Take , solve for and pick the nonnegative root ( denotes euclidean distance).
I just learned the Brouwer Fixed Point Theorem which states that , the 2-sphere, has the fixed point property. I understand most part of the proof by contradiction by assuming that for any map , then for every , but I got stuck on why the function defined by where is the intersection of the ray starting at passing through and leaves , 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 is clear since small pertubations of x produces small pertubations of , hence also small pertubations of the ray through these two points" How can I turn this into an proof?
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 . But, that map, call it , is given by where is the unique representation of an element of as the product of something in and . From there it's clear how to proceed.
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 and , and set its Euclidean distance equals to 1. So, I have . I tempted to say that , 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.