Connected neighborhoods in products
type: Item
instance of: scholarly article
title: Connected neighborhoods in products
zbMATH Open document ID: 1401.54017
author: Alejandro Illanes
author: Jorge M. Martínez-Montejano
author: Karen Villarreal
published in: Topology and its Applications
publication date: 2018-06-25T00:00:00Z
review text: Given a family of metric continua \(\{X_\alpha: \alpha\in J\}\), the product \(X=\prod_{\alpha\in J}\) has the following property: full projection implies arbitrary small connected open neighborhoods, denoted by \textbf{fupcon}, provided that for every subcontinuum \(M\) and open subset \(U\) of \(X\) such that \(M\subset U\) and \(\pi_\alpha(M)=X_\alpha\), for each \(M\subset V\subset U\). This property has been studied, for instance in [\textit{D. P. Bellamy} and \textit{J. M. Lysko}, Topol. Proc. 44, 223--231 (2014; Zbl 1325.54016)], [\textit{A. Illanes}, Acta Math. Hung. 148, No. 1, 73--82 (2016; Zbl 1374.54035); Topology Appl. 230, 506--516 (2017; Zbl 1421.54024)]. The objective of the paper under review is to continue the study of this property. Some results that are included in this paper are: every product of homogeneous continua having the fixed point property has the fupcon property and the product of a chainable Kelley continuum and \([0,1]\) has the fupcon property. Moreover, the authors obtain an example that shows that there exists a Kelley continuum \(X\) such that \(X\times [-1,1]\) does not have the fupcon property.
reviewed by: Alicia Santiago-Santos
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 6894001
zbMATH Keywords: chainable
zbMATH Keywords: fupcon
zbMATH Keywords: product
MaRDI profile type: Publication
full work available at URL: j.topol.2018.03.028
OpenAlex ID: W2795181362
cites work: Q5407078
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DOI: 10.1016/J.TOPOL.2018.03.028
description: scientific article; zbMATH DE number 6894001