Framed continua have unique \(n\)-fold hyperspace suspension
For a metric continuum \(X\), let \(C_n(X)\) denote the hyperspace of nonempty closed subsets of \(X\) with at most \(n\) components and let \(F_n(X)\) denote the hyperspace of nonempty subsets of \(X\) with at most \(n\) points, both equipped with the Hausdorff metric. Notice that \(F_n(X)\) is contained in \(C_n(X)\), so it is possible to define the quotient continuum \(HS_n(X)\) which is obtained from \(C_n(X)\) by identifying the set \(F_n(X)\) into a one-point set. Let \(G(X)\) be the set of points in \(X\) having a neighborhood in \(X\) which is a finite graph. The continuum \(X\) is framed provided that \(G(X)\) is dense in \(X\) and \(X\) has a basis of neighborhoods \(B\) such that for each element \(U\) in \(B\), \(U \cap G(X)\) is connected. Notice that each finite graph is framed. The continuum \(X\) has unique hyperspace suspension \(HS_n(X)\) provided the following implication holds: if \(Y\) is a continuum and \(HS_n(X)\) is homeomorphic to \(HS_n(Y)\), then \(X\) is homeomorphic to \(Y\). Uniqueness of hyperspaces has been widely studied. An account on this topic is in [\textit{A. Illanes}, Quest. Answers Gen. Topology 30, No. 1, 21--44 (2012; Zbl 1275.54006)]. In [\textit{D. Herrera-Carrasco} et al., Topol. Proc. 44, 75--95 (2014; Zbl 1297.54027)] it was shown that if \(X\) is a finite graph, then \(X\) has unique hyperspace suspension \(HS_n(X)\). In the paper under review the authors extend this result by proving that if \(X\) is a framed continuum, then \(X\) has unique hyperspace suspension \(HS_n(X)\).
scientific article; zbMATH DE number 6522019
scientific article