Theorem.
Let B={x∈Rn : ||x||≤1} be the closed unit ball in Rn. Any continuous function f:B→B has a fixed point a=f(a).
Notes
- Shape
-
If D is homeomorphic to B, then Brouwer's theorem also applies to D, and any g:D→D also has some fixed point. This includes any closed balls, squares, rectangles, cubes, boxes, hypercubes, polygons without holes, oranges, tomatoes, triangles, simplices, etc., all of them closed.
- Compactness (i)
-
The theorem doesn't apply to Rn. For instance, any nonzero translation Tv(x)=x+v has no fixed point.
- Compactness (ii)
-
The theorem doesn't apply to the open unit ball B={x∈Rn : ||x||<1}. For instance, f(x) = x/2 + (1/2,0,...,0) has no fixed point on B -- its only fixed point is at (1,0,...,0)∉B.
- Simple connectedness
-
The theorem doesn't apply to torii. These have various "rotations" with no fixed points. For instance, any nonzero rotation of S1={x∈R2 : ||x||=1} has no fixed point. So any topological space T=S1×U, no matter what U is, is a counterexample.
Brouwer's theorem is truly remarkable. It requires next-to-nothing of f (just continuity), and gives a great property of f! So it is probably best regarded as a useful property of topological spaces homeomorphic to closed balls.
It is also nonconstructive. This is of necessity: what could we possibly know about f? But it means we don't even know if the fixed point of f is on the boundary of B or in its interior: it can be anywhere. The theorem is just too generic to yield a construction.
This result must have been deeply unsatisfying to Brouwer. Later, Brouwer founded the constructivist discipline of intuitionism. Constructivists reject Brouwer's theorem(!), although they will accept any particular case of a given function f and its fixed point(s).
Sperner's lemma (which see) yields a nice combinatorical proof of Brouwer's fixed point theorem. There are also proofs from algebraic topology, of course.