Cells in hyperspaces
type: Item
instance of: scholarly article
title: Cells in hyperspaces
zbMATH Open document ID: 1116.54007
published in: Topology and its Applications
publication date: 2007-03-08T00:00:00Z
review text: Let \(C(X)\) denote the hyperspace of subcontinua of a continuum \(X\) equipped with the Hausdorff metric. For \(A,B\in C(X)\) with \(B\subset A\), \(C(B, A)=\{D\in C(A): B\subset D\}\), and if \(p\in X\), \(C(\{p\}, X)\) is denoted by \(C(p, X)\). For a natural number \(k\) , a continuum \(Y\) is called a \(k\)-od, if there exists \(M\in C(Y)\) such that \(Y\setminus M\) has at least \(k\) components, and this \(M\) is called the core of the \(k\)-od. A space \(X\) is called a \(k\)-cell, if \(X\) is homeomorphic to \([0,1]^k\). The author proves the following: Theorem 3.11. For a natural number \(k> 1\) and \(p\in X\), if \(p\) is not in the core of any \(k\)-od, then \(C(p, X)\) contains no \(k\)-cells. Corollary 3.12. The point \(p\in X\) is contained in the core of a \(k\)-cell if and only if \(C(p, X)\) contains a \(k\)-cell. Corollary 3.15. Let \(X\) be a continuum and \(L\in C(X)\). Then \(L\) is contained in the core of a \(k\)-od if and only if \(C(L,X)\) contains a \(k\)-cell. As a consequence, the author shows: For a continuum \(X\) and \(A\in C(X)\), \(A\) is contained in the core of a \(k\)-od if and only if \(C(A, X)\) contains a \(k\)-cell.
reviewed by: Akihiro Okuyama
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 5131629
zbMATH Keywords: continuum
zbMATH Keywords: hyperspace
zbMATH Keywords: Hausdorff metric
zbMATH Keywords: Whitney map
zbMATH Keywords: \(k\)-od, \(k\)-cell
author: Patricia Pellicer-Covarrubias
MaRDI profile type: Publication
full work available at URL: j.topol.2004.12.011
OpenAlex ID: W2073803232
cites work: Intervals of Continua which are Hilbert Cubes
cites work: Induced mappings on hyperspaces
cites work: Q4276152
cites work: Q4230677
cites work: Q3869981
cites work: Q4819421
cites work: Retractions in hyperspaces.
cites work: The hyperspaces \(K(X)\)
cites work: Q5693036
DOI: 10.1016/J.TOPOL.2004.12.011
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description: scientific article; zbMATH DE number 5131629