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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Books about braid theory

I'm looking for books that talk about braid theory, in the sense of braid groups mostly, and not too advanced, if possible. With material understandable for an undergraduate. Thanks for any ...
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Any finite index subgroup of $\mathbb Z_p$ is open [duplicate]

I'm trying to show that every finite index subgroup $H$ of $\mathbb Z_p$ is open. Since $H$ has finite index, it is equivalent (and perhaps easier) to show that it is closed. But I've tried showing ...
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Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
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+100

Theoretical Basis for Eigenvalue transformation on Bessel's Equation

The method I've been taught for finding all of the eigenvalue solutions to Bessel's operator $$b(f)\equiv f''(x)+\frac{1}{x}f'(x)$$ goes as follows. Let $g(a)=f(\sqrt{\lambda}x)$. Then $$b(g)=\lambda ...
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1answer
37 views

Haar measure on locally compact group

Please I need a help to solve two problems in the book of principles of Harmonic analysis of Deitmar and Echterhoff Exercise 1.4 Let $G$ be a locally compact group with Haar measure $\mu$, and let ...
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a interesting question from topological group

$G$ is a locally compact Hausdorff topological group, $m$ is a (left) Haar measure on $X$, $A$ and $B$ are two finite positive measure in $G$, that is $m(A)>0$, $m(B)>0$. My question is: Can ...
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1answer
17 views

Why is $Gal(\Omega/k)$ a topological group under the Krull topology?

For an infinite Galois extension $\Omega/k$ the Krull topology on $G:=Gal(\Omega/k)$ is defined by taking as a basis for the neighbourhood of an element $\sigma \in G$ all cosets of the form $\sigma ...
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1answer
26 views

Relation between open divisible subgroup and the quotient of the group with subgroup

I wanted to prove the following proposition: Let H be an open divisible subgroup of an abelian topological group G. Then G is topologically isomorphic to H x G/H. As for the proof, using extension of ...
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Subgroup of Euclid space either dense or discrete? [closed]

Subgroups of Euclid space $(R^n,+)$ either dense or discrete? In other words, how to find all those subgroups in $R^n$? Or maybe if G is a connected group, H is a subgroup of G, then H is dense or ...
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1answer
41 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
27 views

compact inverse is compact in canonical homomorphism

Let $G$ be locally compact Hausdorff group. Let $N$ be a closed normal subgroup of $G$. Let $f:G\to G/N$ be the canonical homomorphism. I want to show that for every compact subset $C$ of $G/N$, there ...
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42 views

An interesting question about topological group

A subset of a topological space $X$ is called nowhere dense in $X$ if the interior of its closure is empty. A subset of a topological space $X$ is called the first category (or meagre) in $X$ if it ...
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1answer
21 views

Closed subgroups of $\mathbb Q_p$

Let $\mathbb Q_p$ be the field of $p$-adic numbers, $\mathbb Z_p$ the ring of $p$-adic integers. Is there a closed subgroup of $\mathbb Q_p$ other than the following list? 1) 0 2) ...
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21 views

Subgroup $H \leq G$ acting on $G$ by translation is transitive?

In Elementary Topology. Textbook in Problems, by Viro, et al they state the following: Let $G$ be a topological group, $H \leq G$ a subgroup. Then $G$ is a homogeneous $H$-space under the ...
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34 views

Closed subgroup of a locally compact Hausdorff group whose Haar measure is non-zero.

Let $G$ be a locally compact Hausdorff group, $H$ its closed subgroup. To avoid pathologies, we assume the underlying topological space of $G$ has a countable base. Let $\mu$ be a Haar measure on $G$. ...
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1answer
21 views

Free action on space implies that each point has a neighborhood that has an empty intersection with translations

Suppose $G$ is a topological group, $X$ a topological space and $G \times X \rightarrow X$ group action that is continuous. Further, suppose that the action is free ($G_x = \{e\}$, for all $x$). What ...
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1answer
27 views

About connected topological subgroup

I'm trying to understand a proof of a theorem but I didn't understand a point. Let $G$ be an locally compact abelian group. Denote $G_0$ the connected component of $0$ (the identity of $G$). It's an ...
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1answer
26 views

Restriction of topological ring isomorphism

If $\theta: R\to S$ is an isomorphism of topological rings then do we obtain a topological group isomorphism $\theta|_{R^{\times}}:R^{\times}\to S^{\times}$ by restricting to their groups of units? ...
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Weil's definiton of image of Haar measure on homogeneous space $G/\Gamma$ where $\Gamma$ is discrete

Let $G$ be a locally compact Hausdorff group. To simplify matters we assume the underlying topological space of $G$ has a countable base. Let $\Gamma$ be a discrete subgroup of $G$, $G/\Gamma$ the ...
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22 views

Explicit construction of Haar measure on quotient group

Let $G$ be a locally compact Hausdorff group, $H$ a closed normal subgroup. To simplify matters we assume that the underlying topological space of $G$ has a countable base. Suppose a left Harr ...
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1answer
28 views

Question on definition of group acting on a topological space.

