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Given a metric space $ X $, one can forms its completion $ \hat{X} $. However, I have seen that one can also define completeness for topological groups. Can someone explain to me (as simply as possible) what complete topological groups are?

I would also appreciate a more conceptual explanation that is not the most straightforward.

  • They're just topological groups which, when looked at as just a topological space (= forget the group structure), are complete in the usual sense. – Noah Schweber Aug 01 '18 at 23:56

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A Cauchy sequence in a topological group $G$ is a sequence $\{a_n\}$ such that for any neighborhood $U$ of the identity $e\in G$, there exists some $N\in \mathbb{N}$ such that $a_na_m^{-1}\in U$ for all $n,m\geq N$. This generalizes the idea of a Cauchy sequence in a metric space, in the sense that if I go far enough out in the sequence, all terms will be "close" to each other, where here the idea of closeness of two elements is characterized by their difference lying inside a neighborhood of the identity. Then a complete topological group is one in which all Cauchy sequences converge to a point in the space (as usual). If the topology on the group is metrizable, this should coincide with the usual idea of completeness in a metric space.

For more reading, see:

https://en.wikipedia.org/wiki/Cauchy_sequence#Generalizations

https://en.wikipedia.org/wiki/Complete_metric_space#Alternatives_and_generalizations

https://en.wikipedia.org/wiki/Uniform_space

Taylor M
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    Good answer. I like looking at the situation from the uniform-space angle. – Lubin Aug 02 '18 at 00:32
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    I think one should be aware here that the group should be first countable for this to be the correct definition, otherwise one should be using Cauchy filters instead. – Alex Mathers Dec 22 '21 at 08:03