Brouwer fixed-point theorem: Difference between revisions

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In [[mathematics]], the '''Brouwer fixed point theorem''' states that every [[continuous function]] from the closed unit [[Ball (mathematics)|ball]] ''D''^ ''n'' to itself has a [[fixed point (mathematics)|fixed point]]. In this theorem, ''n'' is any positive [[integer]], and the closed unit ball is the set of all points in [[Euclidean space|Euclidean ''n''-space]] '''R'''^''n'' which are at distance at most 1 from the origin. 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, connected, without holes, etcetera).

The theorem has several "real world" illustrations. One common informal version is "you can't comb a hairy ball smooth"; another one works as follows: take two equal size sheets of graph paper with coordinate systems on them, lay one flat on the table and crumple up (but don't rip) the other one and place it any way you like on top of the first. Then there will be at least one point of the crumpled sheet that lies exactly on top of the corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the ''n'' = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet right beneath it. Yet another example: an informational display of a map in, for example, an airport terminal. The function that sends points in real space 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.

The Brouwer fixed point theorem was one of the early achievements of [[algebraic topology]], and is the basis of more general fixed point theorems which are important in [[functional analysis]]. The case ''n'' = 3 was proved by [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] in 1909. [[Jacques Hadamard]] proved the general case in 1910, and Brouwer found a different proof in 1912. Since it must have an essentially non-constructive proof, it ran contrary to Brouwer's [[intuitionist]] ideals.