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May
18
comment Why is the projection map proper?
Thanks. When the degrees (the sum of multiplicities for each point) are the same, we can choose a neighborhood $W$ of $y_0$ such that the cardinality of $f^{-1}(y)$ is just the degree of $y_0$ for $y\in W\setminus\{y_0\}$, then my EDIT1 should work since the local behavior of $f$ is just $z\mapsto z^n$, right? At a glance, I think you're using Rouché theorem just to show the very fact of the local behavior. I don't like go into sequential compactness even for metric spaces.
May
18
revised Why is the projection map proper?
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May
18
accepted Why is the projection map proper?
May
18
revised Why is the projection map proper?
added 49 characters in body
May
18
revised Why is the projection map proper?
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May
18
revised Why is the projection map proper?
added 1304 characters in body
May
18
revised Why is the projection map proper?
added 1304 characters in body
May
18
comment Why is the projection map proper?
The proof seems quite involved. I got a plausible generalization and I hope you can help me to check it, and if fortunate, it could be used to simplify the proof and clarify the logic.
May
18
revised Polynomial that is surjective $\mod n$ for all $n$?
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May
17
answered Polynomial that is surjective $\mod n$ for all $n$?
May
17
comment Polynomial that is surjective $\mod n$ for all $n$?
and $x\mapsto\pm x+m$, $m\in\mathbb Z$.
May
17
revised Why is the projection map proper?
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May
17
asked Why is the projection map proper?
May
17
comment Prove that f is differentiable at $0$! Not continuous though, Right!?
It's differentiable at $0$ since $\lim_{x\to0}f(x)/x=0$, for $0\le\lvert f(x)/x\rvert\le\lvert x\rvert$. It's indeed continuous at $0$.
May
17
accepted Are maps locally preserving collinearity homographies?
May
16
asked Does Caratheodory's method work in Riesz-Markov representation theorem
May
16
asked Are maps locally preserving collinearity homographies?
May
1
comment Should diffeomorphisms preserving arc length be affine?
M.Berger's Geometry I, section 9.5.4.9. I was looking for a reference for projective geometry. A random walk led me to that...
Apr
27
comment Should diffeomorphisms preserving arc length be affine?
It seems that the proof is like the one of Liouville's theorem, i.e., the smooth conformal transformation of Euclidean space is just a similar transformation. The trick you used is so called the braid lemma (symmetric in the first two, anti-symmetric in the last two)?
Apr
25
comment Book suggestions on projective geometry
@user40276 Well, thanks. I'm now reading M.Berger's Geometry, and I found M.Reid's Geometry and Topology. It might not be the ultimate answer, but I decide to read them first. Incidentally, I did search for modular groups, etc, but eventually I found that I don't have enough mathematical maturity, so I postponed such a study process.