Factoring groups into dense subsets

TL;DR

The paper proves that large groups with certain topological properties can be uniquely factorized into dense subsets, but it remains unknown whether this holds for countable groups.

Contribution

It establishes the factorization of uncountable groups into dense subsets with unique representation, extending the understanding of dense factorizations in topological groups.

Findings

Uncountable groups with specified topology can be factorized into dense subsets.

Unique representation of elements in the dense factorization.

Open question remains for countable groups and their factorization.

Abstract

Let $G $ be a group of cardinality $\kappa>\aleph_0 $ endowed with a topology $\tau $ such that $|U|=\kappa$ for every non-empty $U\in\tau$ and $\tau$ has a base of cardinality $\kappa$. We prove that $G$ could be factorized $G=AB$ (i.e. each $g\in G$ has unique representation $g=ab$, $a\in A$, $b\in B$) into dense subsets $A,B$, $|A|=|B|=\kappa$. We do not know if this statement holds for $\kappa = \aleph_0$ even if $G$ is a topological group.