  According to Markov, a subset of an abelian group G of the form {x in G:
nx=a}, for some integer n and some element a 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 bounded) Hausdorff group topology on G.
The family of all algebraic subsets of an abelian group G forms the family of
closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or
verbal, topology of G. 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 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
Zariski closure. As an application, we provide a characterization of the
subsets of G that are dense in some Hausdorff group topology 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.
