Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the sum of two Borel sets


Authors: P. Erdős and A. H. Stone
Journal: Proc. Amer. Math. Soc. 25 (1970), 304-306
MSC: Primary 28.10; Secondary 54.00
Acknowledgment: Proc. Amer. Math. Soc. 29, no. 3 (1971), p. 628.
MathSciNet review: 0260958
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the linear sum of two Borel subsets of the real line need not be Borel, even if one of them is compact and the other is $ {G_\delta }$. This result is extended to a fairly wide class of connected topological groups.


References [Enhancements On Off] (What's this?)

  • [1] C. Kuratowski, Topologie. Vol. 1, 2nd ed., Monografie Mat., vol. 20, PWN, Warsaw, 1948; English transl., Academic Press, New York and PWN, Warsaw, 1966. MR 10, 389.
  • [2] Jan Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147. MR 0173645
  • [3] J. v. Neumann, Ein System algebraisch unabhängiger zahlen, Math. Ann. 99 (1928), no. 1, 134–141 (German). MR 1512442, 10.1007/BF01459089
  • [4] C. A. Rogers, A linear Borel set whose difference set is not a Borel set, Bull. London Math. Soc. (to appear).
  • [5] L. A. Rubel, A pathological Lebesgue-measurable function, J. London Math. Soc. 38 (1963), 1–4. MR 0147608

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28.10, 54.00

Retrieve articles in all journals with MSC: 28.10, 54.00


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0260958-1
Keywords: Borel set, analytic set, complete metric space, Cantor set, algebraically independent, connected topological group, absolute $ {G_\delta }$
Article copyright: © Copyright 1970 American Mathematical Society