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==Brouwer's fixed-point theorem==
Under a continuous mapping $f : S \rightarrow S$ of an $n$-dimensional simplex $S$ into itself there exists at least one point $x \in S$ such that $f(x) = x$; this theorem was proved by L.E.J. Brouwer [[#References|[1]]]. An equivalent theorem had been proved by P.G. Bohl [[#References|[2]]] at a somewhat earlier date. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an $n$-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.

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L.E.J. Brouwer,   "Ueber eineindeutige, stetige Transformationen von Flächen in sich"  ''Math. Ann.'' , '''69'''  (1910)  pp. 176–180</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  P. Bohl,   "Ueber die Beweging eines mechanischen Systems in der Nähe einer Gleichgewichtslage"  ''J. Reine Angew. Math.'' , '''127'''  (1904)  pp. 179–276</TD></TR>
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====Comments====
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. [[#References|[a1]]], Chapt. 4. In 1886, H. Poincaré proved a fixed-point result on continuous mappings $f : \mathbf{E}^n \rightarrow \mathbf{E}^n$ which is now known to be equivalent to the Brouwer fixed-point theorem, [[#References|[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, [[#References|[a1]]]. The first such algorithm was proposed by H. Scarf, [[#References|[a3]]]. Such algorithms later developed in the so-called homotopy or continuation methods for calculating zeros of functions, cf. e.g. [[#References|[a4]]], [[#References|[a5]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Istrăţescu,   "Fixed point theory" , Reidel  (1981)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Poincaré,   "Sur les courbes definies par les équations différentielles"  ''J. de Math.'' , '''2'''  (1886)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Scarf,   "The approximation of fixed points of continuous mappings"  ''SIAM J. Appl. Math.'' , '''15'''  (1967)  pp. 1328–1343</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Karamadian (ed.) , ''Fixed points. Algorithms and applications'' , Acad. Press  (1977)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Allgower,   K. Georg,   "Simplicial and continuation methods for approximating fixed points and solutions to systems of equations"  ''SIAM Rev.'' , '''22'''  (1980)  pp. 28–85</TD></TR>
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==Brouwer's theorem on the invariance of domain==
Under any homeomorphic mapping of a subset $A$ of a Euclidean space $\mathbf{E}^n$ into a subset $B$ of that space any interior point of $A$ (with respect to $\mathbf{E}^n$) is mapped to an interior point of $B$ (with respect to $\mathbf{E}^n$), and any non-interior point is mapped to a non-interior point. It was proved by L.E.J. Brouwer [[#References|[1]]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  L.E.J. Brouwer,   "Ueber Abbildungen von Mannigfaltigkeiten"  ''Math. Ann.'' , '''71'''  (1912)  pp. 97–115</TD></TR>
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''M.I. Voitsekhovskii''

====Comments====
For a modern account of the Brouwer invariance-of-domain theorem cf. [[#References|[a1]]], Chapt. 7, Sect. 3. The result is important for the idea of the topological dimension ($\dim \mathbf{E}^n = n$).

====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,   "Topology" , Allyn &amp; Bacon  (1966)  (Theorem 8.4) {{ZBL|0144.21501}}</TD></TR>
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