Algebraic obstructions to sequential convergence in Hausdorrf abelian groups

Abstract

Given an abelian groupGand a non-trivial sequence inG, when will it be possible to construct a Hausdroff topology onGthat 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 to0in any Hausdroff group topology. We demonstrate that any sequence in the rationals that satisfies a “sparseness” condition will converge to0in uncountably many different Hausdorff group topologies.