International Journal of Mathematics and Mathematical Sciences
Volume 21 (1998), Issue 1, Pages 93-96
doi:10.1155/S0161171298000118
Algebraic obstructions to sequential convergence in Hausdorrf abelian groups
University of Southwestern Louisiana, USA
Received 5 June 1996; Revised 10 October 1996
Copyright © 1998 Bradd Clark and Sharon Cates. 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
Given an abelian group G and a non-trivial sequence in G, when will it be possible
to construct a Hausdroff topology on G that allows the sequence to converge? As one might expect
of such a naive question, the answer is far too complicated for a simple response. The purpose of
this paper is to provide some insights to this question, especially for the integers, the rationals, and
any abelian groups containing these groups as subgroups. We show that the sequence of squares
in the integers cannot converge to 0 in any Hausdroff group topology. We demonstrate that any
sequence in the rationals that satisfies a “sparseness” condition will converge to 0 in uncountably
many different Hausdorff group topologies.