Refining connected topological group topologies on Abelian torsion groups (Q1295327)

There is a vast literature, to which the authors' List of References provides a substantial guide, dealing with questions of the following type: Given topological properties \(\mathbb{P}\) and \(\mathbb{Q}\), does every topological group \((G,{\mathcal T})\in\mathbb{P}\) admit a topological group topology \({\mathcal U}\) such that \({\mathcal U}\supseteq {\mathcal T}\), \({\mathcal U}\neq {\mathcal T}\), and \((G,{\mathcal U})\in \mathbb{Q}\)? Continuing their earlier work concerning Abelian torsion-free groups [J. Pure Appl. Algebra 124, No. 1-3, 281-288 (1998; Zbl 0895.54023)], the authors here establish positive answers to the above-cited questions for every (torsion) Abelian group of bounded exponent in each of the following two cases: (A) \(\mathbb{P}= \mathbb{Q}=\) the class of connected c.c.c. groups \((G,{\mathcal T})\) with \(w(G,{\mathcal T})\leq{\mathfrak c}\). (B) \(\mathbb{P}= \mathbb{Q}=\) the class of connected, separable groups. Several interesting related unsolved problems are posed.