Embedding suspensions into hyperspace suspensions (Q387910)

For a metric continuum \(X\) let \(C(X)\) denote the hyperspace of subcontinua of \(X\) with the Hausdorff metric and let \(F_{1}(X)\) denote the subset of \(C(X)\) consisting of all singletons of \(X\). The hyperspace suspension \(HS(X)\) is the quotient space \(C(X)/F_{1}(X)\). In this paper the authors study the class \(\mathcal{L}\) of continua \(X\) which admit an embedding of the topological suspension \(Sus(X)\) into the hyperspace suspension such that the vertices of \(Sus(X)\) are sent to the two distinguished points \(\{X\}\) and \(\{F_{1}(X)\}\) in \(HS(X)\). It is proved that trees, fans, and smooth dendroids belong to \(\mathcal{L}\); hereditarily indecomposable continua do not belong to \(\mathcal{L}\) and the arc and the simple closed curve are the only two decomposable atriodic continua in \(\mathcal{L}\).