The Markov-Zariski topology of an abelian group

TL;DR

This paper characterizes algebraic sets in abelian groups via their closure properties in precompact Hausdorff topologies, introduces a unique Noetherian Zariski topology, and explores its properties and applications.

Contribution

It establishes the equivalence between algebraic sets and closures in precompact topologies, and constructs a unique Noetherian Zariski topology with specific properties.

Findings

The Zariski topology is hereditarily separable and Frechet-Urysohn.

A precompact metric topology can be constructed to match Zariski closures for countable families.

Characterization of subsets dense in some Hausdorff group topology.

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.