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A theorem of C. Ryll-Nardzewski and metrizable l.c.a. groups


Author: L. Thomas Ramsey
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
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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 [Enhancements On Off] (What's this?)

  • Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
  • C. Ryll-Nardzewski, Concerning almost periodic extensions of functions, Colloq. Math. 12 (1964), 235–237. MR 173129, DOI https://doi.org/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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Keywords: Bohr compactification, l.c.a. groups, <I>I</I>-sets, continuum hypothesis
Article copyright: © Copyright 1980 American Mathematical Society