On symmetric products of continua
type: Item
instance of: scholarly article
title: On symmetric products of continua
zbMATH Open document ID: 0967.54011
DOI: 10.1016/S0166-8641(97)00233-2
published in: Topology and its Applications
publication date: 2001-08-14T00:00:00Z
review text: 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 \(n \geq 3\).
reviewed by: Władysław Makuchowski
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54F55
zbMATH DE Number: 1307880
zbMATH Keywords: hyperspace
zbMATH Keywords: inverse limit
zbMATH Keywords: symmetric product
zbMATH Keywords: Whitney map
author: Sergio Macías
MaRDI profile type: Publication
cites work: On the third symmetric potency of the circumference
cites work: Hyperspaces of finite subsets which are homeomorphic to \(\aleph _ 0\)- dimensional linear metric spaces
cites work: Mapping Theorems for Non-Compact Spaces
cites work: Q4005279
cites work: Q3761263
cites work: Cells and cubes in hyperspaces
cites work: Monotone and open Whitney maps defined in \(2^ X\)
cites work: The space of subcontinua of a 2-dimensional continuum is infinite dimensional
cites work: On symmetric products
cites work: Multicoherence Techniques Applied to Inverse Limits
cites work: Q3869981
cites work: Q5633264
cites work: Concerning Triodic Continua
cites work: Q5625951
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description: scientific article; zbMATH DE number 1307880