Cells and \(n\)-fold hyperspaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1355.54012
author: Javier Camargo
author: Daniel Herrera
author: Sergio Macías
published in: Colloquium Mathematicum
publication date: 2016-11-18T00:00:00Z
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 6654792
zbMATH Keywords: Cantor manifold
zbMATH Keywords: cell
zbMATH Keywords: continuum
zbMATH Keywords: graph
zbMATH Keywords: hereditarily indecomposable continuum
zbMATH Keywords: hyperspace
zbMATH Keywords: hyperspace suspension
zbMATH Keywords: indecomposable continuum
MaRDI profile type: Publication
full work available at URL: cm6527-1-2016
OpenAlex ID: W2343717809
DOI: 10.4064/CM6527-1-2016
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title: Cells and \(n\)-fold hyperspaces
review text: 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; …
reviewed by: Alejandro Illanes
description: scientific article; zbMATH DE number 6654792