A theorem of C. Ryll-Nardzewski and metrizable l.c.a. groups
L. Thomas Ramsey
Proc. Amer. Math. Soc. 78 (1980), 221-224
Primary 43A46; Secondary 03E50
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Abstract: denotes a metrizable locally compact abelian group and its Bohr compactification. Let be a cluster point of some subset E of in the topology of . Then there are two disjoint subsets of E which also cluster at 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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