A solution of a problem about Erd\"os Space

TL;DR

This paper investigates the compatibility of a specific topology generated by clopen sets with the group structure on Erdős space, ultimately showing they are incompatible.

Contribution

It provides a negative answer to whether the topology generated by clopen subsets on Erdős space is compatible with its group structure.

Findings

The topology $ au_{clopen}$ is not compatible with the group structure on Erdős space.

Answers an open question posed by Arhangel'skii and Van Mill.

Contributes to understanding the topological group properties of Erdős space.

Abstract

For Erd\H{o}s space, $\mathfrak{E}$, let us define a topology, $\tau_{clopen}$, which is generated by clopen subsets of $\mathfrak{E}$. A. V. Arhangel'skii and J. Van Mill asked whether the topology $\tau_{clopen}$ is compatible with the group structure on $\mathfrak{E}$. In this paper, we give a negative answer for this question.