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

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