcontinuum of \(\mathcal{H}(X)\).\N\NIn this paper, the properties of the quotient space \(\mathcal{H}(X)/\mathcal{H}(X;K)\) are presented. In particular, when \(X\) is a finite graph, the authors study the conditions which \(X\) and \(K\) should satisfy for \(C_{n}(X)\) and \(C_{n}(X)/C_{n}(X;K)\) to be homeomorphic.