A net in an abelian group is called a T-net if there exists a Hausdorff group topology in which the net converges to 0. This
paper describes a fundamental system for the finest group
topology in which the net converges to 0. The paper uses this
description to develop conditions which insure there exists a
Hausdorff group topology in which a particular subgroup is dense
in a group. Examples given include showing that there are
Hausdorff group topologies on ℝn in which any
particular axis may be dense and Hausdorff group topologies on
the torus in which S1 is dense.