Brouwer Fixed Point Theorem
In mathematics, Brouwer's fixed point theorem is a theorem in topology, named after Luitzen Brouwer. It is one of many fixed point theorems, which state that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself. Among hundreds of fixed point theorems, Brouwer's is particularly well known, due in part to the fact that it is used across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem or the Borsuku2013Ulam theorem. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory, where John Nash used it to prove the existence of a winning strategy for the game Hex.
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