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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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##### References:
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