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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Topological group and neighborhood base
I am reading J.S. Milne’s notes for Fields and Galois theory (v5.10). I have a question about the proof of the following proposition (page 94. Proposition 7.2).
Let $G$ be a topological group and let ...
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When is extension of a Lie group by a discrete group a Lie group?
Suppose we have a group extension:
$$1\to D\to G \to L\to1$$
Where $D$ is a discrete group and $L$ is compatible with a Lie group structure.
What are some minimal conditions one needs to apply to $G$ ...
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Open subgroups of subgroups of profinite groups
The teacher presented me with a fairly easy exercise concerning profinite groups. This is how it goes:
Let $G$ be a profinite group and $H$ be a closed subgroup of $G$.
Show that if $N$ is an open ...
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Topological irreducibility and continuous intertwiners
Consider topological representations $A$ and $A^{\,\prime}$ acting in TVS spaces ${\mathbb V}$ and ${\mathbb V}^{\,\prime}$, and intertwined by a continuous map $M\,$:
$$
M\,A(g)\,=\,A^{\,\prime}(g)\,...
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Are uniform abelian groups the same as topological abelian groups?
Let $\operatorname{Top}$ be the category of topological spaces with continuous maps between them. Let $\operatorname{Unif}$ be the category of uniform spaces with uniformly continuous maps between ...
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In a topological group, the homotopy group operation is the pointwise-product.
I'm coming from this question.
In both the comments and the answers, they says that the Eckmann-Hilton Principle applies, for $[f]\cdot [g] := [fg]$ defines another binary operation on the homotopy ...
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Completions of topological groups by the adjoint functor theorem
Let TG be the category of Hausdorff topological groups and continuous group morphisms and CTG the category of complete Hausdorff topological groups where completeness is with respect to the left ...
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Is the space $\operatorname{ker}(\operatorname{Free}(S^1) \to \mathbb{Z})$ a $K(\mathbb{Z}, 1)$?
In Classifying Spaces Made Easy , John Baez says the following:
Next question: how does $\mathbb{CP}^\infty$ become an abelian topological group? There's a very pretty answer. Consider the space of ...
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Question about the proof of Lemma 0BMV of Stacks Project
I tried asked questions before on Stack Project, but the answers were a bit slow so I will try here. This is a question about the proof of Lemma 0BMV.
Here is the setup. Let $G$ be a topological group ...
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A problem in the reasoning of Bosch's Algera about inifite Galois extesion
Bosch's reasoning on page 144:
To introduce a topology on a set $X$, we can start out from an arbitrary system $\mathfrak{B}$ of subsets of $X$ and look at the topology generated by it. To set up the ...
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Textbooks combining order theory, topology and algebraic structure
It looks like a greedy title.However, when I learn about construction of real numbers and different equivalent ways to do completion like Dedekind completion and Cauchy completion from some papers ...
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Inducing full profinite topology on a subgroup
The profinite topology on a group $G$ is the topology with base of open sets given by all cosets of finite index normal subgroups in $G$. If $H$ is a subgroup of $G$ then $G$ induces the full ...
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Induced topology on subgroup of free pro-$p$ group
I am trying to determine whether the following is true:
Let $H$ be a closed subgroup of a finitely generated free pro-$p$
group $F$. Does $F$ induce on $H$ the full pro-$p$ topology?
Stated in ...
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Action of the fundamental group on the universal cover of a differentiable manifold is differentiable.
Let $M$ be a connected smooth manifold and $\tilde{M}$ its universal cover. I am considering the action of the fundamental group $\pi_1(M,x_0)$ on $\tilde{M}$, given by
$$
A \colon \tilde{M} \times \...
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Definition of local isomorphism for topological groups
Let $G,H$ be topological groups. My definition of local homomorphism is a continuous map
$\phi:U\to H$, where $U$ is an open set of $G$ containing $e_G$, and whenever $x,y,xy$ are all in $U$, it holds ...
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Are finite extensions of a Lie group Lie groups?
Let $H$ be a group with a finite subgroup $F$ so that $H / F$ is a Lie group (when equipped with some topology). Does this guarantee that $H$ admits a topology making it a Lie group, and such that the ...
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What are the connected, locally compact alternative topological fields?
The only connected, locally compact topological fields are isomorphic as topological fields to $\mathbb{R, C}$ with their standard topologies
The only connected, locally compact topological skew ...
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in a normed field, is the inverse function $x\mapsto x^{-1}$ continuous?
A normed field is a pair $(K,|\cdot|)$ where $K$ is a field, and $|\cdot|:K\to \mathbb{R}_{\geq 0}$ s.t.
$|a|=0$ iff $a=0$.
$|a+b|\leq|a|+|b|$.
$|ab|\leq|a|\cdot|b|$. This is sub-multiplicativity ...
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When is $\operatorname {Aut}(X)$ not a paratopological group?
Let $X$ be a topological space and $\operatorname {Aut}(X)$ the set of all homeomorphisms from $X$ to $X$, endowed with the compact-open topology. The composition of functions $\circ : \operatorname {...
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Multiplier that brings group elements close to identity
Given $G$ a Hausdorff topological group, $V$ an open neighborhood of $e$ and $(t_i)_{i\leq N}\subseteq G\backslash \{e\}$ for some $N\in \mathbb{N}$, when can we find $x\in G$ such that $t_ix\in V$ ...
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Is $SO(n, K)$ isomorphic to $O(n, K)_0$ for $K = \mathbb{C}, \mathbb{H}$?
Let K to be a field. I define $O(n, K)_0$ to be the identity component of $O(n, K)$.
Now it is true that $O(n, \mathbb{R})_0 \simeq SO(n, \mathbb{R})$. Is the analagous result true over $\mathbb{C}$ &...
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Metric on the 2-dimensional Torus
I am trying to show that the orbit of the function $T:\mathbb{T}^2\to\mathbb{T}^2$ defined as $T(x,y)=(x+\alpha,y+\beta)$ has interesting properties dependent on your choice of $\alpha$ and $\beta$, ...
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Multiplicativity of Haar measure
Let $H$ be a lattice in $G$, where $G$ is a locally compact abelian group, and let $K$ be a subgroup of $H$ which is also a lattice in $G$. Is it true that the volume of $H/K$ is equal to the volume ...
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Homeomorphism between neighborhoods in topological groups
Let $G$ and $H$ be topological groups and $U$, $V$ neighborhoods of $e$ in $G$ and $H$ respectively. Note we are not assuming neighborhoods are open. Suppose $p:G\to H$ is a continuous group ...
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Reference for closed subgroups in $\mathbb{R}^n$. [closed]
I'm looking for a reference of the following result : every closed subgroups of $\mathbb{R}^n$ is equal to $V \oplus L$ where $V$ is a linear subspace and $L$ is a lattice.
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