Is There A Generalization Of Brouwer's Fixed Point Theorem?

What is the simplest explanation of Brouwer's fixed point theorem?

• Answer:

  Disclaimer: I'm not a mathematician. Statement: Any continuous mapping f of a closed interval into itself has at least one fixed point, , that is, where . [This definition is from the book Fixed Points, by Yu. Shashkin.] Geometric proof: The mapping f is nothing more than a curve which connects a point on the left edge of a square to a point on the right edge. (Why is it a square? Hint: Range = Domain.) Now here's a sweet observation: Any such curve must cut through the line at one point at least. This point is a fixed point for its ordinate equals its abscissa and it lies on the curve f. In simpler terms, to connect the opposite edges of a square by a curve (keeping the curve within the square of course, which is guaranteed by the constraints on f), you must cut through the diagonal at least once (or an odd number of times to be more general). Note: Though the theorem holds for more generalized sets (compact convex sets), I thought the philosophy behind the general proof is pretty much contained in this straightforward example of a function of a single real variable. ----------------------------------------------------------------------------------------------------------------------