Connected open neighborhoods in products
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)].
scientific article; zbMATH DE number 6699158
scientific article