Disk-like products of continua
type: Item
instance of: scholarly article
title: Disk-like products of continua
zbMATH Open document ID: 1122.54017
author: Raúl Escobedo
author: Maria de J. López
author: Jesús F. Tenorio
published in: Topology and its Applications
publication date: 2007-04-18T00:00:00Z
review text: By a continuum is meant a nondegenerate, compact, connected metric space. A continuum \(X\) is called disk-like if for each \(\varepsilon>0\), there is an \(\varepsilon\)-map \(f\) of \(X\) onto \([0,1]\times[0,1]\), i.e., for each \(x\in[0,1]\times[0,1]\), the diameter of \(f^{-1}(x)\) is less than \(\varepsilon\). The authors cite a result due to \textit{C. Hagopian} [Proc. Am. Math. Soc. 51, 448--452 (1975; Zbl 0308.54025)] which states that if \(X\), \(Y\) are continua and \(X\times Y\) is disk-like, then \(X\), \(Y\) are atriodic and hereditarily unicoherent. They now establish more facts about such a disk-like product. One says that a space \(X\) is contractible with respect to another space \(Z\) if every map of \(X\) to \(Z\) is inessential. The main results of the paper are: Theorem 7. If \(X\) and \(Y\) are continua such that \(X\times Y\) is disk-like, then \(X\) is contractible with respect to any \(\roman{ANR}\) and each subcontinuum of \(X\) is contractible with respect to \(S^1\). Theorem 9. If \(X\) and \(Y\) are continua such that \(X\times Y\) is disk-like, then \(X\) is a tree-like continuum that is hereditarily weakly confluent.
reviewed by: Leonard R. Rubin
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 5144522
zbMATH Keywords: continuum
zbMATH Keywords: contractible with respect to
zbMATH Keywords: disk-like
zbMATH Keywords: unicoherent
zbMATH Keywords: tree-like
zbMATH Keywords: confluent
zbMATH Keywords: weakly confluent
MaRDI profile type: Publication
full work available at URL: j.topol.2006.06.006
OpenAlex ID: W1993895052
cites work: Characterizations of tree-like continua
cites work: Atriodic Acyclic Continua and Class W
cites work: Q4200845
cites work: Disk-Like Products of λ Connected Continua. I
cites work: Q4230677
cites work: Q4086402
cites work: ε-Mappings Onto Polyhedra
cites work: Q4004123
cites work: Q4782690
cites work: Q5837858
DOI: 10.1016/J.TOPOL.2006.06.006
Recommended article: Q5288194
Recommended article: Hereditarily weakly confluent maps and a characterization of class \(HW\)
Recommended article: A Characterization of Semispan of Continua
Recommended article: Q3312916
Recommended article: Two Continua Having A Property of J. L. Kelley
Recommended article: 2-To-1 Maps on Hereditarily Indecomposable Continua
Recommended article: Openness of induced mappings
Recommended article: Some classes of locally connected continua
Recommended article: Q3221029
Recommended article: Outlet Points and Homogeneous Continua
description: scientific article; zbMATH DE number 5144522