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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.

2270 questions
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discrete normal subgroup of a connected group

could any one give me hint for this one? $G$ be a connected group, and let $H$ be a discrete normal subgroup of $G$, then we need to show $H$ is contained in the center of $G$ first of all, I have no clear idea what is meant by discrete subgroup and…
Myshkin
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Intersection of all neighborhoods of zero is a subgroup

Let $G$ be a topological abelian group. Let $H$ be the intersection of all neighborhoods of zero. How is $H = \mathrm{cl}(\{0\})$? Isn't the closure of a set $A$ the smallest closed set containing $A$ which is the same as the intersection of all…
Rudy the Reindeer
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Every Lindelöf topological group is isomorphic to a subgroup of the product of second countable topological groups.

I want to show that every Lindelöf topological group is isomorphic to a subgroup of the product of second countable topological groups. I received an answer using the fact that Lindelöf topological groups are $\omega$-narrow, but I want to show it…
Maria
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Is the following proof valid? About the closure of a subgroup, of a topological group, being again a subgroup

I found a proof in the book Fourier Analysis On Number Fields that the closure of any subgroup is a subgroup, using the continuity argument along with the nets. Nevertheless, the following proof seems also plausible, but is it valid? Statement:…
awllower
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Topology on integers making it a topological group

Are there non-trivial topologies (neither discrete nor indiscrete) on the additive group of integers $\mathbb{Z}$, making it into a topological group. Could someone list them all, possibly with some details?
Johnathon
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Finite Haar Measure if and only if Compact

This is an exercise from a book: Let $G$ be a locally compact group with Haar measure $\mu$. $\mu(\{e\})>0$ if and only if $G$ is discrete. $\mu(G)<\infty$ if and only if $G$ is compact. I think I can solve the first part: If $\mu(\{e\})>0$ then…
Gils
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Completion $\widehat{G}$ is a topological group

I want to verify that if $G$ is an abelian topological group then so is its completion $\widehat{G}$. Note that $G$ is not necessarily a metric space. Hence we define a sequence $x_n$ to be Cauchy if for every neighborhood $U$ of $0$ there exists an…
Rudy the Reindeer
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If $G$ is a compact topological group, how to show that a finite index subgroup of $G$ is open?

If $G$ is a compact topological group, how to show that a finite index subgroup of $G$ is open ? I really don't know where/how to start... PS : I precise that by "compact" I mean that it is hausdorff and that any recovering of $G$ by open sets has a…
AynRand
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Why is that $\widehat{\mathbb{R}/\mathbb{Z}}\cong\mathbb{Z}$?

$\widehat{\mathbb{R}/\mathbb{Z}}\cong\mathbb{Z}$, that is, every character of $\mathbb{R}/\mathbb{Z}$ is of the form $x\mapsto e(mx)$ for some integer $m$. I was considering the dual of ${\mathbb{R}/\mathbb{Z}}$. What confused me is why we can…
Qiang Zhang
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$G$ acts transitively on connected space, then so does identity component

Suppose $G$ is a topological group that acts on a connected topological space $X$. Show that if this action is transitive (and continuous), then so is the action of the identity component of the group.
fk44
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Why is any proper closed subgroup of $\mathbb{R}$ necessarily countable?

Possible Duplicate: Subgroup of $\mathbb{R}$ either dense or has a least positive element? If I have $G$ a closed subgroup of $\mathbb{R}$, then why is $G$ necessarily countable, except of course in the case where $G=\mathbb{R}$?
Addison
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Finding a certain subsemigroup of $(\mathbb R,+)$

Is there a subsemigroup $A$ of $(\mathbb R,+)$ such that it has a limit point and has no intersect with its limit points? Thanks for any hints.
Hamid Hoseni
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totally disconnected orbit-stabilizer theorem

So I'm aware that the orbit-stabilizer theorem does not hold for arbitrary spaces with a transitive action by a topological group, but I wonder if it works in the following situation. Let $G$ be a totally disconnected, locally compact Hausdorff…
Justin Campbell
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Strongly complete profinite group

Let $G$ be a profinite group (or equivalently a compact and totally disconnected topological group ) with the property that all of its normal subgroups of finite index are open sets. Does this imply that all of its subgroups of finite index are…
Radu Titiu
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Example of profinite groups

Could someone help me with an simple example of a profinite group that is not the p-adics integers or a finite group? It's my first course on groups and the examples that I've found of profinite groups are very complex and to understand them…
Andres
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