2017/03/12 by Jan Spevak, Spevak, Jan
Mathematics · #FOS: Mathematics #General Topology (math.GN) #math.GN
paper · pdf · doi:10.48550/arxiv.1703.04102
arxiv created 2017/12/07 · arxiv updated 2017/12/08
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 [5] the concept of an independent set in an abelian group was generalized 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 \em 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.