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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Two disjoint compact sets in a topological group

Let $(G, \cdot )$ be a compact (Hausdorff) topological group. If $A$ and $B$ are two disjoint compact subsets of $G$, how can we show that there exists a nonempty open set $V$ such $A\cdot ...
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Explicit construction of Haar measure on a profinite group

Let $G$ be a profinite group. It is known that in $G$, every neighborhood of the identity element contains an open compact subgroup. I would like to explicitly construct the Haar measure on $G$. The ...
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1answer
66 views

Action of $S^1$ on homotopy groups of an $S^1$-space

I am interested in the following question : Let $S^1$ be the $1$-sphere, seen as a topological group by being the unit sphere in the complexe plane $\mathbb{C}$. Let $X$ be a (good, ... etc) pointed ...
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28 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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1answer
57 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
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1answer
20 views

equality in uniform space and topological groups

I wanted to ask the following: If I have a topological group $G$, I know I can create a base for a uniform space as follows: for each $U$ a neighborhood of e, we define $V_u= \{(x,y):x^{-1}y\in U\}$. ...
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53 views

Topology question with closed sets.

Let $ K\subseteq \mathbb{R}^n$ be a compact set and let $E\subseteq \mathbb{R}^n$ be a closed set. ***Its also given that $ \inf \{d(x,y)|x\in K, y\in E\}=0$. $ d(x,y)=\sqrt{\sum_j (x_j-y_j)^2}$ ...
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continuous (smooth) maps and group homomorphism

Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
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24 views

Amenability; topology on power sets

I am currently reading the proof of Proposition 2.2. (a direct limit of discrete amenable groups is amenable) in http://people.maths.ox.ac.uk/kar/amenable.pdf . The proof uses the Tychonoff theorem ...
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1answer
16 views

Symmetric open neighborhood in topological group

I want to prove that ($\ast$) If $G$ is a Hausdorff topological group, then for every neighbourhood $U$ of $e$, there exists a symmetric neighbourhood $V$ of $e$ ($V^{-1}=V$), s.t $V*V \subset U$ ...
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1answer
27 views

Quotient group and kernel of canonical projection

Imagine we have a group $G$ acting properly and freely (as a group action $\Phi: G \times M \rightarrow M$) on a manifold $M$, then $M/G$ is a manifold and there is a smooth submersion $\pi: M ...
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Who are the mathematicians in US who are working on expander graphs right now?

I am familiar with only the "big" names doing this research like Gharan, Nikhil Srivastava, Dan Spielman, Jean Bourgain, Luca Trevisan, Elina Fuchs, Peter Sarnak , Amin Saberi and Terence Tao. I would ...
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1answer
33 views

Topology open balls and open sets

Let $A\subseteq \mathbb{R}^n$ be an open set Prove that A can be written as a countable union of open balls. We thought about it having to do with creating a ball with a radius of $r$ ...
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1answer
42 views

Show that $\widehat{\mathbb Z}\cong \prod_{p\;\text{prime number}}\mathbb Z_p$

Let $\widehat{\mathbb Z}=\varprojlim_n \; \mathbb Z/n\mathbb Z$ be the inverse limit of the inverse system $(\mathbb Z/n\mathbb Z)_{n\in \mathbb N}$ and let $\mathbb Z_p=\;\varprojlim_n\; \mathbb ...
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1answer
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Haar measure - a problem from Folland

I was presented with this question from Folland's real analysis second edition involving Haar measures. It is problem 3 of chapter 11 page 347, which reads as follows: Let G be a locally compact ...
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20 views

Every representation of compact group is a direct sum of irreducible

Recently I asked about (references to) some results concerning representation theory of compact topological groups: here is the discussion Representation theory of locally compact groups In ...
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Question regarding characters and point open topology2

this is a follow-up question for the following one: Dual group of G with point open topology is an intersection of C(G,T) and a closed set In the book of Banaszczyk - "Additive subgroups of ...
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Connected Lie group is second countable?

I know this is true from various sources, unfortunately none of them give the full proof. I already have a start: Let $G$ be connected Lie Group. Choose $K$ to be any compact neighbourhood of the ...
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1answer
19 views

Set of discrete orbits under subgroup of $Isom(\mathbb{E}^n)$ is clopen

I'm trying to solve the following exercise (exercise 1.4 from Szczepanski's "Geometry of Crystallographic Groups"): Let $\Gamma$ be a subgroup of $I(\mathbb{E}^n)$, the group of isometries on ...
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1answer
48 views

$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$

Let's consider a group $M$ (under multiplication) of all matrices $A$ of size $2 \times 2$ over $\mathbb{R}$ so that $\det(A)=1$. How to show that the group is homeomorphic to the $S^{1} \times ...
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Contractible pieces of $GL(n,\mathbb{C})$

Is $GL(n,\mathbb{C})$ contractible for any $n$? My intuition is telling me it is not, because the determinant maps the general linear to $\mathbb{C}\setminus 0$ which is not contractible. If there ...
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1answer
22 views

topological isomorphism between a topological group and the identity component of a topological group

Let $\Bbb R$ be the group of real numbers with the usual topology and $\Bbb Z$ the group of integers with the discrete topology. Is $\Bbb R$ topological isomorphism by the identity component of $(\Bbb ...
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28 views

Open subgroups of quotient topological groups

Let $G$ be a topological abelian group and $H$ a closed subgroup of $G$. Is it true that an open subgroup of $G/H$ has the form $K/H$ where $K$ is an open subgroup of $G$ containing $H$?
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Representation theory of locally compact groups

My knowledge about representation theory of locally compact groups is rather scattered. As I got more interested with this subject, I would like to know some good references, where I could learn the ...
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1answer
35 views

Definition of precompactness in a topological group $G$

I have seen that the definition of precompact sets in a topological group $G$ is a bit tricky. Can someone please explain? I saw that it has to do something with totally bounded sets. Is there a more ...
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1answer
19 views

An inductive limit of amenable groups is amenable

It is a Theorem that an inductive Limit of amenable Groups is amenable. Could someone sketch me a proof of this, or give me a reference? I couldn't find one. Thanks in advance. Edit: I wanted it for ...
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1answer
50 views

False proof that all topological groups are discrete: what went wrong?

