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Connected neighborhoods in Cartesian products of solenoids

Volume 248 / 2020

Jan P. Boroński, Alejandro Illanes, Emanuel R. Márquez Fundamenta Mathematicae 248 (2020), 309-320 MSC: Primary 54F15; Secondary 54F50. DOI: 10.4064/fm678-3-2019 Published online: 7 August 2019

Abstract

Given a collection of pairwise co-prime integers $m_{1},\ldots ,m_{r} \gt 1$, 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, 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).

Authors

  • Jan P. BorońskiNational Supercomputing Centre IT4Innovations
    Division of the University of Ostrava
    Institute for Research and Applications
    of Fuzzy Modeling
    30. Dubna 22
    701 03 Ostrava, Czech Republic
    and Faculty of Applied Mathematics
    AGH University of Science and Technology
    Al. Mickiewicza 30
    30-059 Kraków, Poland
    e-mail
  • Alejandro IllanesInstituto de Matemáticas
    Universidad Nacional Autónoma
    de México
    Circuito Exterior, Cd. Universitaria
    México, D.F., 04510, México
    e-mail
  • Emanuel R. MárquezDepartamento de Matemáticas
    Facultad de Ciencias
    Universidad Nacional Autónoma de México
    Circuito Exterior, Cd. Universitaria
    México, D.F., 04510, México
    e-mail

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