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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225 views

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact.

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact. This seems pretty straight forward but how will I be able to prove this? I saw this ...
3
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2answers
366 views

What topological group is $\mathbb R/\mathbb Z$?

The integers $\mathbb Z$ are a normal subgroup of $(\mathbb R, +)$. The quotient $\mathbb R/\mathbb Z$ is a familiar topological group; what is it? I've found elsewhere on the internet that it is ...
3
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0answers
33 views

Proof of commutativity for topological I-semigroups

A topological semigroup on a closed interval I and order topology is called I-semigroup if 1 acts as an identity and 0 as an annihilator. I've seen in several articles that such I-semigroups should ...
6
votes
1answer
2k views

The fundamental group of a topological group is abelian [duplicate]

I want to show the fundamental group of a topological group is abelian. In fact, the question says the topological group is path connected. I do not know where I should use path-connectedness. I ...
0
votes
1answer
150 views

Neighbourhood base about a point $p$ of a topological group

I am reading topological groups from Van der Waerden. The conventions followed in this book are these. An open set that contains the point $p$ is called an open neighbourhood of $p$. Any set which ...
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0answers
36 views

Non-solvable, closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$

It is mentioned here that non-solvable closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$ are either the entire space or discrete. My question is this: Is there any easy proof of this, or do any of you ...
2
votes
1answer
36 views

Integral of $|f|$ outside a compact set

Let $G$ be a locally compact group. Given $f\in L^1(G)$ and $\epsilon>0$, how to show that there is a compact set $K\subset G$ such that $\int_{G\setminus K}|f|<\epsilon$?
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1answer
33 views

A semitopological qroup which is also quasitopological.

$(G,\mathcal T)$ is a semitopological group with $$(\forall x\in G)(\forall U\in \mathcal T)(\exists V \text{ neighborhood of }1)(x\notin U^cV)$$ Then $G$ is quasitopological (ie inverse is ...
8
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2answers
248 views

Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
2
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1answer
343 views

Finite Haar Measure if and only if Compact

This is an exercise from a book: Let $G$ be a locally compact group with Haar measure $\mu$. $\mu(\{e\})>0$ if and only if $G$ is discrete. $\mu(G)<\infty$ if and only if $G$ is compact. I ...
1
vote
0answers
17 views

Set of $x$ such that $h \mapsto hx$ is proper

Let $X$ be a locally compact second countable space, and $G$ a locally compact second countable group wich operates continuously on $X$. If $x \in X$, let $\rho_x : g \mapsto gx$. I would like to know ...
2
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0answers
29 views

universal nonabelian divisible group

For this post, a group $G$ shall be referred to as generally divisible, in case $\forall{x\in G:}~\forall{n\in\mathbb{N}^{\times}:}~\exists{y\in G:}~y^{n}=x$. Note. Here is no commutativity ...
2
votes
0answers
46 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
0
votes
1answer
68 views

Two basic questions about topological group theory

For a topological group, I'd like to know whether 1.there exist a topological group G which is a Hausdorff space but does not satisfies the first countable axiom or 2.there exist a topological ...
2
votes
1answer
114 views

The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
2
votes
0answers
44 views

Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
3
votes
1answer
152 views

Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
4
votes
1answer
99 views

Quotient group $G/G_0$ in Group Topology

I'm stuck on this (apparently) simple thing: If $G$ is a topological group and $G_0$ is the connected component of $G$ containing the identity then $G/G_0$ is discrete if and only if $G_0$ is open. ...
2
votes
1answer
88 views

Homomorphism Theorem

Let $f$ be an open homomorphism from a topological group $G$ onto a topological group $H.$ We denote $K=Ker(f).$ How can I prove that $\bar f:G/K→H$ is a homeomorphism? I tried to prove it is ...
2
votes
1answer
99 views

Can one visualise the dual groups to Cantor groups?

My question is very simple, and, probably an answer can be found in any harmonic analysis textbook, but it seems I have failed that task. It occurred to me that I don't understand the structure of the ...
1
vote
0answers
71 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 ...
1
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1answer
81 views

Exactness of completion of topological abelian groups

Let $0\to G^{\prime}\to G\to G^{\prime\prime}\to 0$ be an exact sequence of abelian groups. Suppose $G$ is a topological group and then give topologies on $G^{\prime}$ and $G^{\prime\prime}$ which are ...
2
votes
1answer
686 views

Why is the weak* topology not in general metrizable?

A Banach space is a topological group under addition. The dual is a topological group under the weak$^*$ topology. The weak$^*$ topology is weaker than the operator norm topology, so is it ...
3
votes
1answer
107 views

A question related to “Topology induced by the completion of a topological group”

I am sorry if I mistake the answer posted on the question Topology induced by the completion of a topological group. Stated as in that thread, let $G$ be a topological abelian group with a countable ...
0
votes
1answer
49 views

A question on Cauchy sequence in topological abelian group

Let $G$ be a topological abelian group. Recall that a Cauchy sequence $(x_n)$ in $G$ is defined to be a sequence such that for any neighborhood $U$ of $0$, there exists an integer $N$ with ...
2
votes
0answers
81 views

Abstract Fourier Analysis

I am trying to prove that given a locally compact abelian (Hausdorf) topological group, the characters on it parameterize the multiplicative linear functionals on the banach * algebra $L^1(G, d\mu)$ ...
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votes
2answers
265 views

Topological groups question from Munkres

This if from the 'Supplementary Exercises' at the end of Chapter 2 in Munkres' Topology. If $A$ and $B$ are subsets of (a topological group) $G$, let $A \cdot B$ denote the set of all points $a \cdot ...
0
votes
2answers
97 views

