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A net in an abelian group is called aT-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 onℝnin which any particular axis may bedoi:10.1155/s016117120100744x fatcat:ip7zlhrz3fe27l2hvclh3kpcrm