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Math Help - Brouwer fixed point theorem

  1. #1
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    Brouwer fixed point theorem

    I just learned the Brouwer Fixed Point Theorem which states that D^2, the 2-sphere, has the fixed point property. I understand most part of the proof by contradiction by assuming that for any map f: D^2 \rightarrow D^2, then f(x)\neq x for every x\in D^2, but I got stuck on why the function r(x): D^2 \rightarrow S^1 defined by r(x)=p_x where p_x is the intersection of the ray starting at f(x) passing through x and leaves D^2, is a continuous function. The author (Hatcher) gives an explanation for it, but I can't really justify it with a rigorous proof. He said that "continuity of r is clear since small pertubations of x produces small pertubations of f(x), hence also small pertubations of the ray through these two points" How can I turn this into an \epsilon-\delta proof?
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  2. #2
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    Take 1=|(1-s)x+sf(x)|, solve for s\in \mathbb{R} and pick the nonnegative root ( |.| denotes euclidean distance).
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Also, you only need to prove that the map which takes a point of the disc to its corresponding circle point is cont. since your map is the composition of that with f. But, that map, call it R, is given by R:ce^{i\theta}\mapsto e^{i\theta} where ce^{i\theta} is the unique representation of an element of \mathbb{D}^2 as the product of something in c\in(0,1) and e^{i\theta}\in\mathbb{S}^1. From there it's clear how to proceed.
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  4. #4
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    Quote Originally Posted by Jose27 View Post
    Take 1=|(1-s)x+sf(x)|, solve for s\in \mathbb{R} and pick the nonnegative root ( |.| denotes euclidean distance).
    Thank you very much for your help, Jose, but I don't really follow the hint (Sorry. I'm slow on this). So you write the equation of the ray passing through x and f(x), and set its Euclidean distance equals to 1. So, I have 1=\mid s(f(x)-x)+x\mid. I tempted to say that s=\frac{1-x}{f(x)-x}, but that does not seem right since the expression is Euclidean distance.

    Thanks a lot for your help too, Drexel. I'll try to work on your hint tomorrow morning. My brain is not working now.
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