J. Group Theory 3 (2000) 293±321 Journal of Group Theory ( de Gruyter 2000 Suitable sets in some topological groups Dikran Dikranjan and F. Javier Trigos-Arrieta (Communicated by S. A. Morris) Abstract. If G is a topological group with identity 1 and S J Gnf1g, then we say that S is a suitable set for G if (a) the subgroup hSi of G generated by S is dense in G, (b) S is relatively discrete, and (c) S U f1g is closed in G. We solve several open problems regarding the existence of (closed) suitable sets in some locally compact groups equipped with their Bohr topology and characterize the p-adic integers as well as other known compact ®nite-dimensional metrizable groups (Prodanov's class) within the class of countably compact Abelian groups by means of suitable sets. 1 Introduction Topological groups o¨er a rich choice of di¨erent ways of `generation' because of their two-fold nature. In the ®rst place they are groups, so that one can consider the usual notion of generationÐa subset S of a group G is said to generate G if the smallest subgroup hSi of G containing S coincides with G. In the second place G carries a topological structure, so that one can replace the equality G  hSi by the weaker condition that hSi is dense in G. In such a case we refer to S as to a set of topological generators and we say that S topologically generates the group G. Suitable sets of topological groups were introduced by Hofmann and Morris [21]. They naturally generalize ®nite sets of generators as well as sets of topological generators that form a convergent sequence. The latter appeared in the papers of Douady [12], Mel'nikov [26] and Tate (as quoted in [12]). They proved that every countably based pro®nite group can be topologically generated by a convergent sequence. Later, Hofmann and Morris [21] proved that every locally compact group has a suitable set. Further information on suitable sets in locally compact groups and related matters can be found in Chapter 12 of their recent book [23] and in Cleary and Morris [2]. Theorem 6.6 of Comfort, Morris, Robbie, Svetlichny and Tkac Ïenko [5] proves that every metrizable topological group G has a suitable set; this has been generalized recently to almost metrizable groups by Okunev and Tkac Ïenko [29]. They also found the ®rst examples of groups without suitable sets. Further examples were given by Dikranjan, Tkac Ïenko and Tkachuk [10] and [11], and Tomita and Trigos-Arrieta [42]. In [11] and [42] closed suitable sets are studied, the former Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 6/19/15 12:04 AM