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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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Why is the quotient group a topological group?

I agree with the claim that translation by an element of the quotient group, $T_{[g]}:G/H \longrightarrow G/H$ Is a continuous function from $G/H$ to $G/H$, when $H$ is a normal subgroup of $G$. However, for $G/H$ to be a topological group with that…
Santiago Estupiñán
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For syndetic set $S\subseteq G$, is there $e\neq g\in G$ such that $gS\cap S$ is a syndetic set?

Let $G$ be a topological group, $S\subseteq G$ is a syndetic set if there is a compact set $K\subseteq G$ with $G=KS$. Let $\mathcal{P}$ be family of all syndetic sets in $G$, Question. For every $S\in \mathcal{P}$, Is there $e\neq g\in G$ with…
Ali Barzanouni
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The topology of a topological group is determined by the collection of neighborhoods of the identity element of the group

In Jacques Faraut's book "Analysis of Lie Groups", the author says that the topology of a topological group $G$ is determined by the collection of neighborhoods of the identity element of the group, i.e. is determined by the collection $\Omega =…
fer6268
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Proving that an embedding $G \hookrightarrow BC(G)$ is continuous.

I'm going over Professor Tao's presentation of the Birkhoff-Kakutani theorem and I don't see how it follows that $j = (g \mapsto \tau_gf)$ (between "Lemma 2" and "Remark 2") is continuous. I don't even see how you'd apply that uniform continuity…
kahen
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Classifying topologically cyclic groups

The group of p-adic integers has a dense cyclic subgroup, i.e. it is topologically cyclic What are some (or all) other non-trivial (i.e. non-discrete) examples of such groups?
user270792
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Open subgroups of quotient topological groups

Let $G$ be a topological abelian group and $H$ a closed subgroup of $G$. Is it true that an open subgroup of $G/H$ has the form $K/H$ where $K$ is an open subgroup of $G$ containing $H$?
Aliakbar
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Example for algebraic homomorphism between topological groups which is not continuous

I am quite sure there should be an easy example of: (algebraic) homomorphism between topological groups which is not continuous. However, I do not see one immediately.
Asaf Shachar
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Is a discrete subgroup of a Hausdorff group closed?

Let $G$ be a Hausdorff topological group. Let $H$ be a subgroup of $G$ such that $H$ is a discrete subspace of $G$. Is $H$ a closed subgroup of $G$? I thought this is obviously true, but I failed to prove it.
Makoto Kato
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Groups with no nontrivial topology

Does there exist a group $G$ such that $G$ has no topology on it such that $G$ is a topological group apart from the (in)discrete topology (or other such trivalish topologies)? I am asking as interested in the general methods that one construct a…
Meow
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In the quotient $G/H$, why we must suppose that $H$ is closed?

We have the following known statement: Theorem: If $G$ is a topological group and $H\subseteq G$ is a closed invariant subgroup of $G$, then $G/H$ (of course with the quotient topology) is a topological group. I'm reading the proof of this over and…
Surtan
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Closure of the connected component of the unity is connected: is my proof valid?

I have tried to prove that a closure of a connected component of the unity in a topological group is closed, but am not sure of its validity. Since it arose from a sentence in a book on the subject,* while the fact that any open subgroup is also…
awllower
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A paratopological group with intersection of neighborhoods of a point non-closed

Do you have an example of a (para)topological group $(G,\mathcal T)$ such that $\bigcap_{V\in \mathcal N_1} V$ is not closed? $\mathcal N _1$ is the set of all neighborbood of the identity element of $G$.
Minimus Heximus
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property of a topological group

This should be easy, but apparantly not for me. Let G be a topological group, and let $\mathcal{N}$ be a neighbourhood base for the identity element $e$ of $G$. Then for all $N_1,N_2 \in \mathcal{N}$, there exists an $N' \in \mathcal{N}$ such that…
Nadori
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Connected component of identity of an automorphism group is a subgroup

Let $D$ be a bounded domain in $\mathbb{C}^n$ and let Aut$(D)$ be the set of biholomorphic functions from $D$ to $D$. Define a metric on Aut$(D)$ by the supremum norm, $d(f,g) = \sup_{z\in D}|f(z)-g(z)|_{\mathbb{C}^n}$ and let Aut$^{Id}(D)$ to be…
mez
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Topological groups question from Munkres

This if from the 'Supplementary Exercises' at the end of Chapter 2 in Munkres' Topology. If $A$ and $B$ are subsets of (a topological group) $G$, let $A \cdot B$ denote the set of all points $a \cdot b$ for $a \in A$ and $b \in B$. Let $A^{-1}$…
user34832
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