Topologically independent sets in precompact groups

Abstract

It is a simple fact that a subgroup generated by a subset A of an abelian group is the direct sum of the cyclic groups 〈a〉, a∈A if and only if the set A is independent. In [2] the concept of an independent set in an abelian group was extended to a topologically independent set in a topological abelian group (these two notions coincide in discrete abelian groups). It was proved that a topological subgroup generated by a subset A of an abelian topological group is the Tychonoff direct sum of the cyclic topological groups 〈a〉, a∈A if and only if the set A is topologically independent and absolutely Cauchy summable. Further, it was shown, that the assumption of absolute Cauchy summability of A can not be removed in general in this result. In our paper we show that it can be removed in precompact groups.In other words, we prove that if A is a subset of a precompact abelian group, then the topological subgroup generated by A is the Tychonoff direct sum of the topological cyclic subgroups 〈a〉, a∈A if and only if A is topologically independent. We show that precompactness can not be replaced by local compactness in this result.