Connected open neighborhoods in products
type: Item
instance of: scholarly article
title: Connected open neighborhoods in products
zbMATH Open document ID: 1374.54035
author: Alejandro Illanes
published in: Acta Mathematica Hungarica
publication date: 2017-03-31T00:00:00Z
review text: Given a family of metric continua \(\{X_{\alpha}: \alpha \in J \}\), the product \(X = \prod_{\alpha \in J} X_{\alpha}\) has the following property: \textit{full projection implies arbitrary small connected open neighborhoods (fupcon)} provided that for every subcontinuum \(M\) and open subset \(W\) of \(X\) such that \(M \subset W\) and \(\pi_{\alpha}(M) = X_{\alpha}\) for each \(\alpha \in J\) (\(\pi_{\alpha}\) is the \(\alpha^{th}\)-projection), there exists an open connected subset \(U\) of \(X\) such that \(M \subset U \subset W\). The author proves that (a) the product of a pseudoarc with any product of Knaster continua has the fupcon property and (b) there exist indecomposable chainable continua whose product does not have the fupcon property. This is an answer to questions by \textit{D. P. Bellamy} and \textit{J. M. Łysko} [Topol. Proc. 44, 223--231 (2014; Zbl 1325.54016)].
reviewed by: Władysław Makuchowski
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 6699158
zbMATH Keywords: Kelley continuum
zbMATH Keywords: Knaster continuum
zbMATH Keywords: product continuum
zbMATH Keywords: pseudoarc
MaRDI profile type: Publication
full work available at URL: s10474-015-0568-8
OpenAlex ID: W2292948183
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DOI: 10.1007/S10474-015-0568-8
description: scientific article; zbMATH DE number 6699158