Take $G$ an abelian topological group additively written. Now, following the Atiyah Macdonald, I define its completion as the set of Cauchy sequences quotiented for the usual equivalent relation. Now, in the Atiyah Macdonald they doesn't explicit the topology so I've tried to descrive the neighborhoods of the origin.
Take $U$ an neighborhood of $0$ in $G$ and define $\hat{U}$ as the Cauchy sequences which (have a representant such that) are definitively in $U$. In other terms: the set in $\hat{G}$ of class of Cauchy sequences $(x_n)$ such that for $n$ sufficiently great $x_n \in U$.
Is this right?
Is this the topology induced by the map $\phi : G \rightarrow \hat{G}$ which send $g$ in the constant sequence $(g)$? If yes, why?