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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What is the algebraic structure of $\Bbb Q_p/\Bbb Z_p$? [on hold]

I am curious about the algebraic structure of $\Bbb Q_p/\Bbb Z_p$. Is there any result in this direction? Thanks!
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5 views

Haar measure, can image of modular function be any subgroup of $(0,\infty)$?

It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...
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1answer
24 views

Corollary of the Birkhoff Kakutani Theorem: first countable topological vector spaces

http://planetmath.org/birkhoffkakutanitheorem A topological group $(G,*,e)$ is metrizable if and only if $G$ is Hausdorff and the identity $e$ of $G$ has a countable neighborhood basis. In ...
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51 views

Semi-direct product of groups

The situation is the following: Let be $G$ a locally compact (Hausdorff) group such that $G = H \rtimes_{\alpha} N$ is the semi direct product of locally compact groups $N$ and $H$. Let $A$ be a C$^*$-...
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48 views

Extending the topology on a set to the group it generates

The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form $2^{n_2} \cdot 3^{n_3} \cdot 5^{n_5} \cdot ...$ The tuple ...
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25 views

A quotient by a discrete normal subgroup is locally isomorphic to the group itself

Let $G$ be a connected topological group and let $\Gamma$ be a discrete normal subgroup of $G$. Then why $G$ and $G/\Gamma$ are locally isomorphic?
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1answer
20 views

Definition of $\hat{G_1}\times \hat{G_2}$ where $G_1,G_2$ are abelian groups and $\hat{G}$ is the dual of $G$

The question is in the title. I want to know what happens to $(\chi_G,\chi_H)\in\hat{G}\times \hat{H}$. Are they just passively sitting there as a pair or they give something when applied on $(g,h)$? ...
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27 views

Existence of open subgroup extending a smaller one

Let $G$ be an abelian topological group and $H \subseteq G$ a dense subgroup (equipped with the subset topology). Furthermore let $V \subseteq H$ be a subgroup that is open in $H$. Does there exist a ...
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11 views

Fundamental group of a topological group [duplicate]

Given $(G,\cdot)$ a topological group with identity $e$, is it always true that $\pi_1(G,e)$ is abelian? For what it's worth: I have already shown that if $f,g\in\Omega(X,e)$, then $[f \times g]=[f\...
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1answer
42 views

A problem of nets in topology

Let $G$ be a topological group with neutral element $e$. Let $\pi \colon G \to B(E)$ a (non-continuous) representation of $G$ on a Banach space $E$ by bounded linear operators. Let $T$ an element of ...
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Halmos Measure Thoery section 62 exercise 3

Is there a locally compact group $G$ and a Borel measure $\mu$ on $G$ such that \begin{equation*} H=\{g\in G\mid \mu(gE)=\mu(E) \: \text{for all measurable} \: E\} \end{equation*} is not a closed ...
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16 views

connected or compact Dirichlet domain

Let $G$ be a second-countable locally compact group and $d$ be a proper (i.e. bounded closed sets are compact) left invariant metric on $G$. Let $\Gamma$ be a lattice subgroup of $G$. Consider the ...
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1answer
30 views

Examples of nilpotent connected locally compact groups which are not Lie groups

I am looking for examples of nilpotent connected (or at least almost connected) locally compact groups which are not Lie groups. Do you know of such examples ?
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89 views

Order topology on $\Bbb Q^+$ as a Z-module

Consider the multiplicative group of strictly positive rationals, denoted $\Bbb Q^+$. This can be viewed as a $\Bbb Z$-module, with the primes serving as a basis. If we place the order topology on $\...
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1answer
28 views

canonical quotient map on lie group is proper?

Let $G$ be Lie group and $K \subset G$ a compact Lie subgroup of $G$. Let $\pi \colon G \to G/K , \quad g \mapsto g.K=[g]$ denote the canonical projection on the quotient and endow $G/K$ with the ...
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6 views

On volume of arithmetic subgroups

I deal a lot with volumes of arithmetic subgroups, mainly in $SL_2(\mathbf{Z)}$. But I remain not at ease with them, making rough explicit calculations case by case instead of having a general method. ...
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3answers
56 views

Show that the sphere, S, and $\mathbb{R}^2$ is not homeomorphic

I am trying to show that the sphere $S^2$ and $\mathbb{R}^2$ are not homeomorphic.I understand that you can't 'compress' a 3D shape into a 2D plane but I don't know how I would express this formally. ...
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1answer
29 views

A continuous action of a compact group on a uniform space is equicontinuous?

I am stuck on how to prove the statement "A continuous action of a compact group $G$ on a uniform space $X$ is equicontinuous." So, essentially we want to show that for every entourage $\alpha$ of $...
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1answer
34 views

Conjugacy classes in topological groups are closed?

