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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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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12 views

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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1answer
14 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
26 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
25 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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86 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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20 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
126 views

Haar measure, convolution and involutions

I have some problems to follow the proof of the anti commutativity property of the convolution and involution operations defined using a Haar measure as presented in Pedersen's book Analysis Now, ...
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1answer
24 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
131 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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26 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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25 views

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
20 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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16 views

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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18 views

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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31 views

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
33 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
24 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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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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1answer
30 views

Is every closed subgroup of dual group an annihilator?

This is a naive question as I am new to topological group theory. Let $G$ be a Locally Compact Abelian group (LCA group). I know that to every closed subgroup $H$ of $G$ correspond a closed subgroup ...
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43 views

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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69 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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75 views

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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1answer
280 views

Product of totally disconnected space is totally disconnected?

I read that the cartesian product with the product topology of a family of totally disconnected topological spaces is totally disconnected, too. Is that true? How are the connected components in the ...
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178 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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1answer
24 views

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
42 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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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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1answer
28 views

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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14 views

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
23 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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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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1answer
7 views

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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1answer
33 views

Prove that $\mathbf{SO}_3(\mathbf{R})$ is simple by topological methods.

I am trying to solve an exercise consisting in showing that the group $\mathbf{SO}_3(\mathbf{R})$ is simple by using topological methods. A hint say that I should use the trace map, but I have still ...
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1answer
140 views

Uniformly continuous maps between topological groups

Let $G$ be a topological group. For every neighbourhood $U$ of the identity, let $L_U$ be the set of all pairs $(x,y) \in G \times G$ such that $x^{-1} y \in U$. For topological groups $G$ and $H$, a ...
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3answers
52 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
108 views

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ [duplicate]

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ where $(f\otimes g)(s) = f(s) \cdot g(s)$ where $\cdot$ is the group operation on the topological group $G. $ This is a question from ...
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1answer
26 views

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 ...
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27 views

Why is $\mathcal{C}(X,G)$ a top. group via the co. topology?

Let $X$ be an arbitrary topological space and $G$ a topological group. Let $\mathcal{C}(X,G)$ be the group of continuous maps from $X$ to $G$, endowed with the compact-open topology as a topological ...
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1answer
28 views

Countable LCA groups

Is it true that a countable LCA group can only be discrete ? This question is related to a comment here : A theorem on LCA group - is the uncountability necessary?
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1answer
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Path-connectedness of a certain set of invertible complex matrices

We know that $GL_n(\mathbb{C})$ is connected. I am considering the following variant that has been bothering me for quite some time. Let $u\in M_n(\mathbb{C})$ be such that $||u||\leq N$, ...
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1answer
37 views

Connected subgroup of a topological group.

If $G$ is a connected topological group, and $H$ is a closed subgroup of $G$, such that $G/H$ is simply connected, does it follow that $H$ is connected? (In the particular class of cases I have in ...
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20 views

What are the differences between the three editions of the book “The Structure of Compact Groups”?

meta pre-clarification: I looked into another question like this but the guy didn't mark any specific tags for this type of question. Here's a link to the amazon book: ...
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240 views

Lie Groups which are not Hausdorff

I suspect this isn't a terribly difficult question, but I don't know the answer and I'd guess someone has already looked into it. Is it possible for a Lie group on a non-Hausdorff manifold to exist? ...
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1answer
53 views

are surjective group scheme homomorphisms also surjective on global sections?

Suppose $P,G$ are group schemes over $S$ (not necessarily commutative), where $G$ is finite and constant (so isomorphic to $n$ disjoint copies of $S$) Suppose there is a surjection homomorphism $f : ...