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\(n\)-fold hyperspaces, cones and products. (English) Zbl 1030.54021

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.

MSC:

54F15 Continua and generalizations
54B20 Hyperspaces in general topology
54B15 Quotient spaces, decompositions in general topology
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