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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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n-copies of integers as an algebraic subgroup of p-adic integers

Let $\mathbb{Z}_p$ denote the additive group of $p$-adic integers. The group of integers $\mathbb{Z}$ can be viewed as a dense subgroup of $\mathbb{Z}_p$ generated by the element that projects to $1$ mod $p^n$ for all $n$. The same should be…
user294993
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Show that $\hat{G_1} \times \hat{G_2} \cong \widehat {G_1 \times G_2}$

Given a topological group $G$, I need to prove that $\hat{G_1} \times \hat{G_2} \cong \widehat {G_1 \times G_2}$ where $\hat{G} = \{ \chi | \ \chi : G \rightarrow S^1 \}$ $ \ \ , $ ($\chi $ is a continuous group homomorphism and $S^1$ is the…
Dark_Knight
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Why is the topology of characters determined by the open sets containing the trivial character?

Let $G$ be an abelian topological group, and let $\hat G$ denote the set of characters on $G$. Why is it true that if one has a topological basis of for the trivial character (say the topological of uniform convergence on compact sets), one has a…
Helmut
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Question about images of R under injective homomorphisms.

I'm studying for my topology comp and I'm at a bit of a loss on this question. (My experience with Algebra is very limited and my experience with topological groups in particular is almost non-existent.) Let $G$ be a topological group and $f: R \to…
Cros
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Why the multiplicative group $G_m$ is called a 1 dimensional torus?

I am reading a definition saying that an algebraic group over a field $K$ is called a torus if it is isomorphic to product of copies of the multiplicative group $G_m = K^*$. I don't understand why this definition, because $K^*$ is the affine line…
Long
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A topological group is embeddble in a product of a family of second-countable topological groups if and only if it is $\omega$-narrow

How to prove the following property: a topological group is topologically isomorphic to a subgroup of the product of some family of second-countable topological groups if and only if it is ω-narrow
lihong sheng
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Is the projection onto a quotient by a compact normal subgroup proper?

My ultimate goal is to prove the following: If $G$ is a locally compact group and $K_\alpha$ a net of compact normal subgroups with trivial intersection, then the inverse limit $\projlim G/K_\alpha$ is locally compact. (I want to prove this because…
Cronus
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There are infinitely many continuous characters on an infinite abelian topological group

Let $(G,\mathcal T)$ be an infinite abelian group and $\Bbb T$ be the circle group. Why there are infinitely many continuous homorphisms $f:G\to \Bbb T$? Is there a simple proof without using Pontryagin-Van Kampen theorem?
Minimus Heximus
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A separable locally compact group that is not metrizable and not compact

Hausdorffness assumed. All the usual suspects don't work: $\mathbb{Z}^\mathbb{R}$, $2^\mathbb{R}$, discrete $\mathbb{R}$, etc. My reasoning so far: if it is locally compact, then there are separable compact neighborhoods; of these, the ones that I…
Chrystomath
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what is the meaning of a power set of topological vector space?

Given a topological vector space, what is the power set of this space meaning? thanks a lot. and I really appreciate if a straightforward and simple explanation of topological vector space is illustrated too.
iDirk
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A semitopological qroup which is also quasitopological.

$(G,\mathcal T)$ is a semitopological group with $$(\forall x\in G)(\forall U\in \mathcal T)(\exists V \text{ neighborhood of }1)(x\notin U^cV)$$ Then $G$ is quasitopological (ie inverse is continuous). Any idea to prove it or to find missing…
user129173
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Two basic questions about topological group theory

For a topological group, I'd like to know whether 1.there exist a topological group G which is a Hausdorff space but does not satisfies the first countable axiom or 2.there exist a topological group G which is not a Hausdorff space and does not…
David Chan
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example of a discontinuous group operation in $\mathbb{R}$, under usual metric

someone has an example of that? I know that if G is a topological connectedness group ( operation continuous under the topology) then every neighborhood containing the identity elemental, generates the group, but i could not, find an example that…
Daniel
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References on Lie Group Double Coverings

I am currently studying spin geometry and when dealing with the spin groups I came across the notion of a "Lie group double covering", but the book I am following does not define this notion and I couldn't find a reference on this topic. Could…
Níckolas Alves
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$\phi$ homomorphism continouos injective to $S^1$. Then $\phi$ is identity or conjugate function.

Let $\phi: S^1 \rightarrow S^1$ homomorphism continouos injective. Then $\phi(x)= x$ or $\phi(x)= \overline{x}$ for all $x \in S^1$, where $\overline{x}$ is conjugate. Attempt. I proved $\phi$ is bijective function then $\phi$ topological…
PSW
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