Mathematics – General Topology
Scientific paper
2009-12-02
Proceedings of the American Mathematical Society, 138 (2010), 2979-2990
Mathematics
General Topology
Scientific paper
10.1090/S0002-9939-10-10302-5
For an uncountable cardinal \tau and a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns:s\in S}|\ge \tau for all integers n\ge 1; (ii) there exists a group homomorphism \pi:G\to T^{2^\tau} such that \pi(S) is dense in T^{2^\tau}. Moreover, if |G|\le 2^{2^\tau}, then the following item can be added to this list: (iii) there exists an isomorphism \pi:G\to G' between G and a subgroup G' of T^{2^\tau} such that \pi(S) is dense in T^{2^\tau}. We prove that the following conditions are equivalent for an uncountable subset S of an abelian group G that is either (almost) torsion-free or divisible: (a) S is T-dense in G for some Hausdorff group topology T on G; (b) S is T-dense in some precompact Hausdorff group topology T on G; (c) |{ns:s\in S}|\ge \min{\tau:|G|\le 2^{2^\tau}} for every integer n\ge 1. This partially resolves a question of Markov going back to 1946.
Dikranjan Dikran
Shakhmatov Dmitri
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