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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Is there any standard terminology for the quotient of a topological group by the connected component of the identity?

If $G$ is any topological group, then the connected component of its identity is a closed normal subgroup $H$. It follows that $G/H$ is a totally disconnected topological group. Often, $G$ will be ...
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Group Structure on $\Bbb R$

$(\Bbb R,+)$ is a topological group. Is there any other group structure on $\Bbb R$ such that it is still a topological group and this group is not isomorphic to $(\Bbb R,+)$ ? Refer to ...
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Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
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60 views

Constructing Topological Groups [closed]

In general, is there a way to construct topological groups? That is, given two topological groups $X$ and $Y,$ can I construct a topological group $Z$ using $X$ and $Y$ in said construction? I have ...
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Unique Square Root Neighbourhood in Topological Group

For a Lie Group $\mathfrak{G}$ and any neighbourhood $\mathcal{V}\subset\mathfrak{G}$ of the identity $\mathrm{id}\in\mathfrak{G}$, $\exists$ neighbourhood $\mathcal{U}\subset\mathcal{V}$ of ...
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Given a basis for $\mathbb{R}$, show that it constructs the standard topology on $\mathbb{R}$

Let $q_1, q_2, ...,$ be the rational numbers enumerated. Consider the countable collection $$\mathcal{B} = \{ B_{\frac{1}{n}}(q_i) \ | \ i,n \in \mathbb{N} \}$$ of open balls centered at rational ...
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3answers
60 views

Find a locally compact space $X$ with a subspace $A$ that is NOT locally compact.

I'd like to find a locally compact space $X$ with a subspace $A$ that is NOT locally compact. As from here, I know that if $A$ is closed and $X$ is Hausdorff, then $A$ is locally compact. Anyone ...
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79 views

Closed subspace of a compact topological space is compact

Let $X$ be a compact topological space, and $A$ a closed subspace. Show that $A$ is compact. How does this look? Proof: In order to show that $A$ is compact. We need to show that for any open ...
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Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
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46 views

what is the meaning of a power set of topological vector space?

Given a topological vector space, what is the power set of this space meaning? thanks a lot. and I really appreciate if a straightforward and simple explanation of topological vector space is ...
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228 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 ...
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375 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 ...
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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 ...
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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 ...
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1answer
152 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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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 ...
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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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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 ...
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Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
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1answer
352 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 ...
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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 ...
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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 ...
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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 $G_u$ ...
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70 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 ...
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115 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 ...
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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 ...
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1answer
155 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 ...
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1answer
100 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. ...
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1answer
90 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 ...
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1answer
101 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 ...
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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
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 ...
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1answer
706 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
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1answer
108 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 ...
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1answer
51 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 ...
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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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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 ...
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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
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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 ...
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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, ...
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1answer
46 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 ...
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1answer
58 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
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1answer
214 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, ...
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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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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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1answer
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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 ...
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2answers
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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?
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3answers
194 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 ...
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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?
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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 ...