# Topologically independent sets in precompact groups

## TL;DR

This paper proves that in precompact abelian groups, a subset generates a subgroup as a Tychonoff direct sum of cyclic groups if and only if it is topologically independent, removing the need for absolute Cauchy summability.

## Contribution

It establishes that absolute Cauchy summability is unnecessary for the main result in precompact groups, extending previous work on topological independence.

## Key findings

- In precompact abelian groups, topological independence characterizes subgroup generation as a Tychonoff direct sum.
- Absolute Cauchy summability can be omitted in the precompact setting.
- The result does not extend to locally compact groups, only precompact ones.

## 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 $\langle a\rangle$, $a\in A$ 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 $\langle a\rangle$, $a\in A$ 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 $\langle a\rangle$, $a\in A$ if and only if $A$ is topologically independent. We show that precompactness can not be replaced by local compactness in this result.