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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0260958-1
Acknowledgment:
Proc. Amer. Math. Soc. 29, no. 3 (1971), p. 628.
MathSciNet review:
0260958
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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.
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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.
- Jan Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147. MR 173645, DOI https://doi.org/10.4064/fm-55-2-139-147
- J. v. Neumann, Ein System algebraisch unabhängiger Zahlen, Math. Ann. 99 (1928), no. 1, 134–141 (German). MR 1512442, DOI https://doi.org/10.1007/BF01459089 C. A. Rogers, A linear Borel set whose difference set is not a Borel set, Bull. London Math. Soc. (to appear).
- L. A. Rubel, A pathological Lebesgue-measurable function, J. London Math. Soc. 38 (1963), 1–4. MR 147608, DOI https://doi.org/10.1112/jlms/s1-38.1.1
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Additional Information
Keywords:
Borel set,
analytic set,
complete metric space,
Cantor set,
algebraically independent,
connected topological group,
absolute <!– MATH ${G_\delta }$ –> <IMG WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${G_\delta }$">
Article copyright:
© Copyright 1970
American Mathematical Society