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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A question about compact abelian groups

After learning about the duality between compact Abelian groups and discrete Abelian groups, I decided to look at exercises from various sources. One question that stood out was the following: If ...
9
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1answer
236 views

The Group of Homeomorphisms

I have been looking at Topological Groups, and I recently read about the group $\operatorname{Homeo}(X)$ of all homeomorphisms of $X$ onto itself. In particular, when $X$ is a metric space. The ...
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0answers
383 views

Haar-measure on the torus

Good evening! Let $ \mathbb{T}:=\{ z \in \mathbb{C} ; \vert z \vert =1 \} $ be the unit circle in the complex plane. We denote the trace Borel-$\sigma$-algebra on $\mathbb{T}$ by ...
4
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1answer
104 views

Finding an open set for a topological group

Let $G$ be a locally compact topological group, $K$ a compact subgroup and $\Gamma$ a discrete subgroup. I try to find a neighbourhood $U$ of the identity such that $\Gamma \cap UK = \Gamma \cap K$. ...
6
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1answer
113 views

Uniqueness of compact topology for a group

Suppose $G$ is a compact $T_2$ group. Can there be other compact $T_2$ topologies on $G$ which also turn $G$ into a topological group? ($T_2$ refers to the Hausdorff separation axiom)
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1answer
427 views

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of ...
1
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1answer
62 views

unitary representation and denseness.

I have the next unitary representation, $\pi : G\rightarrow \mathcal{U}(H)$, where G is a closed subgroup of $S_{\infty}$ (the group of bijective functions from $\mathbb{N}\rightarrow \mathbb{N}$), ...
2
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1answer
121 views

When is $\: \pi_1(\langle X,\mathcal{T}\hspace{.01 in}\rangle,x_0) \:$ a topological group?

(Although I am taking an algebraic topology class, this is not homework; we have not gotten to this yet.) Let $\langle X,\mathcal{T}_X\rangle$ be a path-connected Hausdorff space. $\:$ Let $x_0$ be ...
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1answer
137 views

Which of the following topological groups are polish or locally compact?

I want to show that the next groups are polish topological groups, which criteria should I use here? And also which are locally compact (same question)? The groups are: The group of permutations ...
10
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1answer
391 views

Is a topological group action continuous if and only if all the stabilizers are open?

Let $G$ be a topological group and $(X,\mu)$ be a $G$-set, i.e. $\mu$ defines an action $X \times G \rightarrow X$. Is it then true that $\mu$ is continuous if and only if for every $x \in X$ the ...
2
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2answers
319 views

Locally compact topological group is Normal

How can I prove directly that a locally compact topological group G is normal? I have done this by showing that every locally compact topological group is strongly Paracompact. But I could not prove ...
4
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1answer
241 views

Compact subgroups of the general linear group

Let $V$ be a finite-dimensional real linear space, and let $K$ be a compact subgroup of $GL(V)$ (with the usual topology); then is there a basis of $V$ such that every $f\in K$ is an orthogonal matrix ...
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1answer
86 views

Centralizers in reductive Liegroups = unimodular?

Let $G$ be a real reductive group. Why is the centralizer of an element unimodular? What is a reference?
12
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2answers
367 views

$(x,y)\to xy$ continuous but $x\to x^{-1}$ not

In the definition of topological groups we impose both $(x,y)\to xy$ and $x\to x^{-1}$ to be continuous. However, I cannot find an example where the first condition holds but the second fails. Is ...
6
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0answers
132 views

Why is any proper closed subgroup of $\mathbb{R}$ necessarily countable? [duplicate]

Possible Duplicate: Subgroup of $\mathbb{R}$ either dense or has a least positive element? If I have $G$ a closed subgroup of $\mathbb{R}$, then why is $G$ necessarily countable, except of ...
3
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1answer
159 views

$\sigma$- compact clopen subgroup.

I am given $G$ locally compact group, and I want to show that there exists a clopen subgroup $H$ of $G$ that is $\sigma$-compact. So here's what I did so far: for $e \in U$, where $U$ is a nbhd of ...
2
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0answers
81 views

Induced representations of topological groups

Sorry if this is a naive question-- I'm trying to learn this stuff. If $G$ is a group with subgroup $H$, then we have the restriction functor $\operatorname{Res}$ from $G-\operatorname{mod}$ to ...
8
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2answers
386 views

Exact sequence in a nonabelian category [previously: “Exact sequence for topological groups?”]

