Brouwer's fixed-point theorem: Under a continuous mapping of an -dimensional simplex into itself there exists at least one point such that ; this theorem was proved by L.E.J. Brouwer [1]. An equivalent theorem had been proved by P.G. Bohl [2] at a somewhat earlier date. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an -dimensional topological vector space and is extensively employed in proofs of theorems on the existence of solutions of various equations. Brouwer's theorem can be generalized to infinite-dimensional topological vector spaces.

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There are many different proofs of the Brouwer fixed-point theorem. The shortest and conceptually easiest, however, use algebraic topology. Completely-elementary proofs also exist. Cf. e.g. [a1], Chapt. 4. In 1886, H. Poincaré proved a fixed-point result on continuous mappings which is now known to be equivalent to the Brouwer fixed-point theorem, [a2]. There are effective ways to calculate (approximate) Brouwer fixed points and these techniques are important in a multitude of applications including the calculation of economic equilibria, [a1]. The first such algorithm was proposed by H. Scarf, [a3]. Such algorithms later developed in the so-called homotopy or continuation methods for calculating zeros of functions, cf. e.g. [a4], [a5].

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Brouwer's theorem on the invariance of domain: Under any homeomorphic mapping of a subset of a Euclidean space into a subset of that space any interior point of (with respect to ) is mapped to an interior point of (with respect to ), and any non-interior point is mapped to a non-interior point. It was proved by L.E.J. Brouwer [1].

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For a modern account of the Brouwer invariance-of-domain theorem cf. [a1], Chapt. 7, Sect. 3. The result is important for the idea of the topological dimension ().

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