Connected neighborhoods in Cartesian products of solenoids
Given a collection of pairwise co-prime integers $% m_{1},\ldots ,m_{r}$, greater than $1$, we consider the product $Σ=Σ_{m_{1}}\times \cdots \times Σ_{m_{r}}$, where each $Σ_{m_{i}}$ is the $m_{i}$-adic solenoid. Answering a question of D. P. Bellamy and J. M. Łysko, in this paper we prove that if $M$ is a subcontinuum of $Σ$ such that the projections of $M$ on each $Σ_{m_{i}}$ are onto, then for each open subset $U$ in $Σ$ with $M\subset U$, there exists an open connected subset $V$ of $Σ$ such that $M\subset V\subset U$; i.e. any such $M$ is ample in the sense of Prajs and Whittington [10]. This contrasts with the property of Cartesian squares of fixed solenoids $Σ_{m_{i}}\times Σ_{m_{i}}$, whose diagonals are never ample [1].
math.GN↗