Cells and \(n\)-fold hyperspaces (Q2833635)

For a metric continuum \(X\), \(C_{n}(X)\) denotes the hyperspace of nonempty closed subsets of \(X\) having at most \(n\) components. An \(n\)-cell is a continuum homeomorphic to the Cartesian product \([0,1] ^n\), and an \(n\)-od is a continuum \(B\) containing a subcontinuum \(A\) such that \(B \setminus A\) has at least \(n\) components.NEWLINENEWLINELocating cells in hyperspaces has been a useful tool to study them. In Theorem 3.4 of [Topology Appl. 109, No. 2, 237--256 (2001; Zbl 0979.54013)], the third named author showed that \(C_{n}(X)\) always contains \(n\)-cells. In [Fundam. Math. 130, No. 1, 57--65 (1988; Zbl 0662.54005)], the reviewer proved the equivalence: \(C_{1}(X)\) contains an \(m\)-cell if and only if \(X\) contains an \(m\)-od.NEWLINENEWLINEIn the paper under review the authors solve the simplest case of the following question: if \(n<m\), when does \(C_{n}(X)\) contain an \(m\)-cell? Namely, they prove that \(C_{n}(X)\) contains an \((n+1)\)-cell if and only if \(X\) contains a \(2\)-od (equivalently, \(X\) is not hereditarily indecomposable). Very recently, the reviewer and Verónica Martínez-de-la-Vega have found a complete answer to the mentioned question by proving that \(C_{n}(X)\) contains an \(m\)-cell if and only if there exist positive integers \(k_{1},\ldots ,k_{n}\) and pairwise disjoint subcontinua \(B_{1},\ldots ,B_{n}\) of \(X\) such that for each \(i\), \(B_{i}\) is a \(k_{i}\)-od in \(X\) and \(k_{1}+\cdots +k_{n}=m\).NEWLINENEWLINEThe authors also find a characterization of continua for which \(C_{n}(X)\) is an \(m\)-cell. They prove the following result.NEWLINENEWLINETHEOREM. \(C_{n}(X)\) is an \(m\)-cell if and only if either \(X\) is an arc, \(n\) belongs to {1,2} and \(m\) belongs to {2,4} or \(X\) is a simple closed curve, \(n=1\) and \(m=2\).NEWLINENEWLINEThe third mentioned author previously asked the question: Is \(C_{3}([0,1])\) a \(6\)-cell?, in his book [Topics on continua. Boca Raton, FL: Chapman \& Hall/CRC (2005; Zbl 1081.54002)], and in the paper under review the authors mention that they provide a negative answer to this. However, this fact was known since the year 2003 as was mentioned in the review of the cited book in Zentralblatt.