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A solution to a problem about the Erdős space

Volume 259 / 2022

Süleyman Önal, Servet Soyarslan Fundamenta Mathematicae 259 (2022), 207-211 MSC: Primary 54H11; Secondary 54F50, 46B45, 46A45. DOI: 10.4064/fm192-4-2022 Published online: 28 June 2022

Abstract

For the Erdős space, ($\mathfrak {E}, \tau $), let us define a new topology, $\tau _{\rm clopen}$, generated by all clopen subsets of $\mathfrak {E}$. A. V. Arhangel’skii and J. van Mill asked whether the topology $\tau _{\rm clopen}$ is compatible with the group structure on $\mathfrak {E}$. In this paper, we give a negative answer to this question by showing that there exists a clopen subset $O$ of $\mathfrak {E}$ such that $0\in O$ and $K + U \nsubseteq O$ for every unbounded set $K$ of $\mathfrak {E}$ and every set $U\in \tau $ containing 0.

Authors

  • Süleyman ÖnalMiddle East Technical University
    Department of Mathematics
    06531 Ankara, Turkey
    e-mail
  • Servet Soyarslan
    e-mail

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