Brouwer fixed-point theorem

3430: 879: 1042:, there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge." He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision". 1206: 5368:. The basic theorem regarding Hex, first proven by John Nash, is that no game of Hex can end in a draw; the first player always has a winning strategy (although this theorem is nonconstructive, and explicit strategies have not been fully developed for board sizes of dimensions 10 x 10 or greater). This turns out to be equivalent to the Brouwer fixed-point theorem for dimension 2. By considering 2977: 5058:, which is also non-singular for the restriction to the boundary (which is just the identity). Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction. 861:

The theorem is supposed to have originated from Brouwer's observation of a cup of gourmet coffee. If one stirs to dissolve a lump of sugar, it appears there is always a point without motion. He drew the conclusion that at any moment, there is a point on the surface that is not moving. The fixed point

1084:

defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing". In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem, although the connection with the subject of this article was not yet

865:

Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet." Brouwer "flattens" his sheet as with a

944:

Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with

841:

In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a delicious cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any

1167:. 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 846:

defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid). In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state.

830:

Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at

1171:

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. These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem. The revolutionary

1069:. Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval 6003:

variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same

3924: 842:

action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume. Ordering a cocktail

4172: 3236: 889:

is defined on a closed interval and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval which maps

6643:"concerne les propriétés invariantes d'une figure lorsqu'on la déforme de manière continue quelconque, sans déchirure (par exemple, dans le cas de la déformation de la sphère, les propriétés corrélatives des objets tracés sur sa surface". From C. Houzel M. Paty 2836: 1180:

comments on the respective roles as follows: "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."

5732: 866:

flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as

1854: 672:

which is a continuous function from the open interval (−1,1) to itself. Since x = 1 is not part of the interval, there is not a fixed point of f(x) = x. The space (−1,1) is convex and bounded, but not closed. On the other hand, the function

1184:

Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension, as well as other key theorems such as the invariance of dimension. In the context of this work, Brouwer also generalized the

1121: 234:, Brouwer's is particularly well known, due in part to its use 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 1239:

functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the

862:

is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit. The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.

5480: 3744: 2364: 1295:

there is a winning strategy for white. In economics, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria

2245: 6388:

The interest of this anecdote rests in its intuitive and didactic character, but its accuracy is dubious. As the history section shows, the origin of the theorem is not Brouwer's work. More than 20 years earlier

4039: 4360: 838:

Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.

3126: 1142:

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

6152:

Il en a démontré l'un des plus beaux théorèmes, le théorème du point fixe, dont les applications et généralisations, de la théorie des jeux aux équations différentielles, se sont révélées fondamentales.

5280: 835:= 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 immediately beneath it. 1131:

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.

1176:, 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. 5914: 2972:{\displaystyle {\mathbf {w} }({\mathbf {x} })=(1-{\mathbf {x} }\cdot {\mathbf {f} }({\mathbf {x} }))\,{\mathbf {x} }-(1-{\mathbf {x} }\cdot {\mathbf {x} })\,{\mathbf {f} }({\mathbf {x} }).} 5081:

to the fixed point so the method is essentially computable. gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.

6287:

This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see

5868: 5200: 5122: 1139:

mathematician, applied topological methods to the study of differential equations. In 1904 he proved the three-dimensional case of our theorem, but his publication was not noticed.

667: 297:

requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the

2117: 5740:

The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and also often an assumption of

4443: 4256: 1389: 4771: 4562: 2368:

defines a homotopy from the identity function to it. The identity function has degree one at every point. In particular, the identity function has degree one at the origin, so

831:

least one point of the crumpled sheet that lies directly above its corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the

788:-dimensional sphere (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (and so it is not convex) . The function 5626: 4860: 1982: 4399: 4211: 5801: 2531:.) Sometimes the theorem is expressed by the statement that "there is always a place on the globe with no wind". An elementary proof of the hairy ball theorem can be found in 1418: 5745: 3514:

sending each point in the disk to its corresponding intersection point on the boundary. As a special case, whenever x itself is on the boundary, then the intersection point

1261: 138: 2014: 1753: 1045:

He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant

6587: 4679: 4619: 4031: 975: 603: 581: 5956: 901:

Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function

2464: 2425: 1681: 1509: 556: 5531: 5336:, taking into account multiplicity and orientation), and should remain constant (as it is very clear in the one-dimensional case). On the other hand, as the parameter 1450: 498: 4975: 774: 7222:

Later it would be shown that the formalism that was combatted by Brouwer can also serve to formalise intuitionism, with some modifications. For further details see

2061: 721: 89: 6328: 6115: 4934: 1606: 1023: 1015: 6031: 5021: 4883: 1272:, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the 1220:

The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called

315:

The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows:

5769: 4998: 4905: 4811: 4719: 4699: 4639: 4585: 4503: 4483: 4463: 2386: 2140: 2034: 1916: 1896: 1745: 1725: 1705: 1646: 1626: 1577: 1553: 1533: 1474: 221: 201: 178: 158: 55: 6648: 5392:

to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number

5073:, on the boundary, (assuming it is not a fixed point) the one manifold with boundary mentioned above does exist and the only possibility is that it leads from 3643:. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group. 692:

Convexity is not strictly necessary for Brouwer's fixed-point theorem. Because the properties involved (continuity, being a fixed point) are invariant under

6072: 3919:{\displaystyle 0<\int _{\partial B}\omega =\int _{\partial B}F^{*}(\omega )=\int _{B}dF^{*}(\omega )=\int _{B}F^{*}(d\omega )=\int _{B}F^{*}(0)=0,} 6501:

L'identité algébrique d'une pratique portée par la discussion sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des planètes

3686:

As in the proof of Brouwer's fixed-point theorem for continuous maps using homology, it is reduced to proving that there is no continuous retraction

6797:[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]. 5398: 8134: 2255: 1193:. This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became 8093: 2148: 7940: 7852: 7825: 7774: 6738: 6483: 6458: 4167:{\displaystyle \Delta ^{n}=\left\{P\in \mathbb {R} ^{n+1}\mid \sum _{i=0}^{n}{P_{i}}=1{\text{ and }}P_{i}\geq 0{\text{ for all }}i\right\}.} 8144: 6409: 6251: 3231:{\displaystyle {\mathbf {X} }({\mathbf {x} },t)=(-t\,{\mathbf {w} }({\mathbf {x} }),{\mathbf {x} }\cdot {\mathbf {w} }({\mathbf {x} })).} 6761: 4264: 6348: 6048: 5958:. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers. 7868: 4793:

is continuous, this simplex can be made arbitrarily small by choosing an arbitrarily fine triangulation. Hence, there must be a point

6606: 5590:

instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not

5348:, which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area of 294: 7959: 7715: 7684: 7616: 7585: 7492: 7265: 7043: 7022: 6705: 6674: 6300: 6275: 6139: 5043: 3711: 2710: 6795:"Первое доказательство теоремы о неподвижной точке при непрерывном отображении шара в себя, данное латышским математиком П.Г.Болем" 5737:

It is not difficult to check that this map is continuous, has its image in the unit sphere of ℓ, but does not have a fixed point.

5054:. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by 1085:

apparent. A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the

8037: 7139: 6082: 7928: 5208: 1512: 7173: 2800:

The continuous version of the hairy ball theorem can now be used to prove the Brouwer fixed point theorem. First suppose that

7798: 5976:(functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set. 1333: 1190: 325: 738:

The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function

250:. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about 6566: 8139: 8129: 8059: 7261: 7193: 6697: 5980: 1277: 1253: 1031: 987:

proved the general case in 1910, and Brouwer found a different proof in the same year. Since these early proofs were all

799: 1097: 7987: 6201: 5962: 1308: 999: 980: 400: 263: 31: 6759:

Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires

8054: 6692: 5069:

proof by observing that the retract is in fact defined everywhere except at the fixed points. For almost any point,

1303:

Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems are

1108:. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a 1081: 6323: 6108: 5873: 3938:

onto its boundary. The proof using Stokes' theorem is closely related to the proof using homology, because the form

7951: 7790: 7644: 6062: 1152: 1101: 1089:

or sometimes the Poincaré group. This method can be used for a very compact proof of the theorem under discussion.

930: 6645: 3932:

More generally, this shows that there is no smooth retraction from any non-empty smooth oriented compact manifold

1224:. Brouwer's theorem is probably the most important. It is also among the foundational theorems on the topology of 6088: 1256:

provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to

203:

to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset

5987:

that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of

5042:

can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the

2473:

This requires some work to make fully general. The definition of degree must be extended to singular values of

696:, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball 6517: 6393:

had proved an equivalent result, and 5 years before Brouwer P. Bohl had proved the three-dimensional case.

1556: 6529: 6038: 5568: 5554: 1241: 247: 7034:"... Brouwer's fixed point theorem, perhaps the most important fixed point theorem." p xiii V. I. Istratescu 5821: 5127: 5092: 7223: 6897:... cette dernière propriété, bien que sous des hypothèses plus grossières, ait été démontré par H. Poincaré 6754: 6067: 3429: 3386:),0). The advantage of this proof is that it uses only elementary techniques; more general results like the 1269: 1093: 1022: 1014: 619: 290: 285:

The theorem was first studied in view of work on differential equations by the French mathematicians around

243: 5727:{\displaystyle y_{0}={\sqrt {1-\|x\|_{2}^{2}}}\quad {\text{ and}}\quad y_{n}=x_{n-1}{\text{ for }}n\geq 1.} 2070: 267: 4404: 4216: 3608: 3607:> 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of 1350: 5493:

is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zero

4724: 4515: 798:

A formal generalization of Brouwer's fixed-point theorem for "hole-free" domains can be derived from the

8113: 8081: 6210: 6174: 4816: 3530: 1921: 1337: 1273: 878: 255: 251: 6981: 6817: 4368: 4180: 723:. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also 6430:

Bohl, P. (1904). "Über die Bewegung eines mechanischen Systems in der Nähe einer Gleichgewichtslage".

5774: 1849:{\displaystyle \operatorname {deg} _{p}(f)=\sum _{x\in f^{-1}(p)}\operatorname {sign} \,\det(df_{x}).} 1394: 1252:

has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the

1076:

To understand differential equations better, a new branch of mathematics was born. Poincaré called it

1034:

returned into the focus of the mathematical community. Its solution required new methods. As noted by

8049: 7877: 7171:

Une extension discontinue du théorème du point fixe de Schauder, et quelques applications en économie

6918: 6877: 6839: 6687: 2499:

in an odd-dimensional Euclidean space, there is no nowhere-vanishing continuous tangent vector field

1684: 1312: 1311:. He became the originator and zealous defender of a way of formalising mathematics that is known as 1229: 1225: 1186: 1003: 235: 8090: 7513: 6780: 6730: 6724: 5586:

The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary

1867:, with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of 94: 8012: 7539: 5973: 5580: 5561: 5550: 5494: 5372:-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the 3403: 3387: 2995: 2616: 1987: 1265: 1236: 1109: 962: 958: 843: 271: 231: 35: 23: 4654: 4594: 4006: 586: 564: 8021: 7750: 7556: 7509: 7409: 7077: 6521: 6227: 6000: 5919: 3976: 3943: 3647: 3577: 3391: 2490: 2481:

simplifies the construction of the degree, and so has become a standard proof in the literature.

1304: 1221: 1194: 1168: 1144: 1073:. If the area is a circular band, or if it is not closed, then this is not necessarily the case. 1046: 1039: 988: 954: 938: 332: 239: 181: 6026: 3988: 3735: 1297: 1026:

The theorem applies to any disk-shaped area, where it guarantees the existence of a fixed point.

6154: 5089:

A variation of the preceding proof does not employ the Sard's theorem, and goes as follows. If

2430: 2391: 1651: 1479: 517: 7955: 7936: 7848: 7821: 7794: 7770: 7711: 7680: 7612: 7581: 7039: 7018: 6966:, then the open set is never homeomorphic to an open subset of a Euclidean space of dimension 6734: 6720: 6701: 6670: 6558: 6542: 6513: 6479: 6454: 6390: 6296: 6271: 6246: 6135: 6043: 5984: 5803:

has a fixed point, where a chainable continuum is a (usually but in this case not necessarily

5500: 5055: 3573: 1423: 1156: 1086: 1035: 462: 286: 7866:(1976). "A constructive proof of the Brouwer fixed point theorem and computational results". 7781:(see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction) 6205: 4942: 741: 432:(functions that have the same set as the domain and codomain) and for nonempty sets that are 7885: 7742: 7679:. Cahiers Scientifiques (in French). Vol. IX. Paris: Gauthier-Villars. pp. 44–47. 7653: 7548: 7501: 7462: 7401: 7205: 7156: 7115: 7086: 6927: 6909: 6886: 6868: 6848: 6830: 6758: 6345: 6290: 6219: 6169: 6077: 5983:

applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of

5365: 1292: 1288: 1268:. One also meets the theorem and its variants outside topology. It can be used to prove the 1209: 1177: 1160: 1124: 984: 732: 304: 279: 8033: 7995: 7969: 7897: 7835: 7808: 7725: 7694: 7667: 7626: 7595: 7568: 7521: 6574:, on the website of l'Association roumaine des chercheurs francophones en sciences humaines 2039: 1307:, and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of 1018:

For flows in an unbounded area, or in an area with a "hole", the theorem is not applicable.

