Editing Brouwer fixed-point theorem (section)

===First proofs===
At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. [[Piers Bohl]], a [[Latvia]]n mathematician, applied topological methods to the study of differential equations.<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bohl.html Piers Bohl]''</ref> In 1904 he proved the three-dimensional case of our theorem,<ref name="Bohl1904" /> but his publication was not noticed.<ref>{{cite journal |first1=A. D. |last1=Myskis |first2=I. M. |last2=Rabinovic |title=Первое доказательство теоремы о неподвижной точке при непрерывном отображении шара в себя, данное латышским математиком П.Г.Болем |trans-title=The first proof of a fixed-point theorem for a continuous mapping of a sphere into itself, given by the Latvian mathematician P. G. Bohl |language=ru |journal=Успехи математических наук |volume=10 |issue=3 |year=1955 |pages=188–192 |url=http://mi.mathnet.ru/eng/umn/v10/i3/p179 }}</ref>

It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especially [[mathematical logic]] and [[topology]]. His initial interest lay in an attempt to solve [[Hilbert's fifth problem]].<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brouwer.html Luitzen Egbertus Jan Brouwer]''</ref> In 1909, during a voyage to Paris, he met [[Henri Poincaré]], [[Jacques Hadamard]], and [[Émile Borel]]. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved the [[hairy ball theorem]] for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point.<ref>{{cite journal |first=Hans  |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 495] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |doi-access=free }}</ref> These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem.<ref>{{cite journal |first=Hans |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 495] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |quote=... cette dernière propriété, bien que sous des hypothèses plus grossières, ait été démontré par H. Poincaré |doi-access=free }}</ref> The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as [[homotopy]], the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. [[Hans Freudenthal]] comments on the respective roles as follows: <!-- NON-LITERAL QUOTATION! translated back from French -->"Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."<ref>{{cite journal |first=Hans |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 501] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |doi-access=free }}</ref>

Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension,<ref name="brouwer-1910" /> as well as other key theorems such as the invariance of dimension.<ref>If an open subset of a [[manifold]] is [[homeomorphism|homeomorphic]] to an open subset of a Euclidean space of dimension ''n'', and if ''p'' is a positive integer other than ''n'', then the open set is never homeomorphic to an open subset of a Euclidean space of dimension ''p''.</ref> In the context of this work, Brouwer also generalized the [[Jordan curve theorem]] to arbitrary dimension and established the properties connected with the [[degree of a continuous mapping]].<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brouwer.html Luitzen Egbertus Jan Brouwer]''.</ref> This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became [[algebraic topology]].<ref>The term ''algebraic topology'' first appeared 1931 under the pen of David van Dantzig: J. Miller ''[http://jeff560.tripod.com/t.html Topological algebra]'' on the site Earliest Known Uses of Some of the Words of Mathematics (2007)</ref>