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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some properties of Borel subgroups of real numbers


Author: Barthélemy Le Gac
Journal: Proc. Amer. Math. Soc. 87 (1983), 677-680
MSC: Primary 28C10; Secondary 04A15, 22A05, 54H05
MathSciNet review: 687640
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0687640-X
PII: S 0002-9939(1983)0687640-X
Keywords: Borel subgroups of real numbers, analytic sets, Souslin's theorem, quasiinvariant measures, topological groups, locally compact and Polish topologies, vector subspaces of $ {\mathbf{R}}$ over the rationals
Article copyright: © Copyright 1983 American Mathematical Society