Convergent nets in abelian topological groups

Abstract

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.