According to Markov (1946) 1241, a subset of an abelian group G of the form (x is an element of G: nx = a), for some integer n and some element a is an element of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (= totally buounded) Hausdorff group topology on G. The family of all algebraic sets of an abelian group G forms the family of closed subsets of a unique Noetherian T(1) topology 3(G) on G called the Zariski, or verbal, topology of G; see Bryant (1977) [31. We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Frechet-Urysohn.
For a countable family 3 of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology 'T on G such that the T-closure of each member of g coincides with its 3(G)-closure. As an application, we provide a characterization of the subsets of G that are 'Tdense in some Hausdorff group topology T on C. and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a longstanding problem of Markov (1946)124]. (C) 2010 Elsevier Inc. All rights reserved.