I can't seem to find the mistake in this obviously false proof I've thought up while trying to understand topological groups. It's pretends to prove the discreteness of all topological groups. Let ...
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3answers
27 views

Question about images of R under injective homomorphisms.

I'm studying for my topology comp and I'm at a bit of a loss on this question. (My experience with Algebra is very limited and my experience with topological groups in particular is almost ...
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0answers
87 views

The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
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1answer
30 views

topological isomorphism between a group and product of its subgroups

I have stumbled upon the following question: Let $G$ be a $\sigma$-compact, locally compact Hausdorff group with $N$ and $H$ closed normal subgroups of $G$. Also $$N\cap H= \{e\}$$ and $$G=NH .$$ Then ...
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1answer
30 views

support compact modulo subgroup

I am studying (co)-induced representations of topological groups and I came across the following situation: $G$ is a topological group, $H$ a closed subgroup and $f\colon G\to W$ a set-theoretic map, ...
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1answer
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LCA - groups under continuous homomorphisms

can someone help me out with this question? LCA stands for Locally compact Hausdorff abelian group. The question is posted in the attached image Let $A$ and $B$ be LCA-groups and $H$ a (not ...
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2answers
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Equivalence to being a topological group

Just some notation I am using: A topological group $G$ is a group with a topology such that $o : G^2 \to G : (x,y) \mapsto xy$ and $inv : G \to G : x \mapsto x^{-1}$ are continuous in the ...
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0answers
40 views

Why the multiplicative group $G_m$ is called a 1 dimensional torus?

I am reading a definition saying that an algebraic group over a field $K$ is called a torus if it is isomorphic to product of copies of the multiplicative group $G_m = K^*$. I don't understand why ...
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1answer
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What prevents the restriction of a Haar measure to a closed subgroup from being a Haar measure?

Let $\mu$ be a Haar measure on a locally compact Hausdorff topological group $G$, and let $H$ be a closed subgroup of $G$. If we restrict $\mu$ to the Borel sets of $G$ which are contained in $H$ ...
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1answer
25 views

Showing that compact metrizable abelian group has enough characters

I wanted to know if anybody has some clue as why a compact metrizable abelian group has enough characters to sepatare points. (a character on a topological group is a continuous homomorphism from the ...
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0answers
26 views

universal bundle of topological monoids

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such ...
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1answer
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When does a topological space inherit multiplication from a dense subspace?

Suppose $K$ is a compact topological Hausdorff space with a dense subspace $G$. Moreover, let $G$ have a group structure which is compatible with the topology inherited from $K$. i.e. $G$ is a ...
3
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1answer
48 views

Is a compact subset of a topological group G/N closed if G is hausdorff?

I just used this hypothesis when I was proving a theorem.But I was not sure if this hypothesis is correct.
3
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1answer
42 views

Compact paratopological groups are automatically topological groups.

A compact paratopological group is a topological group. How to prove it? An abelian paratopological group is a topological group. Is this right? A paratopological group is a topological semigroup ...
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1answer
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example of a particular topological group

Can someone give an example of a topological group $G$ that is not Hausdorff but that contains a fundamental system of neighbourhoods of $1\in G$ consisting of quasi-compact subgroups? Thanks in ...
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1answer
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Why is there a set $W$ (to be described below) such that $\mathbb{A}_K = W + K$?

To prove the compactness of $\mathbb{A}_{\mathbb{Q}}/ \mathbb{Q}$ (and hence $\mathbb{A}_K/K$ for an arbitrary number field $K$), one finds a set $W \subseteq \mathbb{A}_{\mathbb{Q}}$ of the form $$ ...
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0answers
36 views

suspension foliations on thickened surfaces

I've seen this statement without proof in a peer reviewed journal and I'm looking for a proof: "If $L$ is an oriented surface with boundary($\neq D^2$), and $C$ is a designated boundary component, ...
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Is the unitary group of a pre Hilbert space contractible?

for a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for the strong operator topology (Dixmier and Douady, ...
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1answer
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Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...
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Quotient group of $\mathbb{R}^2$ by irrational line

In a section about topological groups, exercise 4.10 in I.M. James' General Topology and Homotopy Theory asks Show that for irrational values of $\alpha$ the factor group of the real plane ...
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1answer
40 views

The whole group is covered by compact translating of subgroups

$G$ is a locally compact (may not necessarily Hausdorff) group, $H$ is a subgroup in $G$, $G/H$ is compact as a quotient space , then there exist a compact subset $K$ such that $G=KH$(or $G=HK$).
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Is there a non-abelian Lie group which is homeomorphic to an $n$-dimensional torus $\mathbb{T}^n$?

I've learned that a compact connected abelian Lie group must be a torus. Of course, conversely, a torus as a group is abelian. I wonder if 'homeomorphic to a torus' is enough to imply abelian. ...
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3answers
33 views

Example for algebraic homomorphism between topological groups which is not continuous

I am quite sure there should be an easy example of: (algebraic) homomorphism between topological groups which is not continuous. However, I do not see one immediately.
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3answers
49 views

Continuous homorphisms between topological groups.

Let $G,H$ be topological groups and let $\rho: G \rightarrow H$ be a homomorphism of topological groups. I understand this to mean that $\rho$ preserves group structure and is a map between the ...