Tagged Questions

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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Can $S^2$ be turned into a topological group?

I know that $S^1$ and $S^3$ can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that $S^2$ ...
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Conditions for a topological group to be a Lie group.

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157): Let $G$ be a locally compact group....
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Topological group: Multiplying two loops is homotopic to linking these paths?

Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group ...
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Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?

Could you please explain me the reason why they are isomorphic? Thanks, bye!
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Is $[0,1]$ a topological group?

Can one endow the unit interval $[0,1]$ with a group operation to make it a topological group under its natural Euclidean topology?
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Topological rings which are manifolds

Is the following statement true: "Every smooth manifold $M$, which is a ring in the category of manifolds, must be diffeomorphic to $\mathbb{R}^n$."? (Actually, homeomorphic would suffice.) I assume ...
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An equivalent definition of the profinite group

A profinite group is by defination a topological group $G$ which is Hausdorff , compact and totally disconnected. How to prove the following equivalent defination: A compact Hausdorff group is ...
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Topological groups are completely regular

I am studying topological groups, and I have been able to do quite a lot on my own by proving the propositions in this link on my own, but when I read up wikipedia that topological groups are all ...
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The Group of Homeomorphisms

I have been looking at Topological Groups, and I recently read about the group $\operatorname{Homeo}(X)$ of all homeomorphisms of $X$ onto itself. In particular, when $X$ is a metric space. The ...
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Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where $Hom(S^{1},X)$,...
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Theoretical Basis for Eigenvalue transformation on Bessel's Equation

The method I've been taught for finding all of the eigenvalue solutions to Bessel's operator $$b(f)\equiv f''(x)+\frac{1}{x}f'(x)$$ goes as follows. Let $g(a)=f(\sqrt{\lambda}x)$. Then b(g)=\lambda ...
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Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
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Lie Groups which are not Hausdorff

I suspect this isn't a terribly difficult question, but I don't know the answer and I'd guess someone has already looked into it. Is it possible for a Lie group on a non-Hausdorff manifold to exist? ...
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 ...