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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Sum of Cauchy sequences in an abelian topological group (first countability hypothesis)

We know that, given a first countable abelian topological group $G$, the sum of two Cauchy sequences gives yet another Cauchy sequence (see, e.g., this answer). For those wondering, we say that a ...
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40 views

The simply-connectedness of quotient space

If $U$ is a Lie group with a closed subgroup $K$ such that both $U$ and $U/K$ are simply-connected, then can we conclude that $K$ is connected?
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Example of a finite, non-abelian group in which left invariant metric is also right invariant [duplicate]

I need an example of a finite, non-abelian group $(G, \cdot)$ which satisfies the following condition: If $d$ is a metric on $G$ such that $d(ax, ay)=d(x,y), \ \ \ \ \forall a,x,y \in G$, then ...
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52 views

Factor group of profinite group

Wikipedia (http://en.wikipedia.org/wiki/Profinite_group, Properties and Facts) says that the factor group of a profinite group $G$ by a closed normal subgroup $N$ is another profinite group. No proof ...
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445 views

Convex Hull of Precompact Subset is Precompact

I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact. I've come across a ...
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340 views

$G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily ...
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223 views

Metric on a group

Is there a non-abelian finite group $G$ with the property: If metric $d$ on $G$ is left invariant then is also right invariant?
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66 views

On compact topological group

Must a compact topological group be metrizable? If not, is it separable? Thanks for any help.
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179 views

Cardinality of quotient

Given $X$ a topological space, we consider $\mathcal{F}$ the class of all continuous maps $f:X\to H$ where $H$ is a topological group... (edited) and $|H|\le |X|$ If $f,g\in\mathcal{F}$, say $f:X\to ...
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40 views

How to show that the circle group T contains a copy of unit interval [0,1]?

Here, $T$ is the set of all complex numbers of absolute value 1. I want to show that there is a (natural) copy of the interval $[0,1]$. Any hint?
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69 views

An example for a non-precompact minimal topological group.

Do you have an example of a non-precompact minimal topological group? A topological group $(G,\mathcal T)$ is said to be minimal iff it is Hausdorff and for any compatible Hausdorff topology ...
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An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
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108 views

group of homeomorphisms subgroup

(a) Let X be a topological space. Prove that the set $Homeo(X)$ of homeomorphisms $f:X \to X$ becomes a group when endowed with the binary operation $f \circ g$ . (b) Let $G$ be a subgroup of ...
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126 views

From positive definite function to Følner sequence -— a question on amenability and nuclearity

We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
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134 views

Maximal compact subgroups of $GL_n(\mathbb{R})$.

The subgroup $O_n=\{M\in GL_n(\mathbb{R}) | ^tM M = I_n\}$ is closed in $GL_n(\mathbb{R})$ because it's the inverse image of the closed set $\{I_n\}$ by the continuous map $X\mapsto ^tX X$. $O_n$ is ...
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360 views

Every Tychonoff space is an image of a moscow space under a continuous open mapping.

Every Tychonoff space is an image of a moscow space under a continuous open mapping. A space $X$ is called Moscow if the closure of every open set $U\subset X$ is the union of a family of ...
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136 views

A question about quotient under group action

Let $X$ be a Hausdorff space, and $G$ a group acting on $X$ by homeomorphisms. Let $H$ be a normal subgroup of $G$. Is it true that $X/G$ is homeomorphic to $(X/H)/(G/H)$ ? If so, can you please ...
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172 views

Question about pointwise canonically weakly pseudocompact space.

A point $x$ of a space $X$ is said to be a point of canonical weak pseudocompactness if the following condition is satisfied: For every canonical open subset $U$ of $X$ such that ...
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206 views

Is there an example of a non-orientable group manifold?

Basically what I'm looking for is a topological group that is also a non-orientable, n-dimensional manifold
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156 views

Noncommutative dual group

If $G$ is a locally compact group, we can define its dual group $\hat G$. That is set of continuous homomorphism from $G$ to circle group $\mathbb T$. My question is how to define dual group $\hat G$ ...
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83 views

Cauchy product on topological rings

Let $R$ be any commutative Hausdorff topological ring. I am looking for a preferably general condition on sequences $(x_n)_{n \in \mathbb{N}}$, $(y_n)_{n \in \mathbb{N}}$ such that the equation $$ ...
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143 views

orthogonal group of a quadratic vector space

I am reading about the orthogonal group $O(V)$ of a real finite dimensional quadratic vector space $(V,Q)$ with $Q$ nondegenerate. By definition $$O(V)=\{f:V\mapsto V |\quad Q(f(v))=Q(v) \quad \forall ...
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156 views

The intersection of open normal subgroups in a compact, totally disconnected topological group is trivial.

I am currently doing self-study on profinite groups and I'm stuck trying to prove the following lemma. If a topological group $G$ is compact and totally disconnected, then the open normal ...
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188 views

Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?

I heard that $\Bbb R$ and $\Bbb C$ are the only connected, locally compact fields. Does anyone know a proof for this result?
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1answer
102 views

fundamental group of a graph

let $G$ be a connected graph and $\Omega$ its universal covering. Let $\gamma_1,\dots,\gamma_r$ be free generators of $\Gamma:=\pi_1(G)$, $v\in\Omega$ be a vertex and $s_i$ a path from $v$ to ...
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Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group. Is this conjecture still unsolved? Is ...
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25 views

inverse on topological semigroup

Assume $(G,\cdot)$ denotes a topological semigroup (no id, non-commutative). Let $V\subset G$ be open and take some arbitrary $g\in G$. Define $$gV^{-1}:= \bigcup_{x\in V}gx^{-1} = \bigcup_{x\in ...
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what are all the open subgroups of $(\mathbb{R},+)$

I am not able to find out what are all the open subgroups of $(\mathbb{R},+)$, open as a set in usual topology and also subgroup.
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1answer
102 views

If $H$ and $G/H$ are compact, then $G$ is compact.