If $f$ is a positive function and $\int_{E}f d\lambda = 0$ then $\lambda (E) = 0$

If $f$ is a positive function and $$ \int_{E}f d\lambda = 0, $$ then show that $\lambda (E) = 0$ where $\lambda $ is a Haar (Radon) measure. I know that if $f$ is a positive function and ...
2
votes
2answers
130 views

E Hausdorff topological space, G acts properly discontinous

Let $E$ be a Hausdorff topological space, $G$ a homeomorphism group that acts on $E$ properly discontinous, i.e. $\forall e\in E$ exists a neighborhood $U$ of $e$ such that $gU\cap U = \emptyset $ for ...
3
votes
1answer
75 views

A question on a countable discrete closed set

Let $X$ be a topological group and let $D$ is a countable discrete closed subset of $X$. We also let $ \mathcal U= \{U_d: d\in D\}$ of open sets of $X$ such that witnesses that $D$ is closed discrete, ...
1
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1answer
44 views

A question on the right translation

Here is a claim: Let $G$ be a right topological group and $g$ be any element of $G$. Then the right translation $R_g$ of $G$ by $g$ is a homeomorphism of the space $G$ onto itself. How can I ...
4
votes
1answer
57 views

Is true that $Z(G)/N = Z(G/N)$ for connected topological groups?

Let $G$ be a connected topological group and $N$ a discrete normal subgroup of $G$. Is it true that $Z(G)/N = Z(G/N)$, where $Z(G)$ denotes the center of $G$? I know that every discrete normal ...
3
votes
1answer
213 views

Haar measure, convolution and involutions

I have some problems to follow the proof of the anti commutativity property of the convolution and involution operations defined using a Haar measure as presented in Pedersen's book Analysis Now, ...
2
votes
1answer
76 views

Combining the axioms of a topological group

According to Wikipedia, a topological group $G$ is a topological space and a group, such that the functions $$(x,y) \mapsto x\cdot y\\x\mapsto x^{-1} $$are continuous. Is the single requirement that ...
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0answers
41 views

Characters of Topological Group of $\mathbb{R}^n$

I am seeking to show that if $\phi :\mathbb{R}^n\rightarrow\mathbb{C}$ is a character of the topological group $\mathbb{R}^n$ then $\phi$ must have the form $\phi(x)=e^{ix\cdot\xi}$ for some $\xi$ in ...
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votes
1answer
32 views

Action of discrete subgroups E(n) on $\Bbb{R}^n$

Isometry group of euclidean space $\Bbb{R}^n$ is displayed by E(n). We say that a subgroup G of E(n) is discrete if and only if the subspace topology (from E(n)) on G is discrete. If X and Y are ...
0
votes
2answers
34 views

H = {A $\in$ G| there exists f:[0,1]$\to$G continuous, such that f(0)=A, f(1)=I}, Is H normal in G?

If G is a subgroup of GL(n;$\mathbb R$) and H = {A $\in$ G| there exists f:[0,1]$\to$G continuous, such that f(0)=A, f(1)=I}, Is H normal in G?
2
votes
3answers
193 views

Book/article recommendations for an introduction to hypergroups and subsequent research

I'm a grad student and I'm choosing an area to follow on my doctorate (in?) and I've been thinking about extension of topological group theory results to topological hypergroups, but for that I need ...
3
votes
2answers
50 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?
4
votes
0answers
83 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
14
votes
1answer
221 views

What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
1
vote
1answer
245 views

Discrete subgroups of isometry group $\mathbb{R}^n$

Let $G$ be a Hausdorff topological group. We say that a subgroup $S$ of $G$ is discrete if and only if the subspace topology (from $G$) on $S$ is discrete. Note that isometry group of euclidean space ...
2
votes
1answer
259 views

Is a regular Borel measure on a locally compact space necessarily $\sigma$-finite?

I am trying to compile a proof of the uniqueness of Haar measure. Usually this is done by multiple-integral mumbo-jumbo, abusing left and right invariance of two potential measures and invoking ...
4
votes
1answer
42 views

Neighborhood in topological groups

Let $G$ be a topological group, $e$ the neutral element and $U$ a neighborhood of $e$. Claim: Then there exists a neighborhood $V$ of $e$, such that $V^2 \subseteq U$. This should follow easily from ...
5
votes
2answers
224 views

equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous ...
2
votes
1answer
71 views

Is there a local group that is not locally isomorphic to a topological group?

Let $\:\langle \hspace{.02 in}\mathbf{U},\hspace{-0.03 in}\mathcal{T}_u\rangle\:$ be a Hausdorff space. $\;\;\;$ Let $\hspace{.02 in}\Gamma\hspace{.02 in}$ be a closed subset of $\hspace{.04 ...
2
votes
2answers
75 views

extension of group operation from $\mathbb{Q}$ to $\mathbb{R}$

I'm having a hard time with this (seems easy, but could be misleading) problem: Let $A \subseteq \mathbb{Q}$ be a convex subset, and let $+$ group operation on $A$. Let $\overline{A} := \{x \in ...
2
votes
1answer
63 views

Familiarizing with the Grothendieck topos $\mathbf{B}G$.

I am trying to familiarize with the Grothendieck topos $\mathbf{B}G$ of continuous $G$-sets, where $G$ is a topological group. I am unfortunately not very familiar with working with different ...
5
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0answers
335 views

Is $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?

Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is ...
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vote
0answers
118 views

Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...