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Questions tagged [topological-groups]

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.

A topological group is a group endowed with a topology such that both the group operation and inversion are continuous maps. Every group can be understood as a topological group, if we take the discrete topology.

Topological groups are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis, see e.g. Pontryagin duality.

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Is the space of left cosets of a topological group completely regular?

Let $G$ be a (Hausdorff) topological group and $H$ its closed subgroup. Then it's known that $G/H$ is a regular topological space. Is it always completely regular? A reference is good too. In the case that $H$ is compact, we know that the quotient…
Jakobian
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Writing G as a product of groups

$G$ is a connected, locally compact group satisfying the second axiom of countability, and $C$ is a discrete central subgroup of $G$ such that $G/C$ is compact. In the book I am reading (Varadarajan, Lie Groups, Lie Algebras, and their…
Not a grad student
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Center of a topological group is closed?

Is it true that the center $Z$ of a topological group $G$ is closed?(maybe we need the space to be Hausdorff or something like that...) I was thinking I can just show it is opened. So if I pick $x\in Z$ then I need to find an open $U \ni x$ such…
roi_saumon
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$\phi^{-1}(\phi(U))=\bigcup_{k\in Ker(\phi)}kU$?

Let $G$, $H$ be topological groups and let $\phi$ be a homomorphism between these two groups. Then is it true that if the image of an open $U$ by $\phi$ is $\phi(U)$ then we have $\phi^{-1}(\phi(U))=\bigcup_{k\in Ker(\phi)}kU$?
roi_saumon
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Subgroup of a topological group

Let $G$ be a topological group and $K\subset G$ a subgroup of G. Is it true that $K = \bigcup_{g\in K}gK$ ? I'm asking this because in my notes I have that $K^{C} = \bigcup_{g\in G-K}gK$ Thank you in advance for your answer!
maramath
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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?
user73564
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Topology on integers without open ideals

I am looking for a topology $\mathcal{T}$ on $\mathbb{Z}$, such that $(\mathbb{Z},+)$ is an abelian topological group, $(\mathbb{Z},\mathcal{T})$ is Hausdorff, there is no proper ideal $I:=n\mathbb{Z}, 1
Some Math Student
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Find an example of an infinitely generated discrete subgroup of ${\rm SL}(2,\Bbb R)$.

I want to find an example of an infinitely generated discrete subgroup of ${\rm SL}(2,\Bbb R)$ but I haven't managed to find such a subgroup. Please would someone help me to find such subgroup?
Roi
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The orbit map is open

Let a locally compact second countable Hausdorff Group $G$ acting continuously and transitively on a locally compact second countable Hausdorff topological space $X$ and let $y$ be an element in $X$ and let $H$ be the stabilizer of $y$ in $G$ and…
Roi
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Locally closed dense subgroup of a topological group coincides with the whole group

This question is very simple, but I don't get the right idea. Assume $H$ be a locally closed dense subgroup of a topological group $G$. Prove that $H=G$. I need to prove that $gH\cap H\ne\emptyset$ for all $g\in G$. I know that $H$ is open…
LBJFS
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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 may be $\phi:O(n,\mathbb{R})\times S^n\rightarrow…
Myshkin
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Is there a specific name for non-homotopic loops encircling same point?

A plane removes several points, for example, here P1, P2, P3. Loop 1 and Loop 2 are freely homotopic(or called homotopic if have same base point). Loops 3 are not homotopic with loop 1 and 2, but it also encircle point P1 and P2, like loop 1 and…
Qi Zhong
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Totally disconnected space and discrete topology

A topological space is totally disconnected if its only non empty connected sets are one point set. Is every totally disconnected space discrete topology?
Samiron Parui
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In topological groups. Is every neighborhood of $e$ supset of a neighborhood subgroup?

Let $G$ be a topological group, $U$ is a neighborhood of $e$ which is the unit element of $G$. My question is does there exist a neighborhood $H \subseteq U$ of $e$ s.t. $H$ is a subgroup of $G$? $H$ is a normal subgroup of $G$? If 1 is true, then…
Popopo
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Show that $\mathcal T\subset\mathcal P(\mathbb Z)$ is topology

Let $\mathcal T$ be a topology on $\mathbb Z:$ $\mathcal T\subset\mathcal P(\mathbb Z)$ with $\emptyset$ and all the unions of $S(a,b)=\{an+b|n\in \mathbb Z\}$ for $a\neq 0$ prove that $\mathcal T$ is topology on the integers. I know that I should…
Error 404
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