699: 67: 8097: 8029: 7991: 7965: 7893: 7831: 7804: 7721: 7690: 7663: 7622: 7608: 7591: 7564: 7517: 7252: 7177: 6765: 6652: 6570: 6352: 6332: 6255: 6119: 5970: 5811: 4910: 2478: 1582: 728: 583:

to itself. As it shifts every point to the right, it cannot have a fixed point. The space

355: 224: 7505: 7881: 7057: 5003: 4936:

must be equal, all these inequalities must actually be equalities. But this means that:

4865: 1172:

aspect of Brouwer's approach was his systematic use of recently developed tools such as

8004: 7981: 7863: 7635: 5754: 5062: 5035: 4983: 4890: 4796: 4704: 4684: 4624: 4570: 4488: 4468: 4448: 2371: 2125: 2019: 1901: 1881: 1868: 1730: 1710: 1690: 1631: 1611: 1562: 1538: 1518: 1459: 1205: 1164: 991: 206: 186: 163: 143: 40: 8107: 7658: 7639: 7487: 7467: 7450: 885:

In one dimension, the result is intuitive and easy to prove. The continuous function

8123: 7762: 7707: 7640:"Finding zeroes of maps: Homotopy methods that are constructive with probability one" 7248: 7134: 6951: 6932: 6913: 6891: 6872: 6853: 6834: 6231: 6179: 5808: 5591: 5587: 5579:

The Brouwer fixed-point theorem forms the starting point of a number of more general

5066: 5051: 5031: 3719: 1453: 1105: 1054: 867: 693: 441: 377: 275: 4775:

By construction, this is a Sperner coloring. Hence, by Sperner's lemma, there is an

2725:

into Euclidean space. The orthogonal projection on to the tangent space is given by

998:

ideals. Although the existence of a fixed point is not constructive in the sense of

8085: 7170: 6585:

Poincaré, H. (1886). "Sur les courbes définies par les équations différentielles".

6525: 5999:

There are several fixed-point theorems which come in three equivalent variants: an

5804: 5352:

is necessarily 0, as its image is the boundary of the ball, a set of null measure.

3640: 3597: 1257: 995: 410: 7820:. Mathematics and its Applications. Vol. 7. Dordrecht–Boston, MA: D. Reidel. 7119: 5965:

generalizes the Brouwer fixed-point theorem in a different direction: it stays in

4651:

We now use this fact to construct a Sperner coloring. For every triangulation of

8005:"Analytic proofs of the 'hairy ball theorem' and the Brouwer fixed-point theorem" 7580:. Pure and Applied Mathematics. Vol. 120 (Second ed.). Academic Press. 5771:

is a product of finitely many chainable continua, then every continuous function

1276:. The theorem can also be found in existence proofs for the solutions of certain 7977: 6186:(Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477 (French) 6113: 5751:

There is also finite-dimensional generalization to a larger class of spaces: If

5373: 5047: 3731: 3715: 1284: 1213: 1066: 1062: 351: 259: 58: 7537:

Boothby, William M. (1971). "On two classical theorems of algebraic topology".

6563: 6292:

General Equilibrium Analysis: Existence and Optimality Properties of Equilibria

3596:= 2 can also be proven by contradiction based on a theorem about non-vanishing 1340:. Several modern accounts of the proof can be found in the literature, notably 274:

in market economies as developed in the 1950s by economics Nobel prize winners

8101: 7483: 5815: 5741: 5475:{\displaystyle \displaystyle \sum _{n}(-1)^{n}\operatorname {Tr} (f|H_{n}(B))} 5361: 3572:= 2 is less obvious, but can be proven by using basic arguments involving the 1316: 1132: 1058: 970: 815: 724: 444:

to convex). The following examples show why the pre-conditions are important.

374: 329: 61: 5344:

transforms continuously from the identity map of the ball, to the retraction

5285:

Differentiating under the sign of integral it is not difficult to check that

3331:) are both non-zero). This contradiction proves the fixed point theorem when 503:

with domain . The range of the function is . Thus, f is not an endomorphism.

6794: 5571:, so this gives a precise description of the strength of Brouwer's theorem. 3723: 2359:{\displaystyle H(t,x)={\frac {x-tf(x)}{\sup _{y\in K}\left|y-tf(y)\right|}}} 826:

The theorem has several "real world" illustrations. Here are some examples.

811: 7091: 7072: 1100:. Picard's approach is based on a result that would later be formalised by 1049:? Poincaré discovered that the answer can be found in what we now call the 795:

have a fixed point for the unit disc, since it takes the origin to itself.

7948:

From calculus to cohomology: de Rham cohomology and characteristic classes

7209: 7106:

Kakutani, S. (1941). "A generalization of Brouwer's Fixed Point Theorem".

7004:

on the site Earliest Known Uses of Some of the Words of Mathematics (2007)

6548:

T Gauthier-Villars, Vol 3 p 389 (1892) new edition Paris: Blanchard, 1987.

6367: 5077:

to a fixed point. It is an easy numerical task to follow such a path from

6947: 5595: 3534: 1189:

to arbitrary dimension and established the properties connected with the

1173: 1148: 1050: 27: 2240:{\displaystyle g(x)={\frac {x-f(x)}{\sup _{y\in K}\left|y-f(y)\right|}}} 1859:

The degree is, roughly speaking, the number of "sheets" of the preimage

8025: 7754: 7560: 7413: 7143: 6223: 4779:-dimensional simplex whose vertices are colored with the entire set of 4000: 776:, which is a continuous function from the unit circle to itself. Since 140:. The simplest forms of Brouwer's theorem are for continuous functions 7392:

David Gale (1979). "The Game of Hex and Brouwer Fixed-Point Theorem".

1120: 6346:

Théorèmes du Point Fixe et Applications aux Equations Différentielles

3423: 1136: 953:

The Brouwer fixed point theorem was one of the early achievements of

262:. In economics, Brouwer's fixed-point theorem and its extension, the 7923: 7889: 7746: 7733:

Gale, D. (1979). "The Game of Hex and Brouwer Fixed-Point Theorem".

7552: 7405: 3991:. We now give an outline of the proof for the special case in which 3529:

Consequently, F is a special type of continuous function known as a

1319:. Brouwer disavowed his original proof of the fixed-point theorem. 1065:, then the trajectory either becomes stationary, or it approaches a 7578:

An introduction to differentiable manifolds and Riemannian geometry

7000: 6656:

Encyclopædia Universalis Albin Michel, Paris, 1999, p. 696–706

3674:

would have to be contractible and its de Rham cohomology in degree

3576:

of the respective spaces: the retraction would induce a surjective

3343:

odd, one can apply the fixed point theorem to the closed unit ball

2388:

also has degree one at the origin. As a consequence, the preimage

7907:"An integral theorem and its applications to coincidence theorems" 6998:

first appeared 1931 under the pen of David van Dantzig: J. Miller

6500: 5034:, based on the impossibility of a differentiable retraction. The 4355:{\displaystyle \sum _{i=0}^{n}{P_{i}}=1=\sum _{i=0}^{n}{f(P)_{i}}} 3592:

while the first group is trivial, so this is impossible. The case

3548:

Intuitively it seems unlikely that there could be a retraction of

3428: 2806:

is even. If there were a fixed-point-free continuous self-mapping

1204: 1119: 1053:

properties in the area containing the trajectory. If this area is

1021: 1013: 394:

An even more general form is better known under a different name:

5598:

of square-summable real (or complex) sequences, consider the map

2578:

sufficiently small, a routine computation shows that the mapping

307:

and the general case for continuous mappings by Brouwer in 1911.

7488:"A Borsuk–Ulam equivalent that directly implies Sperner's lemma" 6520:'s mathematical competition in 1889 for his work on the related 6414: 5384:

The Lefschetz fixed-point theorem says that if a continuous map

3474:) are distinct. Because they are distinct, for every point x in 405:

Every continuous function from a nonempty convex compact subset

6914:"The cradle of modern topology, according to Brouwer's inedita" 6873:"The cradle of modern topology, according to Brouwer's inedita" 6835:"The cradle of modern topology, according to Brouwer's inedita" 6504:

CNRS Fédération de Recherche Mathématique du Nord-Pas-de-Calais

2477:, and then to continuous functions. The more modern advent of 1228:

and is often used to prove other important results such as the

5332:) (that is, the Lebesgue measure of the image of the ball via 3646:

The impossibility of a retraction can also be shown using the

810:

The continuous function in this theorem is not required to be

7906: 5124:

is a smooth retraction, one considers the smooth deformation

6407:

This citation comes originally from a television broadcast:

1006:

fixed points guaranteed by Brouwer's theorem are now known.

877: 7704:

A history of algebraic and differential topology, 1900–1960

6726:

A History of Algebraic and Differential Topology, 1900–1960

4813:

which satisfies the labeling condition in all coordinates:

3670:- 1, and vanishes otherwise. If a retraction existed, then 7058:

Brouwer's Fixed Point Theorem and the Jordan Curve Theorem

6368:"Why is convexity a requirement for Brouwer fixed points?" 5485:

and in particular if the Lefschetz number is nonzero then

1332:

Brouwer's original 1911 proof relied on the notion of the

1216:

to prove the existence of an equilibrium strategy profile.

945:

respect to its original position on the unfolded string."

341:

This can be generalized to an arbitrary finite dimension:

5304:

is a constant function, which is a contradiction because

5275:{\displaystyle \varphi (t):=\int _{B}\det Dg^{t}(x)\,dx.} 2560:. It can be extended radially to a small spherical shell 2678:. This gives a contradiction, because, if the dimension 2554:

is a continuously differentiable unit tangent vector on

2466:

are precisely the fixed points of the original function

784:

has no fixed point. The analogous example works for the

7675:

Dieudonné, Jean (1982). "8. Les théorèmes de Brouwer".

6175:

Note sur quelques applications de l'indice de Kronecker

3494:(see illustration). By calling this intersection point 3440:

Suppose, for contradiction, that a continuous function

2713:, it can be uniformly approximated by a polynomial map 1030:

At the end of the 19th century, the old problem of the

7787:

Homology theory: An introduction to algebraic topology

7194:"L. J. E. Brouwer : Topologie et constructivisme" 6184:

Introduction à la théorie des fonctions d'une variable

7368: 6669:

Kluwer Academic Publishers (réédition de 2001) p 113

6150:

More exactly, according to Encyclopédie Universalis:

5922: 5876: 5824: 5777: 5757: 5629: 5503: 5402: 5401: 5211: 5130: 5095: 5006: 4986: 4945: 4913: 4893: 4868: 4819: 4799: 4727: 4707: 4687: 4657: 4627: 4597: 4573: 4518: 4491: 4471: 4451: 4407: 4371: 4267: 4219: 4183: 4042: 4009: 3747: 3722:

of sufficiently small support and integral one (i.e.

3129: 2839: 2433: 2394: 2374: 2258: 2151: 2128: 2073: 2042: 2022: 1990: 1924: 1904: 1884: 1756: 1733: 1713: 1693: 1654: 1634: 1614: 1585: 1565: 1541: 1521: 1482: 1462: 1426: 1420:

centered at the origin. Suppose for simplicity that

1397: 1353: 744: 702: 622: 589: 567: 520: 465: 303:-dimensional closed ball was first proved in 1910 by 209: 189: 166: 146: 97: 70: 43: 7911:

Acta Universitatis Carolinae. Mathematica et Physica

6954:

to an open subset of a Euclidean space of dimension

6478:. Dordrecht-Boston, Mass.: D. Reidel Publishing Co. 3490:

and follow the ray until it intersects the boundary

2625:

and that the volume of its image is a polynomial in

680:

have a fixed point for the closed interval , namely

6247:

Applications du lemme de Sperner pour les triangles

5746:

