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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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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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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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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24 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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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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16 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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38 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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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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54 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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47 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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39 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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66 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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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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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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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: ...
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14 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 ...
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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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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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32 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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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 $C[a,b]$ ...
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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 ...
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1answer
23 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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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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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
26 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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73 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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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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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 ...
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40 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 ...
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83 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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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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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 ...
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1answer
118 views

Is this a different proof of the fundamental group being abelian?

I have proved the fundamental group of a topological group is abelian. But I've found nowhere the similar proof as mine. Everywhere I looked up, it was done either exploiting categorical properties or ...
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Sectional category (Schwarz genus) of the Milnor join construction

Assume topological spaces to be normal and paracompact. Following the article: "The genus of a fiber space" by A. Schwarz, we call the sectional category (or Schwarz genus) of a locally trivial fiber ...
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$R^2-\{x_1,x_2,\dots,x_n\}$ does not have the structure of topological group

Let $n>1$. I need to show that the space $X=\mathbb{R}^2-\{x_1,x_2,\dots,x_n\}$ does not have the structure of topological group. This is an exercise about the Van Kampen theorem. Certainly, we ...
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Quotients by simply connected closed subgroups

I have come across an exercise asking for a proof of something that is definitely false: If $G$ is a Lie group, $H$ a connected closed subgroup and $G/H$ simply connected, then $G$ is itself simply ...
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Left shift continuity

Let $G$ be a topological group. It says that left shift $L_g: G \rightarrow G$ ($ x\mapsto g\cdot x$) is continious map, but i am confused about proving it. As I expect, it somehow follows from the ...
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Haar Measure of a Topological Ring

A topological ring is a (not necessarily unital) ring $(R,+,\cdot)$ equipped with a topology $\mathcal{T}$ such that, with respect to $\mathcal{T}$, both $(R,+)$ is a topological group and ...
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14 views

Example for converges series in the metric space

Give example for converges series in the metric space: $$ \quad\quad\quad(\mathrm {R}^n,d_{\infty}),d_\infty=\max\mid x_i-y_i\mid$$ My attempt: Let ...
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subgroup of an $FC^−$ nilpotent group is nilpotent group?

Let $G$ be a compactly generated $\overline{FC}$-nilpotent group. For the definition of $\overline{FC}$-nilpotent group, we refer to http://dx.doi.org/10.1017/S0024610701001983 . G is a compactly ...
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51 views

Unique way to show $S^n$, $n \geq 2$ is simply connected.

This questions is asked in Armstrong's Topology book, and I am totally stuck.... I could really use a major hint: Think of $S^n \subset \mathbb{E}^{n+1}$. Given a loop $\alpha \in \pi_1 (S^n , ...
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Is this statement true? (characterize elements of dual group)

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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47 views

Example of building a classifying space

I'm reading some things about algebraic topology, and they mention the classifying space of a group $G$ as $BG$, but they doesn't build one, so I want to ask if someone knows where can I find the way ...
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44 views

Books and sources concerning $G$-spaces

A $G$-space is (generally) a topological space $X$ equipped with a continuous action by a topological group $G$. I mean generally because, I've never studied before $G$-spaces and after I read a ...
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Is there a non-trivial character on any locally compact Abelian group?

Let $G$ be a locally compact Abelian group. Is there a non-trivial character on $G$? indeed, i want the proof of the existence of a non-trivial character on a locally compact Abelian group.