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I was reading the completion chapter of Atiyah-Macdonald. I have the following questions:

(i) What is the topology in the completion group of the topological abelian group? I saw an answer here. But I don't know how he guessed it. My guess is that the topology is given by the map $\phi:G\rightarrow \hat{G}$ to be continuous, i.e. $U$ in $\hat{G}$ is open iff $\phi^{-1}(U)$ is open in $G$. (Because in the book it is claimed that if $f: G\rightarrow H$ is continuous then $\hat{f}:\hat{G}\rightarrow \hat{H}$ is also continuous.) I think the topology in the link given and this topology coincide. Although I could not see it.

(ii) If the topology of $\hat{G}$ is given by above formulation then why it is complete i.e. why every Cauchy sequence in $\hat{G}$ converges? I need detailed clarification in this context. Any help is highly appreciated. Thank you in advance.

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  • I suppose that for the second part of the question it is necessary to have some kind of metric in G so that we can define metric in $\hat G$. In fact in the book it is defined that if the map $\phi$ is an isomorphism then G is said to be complete. And it proved that the map ($\hat{G}\rightarrow \hat{\hat{G}}$) is an isomorphism so that $\hat{G}$ is complete. –  Feb 23 '16 at 16:14

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