I know that a group can act on a graph by acting on the set of vertices on a graph. I also know that a graph can be viewed as a CW complex and therefore a topological space and i am trying to bridge ...
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2answers
92 views

Is division by two continuous in topological groups?

Assume that $(G, +)$ is a Hausdorff topological abelian group which is uniquely divisible by two, i.e. the function $x \to 2x = x+ x$ is a bijection. Clearly, it is also continuous. My question is if ...
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Yet another proof of uniquness of Haar measure

I'm trying to prove the uniqueness of Haar measure in my way. Let $G$ be a locally compact Hausdorff group. To simplify matters we assume the underlying topological space of $G$ has a countable base. ...
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1answer
25 views

Is a discrete subgroup of a Hausdorff group closed?

Let $G$ be a Hausdorff topological group. Let $H$ be a subgroup of $G$ such that $H$ is a discrete subspace of $G$. Is $H$ a closed subgroup of $G$? I thought this is obviously true, but I ...
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33 views

Is the projection onto a quotient by a compact normal subgroup proper?

My ultimate goal is to prove the following: If $G$ is a locally compact group and $K_\alpha$ a net of compact normal subgroups with trivial intersection, then the inverse limit $\projlim G/K_\alpha$ ...
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Sufficient to check intersection of sub base elements with a dense set in compact open topology

I am reading a book about duality and there was this following claim: If I have a compact group G* (dual group of G, and G is discrete) with the compact open topology, then for any A, a subset of G* ...
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2answers
34 views

a question about functional analysis conclusion,and I am not sure whether it is true or not?

we have $R^n$,$R^m$ spaces, suppose open set $O_{1}\subset R^n $ and $O_{2}\subset R^m$, $f:O_{1}->O_{2} $ is k-times differentiable$(1<=k<=\infty)$,then at $x_{0}\in O_{1}$,$rank(f)(x_{0})$ ...
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1answer
45 views

$\mathbb{R}^2$ as a quotient of a group

I am looking for a locally compact group $G$ with a closed subgroup $H$ such that $G/H$ is homeomorphic with $\mathbb{R}^2$ but $G$ does decompose into a semidirect and/or direct product which ...
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1answer
25 views

Intuition for theorem about compact subsets of topological groups

Let $G$ be a topological group, let $K$ be a compact subset of $G$, and let $U$ be an open subset of $G$ such that $K \subset U$. Then, there is an open set $V$ containing the identity such that $KV ...
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71 views

Orbits of properly discontinuous actions

Definition Let $G$ be a group and $X$ a topological space. Let $G\curvearrowright X$ by homeomorphisms. We call the action properly discontinuous if for all $x\in X$ there exists an open neighborhood ...
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12.16 in Lee's Introduction To Topological Manifolds

Reading through Lee's Introduction To Topological Manifolds. Theorem 12.16 says the following: Suppose G and H are connected, locally path-connected topological groups, and $\phi:G \to H$ is a ...
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180 views

What are the properties of the topological group $e^{i\mathbb{Q}}$?

What are its properties as a topological group? It is not $\mathbb{Q}/\mathbb{Z}$ but resembles it, it contains the subgroups $e^{i\mathbb{Z}}\supseteq e^{ip\mathbb{Z}}\supseteq ...
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132 views

approximate vanishing in Pontryagin dual

Let $\{n_k\}\subseteq \mathbb{Z}$ to be any given sequence of integers, and suppose it satisfies the following property: (*) For any $\lambda\in A\subseteq \mathbb{T}$(the unit circle), ...
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kth root of an open set in the circle toplogical group

My intuition tells me that in the topological group of the circle, if I take an open set U, then its kth root (where k is some natural number) in the circle is also an open set. In order to show it I ...
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1answer
44 views

Definition of topological group acting on a topological space

The definition of a topological group $G$ acting on a topological space $X$ is there exists a continuous map from $G\times X \rightarrow X$ such that $e_G.x=x$ for all $x\in X$ and ...
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1answer
81 views

If $f:\mathbb R \to \mathbb R$ is an additive function whose graph is $G_{\delta}$ in $\mathbb R^2$ , then the graph is closed in $\mathbb R^2$?

