On a class of abelian groups with sufficiently many T-sequences

Abstract

We describe the one-to-one sequences \(\textbf{u}\) in an abelian group G that have a T-subsequence, and we show that they are the same as those that admit a TB-subsequence. In terms of the Zariski topology, these are precisely the sequences \(\textbf{u}\) such that 0 belongs to the Zariski closure of the supporting set of \(\textbf{u}\). Additionally, we extend this result to \(T^\mathcal {I}\)- and \(TB^\mathcal {I}\)-subsequences defined using \(\mathcal {I}\)-convergence where \(\mathcal {I}\) is an ideal of \(\mathbb {N}_0\). As a corollary, we describe completely the class \(\mathfrak T\) (resp., \(\mathfrak T_B\), \(\mathfrak T^\mathcal {I}\), \(\mathfrak T_B^\mathcal {I}\)) of abelian groups in which every one-to-one sequence has a T-subsequence (resp., TB-, \(T^\mathcal {I}\)-, \(TB^\mathcal {I}\)-subsequence). These are precisely the abelian groups that are either almost torsion-free or of prime exponent, in other words, the groups with cofinite Zariski topology [5].