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Questions tagged [topological-groups]

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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Defining the completion of a group can be done only using Cauchy sequences

Let $G$ be a group, in Atiyah & MacDonald's Commutative Algebra it says that Assume for simplicity that $0\in G$ has a countable fundamental system of neighborhoods. The completion $\hat G$ of $G$...
ephe's user avatar
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Which commutative rings are endomorphism rings of some topological Abelian group?

I was thinking earlier today about how many commutative unital rings I could hit by considering the endomorphism rings of topological Abelian groups. For example, the group $(\mathbb{R}, +, \tau^R)$ ...
Greg Nisbet's user avatar
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Every simple topological group is either discrete or connected [closed]

I read the claim in a preprint that "A simple topological group is either discrete or connected". However, the explanation given was "a connected component of a topological group that ...
Anguepa's user avatar
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2 votes
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167 views

How to show isomorphism between an adele ring on a number field and its Pontryagin dual?

A book that I'm currently reading (Fourier Analysis on Number Fields by Dinakar Ramakrishnana, Robert J. Valenza) claims that a continuous homomorphism $\mathbb{A}_k \to \widehat{\mathbb{A}_k}$ given ...
Dawid's user avatar
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1 answer
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Necessary condition for an isomorphism of modules to be a homeomorphism

Let $M$ and $N$ be two filtered $A$-modules with filtrations $(M_n)$ and $(N_n)$ and $u\colon M\to N$ be a morphism of modules. We can show that if $u(M_n)=N_n$ and $u$ is an isomorphism then $u$ is ...
ephe's user avatar
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Every Cauchy sequence in $M/N$ converges.

Let $M$ be a filtered module with filtration $(M_k)$ and $N$ be a submodule of $M$. Then the filtration on $M/N$ is given by $(P_k)$ where $P_k=(M_k+N)/N$. We will show that if $M$ is complete and ...
ephe's user avatar
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1 answer
63 views

Generalization of the fact that for $f: S^1 \rightarrow \Bbb R$ continuous, $\exists x \in S^1 : f(x) = f(-x)$

While going through James Munkres' book on topology I came across the following problem: Let $f: S^1 \rightarrow \Bbb R$ a continuous function. Show that there exists a point $x \in S^1$ st. $f(x)=f(-...
clorx's user avatar
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3 votes
1 answer
140 views

Can a compact group have an infinite sequence of closed subgroups?

If $G$ is a copmact Hausdorff group then can there be, in some case of $G$, a sequence $G_i$, $i\ge0$ such that $G_0=G$ and $G_{n+1}$ is a closed proper subgroup of $G_n$ for all $n\ge0$ (proper ...
cnikbesku's user avatar
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Closed subsets in Fell topology on $\mathcal{P}(G)$

I am currently working with a discrete group $G$. I am considering a set $A \subseteq \mathcal{P}(G)$ of finite subsets $F \subseteq G$ whose cardinalities are uniformly bounded. I am having troubles ...
Alice in Wonderland's user avatar
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Profinite completion of profinite groups

I was trying to prove that $\mathbb{R}/\mathbb{Z}$ cannot be a galoisgroup for any extension. My plan for this was to show that $\mathbb{R}$ is a divisible group and that every quotient of a divisible ...
potenzenpaul's user avatar
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47 views

Topology groups

Is there any local topological group, that cannot be obtained through taking sufficiently small neighborhood of unit in any global topological group?
Kirill Zhilitch's user avatar
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Indecomposable complex finite dimensional representations of a compact Lie group are Irreducibles?

I am studying the book "Representations of Compact Lie Groups" by Theodor Brocker and Tammo tom Dieck. At page 68 they prove the following proposition: Let $G$ be a compact group. If $V$ is ...
Don Abbondio's user avatar
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Is the Haar Measure on a Group the Only Reasonable Way to Define Randomness?

