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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Every Lindelöf topological group is isomorphic to a subgroup of the product of second countable topological groups.

I want to show that every Lindelöf topological group is isomorphic to a subgroup of the product of second countable topological groups. I received an answer using the fact that Lindelöf topological ...
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1answer
242 views

Properties of Topological Groups

I'm working though William Basener's Topology and Its Applications and I have come across a problem I can't solve. The book defines a topological group as a group equipped with a topology where for ...
2
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1answer
63 views

An example of finite, connected topological group

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think ...
2
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2answers
132 views

Compact connected group and torsion-free dual

Here's yet another exercise that stumped me: Let $G$ be a compact abelian topological group. Then $G$ is connected iff its dual $\hat G$ is torsion-free. Any hints/solutions will be appreciated. ...
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0answers
17 views

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ becomes a LCA group?

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ itself becomes a LCA group ? I would really really appreciate if I can get a step by step ...
2
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0answers
35 views

Proof that a particular subgroup is proper

I've been stuck on this for a long time ... I'm reading a textbook which simply states "this subgroup is proper" but it doesn't make sense to me. Context: I have a pro-$p$ group $G$, which just means ...
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1answer
27 views

In the quotient $G/H$, why we must suppose that $H$ is closed?

We have the following known statement: Theorem: If $G$ is a topological group and $H\subseteq G$ is a closed invariant subgroup of $G$, then $G/H$ (of course with the quotient topology) is a ...
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0answers
35 views

Question about Modular function in Haar measure

I'm reading the book "Basic Lie Theory" (http://guests.mpim-bonn.mpg.de/abbaspou/Lie-Book_verrouille.pdf) and I'm trying to understand the proof of Lemma 2.3.4 which states that: Let $G$ be a locally ...
9
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174 views

On infinite groups admitting finitely many group topologies

It has been proved there is an infinite group which admits exactly two group topologies [1]. For which $n$, is there an infinite group $G$ which admits exactly $n$ group topologies ordered linearly ...
3
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2answers
75 views

A proof for $\widehat{\Bbb Z_{p^\infty}}\cong Z_p$

According to wikipedia, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers. Where can I find a proof for it on the internet?
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9 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
2
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2answers
108 views

Open subgroups of a topological group are closed

Let $G$ be a topological group such that for each $x \in G$ the mapping $x\mapsto xy$ is a homeomorphism. If $H$ is a open subgroup of $G$, prove that $H$ is also closed Could anyone just give ...
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1answer
28 views

Isomorphisms of LCA Groups

From what I understand, in the category $\mathsf {LCA}$ of lca groups, isomorphisms should respect both topology and group structure, hence they are continuous homomorphisms. I'm trying to learn ...
2
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0answers
26 views

Intersection of invariant subsets of a local group action

I don't understand some facts about invariant subsets of a local group action. Basically (to save you reading definitions) local actions are germs of partial actions which in turn are just like ...
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0answers
55 views

Slice at a point of a topological space

The definition is from the following link -Slice at a point of a topological space Let $G$ be a topological transformation group of a Hausdorff space $X$. A subspace $S$ of is called a slice at a ...
14
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288 views

On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the additive group of all rational numbers) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a ...
2
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1answer
20 views

Weyl Groups/Borel

Could someone tell me where to find a proof of the following statement that I found in some notes about characteristic classes I was reading? If $G$ is a compact connected Lie group with maximal ...
1
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2answers
27 views

Smallest open, dense, G-invariant subset of a metric space

Let $X$ be a metric space and $G$ be a topological group acting continuously on $X$. Let $ \mathcal S $ be the set of open, dense and $G$-invariant subsets of $X$. I need to take inverse limit (of ...
2
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1answer
67 views

In what topological abelian groups does convergence to zero imply summability?

Let $\; \langle G\hspace{-0.02 in},\hspace{-0.04 in}+,\hspace{-0.04 in}\mathcal{T}\hspace{.03 in}\rangle \;$ be a $\hspace{.02 in}\big(\hspace{-0.03 in}$$\text{T}_{\hspace{-0.02 in}0}$$\hspace{-0.03 ...
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0answers
16 views

A compact group with a finite dimensional faithful representation [duplicate]

Theorem: If $G$ a compact group has a finite dimensional faithful representation $W$, then any irreducible representation $V$ is contained in $W(k,l) = W^{\otimes k} \otimes (W^*)^{\otimes l}$ for ...
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1answer
46 views

The natural isomorphism in the Pontryagin Duality

Pontryagin duality is the statement that there's a natural isomorphism between the identity functor on $\mathsf{LCA}$, the category of locally compact (Hausdorff) abelian groups with continuous ...
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1answer
51 views

Prove that $SL(n,R)$ is connected.

Prove that $SL(n,R)$ is connected. The problem is I know only topological groups from Munkres only. Again Just started fundamental groups. So if anyone can explain me how it is true in a lucid ...
2
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2answers
46 views

Show that every topological group is $T_3$

I know that it is sufficient to show that for one point ($e$) and any neighbourhood $U$ of $e$, we have a neighbouhood $V$ with $\bar{V} \subseteq U$. Since $x \to x^{-1}$ is continuous, it follows ...
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1answer
56 views

Why is the character group defined as $\mathsf{Hom}(G,\mathbb T)$, i.e why is the codomain specifically $\mathbb T$?

In the paper Category Theory Applied to Pontryagin Duality by Roeder, the character group of an lca group is defined as the topological (under the compact-open topology) abelian group of continuous ...
3
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0answers
64 views

Prove that the quotient map $P:G \to G/H$ is a covering space.

Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space. My ...
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3answers
859 views

How to show that topological groups are automatically hausdorff?

On page 146, James Munkres' textbook Topology(2ed), Show that $G$(a topological group) is Hausdorff. In fact, show that if $x \neq y$, there is a neighborhood $V$ of $e$ such that $V \cdot x$ and ...
4
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1answer
67 views

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
3
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0answers
32 views

Cech-complete separable groups

It is well known that a Baire measurable homomorphism between Polish groups is continuous. Is the same true if we replace Polish groups by Cech-complete, separable groups? It is not true ...
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0answers
21 views

What's the significance to the $m$ in the notation $L(n,m)$ for the Lens space?

I'm reading a quick example (Example 12.13 of Topological Manifolds by John Lee) of the construction of the lens space $L(n,m)$. Basically, let $$S^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\}$$ ...
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1answer
56 views

Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
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0answers
37 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
3
votes
1answer
41 views

question on subgroup of compact group

Suppose $G$ is a compact metrizable abelian group, is it true that $G$ has no finite index subgroups iff $G$ is connected? Any reference or help is appreciated. Thanks in advance! Here are my ...
7
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3answers
265 views

Does every homogeneous space allow a group structure?

Let $(X,\tau)$ be a homogeneous space, that is for all $x,y \in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there a group operation $*:X\times X\to X$ such that ...
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0answers
53 views

looking for books on topological semigroup:

I'm looking for several books on topological semigroup: Topological semigroups: history, theory, applications. Karl Heinrich Hofmann Mathematics Research Library, Tulane University. The ...
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1answer
50 views

Counterexample of “the product of open subsets is open in a topological ring”?

Given a topological ring $R$ and $U,V$ open subsets, we can show that $U+V$ is an open subset due to the fact that $x\mapsto x+y$ is a homeomorphism for every $y \in R$. Since, in general, $R$ is not ...
2
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1answer
42 views

Translation invariant metrics and topological groups

During a lecture was pointed out that one of the main feature, from a topological perspective, of normed vector spaces is the translational invariance, that is that one can study the topological ...
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2answers
68 views

if $\pi_1(G)$ is trivial, how to prove $ \pi_1(G/H)=\pi_0(H)/\pi_0(G) $?

Let $G$ be a topological group, $H$ be its normal subgroup of $G$, and $G/H$ be the quotient space induced by the natural map.(We know that $G/H$ is again a topological group) If $\pi_1(G)$ is ...
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0answers
23 views

cohomology of unordered configuration spaces of sphere

Let $F(X,n)$ be the configuration space of order $n$. Let $F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. What is $H^*(F(S^2,n)/\Sigma_n;\mathbb{Z}_2)$? I did not find the answer ...
2
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1answer
43 views

Reference for Topological Groups

Topological groups were a topic that were covered minimally at my undergraduate institution but it's a topic that I'm finding a need quite a bit in the number theory I'm reading (class field theory). ...
14
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6answers
460 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
0
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1answer
125 views

Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ ...
2
votes
1answer
67 views

Largest ideal of a local field on which a character is trivial

Let $K$ be a nondiscrete locally compact field. Then fixing a character $\chi$ on $K$, any character on $K$ can be written as $t \mapsto \chi(xt)$ for some $x \in K$. For $E \leq K$ a closed ...
2
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1answer
37 views

Existence of an open normal subgroup of a neighborhood of 1 in a compact Hausdorff and totally disconnected topological group

Let $G$ a compact Hausdorff and totally disconnected topological group. Then every neighborhood of 1 contains an open normal subgroup of finite index in $G$. I need this lemma to prove that every ...
3
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2answers
38 views

Is every regular paratopological group completely regular?

This problem is presented as an open problem 1.31. on p.26 of Arhangel'skii-Tkachenko, Topological groups and related structures. Is this problem still open?
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1answer
93 views

A sequence of subsets of $\Bbb Z$ [closed]

Is there a sequence $(A_n)$ of nonempty subsets of $\Bbb Z$ such that for each $n$, $$\{a-b\mid a,b\in A_{n+1}\}\subsetneqq A_n$$ ?
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1answer
82 views

Is a partially topological group completely regular

Let $G$ be a group and $\mathcal T$ be a topology on $G$ and the function $$ \begin{align*} &f:G\times G\to G\\ &f(x,y)=xy^{-1} \end{align*} $$ be continuous at $(1,1)$. Is $(G,\mathcal T)$ ...
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0answers
23 views

Component of the identity is generated by any connected neighborhood of the identity when the group is locally connected?

I read a theorem that if $G$ is a locally connected group, then the component of the identity $G_0$ is generated by any connected neighborhood of $e$. It goes like: Let $V$ be a connected nbhd of ...
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1answer
44 views

Non-discrete locally compact Hausdorff groups which do not satisfy the second axiom of countability

I think non-discrete locally compact Hausdorff groups which do not satisfy the second axiom of countability are pain in the ass. The major trouble is that the Haar measures on them are not necessarily ...
3
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1answer
54 views

Inducing a surface area measure on $S^2$ from the Haar measure on $SO(3)$

I'm reading the book "Random Matrices: High Dimensional Phenomena" by G. Blower. There is an example that I've been struggled for a long time. For those who have access to the book, it's the Example ...
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1answer
44 views

Can topological groups be smoothed into lie groups?

I've been thinking about this for the past couple of days and I'm really not sure of the answer.. By "smoothed" I mean that for any arbitrary precision we can find a Lie group which approximates the ...