Remarks on $d$-independent topological groups

TL;DR

This paper proves that all separable locally compact abelian $M$-groups are $d$-independent, extending previous results beyond metrizable groups, and also shows that all separable connected compact groups are $d$-independent.

Contribution

It removes the metrizability assumption in the characterization of $d$-independent groups and extends the concept to non-abelian groups, providing new insights into their structure.

Findings

Separable locally compact abelian $M$-groups are $d$-independent.

All separable connected compact groups are $d$-independent.

The notion of $d$-independence is extended to non-abelian groups.

Abstract

A non-trivial topological group is called \emph{$d$-independent} if for every subgroup of cardinality less than the continuum there exists a countable dense subgroup intersecting it trivially. This notion was introduced by M\'arquez and Tkachenko and has been intensively studied in the metrizable setting. In particular, they proved that a second-countable locally compact abelian group is $d$-independent if and only if it is algebraically an $M$-group, and asked whether the same conclusion holds for all separable locally compact groups. In this paper we give an affirmative answer to this question. We show that every separable locally compact abelian $M$-group is $d$-independent, thereby removing the metrizability assumption from the result of M\'arquez and Tkachenko. In addition, we investigate several further aspects of $d$-independence. We study its behaviour under taking powers of topological groups and extend the notion of $d$-independence to the non-abelian setting. Moreover, we prove that every separable connected compact group is $d$-independent, thereby answering another question posed by M\'arquez and Tkachenko.