Given a compact topological group $G$ and a closed subgroup $H$. Let $d \mu_H$ and $d \mu_G$ be the respective unique Haar measures. In general, when we say "pick a random element of $H$," ...
a.e's user avatar
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Natural map of automorphism groups

Question: Write $\mathbb{Q}(\zeta_{\infty}) = \mathbb{Q}(E)$, where $E$ is the group of roots of unity in $\mathbb{Q}^{*}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\zeta_{\infty})$ is Galois, and ...
ByteBlitzer's user avatar
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Compostitum is Galois and isomorphism of Galois groups

Question: Let $K, L, F$ be subfields of a field $\Omega$, and suppose that $K \subset L$ is Galois and that $K \subset F$. Prove that $F \subset LF$ is Galois and that $\text{Gal}(LF/F) \cong \text{...
ByteBlitzer's user avatar
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Confusion on isomorphism profinite completion integers

Question: Prove that $\hat{\mathbb{Z}} \cong \prod_{p} \mathbb{Z}_{p}$ is an isomorphism of topological rings. Own attempts: Although this specific question has been asked quite a few times here, ...
ByteBlitzer's user avatar
2 votes
1 answer
51 views

Is the inverse of an isotopy of embeddings continuous?

I was dealing with isotopies and monodromies when the following question arose, which is more subtle than it might seem at first sight. Let $X$ and $Y$ be topological spaces. Suppose that $\Phi: [0,1] ...
Don's user avatar
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2 votes
1 answer
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Help understanding a step of the proof on the existence of a left invariant pseudometric on a topological group

The following theorem can be found in Hewitt-Ross Abstract Harmonic Analysis I, in chapter II Theorem 8.2: Let $(U_k)_{k=1}^{\infty}$ be a sequence of symmetrics neighborhoods of $e\in G$ such that $ ...
none's user avatar
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Isomorphism of topological groups

Question: Let $\mathbb{Q}(\sqrt{\mathbb{Q}})$ be the subfield of $\overline{\mathbb{Q}}$ generated by $\{\sqrt{x} : x \in \mathbb{Q}\}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\sqrt{\mathbb{Q}})$ ...
ByteBlitzer's user avatar
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Is the map $M\mapsto P$ smooth, where $P$ satisfies $PMP^{-1} = M^T$?

From this post (and other similar ones) we know, in particular, that $$ \forall M\in{\rm GL}(n,\Bbb R),\ \exists P\in{\rm GL}(n,\Bbb R)\ \text{s.t. } PMP^{-1} = M^T\ . \tag{$*$} $$ My question is ...
math-physicist's user avatar
2 votes
2 answers
110 views

Find a sequence $x_n \to \infty$ such that $(e^{i\lambda_1 x_n }, \ldots , e^{i\lambda_N x_n }) \to (1,\ldots, 1)$

Let $\lambda_1,\ldots, \lambda_N \in \mathbb R$ be distinct. Let $\mathbb T \subset \mathbb C$ be the unit circle and $\mathbb T^n$ be the $n$-dimensional torus. This is a group under componentwise ...
Daron's user avatar
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7 votes
1 answer
152 views

Are Hausdorff countably compact topological groups always normal?

A colleague and I have a result that shows that for Hausdorff countably compact W-spaces, being a topological group implies normality. But it occurred to us that (being not super experienced working ...
Steven Clontz's user avatar
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What is the topological interpretation of Euler's identity?

The picture is from the Wikipedia page of Euler's formula. I don't understand topology at all, but I'm curious about the reasoning behind this interpretation. May someone provide references that might ...
Tong Su's user avatar
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Algebraic Topology: Deformation Retraction and Quotient Spaces (using Mobius )

Could someone please help me with this question (and its solution)? Thanks!! Solution Part 1 (which shows that [0,1] x {1/2} deformation retract of [0,1] x [0,1]): I don't get how the homotopy gets ...
user1325970's user avatar
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0 answers
31 views

Proof by induction of $V(r) \subset V(s)$, where the Vs are neighboords of $e$ in a topological group.

Some context: I'm trying to fill the details of a proof concerning neighborhoods indexed by dyadic rational numbers. Suppose we have a sequence $U_n$ of symmetric neighborhoods of $e \in G$ such that $...
none's user avatar
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3 votes
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60 views

Is there a (unique) way to make a ring with a topology into a topological ring?

