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 Hausdorff abelian groups

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.