Weakening topologies on a countable Abelian group of finite exponent (Q2202726)

The notion of a locally minimal topological group was introduced by Morris and Pestov. A topological group \((G,\tau)\) is called \textit{locally minimal} if there exists a neighborhood \(V\) of identity such that if \(\sigma\subseteq \tau\) is a group topology on \(G\) such that \(V\) is a \(\sigma\)-neighborhood of identity, then \(\sigma=\tau\). The author proves the following result which solves the Question 7.35(b) from [\textit{D. Dikranjan} and \textit{M. Megrelishvili}, in: Recent progress in general topology III. Based on the presentations at the Prague symposium, Prague, Czech Republic, 2001. Amsterdam: Atlantis Press, 229--327 (2014; Zbl 1308.54002)]: Any countable abelian group of finite exponent does not admit a locally minimal group topology.