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 $\hat{G}$?
Topology induced by the completion of a topological group
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$\begingroup$
abstract-algebra
general-topology
topological-groups
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0See 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
1 Answers
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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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0@GEdgar: Under this topology can we prove that $\hat{G}$ is complete ? i.e., every Cauchy sequence is convergent ? – 2018-11-20