fixed-point theorems in infinite-dimensional spaces

3458:fixed point. This means that, for every point x in 3247:is a continuous vector field on the unit sphere of 1096:, a contemporary mathematician who generalized the 7254:Topics in Linear and Nonlinear Functional Analysis 6665:Poincaré's theorem is stated in: V. I. Istratescu 5950: 5908: 5862: 5795: 5763: 5726: 5611:) from the closed unit ball of ℓ to the sequence ( 5525: 5474: 5274: 5194: 5116: 5015: 4992: 4969: 4928: 4899: 4877: 4854: 4805: 4765: 4713: 4693: 4673: 4633: 4613: 4579: 4556: 4497: 4477: 4457: 4437: 4393: 4354: 4250: 4205: 4166: 4025: 3918: 3710:is smooth, since it can be approximated using the 3230: 2971: 2458: 2419: 2380: 2358: 2239: 2134: 2111: 2055: 2028: 2008: 1976: 1918:be two continuously differentiable functions, and 1910: 1890: 1848: 1739: 1719: 1699: 1675: 1640: 1620: 1600: 1571: 1547: 1535:is non-singular at every point of the preimage of 1527: 1503: 1468: 1444: 1412: 1383: 1235:Besides the fixed-point theorems for more or less 1151:. His initial interest lay in an attempt to solve 768: 715: 661: 597: 575: 550: 492: 215: 195: 172: 152: 132: 83: 49: 4648:coordinates which are not zero on this sub-face. 1264:, a result generalized further by S. Kakutani to 7924:A First Course in Sobolev Spaces: Second Edition 7607:. Graduate Texts in Mathematics. Vol. 139. 7455:Proceedings of the American Mathematical Society 5553:, Brouwer's theorem can be proved in the system 5237: 2307: 2191: 1821: 1300:), financial equilibria and incomplete markets. 7935:. American Mathematical Society. pp. 734. 7300: 6165: 6163: 6085:– equivalent to the Brouwer fixed-point theorem 5380:A proof using the Lefschetz fixed-point theorem 2631:. On the other hand, as a contraction mapping, 976:Journal für die reine und angewandte Mathematik 364:A slightly more general version is as follows: 254:and is covered in most introductory courses on 180:in the real numbers to itself or from a closed 7356: 7038:Kluwer Academic Publishers (new edition 2001) 7017:Kluwer Academic Publishers (new edition 2001) 6546:Les méthodes nouvelles de la mécanique céleste 6196: 6194: 6192: 5909:{\displaystyle U_{i}\cap U_{j}\neq \emptyset } 4365:Hence, by the pigeonhole principle, for every 3560:= 1, the impossibility is more basic, because 2122:If there is no fixed point of the boundary of 1291:used the theorem to prove that in the game of 436:(thus, in particular, bounded and closed) and 428:The theorem holds only for functions that are 4258:Hence the sum of their coordinates is equal: 8: 6588:Journal de Mathématiques Pures et Appliquées 5857: 5825: 5658: 5651: 4432: 4414: 3957:) which is isomorphic to the homology group 3564:(i.e., the endpoints of the closed interval 2791:|| is a smooth unit tangent vector field on 1155:. In 1909, during a voyage to Paris, he met 8082:Brouwer's Fixed Point Theorem for Triangles 7426: 7155:For context and references see the article 6425: 6423: 6073:Infinite compositions of analytic functions 5567:Brouwer's theorem for a square implies the 3678:- 1 would have to vanish, a contradiction. 1092:Poincaré's method was analogous to that of 7983:Topology from the differentiable viewpoint 7251:(2019). "10. The Brouwer mapping degree". 6533:Site du Ministère Culture et Communication 3065:) is strictly positive. From the original 2988:has no fixed points, it follows that, for 1683:is defined as the sum of the signs of the 373:Every continuous function from a nonempty 7845:Fixed Points: Algorithms and Applications 7657: 7466: 7334: 7090: 6931: 6890: 6852: 6451:Fixed points: algorithms and applications 6250:Bulletin AMQ, V. XLVI N° 4, (2006) p 17. 6206:"Über Abbildungen von Mannigfaltigkeiten" 5937: 5923: 5921: 5894: 5881: 5875: 5851: 5832: 5823: 5776: 5756: 5710: 5698: 5685: 5675: 5666: 5661: 5643: 5634: 5628: 5508: 5502: 5453: 5444: 5426: 5407: 5400: 5316:(1) is zero. The geometric idea is that 5262: 5247: 5231: 5210: 5135: 5129: 5094: 5005: 4985: 4944: 4912: 4892: 4867: 4846: 4833: 4818: 4798: 4754: 4741: 4726: 4706: 4686: 4662: 4656: 4626: 4602: 4596: 4572: 4545: 4523: 4517: 4490: 4470: 4450: 4406: 4382: 4370: 4345: 4331: 4325: 4314: 4294: 4289: 4283: 4272: 4266: 4239: 4218: 4194: 4182: 4148: 4136: 4127: 4114: 4109: 4103: 4092: 4073: 4069: 4068: 4047: 4041: 4014: 4008: 3892: 3882: 3857: 3847: 3825: 3812: 3790: 3777: 3758: 3746: 3666:– (0) is one-dimensional in degree 0 and 3213: 3212: 3203: 3202: 3193: 3192: 3180: 3179: 3170: 3169: 3168: 3141: 3140: 3131: 3130: 3128: 2957: 2956: 2947: 2946: 2945: 2936: 2935: 2926: 2925: 2907: 2906: 2905: 2893: 2892: 2883: 2882: 2873: 2872: 2851: 2850: 2841: 2840: 2838: 2438: 2432: 2399: 2393: 2373: 2310: 2280: 2257: 2194: 2167: 2150: 2127: 2097: 2078: 2072: 2047: 2041: 2021: 1989: 1929: 1923: 1903: 1883: 1834: 1820: 1797: 1786: 1761: 1755: 1732: 1712: 1692: 1653: 1633: 1613: 1584: 1564: 1540: 1520: 1481: 1461: 1425: 1404: 1400: 1399: 1396: 1360: 1352: 1315:, which at the time made a stand against 743: 707: 701: 638: 621: 591: 590: 588: 569: 568: 566: 519: 464: 208: 188: 165: 145: 124: 108: 96: 75: 69: 42: 7138:CMI Université Paul Cézanne (2008–2009) 7132:These examples are taken from: F. Boyer 6793:Myskis, A. D.; Rabinovic, I. M. (1955). 6006: 3588:, but the latter group is isomorphic to 3402:The proof uses the observation that the 780:holds for any point of the unit circle, 258:. It appears in unlikely fields such as 8070:. New York-Toronto-London: McGraw-Hill. 7437: 7323: 7312: 7135:Théorèmes de point fixe et applications 6403: 6401: 6399: 6324:Point fixe, et théorèmes du point fixe 6099: 5863:{\displaystyle \{U_{1},\ldots ,U_{m}\}} 5537:acts as the identity on this group, so 5312:-dimensional volume of the ball, while 5195:{\displaystyle g^{t}(x):=tr(x)+(1-t)x,} 5117:{\displaystyle r\colon B\to \partial B} 2773:is polynomial and nowhere vanishing on 605:is convex and closed, but not bounded. 335:to itself has at least one fixed point. 7946:Madsen, Ib; Tornehave, Jørgen (1997). 7785:Hilton, Peter J.; Wylie, Sean (1960). 7345: 7289: 7235: 5602: : ℓ → ℓ which sends a sequence ( 5560:, and conversely over the base system 4887:Because the sum of the coordinates of 3704:. In that case it can be assumed that 2532: 1341: 1127:helped Brouwer to formalize his ideas. 662:{\displaystyle f(x)={\frac {x+1}{2}},} 7380: 7181:Institut Henri Poincaré, Paris (2007) 7073:"Der Fixpunktsatz in Funktionsräumen" 6769:Journal de Mathématiques p 217 (1893) 6032:Knaster–Kuratowski–Mazurkiewicz lemma 4621:then by the same argument, the index 3478:, we can construct a unique ray from 2548:. By scaling, it can be assumed that 2511:. (The tangency condition means that 2112:{\displaystyle \deg _{p}f=\deg _{p}g} 1874:The degree satisfies the property of 7: 7634:Chow, Shui Nee; Mallet-Paret, John; 7514:10.4169/amer.math.monthly.120.04.346 7506:10.4169/amer.math.monthly.120.04.346 7061:University of Auckland, New Zealand. 6356:Université de Nice-Sophia Antipolis. 5748:for a discussion of these theorems. 5594:. For example, in the Hilbert space 5061:R. Bruce Kellogg, Tien-Yien Li, and 4438:{\displaystyle j\in \{0,\ldots ,n\}} 4251:{\displaystyle f(P)\in \Delta ^{n}.} 3398:A proof using homology or cohomology 3313:has norm strictly less than 1, then 3283:) is nowhere vanishing (because, if 3253:, satisfying the tangency condition 2684:of the Euclidean space is odd, (1 + 2642:must restrict to a homeomorphism of 2485:A proof using the hairy ball theorem 1384:{\displaystyle K={\overline {B(0)}}} 1244:says that a continuous map from the 561:which is a continuous function from 7369:Chow, Mallet-Paret & Yorke 1978 7260:. Graduate Studies in Mathematics. 7015:Fixed Point Theory. An Introduction 4766:{\displaystyle f(P)_{j}\leq P_{j}.} 4557:{\displaystyle P_{j}\geq f(P)_{j}.} 3650:of open subsets of Euclidean space 3071:-dimensional space Euclidean space 1863:lying over a small open set around 1452:is continuously differentiable. A 957:, and is the basis of more general 7869:SIAM Journal on Numerical Analysis 7862:Kellogg, R. Bruce; Li, Tien-Yien; 7036:Fixed Point Theory an Introduction 6667:Fixed Point Theory an Introduction 6634:on ]0, 1[ has no fixed point. 5903: 5108: 4855:{\displaystyle f(P)_{j}\leq P_{j}} 4659: 4599: 4379: 4236: 4191: 4044: 4011: 3778: 3759: 3568:) is not even connected. The case 1977:{\displaystyle H_{t}(x)=tf+(1-t)g} 1260:. This generalization is known as 857:Explanations attributed to Brouwer 14: 7735:The American Mathematical Monthly 7659:10.1090/S0025-5718-1978-0492046-9 7493:The American Mathematical Monthly 7468:10.1090/S0002-9939-1956-0078693-4 7394:The American Mathematical Monthly 6962:is a positive integer other than 5388:from a finite simplicial complex 5360:A quite different proof given by 5044:Weierstrass approximation theorem 4505:th coordinate of its image under 4394:{\displaystyle P\in \Delta ^{n},} 4206:{\displaystyle P\in \Delta ^{n},} 3712:Weierstrass approximation theorem 2711:Weierstrass approximation theorem 1559:, every point of the preimage of 1283:Other areas are also touched. In 994:, they ran contrary to Brouwer's 941:in ; this zero is a fixed point. 350:Every continuous function from a 8043:from the original on 2022-10-09. 6268:Calcul différentiel et géométrie 6132:Calcul différentiel et géométrie 5796:{\displaystyle f:X\rightarrow X} 5545:A proof in a weak logical system 5030:There is also a quick proof, by 4485:is greater than or equal to the 3995:is a function from the standard 3214: 3204: 3194: 3181: 3171: 3142: 3132: 2958: 2948: 2937: 2927: 2908: 2894: 2884: 2874: 2852: 2842: 1413:{\displaystyle \mathbb {R} ^{n}} 424:Importance of the pre-conditions 7929:Graduate Studies in Mathematics 7271:from the original on 2022-10-09 6979:J. J. O'Connor E. F. Robertson 6815:J. J. O'Connor E. F. Robertson 6778:J. J. O'Connor E. F. Robertson 6729:. Boston: Birkhäuser. pp. 5680: 5674: 4641:can be selected from among the 3658:≥ 2, the de Rham cohomology of 3433:Illustration of the retraction 2493:states that on the unit sphere 2427:is not empty. The elements of 1391:denote the closed unit ball in 870:'s, that guarantee uniqueness. 