Reflection principle characterizing groups in which unconditionally closed sets are algebraic

Abstract

We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63 years old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G.According to Markov [6], a subset S of a group G is called:(a) elementary algebraic if there exist an integer n > 0, a 1 , . . . , a n ∈ G and ε 1 , . . . , ε n ∈ {−1, 1} such that S = {x ∈ G : x ε 1 a 1 x ε 2 a 2 . . . a n−1 x εn = a n }, (b) algebraic if S is an intersection of finite unions of elementary algebraic subsets of G, (c) unconditionally closed if S is closed in every Hausdorff group topology of G.Since the family of all finite unions of elementary algebraic subsets of G is closed under finite unions and contains all finite sets, it is a base of closed sets of some T 1 topology Z G on G, called the Zariski topology of G. (This topology is also known under the name verbal topology, see [1].) The family of all unconditionally closed subsets of G coincides with the family of closed subsets of a T 1 topology M G on G, namely the infimum (taken in the lattice of all topologies on G) of all Hausdorff group topologies on G.

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

“…Our next proposition provides an easy algorithm for verifying whether an abelian group is an M -group. Its proof, albeit very short, is based on the fundamental fact that all unconditionally closed subsets of abelian groups are algebraic [10,Corollary 5.7]. Proof.…”

“…Proof. According to [10,Corollary 5.7] and [11, According to the Prüfer theorem [15,Theorem 17.2], a non-trivial abelian group G of finite exponent is a direct sum of cyclic subgroups…”

Topologizations of a set endowed with an action of a monoid

“…The family of unconditionally closed subsets of G coincides with the family of closed subsets in another T 1 topology on G, namely, the infimum of all Hausdorff group topologies on G. This topology was explicitly introduced in [7] under the name Markov topology.…”

Introduction to topological groups

“…In 1944 Markov [76] asked if the equality Z G = M G holds for every group G. He himself showed that the answer is positive in case G is countable [76]. Moreover, in the same manuscript Markov attributes to Perel'man the fact that Z G = M G for every Abelian group G (a proof has never appeared in print until [37]). An example of a group G with Z G = M G was given by Gerchard Hesse [66].…”