On symmetric products of continua
For a given topological space \(X\) and for a positive integer \(n\) the set \({\mathcal F}_n (X) = \{A \subset X : A \) has at most \(n\) points\(\}\) is called the \(n\)-th symmetric product of \(X\). \textit{A. Illanes} [An. Inst. Mat., Univ. Nac. Autón. Méx. 25, 11-24 (1985; Zbl 0623.54002)] proved that if \(X\) is a locally connected normal space then \({\mathcal F}_n (X)\) is unicoherent for \(n \geq 3\). The author shows that for continua the \(n\)-th symmetric product is unicoherent for \(n \geq 3\). \textit{K. Borsuk} and \textit{S. Ulam} [Bull. Am. Math. Soc. 37, 875-882 (1931; Zbl 0003.22402)] showed that \({\mathcal F}_n (I)\) is homeomorphic to \(I^n \) for \(n \in \{1,2,3\}\) (\(I\) denotes the unit interval). The author proves that the arc is the only finite dimensional continuum for which its second symmetric product is homeomorphic to its hyperspace of subcontinua and that for any finite dimensional continuum \(X\) its hyperspace of subcontinua is not homeomorphic to its \(n\)-th symmetric product for any \(n \geq 3\).
scientific article; zbMATH DE number 1307880
scientific article