2012/10/31 by A. A. Klyachko, A. Yu. Olshanskii, D. V. Osin · 1 citation
Mathematics · #math.GR #math.GN
paper · pdf · doi:10.1016/j.topol.2013.08.017
published as Topology and its Applications, 2013, 160:16, 2104-2120
arxiv created 2013/08/09 · arxiv updated 2013/10/02
A group G is called hereditarily non-topologizable if, for every H≤ G, no quotient of H 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 k-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.