Some properties of Borel subgroups of real numbers

Abstract:

As a consequence of Souslin’s theorem, we obtain the following; if $G$ and $H$ both are analytic subgroups of ${\mathbf {R}}$ such that $G + H = {\mathbf {R}}$ and $G \cap H = \{ 0\}$, then either $G = {\mathbf {R}}$ or $G = \{ 0\}$. Next we obtain some measure and topological properties for uncountable proper Borel subgroups of reals. Finally, we prove that if $E$ is a vector subspacc of ${\mathbf {R}}$ over the rationals which admits an uncountable Borel basis, then there exists no Polish topology on $E$ such that $E$ is a topological group with the given Borel structure generated by the open sets.