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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321 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 ...
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2answers
130 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
77 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
529 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
506 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
388 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
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1answer
477 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 ...
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1answer
172 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 $N_1,N_2 \in \mathcal{N}$, ...
8
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2answers
227 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
187 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, ...
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1answer
548 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
135 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
votes
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 $\bar{K_1}$ ...
1
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1answer
107 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
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1answer
103 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
186 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
203 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
264 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
118 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
371 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
votes
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
341 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
167 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
262 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
votes
1answer
323 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
votes
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
449 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 ...
5
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0answers
248 views

Short exact sequences of topological groups and Lie groups

could someone please clarify the definitions of extensions of topological groups and Lie groups. For topological groups, what I see in most papers is as follows: An extension of topological groups $0 ...
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1answer
372 views

Why the discrete subgroup of $R^n$ has finite balls?

Consider a discrete subgroup $G$ of $R^n$. Then let we have an open ball $B\subset G$ (topology in $G$ is induced from $R^n$). Why is $|B|<\infty$? For example I can take ...
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1answer
153 views

example of a discontinuous group operation in $\mathbb{R}$, under usual metric

someone has an example of that? I know that if G is a topological connectedness group ( operation continuous under the topology) then every neighborhood containing the identity elemental, generates ...
2
votes
1answer
133 views

The inverses of open sets

I am not sure if this is already posted, though, I hope I can get some help, and thank in advance. This question arises from the proof of the following. Proposition: Let G be a topological ...
3
votes
1answer
115 views

Generating sets for topological groups

Let G be a compact topological group. Suppose G has a subset X and a normal subgroup N such that the subgroup generated by X is dense in N. Moreover, suppose G has a subset Y such that the subgroup ...
6
votes
1answer
190 views

totally disconnected orbit-stabilizer theorem

So I'm aware that the orbit-stabilizer theorem does not hold for arbitrary spaces with a transitive action by a topological group, but I wonder if it works in the following situation. Let $G$ be a ...
2
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2answers
210 views

does isomorphic groups induce homeomorphic quotients

suppose $X$ is a topological space and $G$ and $H$ are groups acting on it. 1) if $G$ is isomorphic to $H$ do we have necessarely $X/G$ is homeomorphic to $X/H$ 2) suppose $G$ and $H$ are two ...
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0answers
165 views

how to factor a map by a group action

Let $X$ and $Y$ be topological spaces and a surjective map $f:X\rightarrow Y $. Suppose that a group $G$ acts on $X$. and let $\pi:X\rightarrow X/G$ be the quotient map. 1) Under what conditions $f$ ...
10
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2answers
970 views

Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
4
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1answer
114 views

Is there any relation between a group being unimodular and having equivalent uniform structures?

Recall: A topological group is said to have equivalent uniform structures if its left and right uniform structures coincide. A locally compact group is said to be unimodular if left Haar measures and ...
21
votes
1answer
464 views

Shrinking Group Actions

Suppose $H\subset G$ is a subgroup of a topological group $G$, and $Y\subset X$ is a subspace of a topological space $X$. Suppose we are given a continuous group action $\rho : G\times X\rightarrow X$ ...
3
votes
0answers
188 views

If $G$ is a locally compact Hausdorff group, when does $G/Z$ have a probability Haar measure?

I am reading an introductory material about topological groups and the question in the tittle comes up. Due this Proposition Proposition. A locally compact Hausdorff topological group $G$ is ...
3
votes
1answer
263 views

Compact group actions and automatic properness

I am currently re-reading a course on basic algebraic topology, and I am focussing on the parts that I feel I had very little understanding of. There is one exercise in the chapter devoted to groups ...
3
votes
1answer
133 views

Restricted Direct Products in Koch's Number Theory

On p.353 of Number Theory: Algebraic Numbers and Functions by Helmut Koch, he considers a group $G$ which is the restricted direct product of the locally compact abelian groups $G_i$ with respect to ...
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3answers
585 views

Topology on the general linear group of a topological vector space

Let $K$ be a topological field. Let $V$ be a topological vector space over $K$ (if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of ...
4
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2answers
265 views

Does a topological group need to have a uniformity making all group operations uniformly continuous?

Let $G$ be a topological group. $G$ comes equipped with a left (resp. right) uniformity $\mathscr{L}$ (resp. $\mathscr{R}$) which can be characterized as the coarsest uniformity which is compatible ...
7
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2answers
766 views

Why is every discrete subgroup of a Hausdorff group closed?

I have just begun to learn about topological group recently and is still not familiar with combining topology and group theory together. I have read a useful property of discrete group on the ...
24
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1answer
524 views

Useful sufficient conditions for a topological space to be the underlying space of a topological group?

Here is a question that I have had in my head for a little while and was recently reminded of. Let $X$ be a (nonempty!) topological space. What are useful (or even nontrivial) sufficient ...
7
votes
1answer
196 views

Intersection of neighborhoods of 0. Subgroup?

Repeating for my exam in commutative algebra. Let G be a topological abelian group, i.e. such that the mappings $+:G\times G \to G$ and $-:G\to G$ are continuous. Then we have the following Lemma: ...
14
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2answers
1k views

Is addition continuous?

I'm going to ask a very silly question, so I'm begging you to be understanding if it is absolutely trivial, or if it's an exercise in some Bourbaki. I'm afraid of asking you, because the question ...