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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identity for quaternions' group

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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1answer
212 views

Fuchsian groups and topological isomorphism

I have a (finite) presentation of a group and I am wanting to prove that it is not Fuchsian. Because it is given by a presentation, a neat, algebraic description of Fuschian groups would be nice. This ...
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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 $C[a,b]$ ...
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2answers
64 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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1answer
105 views

General Linear Group over the quaternions is a topological group

How to show that General Linear Group over the quaternions is a a topological group?
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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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41 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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15 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: ...
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92 views

A $T_0$ topological groups is $T_{3.5}$ (and consequently $T_3$)

I don't understand: why is this so? I've just seen the proof that a $T_0$ topological group is $T_1$, but don't know how to show that it's $T_{3.5}$. BTW, the fact that $x\overline{V}=\overline{xV}$, ...
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1answer
12 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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0answers
24 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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22 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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2answers
454 views

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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1answer
28 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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0answers
14 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
20 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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23 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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0answers
27 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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1answer
22 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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How can we find and categorize the subgroups of R?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}$ What are all the subgroups of R = $(\R, +)$ and how can we categorize them? I started thinking about this question last night ...
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1answer
68 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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1answer
28 views

Polish groups having finite covering dimension

The paradigm examples of Polish groups that have finite covering dimension are obviously $\mathbb{R}^n$ for any finite $n$. They are also locally compact. Is it possible to construct a sequence $G_n$ ...
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11 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
15 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 ...
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0answers
38 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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1answer
75 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
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
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
111 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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21 views

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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1answer
66 views

$R^2-\{x_1,x_2,\dots,x_n\}$ 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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0answers
23 views

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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5answers
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Closure of a certain subset in a compact topological group

Suppose that $G$ is a compact Hausdorff topological group and that $g\in G$. Consider the set $A=\{g^n : n=0,1,2,\ldots\}$ and let $\bar{A}$ denote the closure of $A$ in $G$. Is it true that ...
4
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0answers
38 views

Proving a version of the Kronecker's Theorem

I am trying to prove the following version of the Kronecker's Theorem: Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then ...
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2answers
22 views

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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929 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
3
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2answers
171 views

Non-trivial group homomorphism from an infinite group to a finite group

Let $G$ be a topological group the underlying set of which is infinite (e.g., $(\mathbb{R}\,;+)$ or $(\mathbb{Z}\,;+)$), and let $H$ be a topological group the underlying set of which is finite (e.g., ...
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1answer
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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16 views

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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1answer
45 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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1answer
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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31 views

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 $\mathbb{K}^+$ ...
5
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1answer
43 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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2answers
49 views

Are all subgroups of a LERF group closed?

A group $G$ is LERF if every finitely generated subgroup of G is closed in the profinite topology of $G$. Let $G$ be LERF group, and let $H$ be a subgroup of $G$. Is $H$ necessarily closed? (I'm not ...
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1answer
84 views

Which definition of Haar measure is correct?

I have encountered several different definitions of left Haar measure that don't all seem to agree. The setting I care about is Locally Compact Groups. The first seems to completely disagree with ...
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29 views

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.
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140 views

Canonical isomorphism between Cauchy sequence completion and inverse limit

I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can ...
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15 views

Infinite Galois theory: every subgroup of finite index is open. Proof-check

Let $\Omega/k$ be a (possibly infinite) Galois extension and let the group $G=G(\Omega/k)$ be equipped with the Krull Topology. My question is about a statement I think its true, but I'm not sure. ...