Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, the theorem equally applies if the domain is not the closed unit ball itself but some set homeomorphic to it (and therefore also closed, bounded, connected, without holes, etc.).
The statement of the theorem is false if formulated for the open unit disk, the set of points with distance strictly less than 1 from the origin. Consider for example the function
In three dimensions the consequence of the Brouwer fixed point theorem is that no matter how much you stir or shake a cocktail in a glass some point in the liquid will remain in the exact same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, and that the liquid after stirring or shaking is contained within the space originally taken up by it.
Another consequence of the case n = 3 is given by an informational display of a map in, for example, an airport terminal. The function that sends points of the terminal to their image on the map is continuous and therefore has a fixed point, usually indicated by a cross or arrow with the text You are here. A similar display outside the terminal would violate the condition that the function is "to itself" and fail to have a fixed point. For this example, the existence of a fixed point is also a consequence of the Banach fixed point theorem, since the function mapping points in space to the display is a contraction mapping.
The argument proceeds by contradiction, supposing that a continuous function f : D n → D n has no fixed point, and then attempting to derive an inconsistency, which proves that the function must in fact have a fixed point. For each x in D n, there is only one straight line that passes through f(x) and x, because it must be the case that f(x) and x are distinct by hypothesis (recall that f having no fixed points means that f(x) ≠ x). Following this line from f(x) through x leads to a point on S n − 1, denoted by F(x). This defines a continuous function F : D n → S n − 1, which is a special type of continuous function known as a retraction: every point of the codomain (in this case S n − 1) is a fixed point of the function.
Intuitively it seems unlikely that there could be a retraction of D n onto S n − 1, and in the case n = 1 it is obviously impossible because S 0 (i.e., the endpoints of the closed interval D 1) is not even connected. The case n = 2 is less obvious, but can be proven by using basic arguments involving the fundamental groups of the respective spaces: the retraction would induce an injective group homomorphism from the fundamental group of S 1 to that of D 2, but the first group is isomorphic to Z while the latter group is trivial, so this is impossible. The case n = 2 can also be proven by contradiction based on a theorem about non-vanishing vector fields.
For n > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of homology groups: it can be shown that the homology Hn − 1(D n) is trivial, while Hn − 1(S n − 1) is infinite cyclic. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.
There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by noting that the map f can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the Weierstrass approximation theorem, for example. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction.
A quite different proof given by David Gale is based on the game of Hex. The basic theorem about Hex is that no game can end in a draw. This is equivalent to the Brouwer fixed point theorem for dimension 2. By considering n-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the determinacy theorem for Hex.
The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary Hilbert space instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not compact. For example, in the Hilbert space ℓ2 of square-summable real (or complex) sequences, consider the map f : ℓ2 → ℓ2 which sends a sequence (xn) from the closed unit ball of ℓ2 to the sequence (yn) defined by
The generalizations of the Brouwer fixed point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and in addition also often an assumption of convexity. See fixed point theorems in infinite-dimensional spaces for a discussion of these theorems.
The Kakutani fixed point theorem generalizes the Brouwer fixed point theorem in a different direction: it stays in Rn, but considers upper semi-continuous correspondences (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.
By using forcing to collapse cardinals, the Brouwer fixed point theorem can be generalized to arbitrary cardinality: if L is a compact, connected order topology, then any continuous function from Ln to itself has a fixed point. Note that if we require L to be separable, this is precisely the Brouwer fixed point theorem.
The Lefschetz fixed-point theorem applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of singular homology that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of D n.