The Markov–Zariski topology of an abelian group

By Dikran Dikranjan, D. Shakhmatov. 2010

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Abstract According to Markov (1946) [24] , a subset of an abelian group G of the form { x ∈ G : n x = a } , 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 T 1 topology Z G 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 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 Z G -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] .

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ccba7bf1639b06f2c5fa6ed2bdb072c22f612873