8135:Theory of continuous functions 8048:Sobolev, Vladimir I. (2001) , 7816:Istrăţescu, Vasile I. (1981). 6686:Voitsekhovskii, M.I. (2001) , 6583:This question was studied in: 6049:Lusternik–Schnirelmann theorem 5938: 5924: 5787: 5520: 5514: 5468: 5465: 5459: 5445: 5438: 5423: 5413: 5259: 5253: 5221: 5215: 5183: 5171: 5165: 5159: 5147: 5141: 5105: 5038:starts by noting that the map 4955: 4949: 4923: 4917: 4830: 4823: 4738: 4731: 4542: 4535: 4342: 4335: 4229: 4223: 3904: 3898: 3872: 3863: 3837: 3831: 3802: 3796: 3580:from the fundamental group of 3222: 3219: 3209: 3186: 3176: 3159: 3153: 3137: 2963: 2953: 2942: 2916: 2902: 2899: 2889: 2863: 2857: 2847: 2453: 2447: 2414: 2408: 2345: 2339: 2301: 2295: 2274: 2262: 2226: 2220: 2185: 2179: 2161: 2155: 1968: 1956: 1941: 1935: 1840: 1824: 1812: 1806: 1776: 1770: 1670: 1664: 1595: 1589: 1498: 1492: 1436: 1372: 1366: 1334:degree of a continuous mapping 1278:partial differential equations 1262:Schauder's fixed-point theorem 1191:degree of a continuous mapping 754: 748: 632: 626: 530: 524: 475: 469: 358:into itself has a fixed point. 293:. Proving results such as the 133:{\displaystyle f(x_{0})=x_{0}} 114: 101: 1: 7843:Karamardian, S., ed. (1977). 7262:American Mathematical Society 7198:Revue d'Histoire des Sciences 7190:For a long explanation, see: 7120:10.1215/S0012-7094-41-00838-4 6449:Karamardian, Stephan (1977). 5981:Lefschetz fixed-point theorem 5818:has a finite open refinement 5065:turned Hirsch's proof into a 3987:The BFPT can be proved using 3682:A proof using Stokes' theorem 2824:-dimensional Euclidean space 2009:{\displaystyle 0\leq t\leq 1} 1254:Lefschetz fixed-point theorem 1032:stability of the solar system 1000:constructivism in mathematics 800:Lefschetz fixed-point theorem 266:, play a central role in the 20:Brouwer's fixed-point theorem 7988:University Press of Virginia 7576:Boothby, William M. (1986). 7357:Kellogg, Li & Yorke 1976 6982:Luitzen Egbertus Jan Brouwer 6933:10.1016/0315-0860(75)90111-1 6892:10.1016/0315-0860(75)90111-1 6854:10.1016/0315-0860(75)90111-1 6818:Luitzen Egbertus Jan Brouwer 6530:Célébrations nationales 2004 6453:. New York: Academic Press. 6289:Florenzano, Monique (2003). 6130:See page 15 of: D. Leborgne 5963:Kakutani fixed point theorem 5489:must have a fixed point. If 4674:{\displaystyle \Delta ^{n},} 4614:{\displaystyle \Delta ^{n},} 4026:{\displaystyle \Delta ^{n},} 3356:dimensions and the mapping 3077:, construct a new auxiliary 2538:In fact, suppose first that 1376: 598:{\displaystyle \mathbb {R} } 576:{\displaystyle \mathbb {R} } 401:Schauder fixed point theorem 264:Kakutani fixed-point theorem 64:to itself, there is a point 8145:Theorems in convex geometry 8114:Brouwer Fixed Point Theorem 8055:Encyclopedia of Mathematics 7301:Madsen & Tornehave 1997 6693:Encyclopedia of Mathematics 6646:Poincaré, Henri (1854–1912) 6474:Istrăţescu, Vasile (1981). 6022:Brouwer fixed-point theorem 5951:{\displaystyle |i-j|\leq 1} 5340:passes from 0 to 1 the map 5085:A proof using oriented area 3271:) = 0. Moreover, 2546:continuously differentiable 1102:another fixed-point theorem 8161: 8066:Spanier, Edwin H. (1966). 7952:Cambridge University Press 7791:Cambridge University Press 7645:Mathematics of Computation 6799:Успехи математических наук 6607:Poincaré–Bendixson theorem 6331:December 26, 2008, at the 6118:December 26, 2008, at the 6063:Banach fixed-point theorem 5356:A proof using the game Hex 5324:) is the oriented area of 4681:the color of every vertex 2525:= 0 for every unit vector 2016:. Suppose that the point 979:). It was later proved by 931:intermediate value theorem 295:Poincaré–Bendixson theorem 7905:Kulpa, Władysław (1989). 7108:Duke Mathematical Journal 7055:E.g.: S. Greenwood J. Cao 6561:taken from: P. A. Miquel 6089:Topological combinatorics 4591:-dimensional sub-face of 3718:with non-negative smooth 2459:{\displaystyle g^{-1}(0)} 2420:{\displaystyle g^{-1}(0)} 1676:{\displaystyle p\in B(0)} 1555:. In particular, by the 1504:{\displaystyle p\in B(0)} 1336:, stemming from ideas in 551:{\displaystyle f(x)=x+1,} 417:itself has a fixed point. 388:itself has a fixed point. 34:. It states that for any 32:L. E. J. (Bertus) Brouwer 8003:Milnor, John W. (1978). 7921:Leoni, Giovanni (2017). 7702:Dieudonné, Jean (1989). 7603:Bredon, Glen E. (1993). 6564:La catégorie de désordre 6266:Page 15 of: D. Leborgne 6083:Poincaré–Miranda theorem 5526:{\displaystyle H_{0}(B)} 5364:is based on the game of 5202:and the smooth function 3944:de Rham cohomology group 3929:giving a contradiction. 3734:on the boundary then by 3502:), we define a function 3307:) is non-zero; while if 2812:of the closed unit ball 2690:) is not a polynomial. 1557:inverse function theorem 1445:{\displaystyle f:K\to K} 1098:Cauchy–Lipschitz theorem 1082:Encyclopædia Universalis 969:= 3 first was proved by 493:{\displaystyle f(x)=x+1} 384:of a Euclidean space to 7451:"A fixed point theorem" 7427:Hilton & Wylie 1960 7224:constructive set theory 6946:If an open subset of a 6295:. Springer. p. 7. 6109:Théorèmes du point fixe 6068:Fixed-point computation 4970:{\displaystyle f(P)=P.} 4401:there must be an index 3028:, the scalar product 3016:) is non-zero; and for 2797:, a contradiction. 2703:unit tangent vector on 1270:Hartman-Grobman theorem 1248:-dimensional sphere to 1153:Hilbert's fifth problem 961:which are important in 769:{\displaystyle f(x)=-x} 244:invariance of dimension 160:from a closed interval 8108:Reconstructing Brouwer 7769:. New York: Springer. 7192:Dubucs, J. P. (1988). 7176:June 11, 2011, at the 7092:10.4064/sm-2-1-171-180 6605:This follows from the 6106:E.g. F & V Bayart 5969:, but considers upper 5952: 5910: 5864: 5797: 5765: 5728: 5527: 5476: 5276: 5196: 5118: 5017: 4994: 4971: 4930: 4901: 4879: 4856: 4807: 4767: 4715: 4695: 4675: 4635: 4615: 4581: 4558: 4499: 4479: 4459: 4439: 4395: 4356: 4330: 4288: 4252: 4207: 4168: 4108: 4027: 3920: 3681: 3541:) is a fixed point of 3437: 3232: 2973: 2460: 2421: 2382: 2360: 2241: 2136: 2113: 2057: 2036:is a regular value of 2030: 2010: 1978: 1912: 1892: 1871:to higher dimensions. 1850: 1741: 1721: 1707:over the preimages of 1701: 1677: 1642: 1622: 1602: 1573: 1549: 1529: 1505: 1470: 1446: 1414: 1385: 1217: 1128: 1027: 1019: 973:in 1904 (published in 882: 770: 717: 663: 613:Consider the function 599: 577: 552: 511:Consider the function 494: 456:Consider the function 252:differential equations 217: 197: 174: 154: 134: 85: 51: 7767:Differential Topology 7605:Topology and geometry 7210:10.3406/rhs.1988.4094 7071:Schauder, J. (1930). 6498:See F. Brechenmacher 6254:June 8, 2011, at the 6211:Mathematische Annalen 5953: 5911: 5865: 5798: 5766: 5729: 5528: 5477: 5277: 5197: 5119: 5018: 4995: 4972: 4931: 4902: 4880: 4857: 4808: 4768: 4716: 4696: 4676: 4636: 4616: 4582: 4559: 4500: 4480: 4460: 4440: 4396: 4357: 4310: 4268: 4253: 4208: 4169: 4088: 4028: 3983:A combinatorial proof 3921: 3533:: every point of the 3432: 3233: 2974: 2461: 2422: 2383: 2361: 2250:is well-defined, and 2242: 2142:, then the function 2137: 2114: 2058: 2056:{\displaystyle H_{t}} 2031: 2011: 1979: 1913: 1893: 1851: 1742: 1722: 1702: 1678: 1643: 1623: 1603: 1574: 1550: 1530: 1506: 1471: 1447: 1415: 1386: 1338:differential topology 1274:Central Limit Theorem 1226:topological manifolds 1208: 1123: 1025: 1017: 925:and ≤ 0 on 881: 771: 718: 716:{\displaystyle D^{n}} 664: 600: 578: 553: 495: 256:differential geometry 218: 198: 175: 155: 135: 86: 84:{\displaystyle x_{0}} 52: 8140:Theorems in topology 8130:Fixed-point theorems 8104:with attached proof. 7710:. pp. 166–203. 6919:Historia Mathematica 6878:Historia Mathematica 6840:Historia Mathematica 6432:J. Reine Angew. Math 6370:. Math StackExchange 6343:C. Minazzo K. Rider 5974:set-valued functions 5920: 5874: 5822: 5775: 5755: 5627: 5581:fixed-point theorems 5501: 5399: 5209: 5128: 5093: 5004: 5000:is a fixed point of 4984: 4943: 4929:{\displaystyle f(P)} 4911: 4891: 4866: 4817: 4797: 4725: 4705: 4685: 4655: 4625: 4595: 4571: 4516: 4489: 4469: 4449: 4405: 4369: 4265: 4217: 4181: 4040: 4007: 3745: 3625:) is trivial, while 3127: 3086:)-dimensional space 2837: 2431: 2392: 2372: 2256: 2149: 2126: 2071: 2040: 2020: 1988: 1922: 1902: 1882: 1754: 1731: 1711: 1691: 1685:Jacobian determinant 1652: 1632: 1612: 1601:{\displaystyle B(0)} 1583: 1563: 1539: 1519: 1480: 1460: 1424: 1395: 1351: 1328:A proof using degree 1266:set-valued functions 1230:Jordan curve theorem 1212:used the theorem in 1187:Jordan curve theorem 1038:, who worked on the 959:fixed point theorems 921:. It is ≥ 0 on 874:One-dimensional case 742: 700: 620: 587: 565: 518: 463: 291:Charles Émile Picard 236:Jordan curve theorem 232:fixed-point theorems 207: 187: 164: 144: 95: 68: 41: 8013:Amer. Math. Monthly 7986:. Charlottesville: 7882:1976SJNA...13..473K 7540:Amer. Math. Monthly 7482:Nyman, Kathryn L.; 7449:Eldon Dyer (1956). 7001:Topological algebra 6417:, 21 septembre 1999 6321:V. & F. Bayart 6039:Borsuk–Ulam theorem 5671: 5551:reverse mathematics 5541:has a fixed point. 5495:simplicial homology 4150: for all 3698:onto its boundary ∂ 3390:require tools from 3388:Borsuk-Ulam theorem 3102:, with coordinates 2617:contraction mapping 1876:homotopy invariance 1648:at a regular value 1242:Borsuk–Ulam theorem 963:functional analysis 844:shaken, not stirred 326:continuous function 272:general equilibrium 248:Borsuk–Ulam theorem 57:mapping a nonempty 36:continuous function 24:fixed-point theorem 16:Theorem in topology 8096:2007-03-19 at the 8068:Algebraic topology 7847:. Academic Press. 