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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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Topology problem on Lie Transformation groups.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and at the page 14 is this lemma: My question is why the uniqueness of that topology is trivial? I just know that i have ...
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Continuous automorphism of a lie group in kobayashi's book

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and i dont understand a thing at page 14. My question is why $A_\varphi$ is continuous? $G$ is a subgroup of ...
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30 views

Sum of closed subgroups and linear topology

Let $(G,+)$ be a $T2$ topological group endowed with a linear topology. It means that there is a local basis around $0$ made of subgroups. Pick two closed subgroups $C,D$; is it true that the sum $...
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30 views

Examples of Abelian Topological Groups

After reading an answer to the relation between Fourier Series and Hilbert Spaces I am curious about interesting examples of Abelian Topological Groups. Specifically, the answer explains that if $f$ ...
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1answer
25 views

Projection homomorphism of an abelian group is closed [on hold]

Let $G$ be a an abelian topological group and let $\Gamma$ be a discrete subgroup. Is the projection homomorphism $\pi:G\to G/\Gamma$ closed?
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How to show that $g^**g$ is a function of positive type, $g \in L^1(G) \cap L^2(G)$?

Let $G$ be a locally compact abelian Hausdorff group (with Haar measure $d\mu$). Call a function $h \in L^\infty(G)$ a function of positive type if $$ \int_G (f^* *f)h \, d\mu \geq 0 \ \ \ \forall f \...
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Trouble understanding topological groups.

I understand that a topological group is a group $G$ endowed with a topology $\tau$ on $G$ such that addition and inverse are continuous on $\tau$. Now, the definition of continuity is that for all $...
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1answer
27 views

Closure of $AB$

I'm trying to understand what the closure of $AB$ looks likes... $AB = \{ab: a \in A, b\in B\}$ So I know the closure of $AB = AB \cup (AB)' = \{ab: a \in A, b\in B\}\cup\{ab: a \in A', b\in B'\} ...
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Hecke algebra of a locally compact, totally disconnected group as a group of distributions and functions.

Let $G$ be a locally compact, totally disconnected topological group, and let $\mathcal D(G)$ be the space of locally constant functions $G \to \mathbb C$ with compact support. The space $\mathcal D'(...
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57 views

Relation Between Nbhd Base at $e$ and the Uniform Structure on a Topological Group

We have the following theorem (from Husain's Introduction to Topologcal Groups), slightly rephrased: The following are true (apologies for the sloppy formatting): (1) in any topological group $G$, ...
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1answer
81 views

Not continuous function $f: \mathbb{Z} \to \mathbb{Z}$ [closed]

Let $\mathbb{Z}$ denote the integers endowed with the cofinite topology. Exhibit an example of a function $f: \mathbb{Z} \to \mathbb{Z}$ which is not continuous. I really need help with this problem; ...
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49 views

Group of homeomorphism of [0,1]

I'm considering the group of homeomorphisms from $[0,1]$ to $[0,1]$. Is this group with functions composition and supremum metric compact or locally compact? I think that this group is not compact, ...
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Find an isomorphism from $H=\{(s,t)\in S^1\times S^1 : s^m=t^l\}$ to $S^1\times C_n$.

Let $m,l\in\mathbb{N}$ be such that $\gcd(m,l)=1$. I denote by $S^1$ the circle (i.e. $S^1 = \{z\in\mathbb{C} : |z|=1\}$) and by $C_n$ the cyclic group of order $n$ (considered as a subgroup of roots ...
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44 views

The fundamental group of the quotient groups

Let $G$ be a simply-connected and connected topological group and let $\Lambda$ be a closed discrete subgroup of $G$. Now, $\pi:G\to G/\Lambda $ is a covering map. What is $\pi_1(G/\Lambda)$? I know ...
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119 views

Exact sequence induced by an exact sequence of groups

Suppose $H\hookrightarrow G$ is an inclusion of topological groups and $H$ is closed in $G$. Then we have an exact sequence $$ 1\to H\to G\xrightarrow \pi G/H \to 1$$ where $G/H$ is just a pointed ...
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35 views

Dual of a complete topological group

Let $(G,+)$ be an Hausdorff topological abelian group. Assume that $G$ is complete, i.e. every Cauchy sequence in $G$ converges in $G$. Let $\widehat G:=\operatorname{Hom}_{\text{cont}}(G, S^{1})$ be ...
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Failure of Schur's lemma for topological group representations

