# Tagged Questions

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### Canonical isomorphism between Cauchy sequence completion and inverse limit

I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can ...
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### Explanation of a passage in Atiyah / Macdonald

On page 105 the authors show that $\hat{\hat{G}} \cong \hat{G}$ (Proposition 10.5) and conclude that the canonical homomorphism $\phi : \hat{G} \to \hat{\hat{G}}$ is an isomorphism. How does the fact ...
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### Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
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### continuous images of Cauchy sequences in topological groups

on page 102 of Atiyah and MacDonald's "Introduction to Commutative Algebra", they state that if $G$ and $H$ are abelian topological groups and $f$ is a continuous homomorphism from $G$ to $H$, then ...
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### Questions about the intersection of all neighborhoods of $0$ in a topological abelian group.

Let $H$ be the intersection of all neighborhoods of $0$ in a topological abelian group. On page 102 of the book introduction to commutative algebra by Atiyah and Macdonald, the fourth line of the ...
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### Closed subgroups of n copies of the p-adic integers

What do closed subgroups of $\mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p$ look like (where there are $n$ summands in the direct sum)?
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### Addition and multiplication are continuous in the $I$-adic topology

Can you tell me if this is correct? Let $R$ be a ring and let it have the $I$-adic topology for some ideal $I$ in $R$. I want to show that $+: R \times R \to R$ is continuous at $(x_0, y_0)$. Proof: ...
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### Adic Completion of a direct sum

This is a question related to this. Let $G= \mathbb Z / p \mathbb Z$ for some prime $p$. Let $A = \bigoplus_{n\in \mathbb N} G$, that is, all sequences in $G$ with all but finitely many terms zero. ...
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### Quick question about Hausdorffness of completion

Let $G$ be the usual: a topological Abelian group with a topology induced by a countable neighbourhood basis $G_n$ of zero such that $G = G_1 \supset G_2 \supset \dots$. Let $\widehat{G}$ denote the ...
Let $G$ be the usual: a topological Abelian group with a topology induced by a countable neighbourhood basis $G_n$ of zero such that $G = G_1 \supset G_2 \supset \dots$. Let $\widehat{G}$ denote the ...