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.

14 pages

References in corpus (2)

• The Markov-Zariski topology of an abelian group
• The number of non-solutions to an equation in a group and non-topologizable torsion-free groups

Cited by in corpus (10)

• The Markov-Zariski topology of an abelian group
• A complete solution of Markov's problem on connected group topologies
• Algebraically determined topologies on permutation groups
• A Kronecker-Weyl theorem for subsets of abelian groups
• Final solution of Protasov-Comfort's problem on minimally almost periodic group topologies
• Topologization of sets endowed with an action of a monoid
• Hereditarily non-topologizable groups
• Zariski topologies on groups
• Characterizing sequences for precompact group topologies
• Markov's problem for free groups