EDIT Just realized that this question Conjugacy classes of a compact matrix group is related, but I think that the answer use specific properties of matrix groups, so it doesn't apply. QUESTION ...
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1answer
18 views

Example of non-amenable group which is the inverse limit of amenable groups

1) Does there exist a non-amenable locally compact group $G$ which is the inverse limit $\varprojlim G_i$ of amenable groups $G_i$? 2) Does there exist a non-amenable locally compact group $G$ which ...
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1answer
43 views

Is a translation in a compact Lie group homotopic to the identity?

The following exercise is from Guillemin and Pollack, Differential Topology. Show that the Euler characteristic of the orthogonal group (or any compact Lie group for that matter) is zero. Hint: ...
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38 views

Is the product of closed subgroups in topological group closed?

Just out of curiosity: If $G$ is a topological group and $H, K$ are closed subgroups, is $H\cdot K$ a closed subgroup? Thanks!
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15 views

Showing some transformation is group isomorphism (topological group).

Let's Mg be a set of all real valued functions defined on topological group G. Assume that $f:G \to R$. Let's $a \in G$, then define $f_a(x):=f(ax)$ for all $x \in G$. Now define $h_a(f):=f_a$ we know ...
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35 views

A dense set in $\mathbb{T}$

Let $$\mathbb{T}=\{ z \in \mathbb{C}: |z|=1\} $$ Consider $\mathbb{T}$ as a topological group under multiplication with it's usual topology, I'm reading a proof wich states that a dense set in $T$ ...
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57 views

Isomorphism theorems for topological groups

I know that the second isomorphism theorem for groups doesn't hold for topological groups, the version that I have for the second isomorphism theorem is: If $G$ is a group, $H$ a subgroup of $G$ and ...
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50 views

Show that $S^1$ acts on $S^3$

$S^3=\{(z_1, z_2) \in \mathbb{C^2} \mid |z_1|^2 + |z_2|^2 = 1 \}$ Show that $S^1$ acts on $S^3$ by $z \cdot (z_1, z_2)=(zz_1, zz_2)$ An action of a topological group $G$ on a topological ...
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41 views

identity for quaternions' group Sp(n)=Sp(2n,C)∩U(2n)

Could you help me for solving this: Let $Sp(n)$ be the group of linear transformations of $H^n$ such that preserve hermitian form $$\sum_{i=1}^n \overline{q_i}r_i,$$ that $H$ is the quaternions _ the ...
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2answers
70 views

Fundamental groups of path connected subspaces

Does every path connected subspace of $\mathbb{R}^2$ have a fundamental group the trivial or an infinity group? For example, for convex subspaces we know that, but if we take only path connected ...
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20 views

Topology of bundle maps in Atiyah-Singer IV

I'm trying to read "The index of elliptic operators IV" by Atiyah and Singer, and I do not understand why the topology on $\mathrm{Diff}(X,E)$ on page 123 is constructed in such a peculiar way. Is ...
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1answer
46 views

Sequences $(U_n)$ of neighborhoods of $0$ in a LCA group with $m(U_n)\to 0$

Let $G$ denote an infinite compact abelian group with Haar measure $m$ (so $m(G)=1$). Given a neighborhood $U_1$ of the unit $0$ in $G$ we can find a symmetric neighborhood $U_2$ of $0$ such that $...
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2answers
23 views

Euclidean Sphere

Consider the Euclidean sphere $S^n = \{x\in \mathbb{R}^{n+1}: ||x||_2=1\}$ of dimension $n\ge1$. Show that for every continuous function $f:S^n\longrightarrow \mathbb{R}$, there exists $x\in S^n$ such ...
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1answer
8 views

If $f(gK) \subset U$ find a compact neighborhood of $g$, $\overline{Q}$, such that $f(\overline{Q} K) \subset U$

sorry for my bad english. I am in a proof and I get stuck in the following step. Let $f : X \to Y$ be a continuous function between two topological spaces, $G$ a locally compact topological group, ...
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1answer
28 views

$T_0$ is equivalent to $T_1$ in a topological group

I need to prove that the separation axiom $T_0$ is equivalent to $T_1$ in a topological group $G$. $T_1$ implies $T_0$. So I only need to prove $T_0$ inplies $T_1$. What I have tried is: Suppose $...
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1answer
19 views

Lie group structures inside of Clifford algebras

I am reading a text by Jean Gallier on Clifford algebras, Pin and Spin groups. I have a problem with one little innocent-looking paragraph establishing Pin and Spin as Lie groups on the page 37. I don'...
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25 views

Closed subgroup implies open?

