The Markov–Zariski topology of an abelian group

Abstract

According to Markov (1946) [24], a subset of an abelian group G of the form {x ∈ G: nx = 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 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 Z G -closure. As an application, we provide a characterization of the subsets of G that are Tdense 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 longstanding problem of Markov (1946) [24]. We use P and N to denote the sets of all prime numbers and all natural numbers, respectively. In this paper, 0 ∈ N. As usual, Z denotes the group of integers, and Z(n) denotes the cyclic group of order n. We use c to denote the cardinality of the continuum. The symbol ω 1 denotes the first uncountable cardinal.

A complete solution of Markov's problem on connected group topologies

“…Since κ j ≥ c by the choice of S, A (κ j ) j is c-homogeneous by Lemma 4.2 (ii). From this, (11) and Lemma 4.2 (iii), we conclude that H is c-homogeneous as well.…”

Some classes of minimally almost periodic topological groups

“…(The group fin(G), whose study was initiated in [7](4.4) and continued in [4]( §2), may be defined by the relation…”

“…When G and A are countable the group G * A also is countable. An equivalent definition and some related consequences can also be found in [7] and in [19] (Definition 2.3.1). Here is the relevant definition.…”

Characterized Subgroups of Topological Abelian Groups

“…Following [18], we say that d ∈ N is a proper divisor of n ∈ N provided that d ∈ {0, n} and dm = n for some m ∈ N. Note that, according to our definition, each d ∈ N \ {0} is a proper divisor of 0. (i) For n ∈ N the group G is said to be of exponent n (denoted by exp(G)) if nG = {0}, but dG = {0} for every proper divisor d of n. We say that G is bounded if exp(G) > 0, and otherwise that G is unbounded.…”