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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A theorem of C. Ryll-Nardzewski and metrizable l.c.a. groups
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by L. Thomas Ramsey PDF
Proc. Amer. Math. Soc. 78 (1980), 221-224 Request permission

Abstract:

$\Gamma$ denotes a metrizable locally compact abelian group and $\bar \Gamma$ its Bohr compactification. Let $\gamma \in \Gamma$ be a cluster point of some subset E of $\Gamma$ in the topology of $\bar \Gamma$. Then there are two disjoint subsets of E which also cluster at $\gamma$ in the Bohr group topology. The proof is elementary and provides a new proof of the theorem of C. Ryll-Nardzewski on cluster points of I-sets in R. Given the continuum hypothesis, either theorem characterizes metrizability in locally compact abelian groups. One of these characterizations is shown to be equivalent to the continuum hypothesis.
References
  • Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
  • C. Ryll-Nardzewski, Concerning almost periodic extensions of functions, Colloq. Math. 12 (1964), 235–237. MR 173129, DOI 10.4064/cm-12-2-235-237
  • Edwin Hewitt and Kenneth A. Ross, Abstract harmonic anaylsis, Academic Press, New York, 1963.
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 78 (1980), 221-224
  • MSC: Primary 43A46; Secondary 03E50
  • DOI: https://doi.org/10.1090/S0002-9939-1980-0550498-0
  • MathSciNet review: 550498