On topologizable and non-topologizable groups
Group Theory
2013-10-02 v3 General Topology
Abstract
A group is called hereditarily non-topologizable if, for every , no quotient of admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked -generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.
Cite
@article{arxiv.1210.7895,
title = {On topologizable and non-topologizable groups},
author = {A. A. Klyachko and A. Yu. Olshanskii and D. V. Osin},
journal= {arXiv preprint arXiv:1210.7895},
year = {2013}
}