Connected neighborhoods in Cartesian products of solenoids

Abstract

Given a collection of pairwise co-prime integers , greater than $1$, 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, in this paper 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 [10]. This contrasts with the property of Cartesian squares of fixed solenoids ${\Sigma }_{m_i}\times {\Sigma }_{m_i}$, whose diagonals are never ample [1].