Let's say we have a ring with a topology that does is not compatible with the ring operations (so addition or multiplication are not continuous). Is there a natural way to extend the topology to make ...
James's user avatar
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3 votes
1 answer
76 views

Lie group structure on exotic $\mathbb{R}^4$

Are there Lie group structures on exotic $\mathbb{R}^4$s? By the theorem that every continuous group homomorphism of two Lie groups is smooth, we can conclude that if $G$ and $H$ are two Lie groups ...
Strichcoder's user avatar
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0 answers
45 views

If the groups $H$ and $\frac{G}{H}$ are $\omega$-narrow then $G$ is also $\omega$-narrow.

Let $H$ be a closed and normal subgroup of topological group $G$. If the groups $H$ and $\frac{G}{H}$ are $\omega$-narrow then $G$ is also $\omega$-narrow. Remember a topological group $G$ is $\omega$-...
Aaron's user avatar
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2 votes
1 answer
54 views

Group $U(1)$ as a principal bundle whose base is $U(1)$ and the fiber is ${\mathbb Z}_2$

Why is the principal bundle with $U(1)$ as its base and ${\mathbb Z}_2$ as a fiber still isomorphic to $U(1)\,$? Stated alternatively, why is it that $U(1) = U(1)/{\mathbb Z}_2\,$? I would greatly ...
Michael_1812's user avatar
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1 vote
0 answers
49 views

Extension topology

I am reading a paper by Goldman and Sah on extension topology, but I am uncertain about the meaning of the sentences. Paraphrasing the paper: Let $X$ be an abelian group and $M$ be a subgroup. ...
YSA's user avatar
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0 answers
19 views

If $V_n$ is a cozero set, then $V_n \cap H = f^{-1}(U_n)$

I'm trying to understand Katchenko's proof about the equivalence between $z$-embedded subgroups and $\mathbb{R}$-factorizable topological groups. Just to know: A subgroup $H$ of a $\mathbb{R}-$...
none's user avatar
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0 answers
64 views

Few Questions about Properties of Exponential Map $\text{exp}: \text{Lie}(G) \to G $ of Compact Complex Lie Group

Let $G$ be compact Riemann surface with the structure of a complex commutative Lie group, ie the multipliciation map $m:G \times G \to G$ is holomorphic (+certain usual diagrams satisfy axiomatic ...
user267839's user avatar
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4 votes
2 answers
112 views

Existence of a uniform measure on the group of permutation of $\mathbb N.$

If $G$ is the group of permutations of $\mathbb N,$ is there a non-trivial probability measure $\mu$ on $G$ such that, when $S\subset G$ is measurable, and $g\in G,$ then $gS=\{gs\mid s\in S\}$ is ...
Thomas Andrews's user avatar
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0 answers
35 views

Compact Riemann Surfactes with Group Structure

Let $X$ be compact Riemann surface with the structure of a complex Lie group, ie the multipliciation map $m:G \times G \to G$ is holomorphic (+certain usual diagrams satisfy axiomatic group law ...
user267839's user avatar
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0 votes
1 answer
24 views

Natural map from tensor product over a ring to the tensor product over an algebra over the ring.

Let $A_1$ be a commutative unital ring, $A_2$ an $A_1$ - algebra via $\varphi:A_1\to A_2$. Let $M$ be an $A_2$ - module. Let the $A_1$ - module structure on $M$ be defined via restriction of scalars i....
Academic's user avatar
  • 307
2 votes
1 answer
86 views

Complete proof for the shape of quasicharacters of $\mathbb{R}^{\times}, \mathbb{C}^{\times}$

Quasicharacters (:=continuous group homomorphism to $\mathbb{C}^{\times}$) of $\mathbb{R}^{\times}, \mathbb{C}^{\times}$ seems to be known to be following forms (the following is quoted from [Raghuram,...
user682141's user avatar
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0 answers
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If $G$ is a locally compact topological group, $H\triangleleft G$ with $H$ and $G/H$ compactly generated, then so is $G$.

