The Markov–Zariski topology of an abelian group

According to Markov (1946) [24], , a subset of an abelian group G of the form , for some integer n and some element a∈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 bounded) 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 T1 topology ZG on G called the Zariski, or verbal, topology of G; see Bryant (1977) [3]. We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Fréchet–Urysohn.For a countable family F 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 F coincides with its ZG-closure. As an application, we provide a characterization of the subsets of G that are T-dense in some Hausdorff group topology T on G, 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 long-standing problem of Markov (1946) [24].