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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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The filters of neighborhood of the identity in a topological group

I'm approaching to topological groups and I was reading "Introduction to topological groups" by Dikranjan: https://users.dimi.uniud.it/~dikran.dikranjan/ITG.pdf . In demonstrating Theorem 3.1.5, he asserts: "Let $\mathcal{V}$ be a filter on $G$…
MJane
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Compact abelian topological group G. Then $G$ is connected iff its dual $\hat G$ is torsion-free.

Here I was trying to prove the following statement. Let $G$ be a compact abelian topological group. Then $G$ is connected iff its dual $\hat G$ is torsion-free. The proof in the link above goes as follows. Suppose $\phi:G\to S^1$ is an element of…
permutation_matrix
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open subset of $G\times G$

If $O$ be an open symmetric subset of topological group $G$ such that $e\in O$, then is $V_O=\{(a,b)\in G\times G: a^{-1}b\in O\}$ open in $G\times G$?
NBhr
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Disconnected closed subgroup implies quotient is not simply connected

I have the following question Let $G$ be a locally connected topological group and $H < G$ a closed locally connected subgroup, show that if $H$ is not connected then $G/H$ is not simply connected. However the book does not provide the definition of…
Daniel Moraes
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Continuous action of topological group and embedding

Let $G$ be a topological group act continuously on a topological space $X$. Why the continuity of the action of $G$ on $X$ implies that $G$ embedded as topological group in $S_{X}$. Here $S_{X}$ is the symetric group on $X$ i.e the group of all self…
Serges
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Continuity of operations in topological groups

I have a question regarding continuity in topological groups. From Wikipedia: A topological group $G$ is a topological space that is also a group such that the group operation: \begin{align*} \mu: G \times G &\rightarrow G\\ (x,y) & \mapsto…
user123456
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Group generated by a connected set is connected.

Consider $G$ a topological group and $S$ a connected subset of this group. I want to show that $$, the group generated by $S$ is also connected. For each $n$, I can construct $2^n$ continuous maps : $$\phi_{ni}:\underbrace{C\times \ldots \times…
roi_saumon
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For each $n\geqslant 1$, there are uncountably many dense subgroups of $\mathbb{F}_p^I$ of index $p^n$.

Let $p$ be a prime and let $I$ be an infinite set. We equip $F^I_p$ with the product topology (where $F_p$ is equipped with the discrete topology). Show that for each $n \geqslant 1$ there are uncountably many dense subgroups of $F^I_p$ of index…
Ryze
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Why is the equivalent Cauchy sequence in a topological group well-defined?

I was reading chapter 10 of Atiyah where I met the notion of equivalent Cauchy sequences for topological groups. Atiyah does not explain the reason why equivalent Cauchy sequences indeed give a equivalence relation. I can manage to prove the…
Yu Ning
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What is $\text{Hom}_{\text{cts}}(\hat{\mathbb{Z}},\mathbb{Z})$?

I am trying to calculate $\text{Hom}_{\text{cts}}(\hat{\mathbb{Z}},\mathbb{Z})$ (i.e., continuous group homomorphisms from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$, viewed as topological groups in the usual way). I know that $\hat{\mathbb{Z}} \simeq…
user514014
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A homomorphism which is trivial on the $n$-torsion must have an $n$'th root?

Suppose that $(G,+)$ is a locally compact abelian group. Let $G_n:=\{g\in G : ng=0_G\}$ and let $\chi:G\rightarrow S^1$ be a continuous character. Assuming that $\chi(G_n) = 1_{S^1}$. Is it necessarily true that there exists a character…
Yanko
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Every element $g$ of $G$ has a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}\subset U$

Let $G$ be a topological group and $g\in G$ , $U$ is a neighborhood of $g$ . Prove that there exists a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}\subset U$. If $g=e$, l have proved it. But if $g\neq e$ , l have no idea. So how to prove ?
water graph
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Is there a LCA group $G$ such that $G/\mathbb{T}\cong\mathbb{R}$?

I'm looking for an example to the following situation: A locally compact abelian (LCA) group $G$ (I assume that the groups are Hausdorff) . A (closed) subgroup $H$ of $G$ which is isomorphic (as topological group) to the circle $\mathbb{T}\cong…
Yanko
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Argument that closure of identity in topological group is normal subgroup.

Let H be the closure of $\{e\}$, where $e$ is the identity element of a topological group $G$. We know that $H$ is a subgroup of $G$. We want to prove that $H$ is normal subgroup of $G$. In Valenza's "Fourier Analysis on Number Fields" the proof is…
HeMan
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If $G$ is a paratopological group, is true that if $K$ is a compact subset of $G$ and $F$ is a closed subset of $G$ then $KF$ is closed in $G$?

A group $G$ endowed with a topology is called a paratopological group if the multiplication $G×G\to G$ is continuous. It is know that that in a topological group $G$ if $K$ is a compact subset of $G$ and $F$ is a closed subset of $G$ then $KF$ and…
Luis Prado
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