I've been studying the next theorem stated in "Grupos Topológicos" by Tkachenko (1997): Let $G$ a locally compact group and $H\triangleleft G$. If $G/H$ and $H$ are compactly generated, then ...
Rolan Rial's user avatar
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0 answers
31 views

An exercise on Haar measures in Lang's Real and Functional Analysis book

I'm trying to understand the statement of the following exercise (Lang, Real and Functional Analysis, page 326): Identify $\mathbb{C}$ with $\mathbb{R}^2$. Let $\mu$ be the Lebesgue (Haar) measure on ...
Matheus Frota's user avatar
4 votes
0 answers
56 views

On the Rationalization of Finite Groups

Let $\Gamma$ be the countable set of finite groups identifying isomorphic ones (formal details below). Then, the direct product and the subgroup relation gives $(\Gamma,\times,\leq)$ the structure of ...
K. Makabre's user avatar
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2 votes
1 answer
41 views

Quotient of a compact Hausdorff and totally disconnected topological group is totally disconnected

If $G$ is a topological group which is compact Hausdorff and totally disconnected, and $H$ is a normal and closed subgroup of $G$, then Weibel's book on homological algebra claims that $G/H$ is a ...
user1008978's user avatar
1 vote
0 answers
36 views

Can all loop groups be realized as direct limits of finite dimensional Lie groups?

The loop group $LG$ is the group of smooth maps from the circle to a general compact Lie group $G$. My question is: can $LG$ always be realized as an appropriately defined "limit" of an ...
Zhengyan Shi's user avatar
6 votes
3 answers
584 views

Continuous addition and multiplication on Euclidean space (dimension > 2) making it into a field?

While TAing a linear algebra class, I happened upon the following question: Question: for $n\geq 3$, are there continuous operations $+, \cdot : \mathbb R^n \times \mathbb R^n \to \mathbb R^n$ that ...
D.R.'s user avatar
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3 votes
1 answer
54 views

Other form to prove $S^2=\{x \in \mathbb{R}^{3}: \lVert x \rVert=1\}$ is not a topological group

Prove $S^2= \{x \in \mathbb{R}^{3}: \lVert x \rVert =1 \}$ is not a topological group, I know that in this answer and lie group can solve it, but I need a form to use a result in differential topology:...
PSW's user avatar
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1 vote
0 answers
33 views

Let $A, B$ closed in a compact group. If $A$ or $B$ is $G_{\delta}$ then $AB$ is $G_{\delta}$

Let $G$ compact group, $A, B \subset G$ closed in G. Prove if $A$ or $B$ is $G_{\delta}$ then $AB$ is $G_{\delta}$. Since, $A$ and $B$ are closed in a compact group, both are compact then $AB$ is ...
PSW's user avatar
  • 307
2 votes
2 answers
136 views

Profinite groups with specific property

I saw that a topological group $G$ is profinite if and only if it is compact, Hausdorff and totally disconnected. In such groups, an open subgroup is closed but not vice-versa. From this, I came to a ...
Maths Rahul's user avatar
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0 answers
17 views

Topological tensor product

Let $A$ and $B$ be topological abelian groups. Consider the following universal property for a pair $(P, \pi)$, where $P$ is a topological abelian group and $\pi: A \times B \to P$ is a jointly ...
Smiley1000's user avatar
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0 votes
1 answer
36 views

Why is shift operator group action of cyclic group $\mathbb{Z}_p$ free depending whether or not $p$ is prime?

While reading one scientific article, I stumbled upon the following statement without explanation: Let the cyclic group $\mathbb{Z}_p$ act on topological space $\left(\prod_{k=1}^p{\mathbb{R}^n}\right)...
toxic's user avatar
  • 329
2 votes
1 answer
89 views

$C(X, A) \otimes C(X, B) \cong C(X, A \otimes B)$?

Consider the category $\mathsf{Ab}$ of abelian groups and $\mathsf{TopAb}$ of topological abelian groups. Given a topological space $X$ and a topological abelian group $A$, define the abelian group $C(...
Smiley1000's user avatar
  • 1,649
3 votes
1 answer
48 views

Every TVS is $T_{3.5}$ (Tychonoff) even if it is not $T_0$

I'm studying the first properties of Topological Vector space, and I'm confused about the separation properties. Is every TVS $T_{3.5}$ even if it is not $T_0$? This is confirmed by this wikipedia ...
marc's user avatar
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0 votes
1 answer
21 views

A question about continuous functions with compact support in locally compact topological groups.

I'm having difficulties proving the following statement: If $G$ is a locally compact Hausdorff topological group and $C_c(G)$ is the space of continuous real functions on $G$ with compact support, ...
Matheus Frota's user avatar

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