PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 54, January 1976 A UNIQUENESS RESULT FOR TOPOLOGICAL GROUPS ROBERT R. KALLMAN1 Abstract. We give a rapid proof of a general result which has an easy corollary that the p-adic integers have a unique topology in which they are a complete separable metric group. In [1] Corwin showed that the /?-adic integers have a unique topology in which they are a nondiscrete locally compact group. The purpose of this note is to give a rapid proof of the following general theorem. It contains most of Corwin's result as a special case. The proof is by methods different than those employed by Corwin. Theorem 1. Let G be a complete separable Abelian metric group. For each integer n, let n ■G = [na\a in G]. Suppose that the translates of the n • G generate the Borel structure of G. Then G has a unique topology in which it is a complete separable metric group. Proof. It is not a priori obvious that the n ■G are Borel subsets of G. However, let P and K be complete separable metric groups, and \p: L -* K a continuous homomorphism. Then ip induces a continuous one-to-one homo- morphism of P/kernel ip onto ip(L). Since P/kernel ip is also a complete separable metric group, Souslin's theorem implies that ip(L) is a Borel subset of K. In particular, the n ■G are Borel subsets of G. Let G' be a complete separable metric group which is isomorphic to G as an abstract group but perhaps has a different topology. Let cb: G' —» G be the natural identification. But for each integer n, n • G' = cp~x(n • G) is a Borel subset of G'. Hence, since the translates of the n ■G generate the Borel structure of G, we have that cf> is a Borel mapping. Hence, by Kuratowski [2, p. 400], there exists a set P of first category in G' such that <j>\G' — P is continuous. The proof of the theorem may now be completed in standard fashion. We claim that cb is actually continuous on all of G'. To show this, let a„ (n > 1) and a be elements of G' such that an -> a (as n t oo). Now if Q is the set which is the union of a~x ■P and a~x ■P (n > 1), Q is again a set of the first category. Hence, G" - Q is nonempty. Let b be an element of G' - Q. Then ab is in G' - P and a„b is in G' - P (n > 1). But a„b -* ab. Hence, <b(anb) -+ <p(ab), and so c?(an) = <p(anb) ■cb(b~x) -> cj>(ab) ■cj>(b~x) = cf>(a). Hence, <p is a continuous one-to-one mapping of G onto G. Hence, since both Received by the editors May 14, 1975. AMS (MOS) subject classifications(1970). Primary 22D05; Secondary 22E35. Key words and phrases. Topological groups, p-adic integers. 1 Supported in part by NSF Grant GP-38023. © American Mathematical Society 1976 439 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use