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International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 11, Pages 645-651
http://dx.doi.org/10.1155/S016117120100744X

Convergent nets in abelian topological groups

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Received 30 April 2001

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 n in which any particular axis may be dense and Hausdorff group topologies on the torus in which S1 is dense.