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Algebraic obstructions to sequential convergence in Hausdorff abelian groups. (English) Zbl 0895.22002
A sequence \((x_k)\) in \(\mathbb{Q}\) is a sparse sequence if \[ \begin{aligned} \frac{x_{k+1}}{x_k} > \frac{k^2+3k} {2(k+1)} &\quad\text{ for every }k\in\mathbb{N} \qquad \text{ and}\tag{1}\\ \frac{x_{k+1}}{x_k} > \frac{k^2+2k} {k+1} &\quad\text{ for infinitely many numbers }k\in\mathbb{N}. \tag{2} \end{aligned} \] If \((x_k)\) is a sparse sequence in \(\mathbb{Q}\) then there exist several uncountably Hausdorff group topologies in \(\mathbb{Q}\) such that \((x_k)\) converges to 0.

MSC:
22A05 Structure of general topological groups
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