If $A$, $B$, and $C$ are topological groups, and $f: A \to B$ and $g: B \to C$ are two continuous group homomorphisms, what does it usually mean for $$1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C ...
6
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2answers
182 views

For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = \overline{[H,H]}$? What about algebraic groups?

When discussing with awllower about this question, I begin to think about another one: For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = ...
1
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1answer
343 views

Closure of the connected component of the unity is connected: is my proof valid?

I have tried to prove that a closure of a connected component of the unity in a topological group is closed, but am not sure of its validity. Since it arose from a sentence in a book on the subject,* ...
4
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1answer
327 views

Completion of Topological Group with Metric

Related to this question, I'm having trouble understanding the construction of the completion of a topological group with metric structure. In particular, under what conditions is the completion also ...
2
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2answers
134 views

Zero Dim Topological group

I have this assertion which looks rather easy (or as always I am missing something): We have $G$ topological group which is zero dimensional, i.e it admits a basis for a topology which consists of ...
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0answers
79 views

A $T_0$ topological groups is $T_{3.5}$ (and consequently $T_3$)

I don't understand why this is so? Iv'e just seen the proof that a $T_0$ topological group is $T_1$, but don't know how to show that it's $T_{3.5}$. BTW, the fact that $x\bar{V}=\bar{xV}$, one ...
3
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1answer
85 views

String and BString

In one of the talks of J.P. May he mentioned some examples of structure groups and their classification spaces (he mentioned: O, U, SO, SU, Sp, Spin, String, Top, STop, F and SF). Most of them are ...
3
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1answer
558 views

Is the following proof valid? About the closure of a subgroup, of a topological group, being again a subgroup

I found a proof in the book Fourier Analysis On Number Fields that the closure of any subgroup is a subgroup, using the continuity argument along with the nets. Nevertheless, the following proof seems ...
7
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1answer
522 views

Sum of Cauchy Sequences Cauchy?

Let $(X,+)$ be an abelian group and $d$ a metric on $X$. Suppose $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences. What conditions on the relation between the group operation and the metric are sufficient ...
2
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1answer
45 views

question on uniform structure

It should be a triviality, I believe. The topology induced by a uniform structure $\mathcal{U}$ with $\cap \mathcal{U} =\Delta$, where $\Delta$ is the diagonal, is Hausdorff. Now I think that if I ...
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2answers
414 views

Is it true that under the Zariski topology, a subset is dense if and only if it is a nonempty open subset

Is it true that under the Zariski topology, a subset is dense if and only if it is a nonempty open subset? I know this is true in one direction, i.e., any nonempty open subset is dense, but how ...
2
votes
1answer
506 views

Topological Groups

I was learning about topological groups from Atiyah-Macdonald's chapter on Completions, and I have the following question: Let $G$ be an abelian topological group. Let $H$ be the intersection of all ...
1
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1answer
178 views

property of a topological group

This should be easy, but apparantly not for me. Let G be a topological group, and let $\mathcal{N}$ be a neighbourhood base for the identity element $e$ of $G$. Then for all $N_1,N_2 \in \mathcal{N}$, ...
9
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2answers
237 views

Topological rings which are manifolds

Is the following statement true: "Every smooth manifold $M$, which is a ring in the category of manifolds, must be diffeomorphic to $\mathbb{R}^n$."? (Actually, homeomorphic would suffice.) I assume ...
3
votes
1answer
192 views

Does the Sorgenfrey Line have a group operation compatible with its order topology?

