 
Summary: The Brouwer Fixed Point Theorem.
Fix a positive integer n and let Dn
= {x Rn
: x 1}. Our goal is to prove
The Brouwer Fixed Point Theorem. Suppose
f : Dn
Dn
is continuous. Then f has a fixed point; that is, there is a Dn
such that f(a) = a.
This will follow quickly from the following
Theorem. You can't retract the ball to its boundary. There exists no continuous retraction
r : Dn
Sn1
.
(We say r : X Y is a retraction if Y X and r(y) = y whenever y Y .)
Indeed, suppose f : Dn
Dn
is continuous but has no fixed point. For each x Dn
let r(x) be the
point in Sn1
