A sequence in an abelian group is called a T-sequence if there
exists a Hausdorff group topology in which the sequence converges
to zero. This paper describes the fundamental system for the finest
group topology in which this sequence converges to zero. A sequence
is a TΩ-sequence if there exist uncountably many different
Hausdorff group topologies in which the sequence converges to zero.
The paper develops a condition which insures that a sequence is a
TΩ-sequence and examples of TΩ-sequences
are given.