Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density

Abstract

For an uncountable cardinal τand a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns:s\in S}|\ge τfor all integers n\ge 1; (ii) there exists a group homomorphism π:G\to T^{2^τ} such that π(S) is dense in T^{2^τ}. Moreover, if |G|\le 2^{2^τ}, then the following item can be added to this list: (iii) there exists an isomorphism π:G\to G' between G and a subgroup G' of T^{2^τ} such that π(S) is dense in T^{2^τ}. 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{τ:|G|\le 2^{2^τ}} for every integer n\ge 1. This partially resolves a question of Markov going back to 1946.

References in corpus (2)

• The Markov-Zariski topology of an abelian group
• A Kronecker-Weyl theorem for subsets of abelian groups

Cited by in corpus (5)

• The Markov-Zariski topology of an abelian group
• A complete solution of Markov's problem on connected group topologies
• A Kronecker-Weyl theorem for subsets of abelian groups
• Final solution of Protasov-Comfort's problem on minimally almost periodic group topologies
• Selectively pseudocompact groups without non-trivial convergent sequences