7818:Fixed Point Theory 7677:Éléments d'analyse 7484:Su, Francis Edward 7078:Studia Mathematica 6996:algebraic topology 6764:2011-07-16 at the 6651:2010-10-08 at the 6618:Multiplication by 6569:2016-03-03 at the 6522:three-body problem 6476:Fixed point theory 6351:2018-04-04 at the 6224:10.1007/BF01456931 6010:Algebraic topology 6001:algebraic topology 5995:Equivalent results 5948: 5906: 5860: 5793: 5761: 5724: 5657: 5569:weak Kőnig's lemma 5523: 5472: 5471: 5412: 5272: 5192: 5114: 5016:{\displaystyle f.} 5013: 4990: 4967: 4926: 4897: 4878:{\displaystyle j.} 4875: 4852: 4803: 4786:available colors. 4763: 4711: 4691: 4671: 4631: 4611: 4577: 4554: 4495: 4475: 4455: 4435: 4391: 4352: 4248: 4203: 4164: 4023: 3916: 3648:de Rham cohomology 3578:group homomorphism 3574:fundamental groups 3556:, and in the case 3438: 3392:algebraic topology 3228: 2969: 2779:; by construction 2491:hairy ball theorem 2456: 2417: 2378: 2356: 2321: 2237: 2205: 2132: 2109: 2053: 2026: 2006: 1974: 1908: 1888: 1846: 1816: 1737: 1717: 1697: 1673: 1638: 1628:). The degree of 1618: 1598: 1569: 1545: 1525: 1501: 1466: 1442: 1410: 1381: 1222:fixed-point theory 1218: 1195:algebraic topology 1169:hairy ball theorem 1145:mathematical logic 1129: 1040:three-body problem 1028: 1020: 955:algebraic topology 883: 852:Intuitive approach 766: 713: 659: 595: 573: 548: 490: 452:as an endomorphism 370:Convex compact set 347:In Euclidean space 268:proof of existence 240:hairy ball theorem 230:Among hundreds of 213: 193: 170: 150: 130: 81: 47: 8050:"Brouwer theorem" 7941:978-1-4704-2921-8 7854:978-0-12-398050-2 7827:978-90-277-1224-0 7776:978-0-387-90148-0 7763:Hirsch, Morris W. 7146:(August 1, 2010). 7013:V. I. Istratescu 6910:Freudenthal, Hans 6869:Freudenthal, Hans 6831:Freudenthal, Hans 6753:See for example: 6740:978-0-8176-3388-2 6688:"Brouwer theorem" 6485:978-90-277-1224-0 6460:978-0-12-398050-2 6327:on Bibmath.net. 6202:Brouwer, L. E. J. 6054: 6053: 5985:singular homology 5764:{\displaystyle X} 5713: 5678: 5672: 5403: 5376:theorem for Hex. 5026:A proof by Hirsch 4993:{\displaystyle P} 4900:{\displaystyle P} 4806:{\displaystyle P} 4714:{\displaystyle j} 4694:{\displaystyle P} 4634:{\displaystyle j} 4580:{\displaystyle P} 4498:{\displaystyle j} 4478:{\displaystyle P} 4465:th coordinate of 4458:{\displaystyle j} 4151: 4130: 4033:to itself, where 3977:de Rham's theorem 3289:has norm 1, then 2381:{\displaystyle g} 2354: 2306: 2235: 2190: 2135:{\displaystyle K} 2029:{\displaystyle p} 1911:{\displaystyle g} 1891:{\displaystyle f} 1782: 1740:{\displaystyle f} 1720:{\displaystyle p} 1700:{\displaystyle f} 1641:{\displaystyle f} 1621:{\displaystyle K} 1608:(the interior of 1572:{\displaystyle f} 1548:{\displaystyle p} 1528:{\displaystyle f} 1469:{\displaystyle f} 1379: 1087:fundamental group 654: 216:{\displaystyle K} 196:{\displaystyle D} 173:{\displaystyle I} 153:{\displaystyle f} 50:{\displaystyle f} 8152: 8071: 8062: 8044: 8042: 8009: 7999: 7973: 7918: 7901: 7858: 7839: 7812: 7780: 7758: 7729: 7698: 7671: 7661: 7652:(143): 887–899. 7630: 7599: 7572: 7525: 7524: 7479: 7473: 7472: 7470: 7446: 7440: 7435: 7429: 7424: 7418: 7417: 7389: 7383: 7378: 7372: 7366: 7360: 7354: 7348: 7343: 7337: 7332: 7326: 7321: 7315: 7310: 7304: 7303:, pp. 39–48 7298: 7292: 7287: 7281: 7280: 7278: 7276: 7270: 7259: 7245: 7239: 7233: 7227: 7220: 7214: 7213: 7188: 7182: 7166: 7160: 7157:Hex (board game) 7153: 7147: 7130: 7124: 7123: 7103: 7097: 7096: 7094: 7068: 7062: 7053: 7047: 7032: 7026: 7011: 7005: 6992: 6986: 6977: 6971: 6944: 6938: 6937: 6935: 6906: 6900: 6899: 6894: 6865: 6859: 6858: 6856: 6827: 6821: 6813: 6807: 6806: 6790: 6784: 6776: 6770: 6751: 6745: 6744: 6717: 6711: 6710: 6683: 6677: 6663: 6657: 6641: 6635: 6633: 6631: 6630: 6627: 6624: 6616: 6610: 6603: 6597: 6596: 6581: 6575: 6555: 6549: 6540: 6534: 6511: 6505: 6496: 6490: 6489: 6471: 6465: 6464: 6446: 6440: 6439: 6427: 6418: 6405: 6394: 6386: 6380: 6379: 6377: 6375: 6363: 6357: 6341: 6335: 6319: 6313: 6312: 6310: 6309: 6285: 6279: 6264: 6258: 6242: 6236: 6235: 6198: 6187: 6170:Jacques Hadamard 6167: 6158: 6148: 6142: 6128: 6122: 6104: 6078:Nash equilibrium 6007: 5957: 5955: 5954: 5949: 5941: 5927: 5915: 5913: 5912: 5907: 5899: 5898: 5886: 5885: 5869: 5867: 5866: 5861: 5856: 5855: 5837: 5836: 5802: 5800: 5799: 5794: 5770: 5768: 5767: 5762: 5733: 5731: 5730: 5725: 5714: 5711: 5709: 5708: 5690: 5689: 5679: 5676: 5673: 5670: 5665: 5644: 5639: 5638: 5532: 5530: 5529: 5524: 5513: 5512: 5481: 5479: 5478: 5473: 5458: 5457: 5448: 5431: 5430: 5411: 5290: 5281: 5279: 5278: 5273: 5252: 5251: 5236: 5235: 5201: 5199: 5198: 5193: 5140: 5139: 5123: 5121: 5120: 5115: 5022: 5020: 5019: 5014: 4999: 4997: 4996: 4991: 4976: 4974: 4973: 4968: 4935: 4933: 4932: 4927: 4906: 4904: 4903: 4898: 4884: 4882: 4881: 4876: 4861: 4859: 4858: 4853: 4851: 4850: 4838: 4837: 4812: 4810: 4809: 4804: 4785: 4772: 4770: 4769: 4764: 4759: 4758: 4746: 4745: 4720: 4718: 4717: 4712: 4700: 4698: 4697: 4692: 4680: 4678: 4677: 4672: 4667: 4666: 4647: 4640: 4638: 4637: 4632: 4620: 4618: 4617: 4612: 4607: 4606: 4586: 4584: 4583: 4578: 4563: 4561: 4560: 4555: 4550: 4549: 4528: 4527: 4504: 4502: 4501: 4496: 4484: 4482: 4481: 4476: 4464: 4462: 4461: 4456: 4444: 4442: 4441: 4436: 4400: 4398: 4397: 4392: 4387: 4386: 4361: 4359: 4358: 4353: 4351: 4350: 4349: 4329: 4324: 4300: 4299: 4298: 4287: 4282: 4257: 4255: 4254: 4249: 4244: 4243: 4212: 4210: 4209: 4204: 4199: 4198: 4177:For every point 4173: 4171: 4170: 4165: 4160: 4156: 4152: 4149: 4141: 4140: 4131: 4128: 4120: 4119: 4118: 4107: 4102: 4084: 4083: 4072: 4052: 4051: 4032: 4030: 4029: 4024: 4019: 4018: 3974: 3968: 3956: 3950: 3941: 3937: 3925: 3923: 3922: 3917: 3897: 3896: 3887: 3886: 3862: 3861: 3852: 3851: 3830: 3829: 3817: 3816: 3795: 3794: 3785: 3784: 3766: 3765: 3729: 3709: 3703: 3697: 3691: 3611:: the homology 3453: 3385: 3379: 3373: 3367: 3361: 3355: 3348: 3342: 3336: 3330: 3324: 3318: 3312: 3306: 3300: 3294: 3282: 3276: 3270: 3264: 3258: 3252: 3246: 3241:By construction 3237: 3235: 3234: 3229: 3218: 3217: 3208: 3207: 3198: 3197: 3185: 3184: 3175: 3174: 3146: 3145: 3136: 3135: 3119: 3113: 3107: 3097: 3091: 3085: 3076: 3070: 3064: 3058: 3052: 3046: 3040: 3034: 3027: 3021: 3015: 3009: 3003: 2993: 2987: 2978: 2976: 2975: 2970: 2962: 2961: 2952: 2951: 2941: 2940: 2931: 2930: 2912: 2911: 2898: 2897: 2888: 2887: 2878: 2877: 2856: 2855: 2846: 2845: 2829: 2823: 2817: 2811: 2805: 2796: 2790: 2784: 2778: 2772: 2766: 2760: 2754: 2748: 2742: 2736: 2730: 2724: 2718: 2708: 2698: 2689: 2683: 2677: 2671: 2665: 2659: 2653: 2647: 2641: 2630: 2624: 2614: 2600: 2594: 2588: 2577: 2571: 2565: 2559: 2553: 2543: 2530: 2524: 2510: 2504: 2498: 2465: 2463: 2462: 2457: 2446: 2445: 2426: 2424: 2423: 2418: 2407: 2406: 2387: 2385: 2384: 2379: 2365: 2363: 2362: 2357: 2355: 2353: 2352: 2348: 2320: 2304: 2281: 2246: 2244: 2243: 2238: 2236: 2234: 2233: 2229: 2204: 2188: 2168: 2141: 2139: 2138: 2133: 2118: 2116: 2115: 2110: 2102: 2101: 2083: 2082: 2062: 2060: 2059: 2054: 2052: 2051: 2035: 2033: 2032: 2027: 2015: 2013: 2012: 2007: 1983: 1981: 1980: 1975: 1934: 1933: 1917: 1915: 1914: 1909: 1897: 1895: 1894: 1889: 1855: 1853: 1852: 1847: 1839: 1838: 1815: 1805: 1804: 1766: 1765: 1746: 1744: 1743: 1738: 1726: 1724: 1723: 1718: 1706: 1704: 1703: 1698: 1682: 1680: 1679: 1674: 1647: 1645: 1644: 1639: 1627: 1625: 1624: 1619: 1607: 1605: 1604: 1599: 1578: 1576: 1575: 1570: 1554: 1552: 1551: 1546: 1534: 1532: 1531: 1526: 1510: 1508: 1507: 1502: 1475: 1473: 1472: 1467: 1451: 1449: 1448: 1443: 1419: 1417: 1416: 1411: 1409: 1408: 1403: 1390: 1388: 1387: 1382: 1380: 1375: 1361: 1305:not constructive 1178:Hans Freudenthal 1161:Jacques Hadamard 1125:Jacques Hadamard 1010:Before discovery 989:non-constructive 985:Jacques Hadamard 981:L. E. J. Brouwer 775: 773: 772: 767: 722: 720: 719: 714: 712: 711: 668: 666: 665: 660: 655: 650: 639: 604: 602: 601: 596: 594: 582: 580: 579: 574: 572: 557: 555: 554: 549: 499: 497: 496: 491: 305:Jacques Hadamard 302: 222: 220: 219: 214: 202: 200: 199: 194: 179: 177: 176: 171: 159: 157: 156: 151: 139: 137: 136: 131: 129: 128: 113: 112: 90: 88: 87: 82: 80: 79: 56: 54: 53: 48: 8160: 8159: 8155: 8154: 8153: 8151: 8150: 8149: 8120: 8119: 8116:at Math Images. 8098:Wayback Machine 8091:Brouwer theorem 8078: 8065: 8047: 8040: 8007: 8002: 7978:Milnor, John W. 7976: 7962: 7945: 7904: 7890:10.1137/0713041 7864:Yorke, James A. 7861: 7855: 7842: 7828: 7815: 7801: 7784: 7777: 7761: 7747:10.2307/2320146 7741:(10): 818–827. 7732: 7718: 7701: 7687: 7674: 7636:Yorke, James A. 7633: 7619: 7609:Springer-Verlag 7602: 7588: 7575: 7553:10.2307/2317520 7536: 7533: 7528: 7481: 7480: 7476: 7448: 7447: 7443: 7436: 7432: 7425: 7421: 7406:10.2307/2320146 7400:(10): 818–827. 7391: 7390: 7386: 7379: 7375: 7367: 7363: 7355: 7351: 7344: 7340: 7333: 7329: 7322: 7318: 7311: 7307: 7299: 7295: 7288: 7284: 7274: 7272: 7268: 7257: 7247: 7246: 7242: 7238:, pp. 1–19 7234: 7230: 7221: 7217: 7191: 7189: 7185: 7178:Wayback Machine 7167: 7163: 7154: 7150: 7131: 7127: 7105: 7104: 7100: 7070: 7069: 7065: 7054: 7050: 7033: 7029: 7012: 7008: 6993: 6989: 6978: 6974: 6945: 6941: 6926:(4): 495–502 . 6908: 6907: 6903: 6885:(4): 495–502 . 6867: 6866: 6862: 6847:(4): 495–502 . 6829: 6828: 6824: 6814: 6810: 6792: 6791: 6787: 6777: 6773: 6766:Wayback Machine 6752: 6748: 6741: 6721:Dieudonné, Jean 6719: 6718: 6714: 6708: 6685: 6684: 6680: 6664: 6660: 6653:Wayback Machine 6642: 6638: 6628: 6625: 6622: 6621: 6619: 6617: 6613: 6604: 6600: 6584: 6582: 6578: 6571:Wayback Machine 6557:Quotation from 6556: 6552: 6541: 6537: 6512: 6508: 6497: 6493: 6486: 6473: 6472: 6468: 6461: 6448: 6447: 6443: 6438:(3/4): 179–276. 