Graphs have unique quotient hyperspace \(\mathcal{C}_1^n(X)\) (Q2074375)

A \textit{continuum} is a compact, connected and nonempty metric space. Given a continuum \(X\) and a positive integer \(n\), the symbol \(C_{n}(X)\) denotes the hyperspace of all non empty, closed subsets of \(X\) having at most \(n\) components, topologized with the Hausdorff metric. Also, the symbol \(C^{1}_{n}(X)\) denotes the quotient space \(C_{n}(X)/C_{1}(X)\). In the paper under review, the authors show that, given a finite graph \(X\) and a continuum \(Y\) such that \(C^{1}_{n}(X)\) and \(C^{1}_{n}(Y)\) are homeomorphic, it holds that \(X\) and \(Y\) are homeomorphic. This result extends work carried out by several authors in the theory of hyperspaces related to the study and characterization of unique hyperspaces.