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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Pro-completion of finite algebras as Stone algebras

Recall that a profinite algebra (e.g. group, monoid, or whatsoever) is a cofiltered/inverse limit of finite algebra. In Johnstone's Stone space, he showed that finite discrete algebras are finitely ...
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Three questions from σ-compact spaces and topological groups

every locally compact subgroup of a Hausdorff group is closed. A Hausdorff and $σ-$compact space X is a Baire space if and only if the set of points at which is $X$ is locally compact is dense in ...
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Prob. 5 (a) in Supplementary Exercises in Munkres' TOPOLOGY, 2nd ed: How to show that this map is a homeomorphism?

Let $G$ be a topplogical group, and let $H$ be a subgroup of $G$. Let $G / H $ denote the collection of all left cosets of $H$ in $G$, and let $a$ be a fixed element of $G$. Let the map $f \colon G / ...
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1answer
50 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
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1answer
13 views

Construction of a given neighbourhood in a locally compact group

Let $G$ be a locally compact group. Why is it possible to select a compact neighbourhood $U$ of $e \in G$ such that $U=U^{-1}$ and $gU^2 \subset V$? This is a construction quickly stated by Helgason, ...
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27 views

When can a group be made into a ring? How `little' of the ring structure must be specified?

Given a (topological) abelian group $G$ and a (bicontinuous) $G$-bilinear map $\mu: G \times G \to G$, clearly $G$ becomes a (topological) ring by specifying $$ x y := \mu(x, y) \quad \forall x, y \in ...
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The appropriate translation of Sets of Positive Measure is positive

$A$ and $B$ are two measurable subset of $ \mathbb{R}$, and $m$ is a Lebesgue measure on $\mathbb{R}$, if $m(A)>0$ and $m(B)>0$, then there exist a $ x\in \mathbb{R}$, such that $m(A \bigcap ...
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an existence question from topological groups

$G$ is a topological group, $A$ and $B$ are the subsets of $G$, we denote $AB$=$\{ab:a \in A, b\in B \}$. Let $G$ be a locally compact Hausdorff topological group, $m$ is a left Haar measure on $G$, ...
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26 views

Topology of $\Bbb{Q}_p$

Let $a\in \Bbb{Q}_p$. Is $ a+p^x\Bbb{Z}_p$ an open set around $a$ in the topology of $\Bbb{Q}_p$. Here $x \in \Bbb{Z}$. Also I have another question. Is $\mathbb{Z}_p$ open in $\Bbb{Q}_p$?
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1answer
41 views

intersection about the second category

$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that ...
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26 views

Topological group, which is second category in itself, is a Baire space.

A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. $G$ is a topological group, if $G$ is of the second ...
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1answer
64 views

Compact subgroups are contained in open compact subgroups in locally profinite groups

Let $G$ be a totally disconnected, Hausdorff, locally compact group. In the wikipedia page about these groups there is a claim that any compact subgroup of $G$ is contained in some compact open ...
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1answer
21 views

Two questions about increasing unions of compact subsets of a locally compact Hausdorff group.

I have two questions to ask related to my research. Question 1. Let $ G $ be a locally compact Hausdorff group. Is it possible that $ G $ is the union of a chain of compact subsets (ordered by ...
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26 views

Choice of a dense subset of a separable Banach space

I recently came across the following statement, and still can't prove it: Statement: Suppose $X$ is a separable,closed subspace of $L^1(G)$, where $G$ is a locally compact group. Since $X$ is ...
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29 views

Closed mapping theorem for almost connected groups

Edit: I originally asked if $\phi$ is a closed map. Clearly it isn't in general, as the real numbers can be mapped to the torus with dense image. Let $G$ and $H$ be connected locally compact groups, ...
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1answer
10 views

Sigma-compact Polish groups

I would like to see an example of a sigma-compact Polish group which is not locally compact. I know that e.g. $l^{\infty}$ is a topological group which is sigma-compact but not locally compact. But ...
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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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84 views

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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52 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
21 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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30 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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46 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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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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1answer
67 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
26 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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23 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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43 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
28 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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26 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
30 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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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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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
36 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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47 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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188 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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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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49 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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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 , ...