# 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.

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### Particular property of an open subsemigroup in a given topological group

In chapter $\rm{III}$ of the "Topologie générale" of Bourbaki -- which is the chapter they dedicate to a brief introduction to generalities concerning topological groups -- I have ...
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### Concept of a topological group [closed]

I have read the definition multiple times and have skimmed a few posts here, yet I still cannot fully grasp the idea of a topological group. Would anyone care to explain it with a few examples?
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### understanding a step in proof of strong approximation for $SL_2$

Let $k$ be a global field and $k_v$ denote its completion at a place $v$. Let $Z$ be the closure of $SL_2(k)$ in $SL_2(\mathbb{A})$; $Z$ is a subgroup. Assume $Z$ contains $SL_2(\mathcal{O}_v)$ for ...
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### $H$ closed implies $G / H$ Hausdorff

Let $G$ be a topological group and $H$ be a subgroup of $G$. It is to show that the closedness of $H$ implies $G/H$ being a Hausdorff space. We denote the projection map $G \to G/H$ with $p$. My book ...
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### Is there a group defined over a set of reals whose elements change the real numbers while preserving the sum of the reals in the set?

Is there a continuous, (maybe smooth?) group that is defined over a finite set of zeros and/or positive real numbers (say the set of $3$ real numbers $\{1.59, 105, 19.42\}$) whose elements are binary ...
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### Sharply $k$-transitive actions on spheres

A nice fact from complex analysis is that the mobius group acts sharply 3-transitively on the Riemann sphere. I am wondering if other sharply k-transitive (continuous) actions are known on any $S^n$, ...
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### Co-compact lattice in locally compact hausdorff groups

Bekka-Mayer in their book below, in II.2 says: If $\Gamma$ is a discrete and cocompact subgroup of a locally compact group $G$, then $\Gamma$ is a lattice in $G$. I can't seem to prove it. Of course ...
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### Subsets of positive measure in a group

Let $G$ be a Lie group (maybe LCH and second countable topological group is enough here) and $\mu$ be a Borel probability measure on $G$ (not necessarily Haar). Suppose $U$ is an open subset of $G$ of ...
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### Is the group action by a topological homeomorphism group continuous?

Let $F$ be a topological space such that $G := Homeo(F)$ is a topological group in the compact-open topology. Let $\phi : G \times F \to F$ sending $(g,x)$ to $g(x)$ be the map for the action of $G$ ...
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### Support of the convolution of two measures

Let $\mu, \nu$ be two Borel Probability measures on a Lie group $G$. I wonder if it is true that $$supp(\mu \ast \nu)=supp(\mu) \cdot supp(\nu),$$ or at least we have one contained in another? Recall ...
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### Uniform parametrisation of $SO(3)$

Does there exist a map from a hypercube of some dimension to $SO(3)$ such that pushforward measure of standard measure on the hypercube is invariant under action of $SO(3)$ on itself?
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### True or false: infinite sequence in a compact topological group is dense. [duplicate]

This is from an exercise in Bredon's Topology and Geometry: Let $G$ be a compact topological group (assumed to be Hausdorff). Let $g\in G$ and define $A=\{g^n:n=0,1,2...\}$. Then show that the ...
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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, ...
### Verify that $f_{n} \rightarrow f_0$, where $f_{n}=0 \ \forall \ x \in[0,1]$ [closed]
Let $C[0,1]$ and $d$ be defined on this set by $$d(f, g)=\int_{0}^{1}|f(x)-g(x)| \, \mathrm d x$$ where $f$ and $g$ are in $C[0,1]$. Define a sequence of functions$f_{1}, f_2, \ldots, f_{n} \ldots$ in ...
Given that $G$ is a linearly ordered group (bi-ordered). I want to try and understand the difference between the “size” of left multiplication vs right multiplication (which I have written below using ...