Some properties of Borel subgroups of real numbers

Abstract

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