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Proceedings of the American Mathematical Society

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



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
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.

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Keywords: Bohr compactification, l.c.a. groups, I-sets, continuum hypothesis
Article copyright: © Copyright 1980 American Mathematical Society