6429: 6428: 6421: 6406: 6397: 6387: 6383: 6373: 6371: 6365: 6364: 6360: 6353:Wayback Machine 6342: 6338: 6333:Wayback Machine 6320: 6316: 6307: 6305: 6303: 6288: 6286: 6282: 6265: 6261: 6256:Wayback Machine 6243: 6239: 6200: 6199: 6190: 6168: 6161: 6149: 6145: 6129: 6125: 6120:Wayback Machine 6105: 6101: 6097: 6059: 6027:Sperner's lemma 5997: 5971:hemi-continuous 5918: 5917: 5916:if and only if 5890: 5877: 5872: 5871: 5847: 5828: 5820: 5819: 5814:of which every 5812:Hausdorff space 5773: 5772: 5753: 5752: 5712: for 5694: 5681: 5630: 5625: 5624: 5619: 5610: 5577: 5575:Generalizations 5565: 5558: 5547: 5504: 5499: 5498: 5449: 5422: 5397: 5396: 5382: 5358: 5288: 5243: 5227: 5207: 5206: 5131: 5126: 5125: 5091: 5090: 5087: 5028: 5002: 5001: 4982: 4981: 4941: 4940: 4909: 4908: 4889: 4888: 4864: 4863: 4842: 4829: 4815: 4814: 4795: 4794: 4780: 4750: 4737: 4723: 4722: 4703: 4702: 4683: 4682: 4658: 4653: 4652: 4642: 4623: 4622: 4598: 4593: 4592: 4569: 4568: 4541: 4519: 4514: 4513: 4487: 4486: 4467: 4466: 4447: 4446: 4403: 4402: 4378: 4367: 4366: 4341: 4290: 4263: 4262: 4235: 4215: 4214: 4190: 4179: 4178: 4132: 4129: and 4110: 4067: 4060: 4056: 4043: 4038: 4037: 4010: 4005: 4004: 3989:Sperner's lemma 3985: 3970: 3967: 3958: 3952: 3946: 3939: 3933: 3888: 3878: 3853: 3843: 3821: 3808: 3786: 3773: 3754: 3743: 3742: 3736:Stokes' theorem 3727: 3705: 3699: 3693: 3687: 3684: 3634: 3620: 3609:homology groups 3441: 3400: 3381: 3375: 3369: 3363: 3357: 3350: 3344: 3338: 3332: 3326: 3320: 3314: 3308: 3302: 3296: 3290: 3288: 3278: 3272: 3266: 3260: 3254: 3248: 3242: 3125: 3124: 3115: 3109: 3103: 3093: 3087: 3080: 3078: 3072: 3066: 3060: 3054: 3048: 3042: 3036: 3030: 3029: 3023: 3017: 3011: 3005: 2999: 2989: 2983: 2835: 2834: 2825: 2819: 2813: 2807: 2801: 2792: 2786: 2780: 2774: 2768: 2762: 2756: 2750: 2744: 2738: 2732: 2726: 2720: 2714: 2704: 2694: 2685: 2679: 2673: 2667: 2661: 2655: 2649: 2643: 2640: 2632: 2626: 2620: 2602: 2596: 2590: 2587: 2579: 2573: 2567: 2561: 2555: 2549: 2539: 2526: 2512: 2506: 2500: 2494: 2487: 2479:homology theory 2434: 2429: 2428: 2395: 2390: 2389: 2370: 2369: 2326: 2322: 2305: 2282: 2254: 2253: 2210: 2206: 2189: 2169: 2147: 2146: 2124: 2123: 2093: 2074: 2069: 2068: 2043: 2038: 2037: 2018: 2017: 1986: 1985: 1925: 1920: 1919: 1900: 1899: 1880: 1879: 1830: 1793: 1757: 1752: 1751: 1729: 1728: 1709: 1708: 1689: 1688: 1650: 1649: 1630: 1629: 1610: 1609: 1581: 1580: 1561: 1560: 1537: 1536: 1517: 1516: 1478: 1477: 1458: 1457: 1422: 1421: 1398: 1393: 1392: 1362: 1349: 1348: 1330: 1325: 1298:Hotelling's law 1203: 1118: 1012: 992:indirect proofs 951: 898:(light green). 876: 859: 854: 824: 808: 740: 739: 703: 698: 697: 690: 640: 618: 617: 611: 585: 584: 563: 562: 516: 515: 509: 461: 460: 454: 426: 356:Euclidean space 313: 298: 225:Euclidean space 205: 204: 185: 184: 162: 161: 142: 141: 120: 104: 93: 92: 71: 66: 65: 39: 38: 17: 12: 11: 5: 8158: 8156: 8148: 8147: 8142: 8137: 8132: 8122: 8121: 8118: 8117: 8111: 8105: 8088: 8077: 8076:External links 8074: 8073: 8072: 8063: 8045: 8020:(7): 521–524. 8000: 7974: 7960: 7943: 7919: 7902: 7876:(4): 473–483. 7859: 7853: 7840: 7826: 7813: 7799: 7782: 7775: 7759: 7730: 7716: 7699: 7685: 7672: 7631: 7617: 7600: 7586: 7573: 7547:(3): 237–249. 7532: 7529: 7527: 7526: 7500:(4): 346–354, 7474: 7461:(4): 662–672. 7441: 7430: 7419: 7384: 7373: 7361: 7349: 7338: 7335:Dieudonné 1982 7327: 7316: 7305: 7293: 7282: 7249:Teschl, Gerald 7240: 7228: 7215: 7204:(2): 133–155. 7183: 7161: 7148: 7125: 7114:(3): 457–459. 7098: 7063: 7048: 7027: 7006: 6987: 6972: 6939: 6901: 6860: 6822: 6808: 6801:(in Russian). 6785: 6771: 6746: 6739: 6712: 6706: 6678: 6658: 6636: 6611: 6598: 6576: 6559:Henri Poincaré 6550: 6543:Henri Poincaré 6535: 6518:King of Sweden 6514:Henri Poincaré 6506: 6491: 6484: 6466: 6459: 6441: 6419: 6395: 6391:Henri Poincaré 6381: 6358: 6336: 6314: 6301: 6280: 6259: 6237: 6188: 6159: 6155:Luizen Brouwer 6143: 6123: 6098: 6096: 6093: 6092: 6091: 6086: 6080: 6075: 6070: 6065: 6058: 6055: 6052: 6051: 6046: 6044:Tucker's lemma 6041: 6035: 6034: 6029: 6024: 6018: 6017: 6014: 6011: 5996: 5993: 5947: 5944: 5940: 5936: 5933: 5930: 5926: 5905: 5902: 5897: 5893: 5889: 5884: 5880: 5859: 5854: 5850: 5846: 5843: 5840: 5835: 5831: 5827: 5792: 5789: 5786: 5783: 5780: 5760: 5735: 5734: 5723: 5720: 5717: 5707: 5704: 5701: 5697: 5693: 5688: 5684: 5669: 5664: 5660: 5656: 5653: 5650: 5647: 5642: 5637: 5633: 5615: 5606: 5576: 5573: 5563: 5556: 5546: 5543: 5522: 5519: 5516: 5511: 5507: 5483: 5482: 5470: 5467: 5464: 5461: 5456: 5452: 5447: 5443: 5440: 5437: 5434: 5429: 5425: 5421: 5418: 5415: 5410: 5406: 5381: 5378: 5357: 5354: 5296:) = 0 for all 5283: 5282: 5271: 5268: 5265: 5261: 5258: 5255: 5250: 5246: 5242: 5239: 5234: 5230: 5226: 5223: 5220: 5217: 5214: 5191: 5188: 5185: 5182: 5179: 5176: 5173: 5170: 5167: 5164: 5161: 5158: 5155: 5152: 5149: 5146: 5143: 5138: 5134: 5113: 5110: 5107: 5104: 5101: 5098: 5086: 5083: 5063:James A. Yorke 5056:Sard's theorem 5052:bump functions 5036:indirect proof 5027: 5024: 5012: 5009: 4989: 4978: 4977: 4966: 4963: 4960: 4957: 4954: 4951: 4948: 4925: 4922: 4919: 4916: 4896: 4874: 4871: 4849: 4845: 4841: 4836: 4832: 4828: 4825: 4822: 4802: 4762: 4757: 4753: 4749: 4744: 4740: 4736: 4733: 4730: 4710: 4690: 4670: 4665: 4661: 4630: 4610: 4605: 4601: 4576: 4565: 4564: 4553: 4548: 4544: 4540: 4537: 4534: 4531: 4526: 4522: 4494: 4474: 4454: 4445:such that the 4434: 4431: 4428: 4425: 4422: 4419: 4416: 4413: 4410: 4390: 4385: 4381: 4377: 4374: 4363: 4362: 4348: 4344: 4340: 4337: 4334: 4328: 4323: 4320: 4317: 4313: 4309: 4306: 4303: 4297: 4293: 4286: 4281: 4278: 4275: 4271: 4247: 4242: 4238: 4234: 4231: 4228: 4225: 4222: 4202: 4197: 4193: 4189: 4186: 4175: 4174: 4163: 4159: 4155: 4147: 4144: 4139: 4135: 4126: 4123: 4117: 4113: 4106: 4101: 4098: 4095: 4091: 4087: 4082: 4079: 4076: 4071: 4066: 4063: 4059: 4055: 4050: 4046: 4022: 4017: 4013: 3984: 3981: 3962: 3942:generates the 3927: 3926: 3915: 3912: 3909: 3906: 3903: 3900: 3895: 3891: 3885: 3881: 3877: 3874: 3871: 3868: 3865: 3860: 3856: 3850: 3846: 3842: 3839: 3836: 3833: 3828: 3824: 3820: 3815: 3811: 3807: 3804: 3801: 3798: 3793: 3789: 3783: 3780: 3776: 3772: 3769: 3764: 3761: 3757: 3753: 3750: 3720:bump functions 3692:from the ball 3683: 3680: 3639:) is infinite 3629: 3615: 3537:(in this case 3399: 3396: 3284: 3239: 3238: 3227: 3224: 3221: 3216: 3211: 3206: 3201: 3196: 3191: 3188: 3183: 3178: 3173: 3167: 3164: 3161: 3158: 3155: 3152: 3149: 3144: 3139: 3134: 2980: 2979: 2968: 2965: 2960: 2955: 2950: 2944: 2939: 2934: 2929: 2924: 2921: 2918: 2915: 2910: 2904: 2901: 2896: 2891: 2886: 2881: 2876: 2871: 2868: 2865: 2862: 2859: 2854: 2849: 2844: 2636: 2583: 2486: 2483: 2455: 2452: 2449: 2444: 2441: 2437: 2416: 2413: 2410: 2405: 2402: 2398: 2377: 2351: 2347: 2344: 2341: 2338: 2335: 2332: 2329: 2325: 2319: 2316: 2313: 2309: 2303: 2300: 2297: 2294: 2291: 2288: 2285: 2279: 2276: 2273: 2270: 2267: 2264: 2261: 2248: 2247: 2232: 2228: 2225: 2222: 2219: 2216: 2213: 2209: 2203: 2200: 2197: 2193: 2187: 2184: 2181: 2178: 2175: 2172: 2166: 2163: 2160: 2157: 2154: 2131: 2108: 2105: 2100: 2096: 2092: 2089: 2086: 2081: 2077: 2050: 2046: 2025: 2005: 2002: 1999: 1996: 1993: 1973: 1970: 1967: 1964: 1961: 1958: 1955: 1952: 1949: 1946: 1943: 1940: 1937: 1932: 1928: 1907: 1887: 1869:winding number 1857: 1856: 1845: 1842: 1837: 1833: 1829: 1826: 1823: 1819: 1814: 1811: 1808: 1803: 1800: 1796: 1792: 1789: 1785: 1781: 1778: 1775: 1772: 1769: 1764: 1760: 1736: 1716: 1696: 1672: 1669: 1666: 1663: 1660: 1657: 1637: 1617: 1597: 1594: 1591: 1588: 1568: 1544: 1524: 1511:such that the 1500: 1497: 1494: 1491: 1488: 1485: 1465: 1441: 1438: 1435: 1432: 1429: 1407: 1402: 1378: 1374: 1371: 1368: 1365: 1359: 1356: 1329: 1326: 1324: 1323:Proof outlines 1321: 1309:constructivity 1202: 1199: 1157:Henri Poincaré 1117: 1114: 1104:, named after 1078:analysis situs 1036:Henri Poincaré 1011: 1008: 950: 947: 917:) − 875: 872: 858: 855: 853: 850: 849: 848: 839: 836: 823: 820: 807: 804: 794: 765: 762: 759: 756: 753: 750: 747: 710: 706: 694:homeomorphisms 689: 686: 679: 670: 669: 658: 653: 649: 646: 643: 637: 634: 631: 628: 625: 610: 607: 593: 571: 559: 558: 547: 544: 541: 538: 535: 532: 529: 526: 523: 508: 505: 501: 500: 489: 486: 483: 480: 477: 474: 471: 468: 453: 446: 425: 422: 421: 420: 419: 418: 403: 392: 391: 390: 389: 371: 362: 361: 360: 359: 348: 339: 338: 337: 336: 322: 312: 309: 287:Henri Poincaré 212: 192: 169: 149: 127: 123: 119: 116: 111: 107: 103: 100: 78: 74: 46: 30:, named after 15: 13: 10: 9: 6: 4: 3: 2: 8157: 8146: 8143: 8141: 8138: 8136: 8133: 8131: 8128: 8127: 8125: 8115: 8112: 8109: 8106: 8103: 8099: 8095: 8092: 8089: 8087: 8083: 8080: 8079: 8075: 8069: 8064: 8061: 8057: 8056: 8051: 8046: 8039: 8035: 8031: 8027: 8023: 8019: 8015: 8014: 8006: 8001: 7997: 7993: 7989: 7985: 7984: 7979: 7975: 7971: 7967: 7963: 7961:0-521-58059-5 7957: 7953: 7949: 7944: 7942: 7938: 7934: 7930: 7926: 7925: 7920: 7916: 7912: 7908: 7903: 7899: 7895: 7891: 7887: 7883: 7879: 7875: 7871: 7870: 7865: 7860: 7856: 7850: 7846: 7841: 7837: 7833: 7829: 7823: 7819: 7814: 7810: 7806: 7802: 7796: 7792: 7788: 7783: 7778: 7772: 7768: 7764: 7760: 7756: 7752: 7748: 7744: 7740: 7736: 7731: 7727: 7723: 7719: 7717:0-8176-3388-X 7713: 7709: 7705: 7700: 7696: 7692: 7688: 7686:2-04-011499-8 7682: 7678: 7673: 7669: 7665: 7660: 7655: 7651: 7647: 7646: 7641: 7637: 7632: 7628: 7624: 7620: 7618:0-387-97926-3 7614: 7610: 7606: 7601: 7597: 7593: 7589: 7587:0-12-116052-1 7583: 7579: 7574: 7570: 7566: 7562: 7558: 7554: 7550: 7546: 7542: 7541: 7535: 7534: 7530: 7523: 7519: 7515: 7511: 7507: 7503: 7499: 7495: 7494: 7489: 7485: 7478: 7475: 7469: 7464: 7460: 7456: 7452: 7445: 7442: 7439: 7434: 7431: 7428: 7423: 7420: 7415: 7411: 7407: 7403: 7399: 7395: 7388: 7385: 7382: 7377: 7374: 7370: 7365: 7362: 7358: 7353: 7350: 7347: 7342: 7339: 7336: 7331: 7328: 7325: 7320: 7317: 7314: 7309: 7306: 7302: 7297: 7294: 7291: 7286: 7283: 7267: 7263: 7256: 7255: 7250: 7244: 7241: 7237: 7232: 7229: 7225: 7219: 7216: 7211: 7207: 7203: 7199: 7195: 7187: 7184: 7180: 7179: 7175: 7172: 7165: 7162: 7158: 7152: 7149: 7145: 7141: 7140:Archived copy 7137: 7136: 7129: 7126: 7121: 7117: 7113: 7109: 7102: 7099: 7093: 7088: 7084: 7080: 7079: 7074: 7067: 7064: 7060: 7059: 7052: 7049: 7045: 7044:1-4020-0301-3 7041: 7037: 7031: 7028: 7024: 7023:1-4020-0301-3 7020: 7016: 7010: 7007: 7003: 7002: 6997: 6991: 6988: 6984: 6983: 6976: 6973: 6969: 6965: 6961: 6957: 6953: 6949: 6943: 6940: 6934: 6929: 6925: 6921: 6920: 6915: 6911: 6905: 6902: 6898: 6893: 6888: 6884: 6880: 6879: 6874: 6870: 6864: 6861: 6855: 6850: 6846: 6842: 6841: 6836: 6832: 6826: 6823: 6820: 6819: 6812: 6809: 6805:(3): 188–192. 