Suppose that $G$ is a topological group and that $H$ is a subgroup of $G$ so that $H$ and $G/H$ are compact. I am trying to show that $G$ must be compact. The first idea is to use the natural map ...
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1answer
133 views

Properties of $ \text{Exp}(A) $, where $ A $ is a Banach algebra.

$ \newcommand{\Exp}{\operatorname{Exp}} $ Let $ A $ be a unital Banach algebra. For $ a \in A $, consider $$ \Exp(A) \stackrel{\text{def}}{=} \{ e^{a_{1}} e^{a_{2}} \cdots e^{a_{n}} ~|~ n \in ...
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wiki's definition of “strongly continuous group action” wrong?

Wikipedia defines strongly continuous group action as follows: A group action of a topological group G on a topological space X is said to be strongly continuous if for all x in X, the map g ↦ ...
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Topological Group, symmetric neighborhood, Hausdorff, disjoint open sets

Let $G$ be a topological group with identity $e$. If $A, B$ are subsets of $G$, we let $A * B$ denote the collection of elements $a * b$ for $a \in A, b \in B$, and we let $A^{-1}$ denote the set of ...
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Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
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364 views

How is the general linear group a topological group?

How to see if the general linear group GL($n$), of non-singular $n$-square matrices over the real (or complex) numbers under matrix multiplication, is a topological group? How to show that matrix ...
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History: continuously differentiable groups over the real numbers

Continuously differentiable groups over the real numbers are all isomorphic to addition, as is well-known, but who proved it and when?
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Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
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1answer
143 views

Action of a subgroup of finite index on a tree induced by an action of a group on a tree

Let $G$ be a group wich acts on a tree $\Gamma$. Then $U$ acts on $\Gamma$ for every $U\leq G$. Question: Why does the following hold? If $|G:U|<\infty$. Then the minimal $U$-invariant subtree ...
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1answer
198 views

Is the Haar measure of a product of finite measure and compact, finite?

Let $G$ be a locally compact group with Haar measure $ \mu $, $K \subset G$ a compact subset and $ F \subset G $ any subset of finite Haar measure $\mu (F) < \infty $. Is the Haar measure of the ...
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1answer
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proof by nuke of the fact that fundamental group of topological group is abelian

"The fundamental group of a topological group is abelian". does this problem admit a proof by nuke. This is inspired by a a question in mathoverflow. The usual proof is by a Eckmann-Hilton ...
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38 views

question about $Y$-homogeneous spaces.

A subspace $Y$ of space $X$ is $h$-dense in $X$, if $Y$ is dense in $X$ and, for each $x\in X$, there exists a homeomorphism $h$ of $X$ onto itself such that $h(x)\in Y$. in this case we say that $X$ ...
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168 views

Moscow space-Examples

A space $X$ is called $Moscow$, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G‎_{δ}$‎‎‎ ‎-subsets of $X$ . For example, Every first countable $T_1$ ...
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1answer
710 views

discrete normal subgroup of a connected group

could any one give me hint for this one? $G$ be a connected group, and let $H$ be a discrete normal subgroup of $G$, then we need to show $H$ is contained in the center of $G$ first of all, I have ...
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1answer
42 views

action of $O(n,\mathbb{R})$ on $S^n$

I need to know what is the action of $O(n,\mathbb{R})$ on $S^n$, and $O(n,\mathbb{R})/O(n-1,\mathbb{R})\cong S^{n-1}$, how does $O(n-1,\mathbb{R})$ sit inside $O(n,\mathbb{R})$? The obvious action ...
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286 views

Continuous homomorphism into locally compact Hausdorff group

Could any one give me hint to solve this one? $f:G\rightarrow H$ is continuous homomorphism into a locally compact Hausdorff group $H$. Then we need to show $f$ is necessarily open. all spaces 2nd ...
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why the orthogonal group $O(k,l)$ is homotopy equivalent to $SO(K)\times SO(l)$

I want to prove that the orthogonal group $O(k,l)$ (http://en.wikipedia.org/wiki/Indefinite_orthogonal_group)is homotopy equivalent to $SO(k)\times SO(l)$, so that ...
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1answer
96 views

topological group operation vs homotopy group operation

Let $X$ be a topological group. Let $\tau_1$ and $\tau_2$ representing elements of $\pi_n(X)$. Is it true that $$ [\tau_1] [\tau_2] = [\tau_1 \tau_2] $$ in $\pi_n(X)$?, where of course "$[\tau_1] ...
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1answer
299 views

Metrizable group

Let $ G $ be a metrizable group. If (i) $ K $ is a closed normal subgroup of $ G $ and (ii) both $ K $ and $ G/K $ are complete, then $ G $ is complete. Here is how I am proceeding: It can be ...
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1answer
114 views

A new group operation on the fudamental group [duplicate]

Suppose $G$ is a topological group with operation $\cdot$ and identity element $x_0$. Let $\Omega (G, x_0)$ denote the set of all loops in $G$ based at $x_0$. For $f, g\in\Omega (G, x_0)$ define ...
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200 views

A couple of questions about closed subgroups of a topological group.

I am reviewing my previous exams, and I completely missed the following two-part question. It deals with closed subgroups of topological groups under certain situations. I am having trouble working it ...
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Why are topological groups semitopological groups?

As part of a casual self-study of topology, I have started fooling around with topological groups. I noticed that the article on Wikipedia mentioned that "weakening the continuity conditions" gives ...