Connected neighborhoods in Cartesian products of solenoids
Tom 248 / 2020
Fundamenta Mathematicae 248 (2020), 309-320
MSC: Primary 54F15; Secondary 54F50.
DOI: 10.4064/fm678-3-2019
Opublikowany online: 7 August 2019
Streszczenie
Given a collection of pairwise co-prime integers , we consider the product \Sigma =\Sigma _{m_{1}}\times \cdots \times \Sigma _{m_{r}}, where each \Sigma _{m_{i}} is the m_{i}-adic solenoid. Answering a question of D. P. Bellamy and J. M. Łysko, we prove that if M is a subcontinuum of \Sigma such that the projections of M on each \Sigma _{m_{i}} are onto, then for each open subset U in \Sigma with M\subset U, there exists an open connected subset V of \Sigma such that M\subset V\subset U, i.e. any such M is ample in the sense of Prajs and Whittington (2007). This contrasts with the property of Cartesian squares of fixed solenoids \Sigma _{m_{i}}\times \Sigma _{m_{i}}, whose diagonals are never ample (Bellamy and Łysko, 2014).
