Question about milnor's proof of hairy ball theorem Here is a link about the proof：
http://people.ucsc.edu/~lewis/Math208/hairyball.pdf
My question is： after lemma 2， Milnor takes the region A to be the region between two concentric spheres. Why can't we simply take the region A to be a closed ball？
Thanks in advance for taking the time to answer my question.
 A: A few years back I decided to have a look at Milnor's lovely proof, and blogged about it here. It's close to impossible to improve on anything that Milnor writes, but it's conceivable that a slightly different arrangement of his proof would clarify certain points. 
To answer the question: if you look carefully at how $f_t$ is defined, you'll see that it doesn't extend to a function defined on the closed ball centered at the origin. Milnor starts by defining $f_t$ on the unit sphere ${\|x\|} = 1$ by the mapping $x \mapsto x + tv(x)$ where $v$ is a unit vector field on the sphere. Then the instruction is to extend this map $f_t$ to a function on the shell $a \leq {\|x\|} \leq b$ (for $0 < a < 1 < b$) by "homothety", meaning we send $x$ in this shell to 
$${\|x\|} \cdot f_t(\frac{x}{{\|x\|}})$$ 
so that $x$ of radius (norm) $r$ in this shell gets mapped out to a point of radius $r(1 + t^2)^{1/2}$. There is just no way you'll be able to extend this to a continuous mapping on the closed ball (ask yourself: where would the origin go?). 
