A Class of Groups in Which All Unconditionally Closed Sets are Algebraic

TL;DR

This paper demonstrates that in specific subgroups of direct products of countable groups, unconditionally closed sets are equivalent to algebraic sets, notably including all Abelian groups, thus unifying these concepts in these contexts.

Contribution

It establishes the equivalence of unconditionally closed and algebraic sets in certain subgroups of direct products of countable groups, especially in Abelian groups.

Findings

Unconditionally closed sets coincide with algebraic sets in these subgroups.

The property holds in all Abelian groups.

Provides a unifying perspective on closed and algebraic sets in these groups.

Abstract

It is proved that, in certain subgroups of direct products of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. In particular, these properties coincide in all Abelian groups.