Editing Brouwer fixed-point theorem (section)

==Generalizations==
The Brouwer fixed-point theorem forms the starting point of a number of more general [[fixed-point theorem]]s.

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 space|compact]]. For example, in the Hilbert space [[Lp space|ℓ<sup>2</sup>]] of square-summable real (or complex) sequences, consider the map ''f'' : ℓ<sup>2</sup> → ℓ<sup>2</sup> which sends a sequence (''x''<sub>''n''</sub>) from the closed unit ball of ℓ<sup>2</sup> to the sequence (''y''<sub>''n''</sub>) defined by
:<math>y_0 = \sqrt{1 - \|x\|_2^2}\quad\text{ and}\quad y_n = x_{n-1} \text{ for } n \geq 1.</math>

It is not difficult to check that this map is continuous, has its image in the unit sphere of ℓ<sup>2</sup>, but does not have a fixed point.

The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and also often an assumption of [[Convex set|convexity]]. See [[fixed-point theorems in infinite-dimensional spaces]] for a discussion of these theorems.

There is also finite-dimensional generalization to a larger class of spaces: If <math>X</math> is a product of finitely many chainable continua, then every continuous function <math>f:X\rightarrow X</math> has a fixed point,<ref>{{cite journal|author=Eldon Dyer |year=1956|title=A fixed point theorem | journal=Proceedings of the American Mathematical Society| volume=7 | pages=662–672|doi=10.1090/S0002-9939-1956-0078693-4|issue=4|doi-access=free}}</ref> where a chainable continuum is a (usually but in this case not necessarily [[Metric space|metric]]) [[Compact space|compact]] [[Hausdorff space]] of which every [[open cover]] has a finite open refinement <math>\{U_1,\ldots,U_m\}</math>, such that <math>U_i \cap U_j \neq \emptyset</math> if and only if <math>|i-j| \leq 1</math>. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.

The [[Kakutani fixed point theorem]] generalizes the Brouwer fixed-point theorem in a different direction: it stays in '''R'''<sup>''n''</sup>, but considers upper [[hemi-continuous]] [[set-valued function]]s (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.

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''<sup>''n''</sup>.