A solution to a problem about the Erdős space
Volume 259 / 2022
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.
