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.
Highlights
Given an abelian group and a non-trivial sequence in G, when will it be possible to construct a Hausdorff group 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
We shall assume as additional hypothesis throughout the paper that G is an abelian group, and that each sequence under consideration is a one-to-one function from the natural numbers into G
The referee has pointed out that in a Hausdorit topological group proposition 2.1 can be derived by considering sequential convergence as a special case of FLUSH-convergence
Summary
Given an abelian group and a non-trivial sequence in G, when will it be possible to construct a Hausdorff group 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. Given an abelian group and a non-trivial sequence in G, when will it be possible to construct a Hausdorff group topology on G that allows the sequence to converge? We shall assume as additional hypothesis throughout the paper that G is an abelian group, and that each sequence under consideration is a one-to-one function from the natural numbers into G.
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