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

We describe the one-to-one sequences 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 u such that 0 belongs to the Zariski closure of the supporting set of u. Additionally, we extend this result to TI- and TBI-subsequences defined using I-convergence where I is an ideal of N0. As a corollary, we describe completely the class T (resp., TB, TI, TBI) of abelian groups in which every one-to-one sequence has a T-subsequence (resp., TB-, TI-, TBI-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].