Tagged Questions

Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

0answers
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Topological manifold but not “smooth”

I would like to know if there are examples of topological manifolds that not admit a smooth atlas, or more generally a differentiable structure of class $C^k$ for $k\geq1$. In the book "Introduction ...
1answer
45 views

Can we have a continuous choice in the mean value theorem

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable. Must there exist a continuous function $g:\{(a,b)\in \mathbb{R}^2: a<b\}\rightarrow \mathbb{R}$ such that: For every two distinct real ...
3answers
287 views

0answers
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Example of a non semisimple Lie Algebra over $\mathbb{C}$ such that $L' = L$ [duplicate]

I wanted to show that if $L$ is a finite dimension semisimple Lie Algebra over $\mathbb{C}$, then $L' = L$ ($L' = [L,L]$) but the converse is not true. I could prove the statement but am unable ...
1answer
25 views

Equidecomposable examples

Decomposable: A set $S \subset \mathbb{R}^n$ is decomposable in $m$ sets $A_1,…,A_m \subset \mathbb{R}^n$ if there exist isometries $\phi_1,…,\phi_m:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that: ...
5answers
3k views