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Let $G$ be an abelian topological group and let $\hat{G}$ be its completion, i.e. the group containing the equivalence classes of all Cauchy sequences of $G$. What exactly is the topology of $\hat{G}$?

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    See also [this thread](http://math.stackexchange.com/questions/311897/completion-as-a-functor-between-topological-rings). It's about topological rings, but the answer of Martin Brandenburg also fits to topological groups. I really recommend the book "General Topology" from Nicolas Bourbaki for further details.2013-03-10

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formerly a remark

For each neighborhood $N$ of zero in G, define a neighborhood $\hat{N}$ in $\hat{G}$ consisting of those equivalence classes for which all sequences in the class are eventually in $N$. This is a base (of neighborhoods of zero) for the new topology.

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    @GEdgar: Under this topology can we prove that $\hat{G}$ is complete ? i.e., every Cauchy sequence is convergent ?2018-11-20