6804: 6800: 6796: 6789: 6786: 6783: 6782: 6775: 6772: 6768: 6767: 6763: 6760: 6756: 6750: 6747: 6742: 6736: 6732: 6728: 6727: 6722: 6716: 6713: 6709: 6707:1-4020-0609-8 6703: 6699: 6695: 6694: 6689: 6682: 6679: 6676: 6675:1-4020-0301-3 6672: 6668: 6662: 6659: 6655: 6654: 6650: 6647: 6640: 6637: 6615: 6612: 6608: 6602: 6599: 6595:(4): 167–244. 6594: 6590: 6589: 6580: 6577: 6573: 6572: 6568: 6565: 6560: 6554: 6551: 6547: 6544: 6539: 6536: 6532: 6531: 6527: 6523: 6519: 6515: 6510: 6507: 6503: 6502: 6495: 6492: 6487: 6481: 6477: 6470: 6467: 6462: 6456: 6452: 6445: 6442: 6437: 6433: 6426: 6424: 6420: 6416: 6412: 6411: 6404: 6402: 6400: 6396: 6392: 6385: 6382: 6369: 6362: 6359: 6355: 6354: 6350: 6347: 6340: 6337: 6334: 6330: 6326: 6325: 6318: 6315: 6304: 6302:9781402075124 6298: 6294: 6293: 6284: 6281: 6277: 6276:2-13-037495-6 6273: 6269: 6263: 6260: 6257: 6253: 6249: 6248: 6241: 6238: 6233: 6229: 6225: 6221: 6217: 6214:(in German). 6213: 6212: 6207: 6203: 6197: 6195: 6193: 6189: 6185: 6181: 6180:Jules Tannery 6177: 6176: 6171: 6166: 6164: 6160: 6157:by G. Sabbagh 6156: 6153: 6147: 6144: 6141: 6140:2-13-037495-6 6137: 6133: 6127: 6124: 6121: 6117: 6114: 6111: 6110: 6103: 6100: 6094: 6090: 6087: 6084: 6081: 6079: 6076: 6074: 6071: 6069: 6066: 6064: 6061: 6060: 6056: 6050: 6047: 6045: 6042: 6040: 6037: 6036: 6033: 6030: 6028: 6025: 6023: 6020: 6019: 6016:Set covering 6015: 6013:Combinatorics 6012: 6009: 6008: 6005: 6002: 5994: 5992: 5990: 5986: 5982: 5977: 5975: 5972: 5968: 5964: 5959: 5945: 5942: 5934: 5931: 5928: 5900: 5895: 5891: 5887: 5882: 5878: 5852: 5848: 5844: 5841: 5838: 5833: 5829: 5817: 5813: 5810: 5806: 5790: 5784: 5781: 5778: 5758: 5749: 5747: 5743: 5738: 5721: 5718: 5715: 5705: 5702: 5699: 5695: 5691: 5686: 5682: 5667: 5662: 5654: 5648: 5645: 5640: 5635: 5631: 5623: 5622: 5621: 5620:) defined by 5618: 5614: 5609: 5605: 5601: 5597: 5593: 5589: 5588:Hilbert space 5584: 5582: 5574: 5572: 5570: 5566: 5559: 5552: 5544: 5542: 5540: 5536: 5517: 5509: 5505: 5496: 5492: 5488: 5462: 5454: 5450: 5441: 5435: 5432: 5427: 5419: 5416: 5408: 5404: 5395: 5394: 5393: 5391: 5387: 5379: 5377: 5375: 5371: 5367: 5363: 5355: 5353: 5351: 5347: 5343: 5339: 5335: 5331: 5327: 5323: 5319: 5315: 5311: 5307: 5303: 5299: 5295: 5291: 5269: 5266: 5263: 5256: 5248: 5244: 5240: 5232: 5228: 5224: 5218: 5212: 5205: 5204: 5203: 5189: 5186: 5180: 5177: 5174: 5168: 5162: 5156: 5153: 5150: 5144: 5136: 5132: 5111: 5102: 5099: 5096: 5084: 5082: 5080: 5076: 5072: 5068: 5064: 5059: 5057: 5053: 5049: 5045: 5041: 5037: 5033: 5032:Morris Hirsch 5025: 5023: 5010: 5007: 4987: 4964: 4961: 4958: 4952: 4946: 4939: 4938: 4937: 4920: 4914: 4894: 4885: 4872: 4869: 4847: 4843: 4839: 4834: 4826: 4820: 4800: 4792: 4787: 4783: 4778: 4773: 4760: 4755: 4751: 4747: 4742: 4734: 4728: 4708: 4688: 4668: 4663: 4649: 4645: 4628: 4608: 4603: 4590: 4574: 4567:Moreover, if 4551: 4546: 4538: 4532: 4529: 4524: 4520: 4512: 4511: 4510: 4508: 4492: 4472: 4452: 4429: 4426: 4423: 4420: 4417: 4411: 4408: 4388: 4383: 4375: 4372: 4346: 4338: 4332: 4326: 4321: 4318: 4315: 4311: 4307: 4304: 4301: 4295: 4291: 4284: 4279: 4276: 4273: 4269: 4261: 4260: 4259: 4245: 4240: 4232: 4226: 4220: 4200: 4195: 4187: 4184: 4161: 4157: 4153: 4145: 4142: 4137: 4133: 4124: 4121: 4115: 4111: 4104: 4099: 4096: 4093: 4089: 4085: 4080: 4077: 4074: 4064: 4061: 4057: 4053: 4048: 4036: 4035: 4034: 4020: 4015: 4002: 3998: 3994: 3990: 3982: 3980: 3978: 3973: 3965: 3961: 3955: 3949: 3945: 3936: 3930: 3913: 3910: 3907: 3901: 3893: 3889: 3883: 3879: 3875: 3869: 3866: 3858: 3854: 3848: 3844: 3840: 3834: 3826: 3822: 3818: 3813: 3809: 3805: 3799: 3791: 3787: 3781: 3774: 3770: 3767: 3762: 3755: 3751: 3748: 3741: 3740: 3739: 3737: 3733: 3725: 3721: 3717: 3713: 3708: 3702: 3696: 3690: 3679: 3677: 3673: 3669: 3665: 3661: 3657: 3653: 3649: 3644: 3642: 3638: 3632: 3628: 3624: 3618: 3614: 3610: 3606: 3601: 3599: 3598:vector fields 3595: 3591: 3587: 3583: 3579: 3575: 3571: 3567: 3563: 3559: 3555: 3551: 3546: 3544: 3540: 3536: 3532: 3527: 3525: 3521: 3517: 3513: 3510: → 3509: 3506: : 3505: 3501: 3497: 3493: 3489: 3485: 3481: 3477: 3473: 3469: 3465: 3462:, the points 3461: 3457: 3452: 3448: 3444: 3436: 3431: 3427: 3425: 3421: 3417: 3413: 3409: 3405: 3397: 3395: 3393: 3389: 3384: 3378: 3372: 3366: 3360: 3353: 3347: 3341: 3337:is even. For 3335: 3329: 3323: 3317: 3311: 3305: 3299: 3293: 3287: 3281: 3275: 3269: 3263: 3257: 3251: 3245: 3225: 3199: 3189: 3165: 3162: 3156: 3150: 3147: 3123: 3122: 3121: 3118: 3112: 3106: 3101: 3096: 3090: 3083: 3075: 3069: 3063: 3057: 3051: 3045: 3039: 3033: 3026: 3020: 3014: 3008: 3004:, the vector 3002: 2997: 2992: 2986: 2966: 2932: 2922: 2919: 2913: 2879: 2869: 2866: 2860: 2833: 2832: 2831: 2828: 2822: 2816: 2810: 2804: 2798: 2795: 2789: 2783: 2777: 2771: 2765: 2759: 2753: 2747: 2741: 2735: 2729: 2723: 2717: 2712: 2707: 2702: 2697: 2691: 2688: 2682: 2676: 2670: 2664: 2658: 2652: 2646: 2639: 2635: 2629: 2623: 2618: 2612: 2608: 2605: 2599: 2593: 2586: 2582: 2576: 2570: 2564: 2558: 2552: 2547: 2542: 2536: 2534: 2533:Milnor (1978) 2529: 2523: 2519: 2515: 2509: 2503: 2497: 2492: 2484: 2482: 2480: 2476: 2471: 2469: 2450: 2442: 2439: 2435: 2411: 2403: 2400: 2396: 2375: 2366: 2349: 2342: 2336: 2333: 2330: 2327: 2323: 2317: 2314: 2311: 2298: 2292: 2289: 2286: 2283: 2277: 2271: 2268: 2265: 2259: 2251: 2230: 2223: 2217: 2214: 2211: 2207: 2201: 2198: 2195: 2182: 2176: 2173: 2170: 2164: 2158: 2152: 2145: 2144: 2143: 2129: 2120: 2106: 2103: 2098: 2094: 2090: 2087: 2084: 2079: 2075: 2066: 2048: 2044: 2023: 2003: 2000: 1997: 1994: 1991: 1971: 1965: 1962: 1959: 1953: 1950: 1947: 1944: 1938: 1930: 1926: 1905: 1885: 1877: 1872: 1870: 1866: 1862: 1843: 1835: 1831: 1827: 1817: 1809: 1801: 1798: 1794: 1790: 1787: 1783: 1779: 1773: 1767: 1762: 1758: 1750: 1749: 1748: 1734: 1714: 1694: 1686: 1667: 1661: 1658: 1655: 1635: 1615: 1592: 1586: 1566: 1558: 1542: 1522: 1514: 1495: 1489: 1486: 1483: 1463: 1455: 1454:regular value 1439: 1433: 1430: 1427: 1405: 1369: 1363: 1357: 1354: 1345: 1343: 1342:Milnor (1965) 1339: 1335: 1327: 1322: 1320: 1318: 1314: 1310: 1306: 1301: 1299: 1294: 1290: 1286: 1281: 1279: 1275: 1271: 1267: 1263: 1259: 1258:Banach spaces 1255: 1251: 1247: 1243: 1238: 1233: 1231: 1227: 1223: 1215: 1211: 1207: 1200: 1198: 1196: 1192: 1188: 1182: 1179: 1175: 1170: 1166: 1162: 1158: 1154: 1150: 1146: 1140: 1138: 1134: 1126: 1122: 1115: 1113: 1111: 1107: 1103: 1099: 1095: 1090: 1088: 1083: 1080:. The French 1079: 1074: 1072: 1068: 1064: 1060: 1056: 1052: 1048: 1043: 1041: 1037: 1033: 1024: 1016: 1009: 1007: 1005: 1002:, methods to 1001: 997: 993: 990: 986: 982: 978: 977: 972: 968: 964: 960: 956: 948: 946: 942: 940: 936: 932: 928: 924: 920: 916: 912: 908: 904: 899: 897: 893: 888: 880: 873: 871: 869: 868:Stefan Banach 863: 856: 851: 845: 840: 837: 834: 829: 828: 827: 822:Illustrations 821: 819: 817: 813: 805: 803: 801: 796: 792: 791: 787: 783: 779: 763: 760: 757: 751: 745: 736: 734: 733:without holes 730: 726: 708: 704: 695: 687: 685: 683: 677: 676: 656: 651: 647: 644: 641: 635: 629: 623: 616: 615: 614: 608: 606: 545: 542: 539: 536: 533: 527: 521: 514: 513: 512: 506: 504: 487: 484: 481: 478: 472: 466: 459: 458: 457: 451: 448:The function 447: 445: 443: 439: 435: 431: 430:endomorphisms 423: 416: 412: 408: 404: 402: 399: 398: 397: 396: 395: 387: 383: 379: 376: 372: 369: 368: 367: 366: 365: 357: 353: 349: 346: 345: 344: 343: 342: 334: 331: 327: 323: 320: 319: 318: 317: 316: 310: 308: 306: 301: 296: 292: 288: 283: 281: 280:Gérard Debreu 277: 276:Kenneth Arrow 273: 269: 265: 261: 257: 253: 249: 245: 241: 237: 233: 228: 226: 210: 190: 183: 167: 147: 125: 121: 117: 109: 105: 98: 76: 72: 63: 60: 44: 37: 33: 29: 25: 21: 8110:at MathPages 8086:cut-the-knot 8067: 8053: 8017: 8011: 7982: 7947: 7932: 7922: 7914: 7910: 7873: 7867: 7844: 7817: 7789:. New York: 7786: 7766: 7738: 7734: 7703: 7676: 7649: 7643: 7604: 7577: 7544: 7538: 7497: 7491: 7477: 7458: 7454: 7444: 7438:Spanier 1966 7433: 7422: 7397: 7393: 7387: 7376: 7364: 7352: 7341: 7330: 7324:Boothby 1986 7319: 7313:Boothby 1971 7308: 7296: 7285: 7273:. Retrieved 7253: 7243: 7231: 7218: 7201: 7197: 7186: 7169: 7164: 7151: 7133: 7128: 7111: 7107: 7101: 7082: 7076: 7066: 7056: 7051: 7035: 7030: 7014: 7009: 6999: 6995: 6990: 6980: 6975: 6967: 6963: 6959: 6955: 6952:homeomorphic 6942: 6923: 6917: 6904: 6896: 6882: 6876: 6863: 6844: 6838: 6825: 6816: 6811: 6802: 6798: 6788: 6779: 6774: 6757: 6755:Émile Picard 6749: 6725: 6715: 6691: 6681: 6666: 6661: 6644: 6639: 6614: 6601: 6592: 6586: 6579: 6562: 6553: 6545: 6538: 6528: 6526:Jacques Tits 6509: 6499: 6494: 6475: 6469: 6450: 6444: 6435: 6431: 6408: 6384: 6372:. Retrieved 6361: 6344: 6339: 6322: 6317: 6306:. Retrieved 6291: 6283: 6267: 6262: 6245: 6244:D. Violette 6240: 6215: 6209: 6183: 6173: 6151: 6146: 6131: 6126: 6107: 6102: 6021: 5998: 5988: 5978: 5966: 5960: 5870:, such that 5750: 5739: 5736: 5616: 5612: 5607: 5603: 5599: 5585: 5578: 5548: 5538: 5534: 5490: 5486: 5484: 5389: 5385: 5383: 5369: 5359: 5349: 5345: 5341: 5337: 5333: 5329: 5325: 5321: 5317: 5313: 5309: 5305: 5301: 5297: 5293: 5286: 5284: 5088: 5078: 5074: 5070: 5060: 5050:with smooth 5039: 5029: 4979: 4886: 4790: 4788: 4781: 4776: 4774: 4701:is an index 4650: 4643: 4588: 4566: 4506: 4364: 4176: 3996: 3992: 3986: 3971: 3963: 3959: 3953: 3947: 3934: 3931: 3928: 3706: 3700: 3694: 3688: 3685: 3675: 3671: 3667: 3663: 3659: 3655: 3651: 3645: 3636: 3630: 3626: 3622: 3616: 3612: 3604: 3602: 