Is there an example of $G$, $\rho$ as below? $G$ is a locally compact group. $\rho$ is a continuous representation of $G$ on a Hilbert space $V$. This means that we have a homomorphism from $G$ to ...
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15 views

compact subsets and filtrations of closed subgroups

Let $G$ be a topological group (T2), and assume that there exists a collection of closed subgroups $\{C_i\}_{i\in\mathbb Z}$ such that: $$\ldots\supset C_{n}\supset C_{n+1}\supset\ldots\supset \{0\}$$...
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32 views

Why the given groups are compact? [on hold]

I was told that $Z_{m}$ is a compact topological group. But I do not know why it is compact ..... could anyone explain this for me please?
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Showing continuity of a function defined as the integral of $\chi \mapsto \chi(x)$ over the dual group $\hat G$

Let $G$ be a topological group. Let's assume $G$ is abelian, locally compact and Hausdorff. Then there exists is a Haar measure $d \nu$ for $G$ and $d \mu$ for $\hat G$. Under the assumption that the ...
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Conditions for groups having homeomorphic Chabauty Spaces to be isomorphic.

Let $G$ and $H$ be locally compact such that their Chabauty spaces $\mathcal{S}(G)$ and $\mathcal{S}(H)$ are homeomorphic. As a reminder, the Chabauty space $\mathcal{S}(G)\subseteq\mathcal{P}(G)$ of ...
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Prerequisites for topological dynamics

I will start reading the book “Lectures on Topological Dynamics” by Ellis and therefore I want to know what the logical prerequisites for this subject of mathematics are in order to understand it well....
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1answer
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The action of the group $\Gamma=\mathbb{Z}$ on the manifold $\mathbb{C}^n-\{0\}$

Let $\Gamma=\mathbb{Z}$ be the additive group of integers and give it the discrete topology. Suppose $\Gamma$ acts continuously on the topological n-manifold $\mathbb{C}^n-\{0\}$ by the map $x \mapsto ...
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Property of smooth functions on the adeles

Let $k$ be a number field, $\mathbb A$ the ring of adeles of $k$, $\mathbb A_f$ the finite adeles, and $\mathbb A_{\infty}$ the infinite adeles. Let $\phi: \mathbb A = \mathbb A_{\infty} \times \...
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What is the profinite completion of a free abelian group of infinite rank?

By definition, profinite completion of a group $G$ is $\widehat{G}=\varprojlim_N G/N$ where $N$ runs through every subgroup of finite index in $G$. Let $M=\bigoplus_{n\ge1} \Bbb{Z}$ be a free abelian ...
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3answers
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Is every $G_\delta $, zero-set?

Zero-set means a set of the form: $Z(f) = \{ x \in X | f(x) = 0 \}\quad\text{for some } f \in C(X)$ $C(X)$ is the ring of continuous function on $X$. I know that every zero-set is $G_\delta $, i....
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34 views

countably compact and compact

I know that compact space is countably compact, and every compact is pseudo-compact. Is there a simple example that show any Compact space is not necessarily compact ? Is there an example ...
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40 views

Show that $\tau$ is a topology on $\mathbb{R}$ [closed]

I really wanted to solve this math problem on my own, but I have absolutely no idea how to attack this exercise and REALLY needs some hints: Let $\tau$ be the system of subsets U in $\mathbb{R}$ ...
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1answer
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About closed subsets of the dual group $\widehat G$

Let $G$ be a topological abelian group (linearly topologized), not necessarilly locally compact and define its dual: $$\widehat{G}:=\operatorname{Hom}(G,S^1)$$ We endow $\widehat{G}$ with the ...
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1answer
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Are normal subgroups of a profinite group with finite index closed?

Let $G$ be a profinite group and $H \subseteq G$ be a normal subgroup with $[G:H] < \infty$, i.e. $H$ has finite index. Question: Is $H$ closed? Problems: I have a lot of trouble to understand ...
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1answer
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Make a short exact sequence of abstract groups into a short exact sequence of topological groups (motivated by the Weil Group)

Let $1 \to H_1 \to G \to H_2 \to 1$ be a short exact sequence of abstract groups. Question: If $H_1$, $H_2$ have fixed topologies, can we endow $G$ with a topology such that the sequence above ...
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Why is the Hausdorff property mentioned in the characterization of profinite groups?

