# Connected neighborhoods in Cartesian products of solenoids

## TL;DR

This paper proves that certain subcontinua in products of co-prime m-adic solenoids are ample, meaning they can be contained in open connected neighborhoods, contrasting with properties of Cartesian squares of fixed solenoids.

## Contribution

It establishes that subcontinua with onto projections in products of co-prime solenoids are ample, answering a question and contrasting with known properties of fixed solenoid squares.

## Key findings

- Subcontinua with onto projections are ample in product solenoids.
- Such subcontinua can be contained in open connected neighborhoods.
- Contrasts with non-ampleness of diagonals in Cartesian squares of fixed solenoids.

## Abstract

Given a collection of pairwise co-prime integers $% m_{1},\ldots ,m_{r}$, 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. \L ysko, 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].