# 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.

2,220 questions
Filter by
Sorted by
Tagged with
35 views

63 views

### $(2)$ is not open in the $(6)$-adic topology

Suppose we have the ring $\mathbb{Z}$ and the ideal $I=(6)$ on it. Thus we can talk about the $(6)$-adic topology. Supposedly, $\mathbb{Z}$ with the $(6)$-adic topology is a topological ring. For that,...
35 views

### Accumulation point in a topological group of orthogonal matrices over R

I would like to ask your opinion on the point that looks simple. Consider the group of orthogonal matrices of order n over the field R of reals, equipped with the topology induced by the Euclidean ...
65 views

### Why is coboundary $\sigma \to \sigma m-m$ automatically continuous?

Let $K$ be a field. Let $G_K := Gal(\overline{K}/K)$ be absolute Galois group of $K$. Let $M$ be a $G_K$-module, that is, $G_K$ (with the Krull topology) acts continuously on $M$ with discrete ...
73 views

### Prove that the closure $\overline{H}$ of a subgroup $H \subseteq G$ of a topological group, is a subgroup. [duplicate]

I'm tackling the problems in `Introduction to Topological Manifolds' by Lee, and this is the first time I've been tackling with topological groups. This is problem 13-9: "Let $G$ be a topological ...
1 vote
14 views

### Ergodic transformations form a $G_\delta$ set in the weak topology of the automorphism group.

If $(X,\mathcal{L},\mu)$ is a Lebesgue-standar space and $G$ is its group of automorphisms, i know that the set of all ergodic transformations $\mathcal{E}$ is a $G_\delta$ set in the weak topology. ...
66 views

### Finite covers of $X$ and continuous profinite group actions

I have a question that is related to the one here, but I found the answer there unsatisfactory and kind of confusing. For context, we are considering the action of the fundamental group $\pi_1(X,x)$ ...
29 views

136 views

### Is there a manifold $M$ with nontrivial $\pi_2(M)$ but trivial $\pi_3(M)$?

Forgive me if the following questions on homotopy groups are trivial - I am not knowledgeable in the subject. Is there a manifold $M$ with nontrivial second homotopy group $\pi_2(M)$ but trivial third ...
84 views

43 views

32 views

### Characterization of the completion of a first-countable topological abelian group

$\def\codom{\operatorname{codom}}$If $G$ is a first-countable abelian topological group, one can find a morphism $G\to\hat{G}$ of abelian topological groups, with $\hat{G}$ first-countable and ...
120 views

### Show that some quotient space of $GL_2(\mathbb{Q}_p)$ is compact

Let $T_{\tau}:=\{\begin{pmatrix}a&b\tau\\b&a\end{pmatrix}:a,b\in\mathbb{Q}_p,a^2-b^2\tau\neq0\}$, where $\tau\in\mathbb{Q}_p^{\star}$ with $\tau\notin(\mathbb{Q}_p^{\star})^2$. Author claimed ...
1 vote
66 views

### Consequences of differing definitions of topological groups

I'm using this document to learn about the representation theory of compact groups. They define a topological group $G$ to be a group with a topology such that The multiplication and inversion ...
1 vote
88 views

### Pontlyagin dual of direct sum,$\widehat{\bigoplus_{i\in \Lambda} M_i} = \prod_{i\in \Lambda} \hat{M_i}.$

Let $M_i$ (for $i \in \Lambda$) be a family of abelian groups. Let $\bigoplus_{i\in \Lambda} M_i$ denote the infinite direct sum of the groups $M_i$. Let $\hat{M}$ denote the Pontryagin dual of the ...
56 views

### Linear topology

Definition: A linear topology $\tau$ on a left $A$-module $M$ is a topology on $M$ that is invariant under translations and admits a fundamental system of neighborhood of $0$ that consists of ...
84 views

### Doubts in the proof that a subgroup $H$ of a Lie group $G$ that is a submanifold of $G$ is (topologically) closed in $G$

I am trying to understand the details in the proof of Let $G$ be a Lie group, $H$ a subgroup. then $H$ is a submanifold of $G \implies H$ is closed in $G$ proof: Since $H$ is a submanifold of $G$, ...
31 views

1 vote
26 views

### When can monothetic groups be turned into rings?

If G is a Hausdorff topological group, saying that G is monothetic is equivalent to saying there exists a homomorphism $f: \mathbb{Z} \to G$ with dense image. A multiplication can be naturally defined ...
I have a locally compact abelian group $G$ with the following properties: It is connected (therefore divisible) and non-compact; It admits a $\mathbb{Q}$-vector space structure and for any $g \neq 0$,...
If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open? If this map happens to be open for all n and ...