The Markov-Zariski topology of an abelian group

Abstract

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.

References in corpus (4)

• Reflection principle characterizing groups in which unconditionally closed sets are algebraic
• The number of non-solutions to an equation in a group and non-topologizable torsion-free groups
• A Kronecker-Weyl theorem for subsets of abelian groups
• Consistent solution of Markov's problem about algebraic sets

Cited by in corpus (6)

• Reflection principle characterizing groups in which unconditionally closed sets are algebraic
• A Kronecker-Weyl theorem for subsets of abelian groups
• Final solution of Protasov-Comfort's problem on minimally almost periodic group topologies
• Characterizing sequences for precompact group topologies
• Some classes of minimally almost periodic topological groups
• Markov's problem for free groups