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Refining connected topological group topologies on Abelian torsion groups. (English) Zbl 0930.54030
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.

MSC:
54H11 Topological groups (topological aspects)
22A05 Structure of general topological groups
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D05 Connected and locally connected spaces (general aspects)
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