If $f:\mathbb R \to \mathbb R$ is an additive function i.e. $f(x+y)=f(x)+f(y) ,\forall x,y \in \mathbb R $ such that $G(f):\{(x,f(x)) : x\in \mathbb R\}$ is a countable intersection of open sets , ...
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1answer
28 views

Intersection between a compact and a locally compact set

I'm trying to understand Rudin's proof of Pontryagin duality theorem, but I still haven't undersood an argument. (Fourier analysis on groups, p29) Let $G$ be a group and denote $\Gamma =\widehat{G}$ ...
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base of open neighborhood for dual group in k-topology

I wanted to ask the following: Suppose I have an abelian topological $G$, and $G^*$ is its dual group (all the continuous homomorphisms from $G$ to the circle group $T$). How can I show that the ...
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There are infinitely many continuous characters on an infinite abelian topological group

Let $(G,\mathcal T)$ be an infinite abelian group and $\Bbb T$ be the circle group. Why there are infinitely many continuous homorphisms $f:G\to \Bbb T$? Is there a simple proof without using ...
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Stabilizer, Cosets, homeomorphism and Compact groups : proving things in The Structure of Compact Groups by Hofmann and Morris

I'm currently struggling trying to prove a few things in the book The Structure of Compact Groups by Hofmann and Morris. The first one would be Proposition 1.10.i (or E1.4) : If the topological ...
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A separable locally compact group that is not metrizable and not compact

Hausdorffness assumed. All the usual suspects don't work: $\mathbb{Z}^\mathbb{R}$, $2^\mathbb{R}$, discrete $\mathbb{R}$, etc. My reasoning so far: if it is locally compact, then there are separable ...
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Compact subset of locally compact $\sigma$-compact

Let $X$ be a non-compact, locally compact space. Also suppose that there is a sequence of compact non-empty sets $\{K_n\}_{n\in N}$ such that $$X=\bigcup_{n\in N} K_n,\quad K_n\subset K_{n+1}.$$ Now ...
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How to define multiplication in covering group?

Let $G$ be a connected topological group and let $p:\tilde{G}\to G$ be a universal covering of $G$. Then $\tilde{G}$ is also a topological group and $p$ is a continuous homomorphism. My question is: ...
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Dimension of a subgroup of a solenoid with measure zero

Let $G$ be a connected compact finite-dimensional abelian group (also called a solenoid). If $H$ is a subgroup of $G$ with Haar measure $0$, can we say something about the connectedness or the ...
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1answer
25 views

support of function in topological group

Let $G$ be a compact Hausdorff topological group. Let $U$ be a neighbourhood of the identity $e$, and let $V = U \cap U^{-1}$ where $U^{-1} = \{x^{-1} : x \in U\}$. Apparently there always exists a ...
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Dual group endowed with the compact-open topology

I wanted to ask a question. Let $G^*$ be the dual group of an abelian topological group $G$ ($G^*$ is defined to be the group of all continuous homomorphisms from $G$ to the circle group $T$). I ...
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1answer
24 views

If $X^n$ is Lindelöf, then so is $\widetilde{X}^n$.

Let $G$ be a topological group $T_2$, and $X$ a closed subspace of $G$. We suppose that, for each $n\in\mathbb{N}$, $X^n$ is Lindelöf. Consider $\widetilde{X}:=X\oplus\{e\}\oplus X^{-1}$ (we will ...
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closure of dual group in pointwise convergence.

I have something which seems kind of trivial but I can't seem to prove it. Let G be an abelian topological group and let T be the circle group. Denote by G* the group of all continuous homomorphisms ...
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3answers
54 views

How to show that the intersection of all neighbourhood of $0$ in a topological group is a subgroup?

Let $H$ be the intersection of all neighbourhood of $0$ in a topological group $G$. How to show that $H$ is a subgroup? I tried to use the continuity of multiplication and inverse. But not ...
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1answer
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Closure of an abelian subgroup of a topological group is abelian

I have proved that if $G$ is a Hausdorff Topological Group and $H \subset G$ is a subgroup, then $\bar H$ is a subgroup, and that if $H$ is abelian, so is $\bar H$. Is it possible to drop the ...