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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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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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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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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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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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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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 $C[a,b]$ ...
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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
21 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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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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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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1answer
23 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
69 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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1answer
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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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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
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 ...
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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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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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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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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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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
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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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 $\mathbb{K}^+$ ...
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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
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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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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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
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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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. ...
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29 views

Profinite Group, Compact Hausdorff Totally disconnected topological group

If G is a compact Hausdorff totally disconnected topological group, and I want to show that G is profinite, I am interested in surjectivity for now. Let $\hat{G}$ be the profinite completion of $G$, ...
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3answers
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Continuous maps between open/closed topological subspaces give a continuous map between topological spaces

Let $A,B$ be topological spaces and $\phi:A\rightarrow B$ a map. Let $A',A''\subseteq A$ with $A=A'\cup A''$ and the maps $\phi|_{A'}:A'\rightarrow B$ en $\phi|_{A''}:A''\rightarrow B$ are continuous. ...
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1answer
29 views

Module over a commutative ring with a topology

Let $M$ be an $R$-module ($R$ commutative ring with unity). Let $M=M_0 \supseteq M_1\supseteq M_2\supseteq\cdots$ be a chain of submodules. The topology in $M$: The open sets in $M$ are arbitrary ...
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Is the Fourier transform a special case of this version of the Yoneda lemma?

The (co)Yoneda lemma tells us that for a presheaf $F\in \hat{\mathbb{C}}$, the following formula holds: $$ F=\int^{c\in\mathbb{C}} Fc\ \times\ h_{c}\ , $$ where $h_c$ is a representable presheaf for ...
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1answer
33 views

Probability of generating group

Let $G$ a finitely generated pro-$p$-group, with $p$ a prime; if $\mu$ is the unique Haar measure on $G$ with $\mu(G)=1$, we can define $P_{G}(k)=\displaystyle \mu\left(G^{k}\setminus \bigcup_{M} ...
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26 views

Free group on 3 generators is a subgroup of free group on 2 generators [duplicate]

I think this true but can it be proven in an explicit way? A group, G, is called Free if there is a subset S of G such that any element of G can be written in one and only one way as a product of ...
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1answer
30 views

Open Profinite Subgroup $\Rightarrow$ Profinite Group

I'm currently struggling with profinite groups. I'm not able to solve many practice exercises. I was wondering if someone could give me some hints on the following: if $G$ is a compact group and $H$ ...
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1answer
43 views

How does this argument show continuity?

I want to show that the unitary group $U(\mathcal H)$ of a Hilbertspace $\mathcal H$ is a topological group wrt the strong operator topology. For the standard proof it is most convenient to use that ...
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1answer
21 views

If $G$ is a discrete topological group, is $(G,X,\theta)$ a $G$-space?

If $G$ is a topological group, $X$ is a topological space and $\theta:G\times X\to X$ is an action, then $X$ is called a $G$-space provided $\theta$ is continuous. This is something I just read in a ...
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1answer
44 views

Topological Groups and Covering Spaces

So the question is suppose $G$ is a topological group and $H$ is a closed, discrete subgroup of $G$, we have to show that the quotient map $p: G\to \frac GH$ is a covering projection. The way I'm ...