Connected neighborhoods in products
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.
scientific article; zbMATH DE number 6894001
scientific article