Profinite groups are usually characterized as compact, totally disconnected, Hausdorff groups. However, as shown here, every totally disconnected topological group already has the Hausdorff property. ...
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1answer
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Characterization for the continuity of Weil representations

Let $K$ be a non-Archimedean local field and $W_K$ be the Weil group of $K$. We consider a representation $\rho: W_K \to \operatorname{GL}_n(\mathbb{C})$ between two topological groups. Here, $\...
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1answer
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Admissibility of representations

Consider a smooth representation $(\pi,V)$ of the group $G=GL_n(\mathbb{Q}_p)$. The representation is said to be admissible if for every open compact subgroup $K \subseteq G$, the space $V^K$ of $K$-...
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1answer
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A question about $C^{*}$-embedded and $C$-embedded

A subspace $S$ of $X$ is $C$-embedded in $X$ if every function in $C(S)$ can be extended to a function in $C(X)$. A subspace $S$ of $X$ is $C^{*}$-embedded in $X$ if every function in $C^{*}(S)$ can ...
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Is every closed $G_{\delta}$ a zero-set? [duplicate]

The set $C(X)$ of all continuous, real-value functions no a topological space $X$ will be provided with an algebraic structure and order structure. zero-set means: zero- set is : $Z(f) = Z_{X} (f) = \...
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1answer
37 views

A question about zero-sets

The set $C(X)$ of all continuous, real-value functions no a topological space $X$ will be provided with an algebraic structure and order structure. zero-set means: $Z(f) = Z_{X} (f) = \{ x \in X : f(...
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0answers
19 views

Bruhat-Schwartz functions on pro-finite groups

Let $X$ be a profinite topological group equipped with its Haar measure $\mu$ of total volume $1$, and $S(X,\mathbf{C})$ be the complex vector space of those continuous functions $f : X\to \mathbf{C}$ ...
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1answer
45 views

One dimensional topological subgroups of the torus

I was recently asked to characterize Lie subgroups of the torus $\mathbb{S}^1\times\mathbb{S}^1$. I found the one-dimensional case more difficult than I imagined, having to appeal to either Lie ...
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1answer
29 views

Is a continuous map between two topological groups homotopic to a homomorphism between them?

Let $G$ and $H$ be two topological groups and $f:G\to H$ be a continuous map. Is there a continuous homomorphism $g:G\to H$ homotopic to $f$?
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1answer
55 views

A more Intuitive proof of regularity of topological group [closed]

Does someone knows a good reference for the following result? "A topological group is Hausdorff if and only if the identity is closed." I have seen proofs in lecture notes of courses on the web, but ...
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1answer
67 views

Yamabe's theorem proof

*I'm trying to make the proof of Yamabe's Theorem that says that an arcwise connected subgroup of a Lie group G is a Lie subgroup of G. I found the proof in Goto's article (https://www.ams.org/...
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1answer
40 views

Homomorphism from $p$-adic to $l$-adic groups

I have seen and heard the statement that the $p$-adic and $l$-adic topologies are incompatible. I would appreciate a proof or references supporting this statement. More precisely, I am interested in ...
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1answer
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If $G$ is profinite, and $A$ is discrete, $f: G \to A$ is continuous $\implies$ $f$ factors through a normal open subgroup

Let $G$ be a profinite group; that is compact, and totally disconnected. Take $A$ a discrete space, and a continuous map $f: G \to A$. $\exists N$ open and normal in $G$ and a continuous map $g: G/...
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1answer
33 views

Simple groups and fundamental groups

What does it mean, topologically, to have a simple fundamental group? For instance, the torus $S^1 \times S^1$ has $\mathbb Z \times \mathbb Z$not simple. The case of $S^1$ is $\mathbb Z$, not simple ...
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34 views

Upper-triangular subgroup is not unimodular

Consider the group $GL_n(\mathbb{Q}_p)$ of $n \times n$ invertible matrices over the $p$-adic field $\mathbb{Q}_p$. My goal is to prove that the subgroup $P_0$ of upper triangular matrices is not ...
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2answers
45 views

Is a finitely generated metrizable group discrete?

The question is in the title. A countable locally compact Hausdorff group is discrete, so saying that a finitely generated metrizable group is locally compact would be enough. What if the group is a ...
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1answer
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Is $\Bbb Q / \Bbb Z$ discrete?

I would like to say that $\Bbb Q / \Bbb Z$ is not discrete (when $\Bbb Q$ has euclidean topology), since $\Bbb Z \subset \Bbb Q$ is not open. But OTOH we have $$\Bbb Q / \Bbb Z \cong \bigoplus_p \Bbb ...
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33 views

A word of length 4 in a local group, with two different values

I've been told that in a local group, one can find a word of length of $n=4$, i.e $w=g_1 g_2 g_3 g_4$, with two different meanings depending on how we put the parentheses. However I've also read that ...
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1answer
37 views

Every element $g$ of $G$ has a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}\subset U$

Let $G$ be a topological group and $g\in G$ , $U$ is a neighborhood of $g$ . Prove that there exists a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}\subset U$. If $g=e$, l have proved it. ...