If $H$ is a closed subgroup of a topological group, $H$ is also open?, I know that an open subgroup of a topological group is also closed, but the converse is true? if isn't, wich could be a ...
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26 views

Show that $ \operatorname{Sp}(n)=\{A \in M_n(\mathbb{H}) \mid AA^*=I=A^*A\} $ is a compact group

Let $M_n(\mathbb{H})$ be the set of all $n \times n$ matrices with entries in the quaternions $\mathbb{H}$. For $A=(a_{ij} ) $ let $ A^*=(a^*_{ij} ) $ be the matrix with $a^*_{ij}=\bar{a}_{ij} $, ...
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1answer
34 views

Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Let $G$ ge a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogenuous space $\frac{G}{H}$ are connected, then $G$ is connected. Remark: For this proof I use the following ...
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3answers
29 views

Proving that $B:=\{f(x)\in C[a,b]:f(a)=0\}$ is close set

Let $B$ be a group of all the continuous functions in the interval $[a,b]$ such that $f(a)=0$. Prove that $A$ is close group in the metric space $C[a,b]$ My attempt: Metric space $C[a,b]$ defined ...
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0answers
15 views

Binary operations being continuous under a topology?

For a set $S$ and some function $f: S \rightarrow S$ and $a\in S$, $f$ is continuous at $a$ under a topology $N$ if for all neighborhoods $N_1(f(a))$ there exists a neighborhood $N_2(a)$ such that $f(...
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1answer
24 views

Proving that Lie groups are locally connected

I'm trying to show that if $G$ is a Lie group, then it is locally connected, i.e., for any point $p$ in an open subset $U$ of $G$, there is a connected neighborhood $V$ of $p$ such that $V\subset U$. ...
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31 views

Metrizability of space of unitary operators on Hilbert space

Let $\mathbb{H}$ be a (complex) Hilbert space. Define $\mathbb{P}$ as a projective space over $\mathbb{H}$, i.e. $\mathbb{P}=\left(\mathbb{H}\setminus\{0\}\right)/\sim$, where $f\sim g$ iff there ...
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26 views

Show that $\|\cdot\|_Q$ is a seminorm on $l_{\infty}$

On the space $l^{\infty}$, define $||.||_Q=\limsup |x_n|,$ $x=(x_n)$ belongs to $l^{\infty}.$ Show that $||.||_Q$ is a seminorm on $l^{\infty}$ and that its null space is precisely $c_0$, where $c_0$ ...
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1answer
33 views

Homeomorphism of $SO(3)$?

I am trying to get a better understanding of the homeomorphism of $SO(3)$ to the Real Projective Plane, so that ultimately I can show that $\pi_1(SO(3)) = \mathbb{Z}_2$. From wikipedia and many other ...
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1answer
75 views

When is $BG$ a topological group?

Let $G$ be a topological group, then it has a classifying space $BG$. When is $BG$ a topological group? My motivation for asking this question is that I was thinking about the $B$-analogue of ...
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13 views

A question about cosets in Raikov complete topological groups

Let $G$ be a topological group and let $G^*$ be its Raikov completion, i.e its completion for its two-sided uniformity. Call $G$ Raikov complete if it is isomorphic to its Raikov completion. Now ...
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1answer
16 views

Give example of congregate serieses in the metric space : $(R^n,d_1),d_1=\sum_{i=1}^{n}\mid x_i-y_i \mid$

Give example of congregate serieses in the metric space : $$(R^n,d_1),d_1=\sum_{i=1}^{n}\mid x_i-y_i \mid$$ What I tried: I think I should find $\{X_n\}\to x$ $\left(\frac{\sin n}{n},\left(1+1/n\...
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41 views

The action of a topological group on the function space is continuous?

Sorry for my bad english. Let $X$ and $Y$ be two topological spaces, and $G$ a topological group, let $\theta : G \times X \to X$ be a continuous action of $G$ on $X$. We defined the action of $G$ on ...
3
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1answer
96 views

Book suggestion: An introduction for topology

I am a graduated student in physics, and interested on topological properties of the matter, as it is a really hot topic. In physics, the topological properties arise from symmetries in the models. ...
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1answer
12 views

Show that every nearly paracompact space is almost paracompact but the converse is not true

I am learning about the nearly para-compact space and almost para-compact space. I know that every nearly para-compact space is almost para-compact space but the converse is not true in general. So i ...
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1answer
21 views

Show that every paracompact space is nearly paracompact but the converse is not true

I am learning about the para-compact space and nearly para-compact space. I know that every nearly para-compact space is para-compact space but the converse is not true in general. So i need an ...