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This is a basic question from Atiyah and Macdonald's book 'Commutative Algebra', page 102.

Let $G$ be a topological abelian group with a countable fundamental system of nbhds of $0$ and $\hat{G}$ the completion of $G$. The book defines it as the abelian group consisting of all equivalence classes of Cauchy sequences in $G$. Two Cauchy sequences $(x_n)$ and $(y_n)$ are s.t.b. equivalent if $x_n-y_n \to 0$ in $G$. In the book they assume that $\hat{G}$ is also a topological group.

However there is no mention of the topology on $\hat{G}$ and I am not sure what it is. Apart from the one induced from the inverse limit construction is there an obvious one (which the authors seem to assume) which I have missed?

Many thanks.

eddie
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    You can define a base of id-neighbourhoods as follows: for every id-neighbourhood $U$ of $G$ let $\hat{U}$ be an id-neighbourhood of $\hat{G}$, where $\hat{U}$ is defined as the set of all equivalence classes of Cauchy nets all of whom are eventually in $U$. This was taken from here: https://math.stackexchange.com/questions/192808/topology-induced-by-the-completion-of-a-topological-group?rq=1 – Cronus Sep 09 '17 at 12:00
  • @Anakhand It has been more that 6 years!! why is it important to find that it is a dub ? even the op is not active rn. – pie Jan 31 '24 at 07:11
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    @pie It's a matter of improving navigation for everyone on the site. A duplicate is not necessarily a bad thing. If after a search you land on this question, and this question is marked as a duplicate, it will be much easier to go to the other question that contains helpful answers as well as links to every other duplicate which might contain even more helpful answers. Closing as duplicate is also not "punishment" to the OP in any way. So, the age of the question and the fact that the OP is no longer active are completely irrelevant. See https://math.stackexchange.com/help/duplicates – Anakhand Feb 04 '24 at 12:54
  • @Anakhand I asked this question on meta about closing old questions – pie Feb 04 '24 at 12:55

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