On a question of Clark and Ledet

TL;DR

This paper proves that for any T-sequence on a countable abelian group, there are an extremely large number of Hausdorff group topologies where the sequence converges to zero, answering a longstanding open question.

Contribution

It establishes the existence of an enormous family of Hausdorff topologies on countable abelian groups where a given T-sequence converges to zero, resolving a question from 2000.

Findings

Existence of 2^{2^{|G|}} such topologies for any T-sequence

Answer to a question posed in 2000 about convergence topologies

Large diversity of topologies with prescribed convergence behavior

Abstract

Given a $T$-sequence on a countable abelian group $G$, we prove that there exists $2^{2^{|G|}}$ Hausdorff group topologies in which this sequence converges to $0$. This answers a question posed in Intern. J. Math. Math. Sci. {\bf 24}(3) (2000), 145-148.