# Questions tagged [topological-groups]

A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.

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### Defining the completion of a group can be done only using Cauchy sequences

Let $G$ be a group, in Atiyah & MacDonald's Commutative Algebra it says that Assume for simplicity that $0\in G$ has a countable fundamental system of neighborhoods. The completion $\hat G$ of $G$...
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### Which commutative rings are endomorphism rings of some topological Abelian group?

I was thinking earlier today about how many commutative unital rings I could hit by considering the endomorphism rings of topological Abelian groups. For example, the group $(\mathbb{R}, +, \tau^R)$ ...
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### Every simple topological group is either discrete or connected [closed]

I read the claim in a preprint that "A simple topological group is either discrete or connected". However, the explanation given was "a connected component of a topological group that ...
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### How to show isomorphism between an adele ring on a number field and its Pontryagin dual?

A book that I'm currently reading (Fourier Analysis on Number Fields by Dinakar Ramakrishnana, Robert J. Valenza) claims that a continuous homomorphism $\mathbb{A}_k \to \widehat{\mathbb{A}_k}$ given ...
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### Necessary condition for an isomorphism of modules to be a homeomorphism

Let $M$ and $N$ be two filtered $A$-modules with filtrations $(M_n)$ and $(N_n)$ and $u\colon M\to N$ be a morphism of modules. We can show that if $u(M_n)=N_n$ and $u$ is an isomorphism then $u$ is ...
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### Every Cauchy sequence in $M/N$ converges.

Let $M$ be a filtered module with filtration $(M_k)$ and $N$ be a submodule of $M$. Then the filtration on $M/N$ is given by $(P_k)$ where $P_k=(M_k+N)/N$. We will show that if $M$ is complete and ...
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### Confusion on isomorphism profinite completion integers

Question: Prove that $\hat{\mathbb{Z}} \cong \prod_{p} \mathbb{Z}_{p}$ is an isomorphism of topological rings. Own attempts: Although this specific question has been asked quite a few times here, ...
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### Isomorphism of topological groups

Question: Let $\mathbb{Q}(\sqrt{\mathbb{Q}})$ be the subfield of $\overline{\mathbb{Q}}$ generated by $\{\sqrt{x} : x \in \mathbb{Q}\}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\sqrt{\mathbb{Q}})$ ...
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### Is the map $M\mapsto P$ smooth, where $P$ satisfies $PMP^{-1} = M^T$?

From this post (and other similar ones) we know, in particular, that $$\forall M\in{\rm GL}(n,\Bbb R),\ \exists P\in{\rm GL}(n,\Bbb R)\ \text{s.t. } PMP^{-1} = M^T\ . \tag{*}$$ My question is ...
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### Find a sequence $x_n \to \infty$ such that $(e^{i\lambda_1 x_n }, \ldots , e^{i\lambda_N x_n }) \to (1,\ldots, 1)$

Let $\lambda_1,\ldots, \lambda_N \in \mathbb R$ be distinct. Let $\mathbb T \subset \mathbb C$ be the unit circle and $\mathbb T^n$ be the $n$-dimensional torus. This is a group under componentwise ...
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### Are Hausdorff countably compact topological groups always normal?

A colleague and I have a result that shows that for Hausdorff countably compact W-spaces, being a topological group implies normality. But it occurred to us that (being not super experienced working ...
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### What is the topological interpretation of Euler's identity?

The picture is from the Wikipedia page of Euler's formula. I don't understand topology at all, but I'm curious about the reasoning behind this interpretation. May someone provide references that might ...
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### Algebraic Topology: Deformation Retraction and Quotient Spaces (using Mobius )

Could someone please help me with this question (and its solution)? Thanks!! Solution Part 1 (which shows that [0,1] x {1/2} deformation retract of [0,1] x [0,1]): I don't get how the homotopy gets ...