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

Abstract We describe the one-to-one sequences $$\textbf{u}$$ 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 $$\textbf{u}$$ u such that 0 belongs to the Zariski closure of the supporting set of $$\textbf{u}$$ u . Additionally, we extend this result to $$T^\mathcal {I}$$ T I - and $$TB^\mathcal {I}$$ T B I -subsequences defined using $$\mathcal {I}$$ I -convergence where $$\mathcal {I}$$ I is an ideal of $$\mathbb {N}_0$$ N 0 . As a corollary, we describe completely the class $$\mathfrak T$$ T (resp., $$\mathfrak T_B$$ T B , $$\mathfrak T^\mathcal {I}$$ T I , $$\mathfrak T_B^\mathcal {I}$$ T B I ) of abelian groups in which every one-to-one sequence has a T -subsequence (resp., TB -, $$T^\mathcal {I}$$ T I -, $$TB^\mathcal {I}$$ 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].