The title is the question, but let me explain. Let $\mathbb{L}$ denote the Sorgenfrey line. I and a friend were trying to develop some of the properties of the sorgenfrey line. (if it's metrizable, ...
6
votes
1answer
565 views

Product of compact and closed in topological group is closed

This could be classified as "homework", but I tried to solve this, made research online, and still failed, so I'll be glad to get some hints. Let $G$ be a topological group, let $A$ be a compact ...
5
votes
1answer
137 views

Measure of a conjugacy class in a compact group

Suppose $G$ is a compact group endowed with Haar measure $\mu$. If $g \in G$, then denote by $g^G$ the conjugacy class of $g$ in $G$. Is there anything that can be said in general about $\mu(g^G)$? ...
2
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2answers
95 views

Direct products of closures of subgroups

Let $H$ be subgroup of a topological group $G$. Suppose $H$ is the (internal) direct product of two of its subgroups $K_1$ and $K_2$. Does it follow that $\bar{H}$ is the direct product of $\bar{K_1}$ ...
2
votes
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. ...
5
votes
1answer
108 views

Separability of a group and its dual

Here is the following exercise: Let $G$ be an abelian topological group. $G$ has a countable topological basis iff its dual $\hat G$ has one. I am running into difficulties with the ...
4
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1answer
197 views

Existence of balanced neighborhoods in a topological vector space

I'm wondering about the following: Let$\ X $ be a topological vector space. Then one could pick balanced neighborhoods$\ W $ and$\ U $ of$\ 0 $ such that $\ \overline{U} + \overline{U} \subset W $, ...
9
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1answer
211 views

Lie Groups which are not Hausdorff

I suspect this isn't a terribly difficult question, but I don't know the answer and I'd guess someone has already looked into it. Is it possible for a Lie group on a non-Hausdorff manifold to exist? ...
3
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1answer
280 views

Example of a locally compact connected Abelian group with non-$\sigma$-finite measure

I look for an example of an Abelian locally compact topological group $G$ such that: $G$ is connected and Haar measure on $G$ is not $\sigma$-finite and $\{0\} \times G \subset \mathbf{R} \times G$ ...
4
votes
1answer
130 views

Decomposition of $\mathbb{C}_p^*$

I'm looking for a topological group decomposition of $\mathbb{C}_p^*$. I know that I can write $\mathbb{C}_p^*\cong p^\mathbb{Q}\times \mathcal{O}_{\mathbb{C}_p}^* \cong p^\mathbb{Q}\times ...
2
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0answers
122 views

Homeomorphism groups as topological groups

As is well known, the homeomorphism group of a compact Hausdorff space is a topological group. The same is true for locally compact locally connected Hausdorff spaces, but it is false in general. Now ...
13
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1answer
375 views

No non-trivial homomorphism to a group

Let $G$ be a compact Hausdorff topological group, and let $H$ be a torsion-free group satisfying the ascending condition, i.e. there are no infinite strictly ascending chains $H_1<H_2<...$ of ...
2
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1answer
82 views

Convex subsets of a group

Assume that $(G,+)$ is an Abelian topological group (maybe locally compact, if necessary) and assume that $V$ is an open connected neighbourhood of zero. Does there exist an open "convex" ...
4
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3answers
366 views

G/H is Hausdorff implies H is closed (General topology, Volume 1 by N. Bourbaki)

I am reading General topology, Volume 1 By Nicolas Bourbaki. I refer to the proof of Proposition 13. Could someone kindly explain the G/H Hausdorff $\implies$ H closed part of the proof? I ...
4
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2answers
168 views

Definition of topological group

Let $U$ be open in a topological group, G. Why then is it necessarily true that $UH$ where $H$ is some subgroup of $G$ open in $G$? (I think I don't quite get the idea of a topological group even ...
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1answer
264 views

Transitive group actions and homogeneous spaces

Given a topological group $G$ and a space $X$ with a transitive $G$ action, let $G_x$ be the isotropy group of a point. In Folland "A course in harmonic analysis", there is a statement that $X$ is ...
8
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1answer
334 views

An equivalent definition of the profinite group

A profinite group is by defination a topological group $G$ which is Hausdorff , compact and totally disconnected. How to prove the following equivalent defination: A compact Hausdorff group is ...
2
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1answer
90 views

What's an example of a group with equivalent uniform structures where multiplication is not uniformly continuous?

Say we have a topological group $G$. It's easy to see that if $\cdot: G \times G \rightarrow G$ is uniformly continuous (with respect to either the right or left uniformity), then $G$ must have ...
8
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1answer
473 views

Topology on integers making it a topological group

Are there non-trivial topologies (neither discrete nor indiscrete) on the additive group of integers $\mathbb{Z}$, making it into a topological group. Could someone list them all, possibly with some ...