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

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.