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r denotes a metrizable locally compact abelian group and 1' its Bohr compactification. Let y E r be a cluster point of some subset E of r in the topology of r. Then there are two disjoint subsets of E which also cluster at y 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. F denotes a locally compact abelian group whose dual is G. F, the Bohr compactification of F, is the l.c.a. group that is dual to Gd, G with the discrete topology. THEOREM 1. Suppose that F is metrizable. Let E be a subset of r which clusters at y E F in the topology of P. Then there are two disjoint subsets of E which likewise cluster at -y. PROOF. A basic neighborhood U = U(y; gl, . .. , g,,; e) of y in r is determined by a finite set of points gl, . . . , gn from G and some e > 0. Precisely, U consists of X in P such that JX(g) -y(gi)l 0, for c > 0, and F a finite subset of E, there exist A1, . . ., Am in E \ F such that (hl, . . ., hn) E Kn implies that for some I < j < , E E U(y; hl, . . . , hn; E). Denote the set {Al, . .. , A,,,) as D(K, n, e, F). We now use the hypothesis that r is metrizable. This occurs exactly when G is a-compact. Let G be U n I K,, with each Kn compact and Kn 5 Kn +1. Define two sequences of finite subsets of E inductively. Set To = S0 = 0. Let Sn+1 = D(Kn+ , n + 1, (n + 1, Fn , 1) where Fn +I = U i< n(Tj u S). Then Tn+ = D(Kn+1, n + 1, (n + 1)-1, Fn+l U Sn+1). Clearly U,T7 and U,S, are disjoint subsets of E which cluster at y in r. EoJ Received by the editor November 29, 1978. AMS (MOS) subject classifications (1970). Primary 43A25, 43A60; Secondary 03E50.

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