Local dendrites with unique hyperspace \(C(X)\)
A continuum \(X\) is a nonempty, compact, connected metric space. The authors are concerned with the hyperspace \(C(X)\) of the subcontinua of \(X\). By definition, \(X\) has unique hyperspace \(C(X)\) if \(C(X)\cong C(Y)\) implies that \(X\cong Y\). Much work has been done on this subject, and the authors provide a list of references including many of these results. A dendrite is a locally connected continuum that contains no simple closed curve. Define \(\mathcal{D}\) to be the class of dendrites whose set of endpoints is closed. It is known, \textit{D. Herrera-Carrasco}, [Houston J. Math. 33, No.~3, 795--805 (2007; Zbl 1147.54005)], that if \(X\in\mathcal{D}\) and \(X\) is not an arc, then \(X\) has unique hyperspace \(C(X)\). Let \(\mathcal{L}\) be the class of local dendrites and \(\mathcal{L}\mathcal{D}\) the class of local dendrites each point of which has a neighborhood in \(\mathcal{D}\). The authors study the structures of \(\mathcal{D}\) and \(\mathcal{L}\mathcal{D}\). They prove, as an application, that if \(X \in\mathcal{L}\mathcal{D}\) and \(X\) is not an arc, then \(X\) has unique hyperspace \(C(X)\).
scientific article; zbMATH DE number 5775726
scientific article