Factoring groups into dense subsets

Abstract

Let $G $ be a group of cardinality $κ>\aleph_0 $ endowed with a topology $τ$ such that $|U|=κ$ for every non-empty $U\inτ$ and $τ$ has a base of cardinality $κ$. 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 $A,B$, $|A|=|B|=κ$. We do not know if this statement holds for $κ= \aleph_0$ even if $G$ is a topological group.