Macías, Sergio; Nadler, Sam B. jun. \(n\)-fold hyperspaces, cones and products. (English) Zbl 1030.54021 Topol. Proc. 26, No. 1, 255-270 (2002). The authors ask if the \(n\)-fold hyperspace \(\mathcal C_n(X)=\{A \mid A\subset X, A\) has at most \(n\) components}, \(n>1\), of a continuum \(X\) is homeomorphic to the cone over \(X\) or a product of continua. They give a partial answer to this question.The authors prove that for any continuum \(X\) and any integer \(n\) greater than one, \(\mathcal C_n(X)\) is not homeomorphic to \(Cone(X)\). On the other hand, under the assumption that \(X\) is a continuum whose \(n\)-fold hyperspace is homeomorphic to the cone over a finite dimensional continuum \(Z\) they present some properties of \(X\) and \(Z\). As to products, the authors show that if the \(n\)-fold hyperspace of a continuum \(X\) is homeomorphic to a product of two continua, then \(X\) must be hereditarily decomposable without nondegenerate proper terminal subcontinua. They consider also the case \(\mathcal C_n(S^1)\) and pose some problems. Reviewer: A.A.Ivanov (St.Peterburg) Cited in 9 Documents MSC: 54F15 Continua and generalizations 54B20 Hyperspaces in general topology 54B15 Quotient spaces, decompositions in general topology Keywords:absolute (neighborhood) retract; Cantor manifold; (\(n\)-fold) hyperspace; decomposable; hereditarily indecomposable; indecomposable; terminal; unicoherent continuum; \(n\)-fold symmetric product PDFBibTeX XMLCite \textit{S. Macías} and \textit{S. B. Nadler jun.}, Topol. Proc. 26, No. 1, 255--270 (2002; Zbl 1030.54021)