3593: 3589: 3585: 3581: 3569: 3565: 3561: 3557: 3553: 3549: 3547: 3542: 3538: 3528: 3523: 3519: 3515: 3511: 3507: 3503: 3499: 3495: 3491: 3487: 3483: 3479: 3475: 3471: 3467: 3463: 3459: 3455: 3450: 3446: 3442: 3439: 3434: 3419: 3415: 3411: 3407: 3401: 3382: 3376: 3370: 3364: 3358: 3351: 3345: 3339: 3333: 3327: 3321: 3315: 3309: 3303: 3297: 3291: 3285: 3279: 3273: 3267: 3261: 3255: 3249: 3243: 3240: 3116: 3110: 3104: 3099: 3094: 3088: 3081: 3073: 3067: 3061: 3055: 3049: 3043: 3037: 3031: 3024: 3018: 3012: 3006: 3000: 2990: 2984: 2981: 2826: 2820: 2814: 2808: 2802: 2799: 2793: 2787: 2781: 2775: 2769: 2763: 2757: 2751: 2745: 2739: 2733: 2727: 2721: 2715: 2705: 2700: 2695: 2692: 2686: 2680: 2674: 2668: 2662: 2656: 2650: 2644: 2637: 2633: 2627: 2621: 2610: 2606: 2603: 2597: 2591: 2584: 2580: 2574: 2568: 2562: 2556: 2550: 2545: 2540: 2537: 2527: 2521: 2517: 2513: 2507: 2501: 2495: 2488: 2474: 2472: 2467: 2367: 2252: 2249: 2121: 2064: 1875: 1873: 1864: 1860: 1858: 1346: 1331: 1313:intuitionism 1302: 1282: 1249: 1245: 1234: 1219: 1183: 1141: 1130: 1116:First proofs 1094:Émile Picard 1091: 1077: 1075: 1070: 1057:, i.e. both 1044: 1029: 996:intuitionist 974: 966: 952: 943: 934: 926: 922: 918: 914: 910: 906: 902: 900: 895: 891: 886: 884: 864: 860: 832: 825: 809: 797: 789: 785: 781: 777: 737: 691: 681: 674: 671: 612: 560: 510: 502: 455: 449: 442:homeomorphic 437: 433: 429: 427: 414: 411:Banach space 406: 393: 385: 381: 363: 340: 321:In the plane 314: 299: 284: 229: 19: 18: 7917:(2): 83–90. 7346:Hirsch 1988 7290:Milnor 1978 7236:Milnor 1965 7085:: 171–180. 6366:Belk, Jim. 6270:Puf (1982) 6134:Puf (1982) 5374:determinacy 5308:(0) is the 3732:volume form 3584:to that of 1476:is a point 1285:game theory 1237:contracting 1214:game theory 1165:Émile Borel 1110:contraction 1067:limit cycle 1051:topological 1004:approximate 965:. The case 905:which maps 727:, bounded, 507:Boundedness 352:closed ball 260:game theory 227:to itself. 8124:Categories 8102:PlanetMath 7800:0521094224 7708:Birkhäuser 7531:References 7381:Kulpa 1989 7275:1 February 6781:Piers Bohl 6308:2016-03-08 6218:: 97–115. 5816:open cover 5497:group is: 5362:David Gale 5067:computable 5048:convolving 4721:such that 4587:lies on a 3724:mollifying 3716:convolving 3531:retraction 3522:) must be 2701:continuous 2699:is only a 2666:onto (1 + 2648:onto (1 + 1317:set theory 1133:Piers Bohl 971:Piers Bohl 929:. By the 816:surjective 609:Closedness 91:such that 62:convex set 8060:EMS Press 6994:The term 6958:, and if 6698:EMS Press 6410:Archimède 6232:177796823 5943:≤ 5932:− 5904:∅ 5901:≠ 5888:∩ 5842:… 5788:→ 5742:convexity 5719:≥ 5703:− 5677: and 5659:‖ 5652:‖ 5649:− 5436:⁡ 5417:− 5405:∑ 5229:∫ 5213:φ 5178:− 5109:∂ 5106:→ 5100:: 4980:That is, 4840:≤ 4748:≤ 4660:Δ 4600:Δ 4530:≥ 4424:… 4412:∈ 4380:Δ 4376:∈ 4312:∑ 4270:∑ 4237:Δ 4233:∈ 4192:Δ 4188:∈ 4143:≥ 4090:∑ 4086:∣ 4065:∈ 4045:Δ 4012:Δ 3894:∗ 3880:∫ 3870:ω 3859:∗ 3845:∫ 3835:ω 3827:∗ 3810:∫ 3800:ω 3792:∗ 3779:∂ 3775:∫ 3768:ω 3760:∂ 3756:∫ 3200:⋅ 3163:− 2933:⋅ 2923:− 2914:− 2880:⋅ 2870:− 2709:, by the 2440:− 2401:− 2331:− 2315:∈ 2287:− 2215:− 2199:∈ 2174:− 2104:⁡ 2085:⁡ 2001:≤ 1995:≤ 1963:− 1799:− 1791:∈ 1784:∑ 1768:⁡ 1659:∈ 1487:∈ 1437:→ 1377:¯ 1289:John Nash 1210:John Nash 1201:Reception 983:in 1909. 812:bijective 761:− 735:, etc.). 729:connected 688:Convexity 684:(1) = 1. 311:Statement 8094:Archived 8038:Archived 7980:(1965). 7765:(1988). 7638:(1978). 7486:(2013), 7266:Archived 7174:Archived 7168:P. Bich 6948:manifold 6912:(1975). 6871:(1975). 6833:(1975). 6762:Archived 6723:(1989). 6649:Archived 6567:Archived 6516:won the 6349:Archived 6329:Archived 6252:Archived 6204:(1911). 6116:Archived 6057:See also 6004:column. 4862:for all 4789:Because 3535:codomain 3445: : 3404:boundary 3047:) = 1 – 2996:interior 2067:. Then 2063:for all 1579:lies in 1513:Jacobian 1174:homotopy 1149:topology 246:and the 28:topology 8100:, from 8034:0505523 8026:2320860 7996:0226651 7970:1454127 7898:0416010 7878:Bibcode 7836:0620639 7809:0115161 7755:2320146 7726:0995842 7695:0658305 7668:0492046 7627:1224675 7596:0861409 7569:0283792 7561:2317520 7522:3035127 7414:2320146 7144:WebCite 6632:⁠ 6620:⁠ 5809:compact 5592:compact 4001:simplex 3418:, the ( 3406:of the 3120:). Set 2994:in the 2818:of the 2767:. Thus 2572:. For 1137:Latvian 1063:bounded 1055:compact 949:History 434:compact 380:subset 378:compact 328:from a 59:compact 8032: 8024: 7994: 7968: 7958: 7939: 7896: 7851: 7834: 7824: 7807: 7797: 7773: 7753: 7724: 7714: 7693: 7683: 7666: 7625: 7615: 7594: 7584: 7567: 7559: 7520: 7512: 7412: 7042: 7021: 6737: 6704: 6673: 6482: 6457: 6374:22 May 6299: 6274: 6230: 6138: 5805:metric 5744:. See 5046:or by 3726:). If 3714:or by 3654:. For 3641:cyclic 3424:sphere 3410:-disk 2982:Since 2830:, set 1878:: let 1727:under 1163:, and 1106:Banach 1059:closed 937:has a 725:closed 438:convex 375:convex 330:closed 324:Every 242:, the 238:, the 8041:(PDF) 8022:JSTOR 8008:(PDF) 7751:JSTOR 7557:JSTOR 7510:JSTOR 7410:JSTOR 7269:(PDF) 7258:(PDF) 6731:17–24 6228:S2CID 6095:Notes 5300:, so 4213:also 3975:) by 3730:is a 3552:onto 3486:) to 3422:− 1)- 3374:) = ( 2660:and 2615:is a 806:Notes 409:of a 354:of a 22:is a 7956:ISBN 7937:ISBN 7849:ISBN 7822:ISBN 7795:ISBN 7771:ISBN 7712:ISBN 7681:ISBN 7613:ISBN 7582:ISBN 7277:2022 7040:ISBN 7019:ISBN 6735:ISBN 6702:ISBN 6671:ISBN 6480:ISBN 6455:ISBN 6415:Arte 6376:2015 6297:ISBN 6272:ISBN 6136:ISBN 5979:The 5961:The 5533:and 4907:and 3752:< 3603:For 3466:and 3454:has 3319:and 2761:) ⋅ 2749:) - 2737:) = 2595:) = 2520:) ⋅ 2489:The 1984:for 1898:and 1818:sign 1347:Let 1147:and 1135:, a 1061:and 1047:flow 939:zero 793:does 778:-x≠x 678:does 440:(or 333:disk 289:and 278:and 182:disk 8084:at 7933:181 7886:doi 7743:doi 7654:doi 7549:doi 7502:doi 7498:120 7463:doi 7402:doi 7206:doi 7142:at 7116:doi 7087:doi 6950:is 6928:doi 6887:doi 6849:doi 6436:127 6220:doi 6178:in 6112:on 5562:RCA 5555:WKL 5549:In 5366:Hex 5238:det 4784:+ 1 4646:+ 1 3414:is 3354:+ 1 3349:in 3108:= ( 3092:= 3084:+ 1 3022:in 2998:of 2785:/|| 2719:of 2693:If 2619:on 2566:of 2544:is 2535:. 2505:on 2308:sup 2192:sup 2095:deg 2076:deg 1822:det 1759:deg 1687:of 1515:of 1456:of 1293:Hex 909:to 894:to 814:or 413:to 270:of 223:of 26:in 8126:: 8058:, 8052:, 8036:. 8030:MR 8028:. 8018:85 8016:. 8010:. 7992:MR 7990:. 7966:MR 7964:. 7954:. 7950:. 7931:. 7927:. 7915:30 7913:. 7909:. 7894:MR 7892:. 7884:. 7874:13 7872:. 7832:MR 7830:. 7805:MR 7803:. 7793:. 7749:. 7739:86 7737:. 7722:MR 7720:. 7706:. 7691:MR 7689:. 7664:MR 7662:. 7650:32 7648:. 7642:. 7623:MR 7621:. 7611:. 7592:MR 7590:. 7565:MR 7563:. 7555:. 7545:78 7543:. 7518:MR 7516:, 7508:, 7496:, 7490:, 7457:. 7453:. 7408:. 7398:86 7396:. 7264:. 7202:41 7200:. 7196:. 7110:. 7081:. 7075:. 6922:. 6916:. 6895:. 6881:. 6875:. 6843:. 6837:. 6803:10 6733:. 6700:, 6696:, 6690:, 6591:. 6524:: 6434:. 6422:^ 6413:, 6398:^ 6226:. 6216:71 6208:. 6191:^ 6182:: 6172:: 6162:^ 5991:. 5807:) 5722:1. 5583:. 5433:Tr 5225::= 5151::= 4509:: 4003:, 3979:. 3969:(∂ 3966:-1 3951:(∂ 3738:, 3662:= 3633:−1 3619:−1 3600:. 3545:. 3526:. 3456:no 3449:→ 3426:. 3394:. 3295:⋅ 3259:⋅ 3114:, 3098:x 3053:⋅ 3035:⋅ 2672:) 2654:) 2601:+ 2470:. 2119:. 1747:: 1344:. 1287:, 1280:. 1232:. 1197:. 1159:, 1112:. 933:, 818:. 802:. 731:, 282:. 7998:. 7972:. 7900:. 7888:: 7880:: 7857:. 7838:. 7811:. 7779:. 7757:. 7745:: 7728:. 7697:. 7670:. 7656:: 7629:. 7598:. 7571:. 7551:: 7504:: 7471:. 7465:: 7459:7 7416:. 7404:: 7371:. 7359:. 7279:. 7226:. 7212:. 7208:: 7159:. 7122:. 7118:: 7112:8 7095:. 7089:: 7083:2 7046:. 7025:. 6985:. 6970:. 6968:p 6964:n 6960:p 6956:n 6936:. 6930:: 6924:2 6889:: 6883:2 6857:. 6851:: 6845:2 6743:. 6629:2 6626:/ 6623:1 6609:. 6593:2 6488:. 6463:. 6378:. 6311:. 6278:. 6234:. 6222:: 5989:D 5967:R 5946:1 5939:| 5935:j 5929:i 5925:| 5896:j 5892:U 5883:i 5879:U 5858:} 5853:m 5849:U 5845:, 5839:, 5834:1 5830:U 5826:{ 5791:X 5785:X 5782:: 5779:f 5759:X 5716:n 5706:1 5700:n 5696:x 5692:= 5687:n 5683:y 5668:2 5663:2 5655:x 5646:1 5641:= 5636:0 5632:y 5617:n 5613:y 5608:n 5604:x 5600:f 5596:ℓ 5564:0 5557:0 5539:f 5535:f 5521:) 5518:B 5515:( 5510:0 5506:H 5491:B 5487:f 5469:) 5466:) 5463:B 5460:( 5455:n 5451:H 5446:| 5442:f 5439:( 5428:n 5424:) 5420:1 5414:( 5409:n 5390:B 5386:f 5370:n 5350:r 5346:r 5342:g 5338:t 5334:g 5330:B 5328:( 5326:g 5322:t 5320:( 5318:φ 5314:φ 5310:n 5306:φ 5302:φ 5298:t 5294:t 5292:( 5289:′ 5287:φ 5270:. 5267:x 5264:d 5260:) 5257:x 5254:( 5249:t 5245:g 5241:D 5233:B 5222:) 5219:t 5216:( 5190:, 5187:x 5184:) 5181:t 5175:1 5172:( 5169:+ 5166:) 5163:x 5160:( 5157:r 5154:t 5148:) 5145:x 5142:( 5137:t 5133:g 5112:B 5103:B 5097:r 5079:q 5075:q 5071:q 5040:f 5011:. 5008:f 4988:P 4965:. 4962:P 4959:= 4956:) 4953:P 4950:( 4947:f 4924:) 4921:P 4918:( 4915:f 4895:P 4873:. 4870:j 4848:j 4844:P 4835:j 4831:) 4827:P 4824:( 4821:f 4801:P 4791:f 4782:n 4777:n 4761:. 4756:j 4752:P 4743:j 4739:) 4735:P 4732:( 4729:f 4709:j 4689:P 4669:, 4664:n 4644:k 4629:j 4609:, 4604:n 4589:k 4575:P 4552:. 4547:j 4543:) 4539:P 4536:( 4533:f 4525:j 4521:P 4507:f 4493:j 4473:P 4453:j 4433:} 4430:n 4427:, 4421:, 4418:0 4415:{ 4409:j 4389:, 4384:n 4373:P 4347:i 4343:) 4339:P 4336:( 4333:f 4327:n 4322:0 4319:= 4316:i 4308:= 4305:1 4302:= 4296:i 4292:P 4285:n 4280:0 4277:= 4274:i 4246:. 4241:n 4230:) 4227:P 4224:( 4221:f 4201:, 4196:n 4185:P 4162:. 4158:} 4154:i 4146:0 4138:i 4134:P 4125:1 4122:= 4116:i 4112:P 4105:n 4100:0 4097:= 4094:i 4081:1 4078:+ 4075:n 4070:R 4062:P 4058:{ 4054:= 4049:n 4021:, 4016:n 3999:- 3997:n 3993:f 3972:M 3964:n 3960:H 3954:M 3948:H 3940:ω 3935:M 3914:, 3911:0 3908:= 3905:) 3902:0 3899:( 3890:F 3884:B 3876:= 3873:) 3867:d 3864:( 3855:F 3849:B 3841:= 3838:) 3832:( 3823:F 3819:d 3814:B 3806:= 3803:) 3797:( 3788:F 3782:B 3771:= 3763:B 3749:0 3728:ω 3707:F 3701:B 3695:B 3689:F 3676:n 3672:U 3668:n 3664:E 3660:U 3656:n 3652:E 3637:S 3635:( 3631:n 3627:H 3623:D 3621:( 3617:n 3613:H 3605:n 3594:n 3590:Z 3586:S 3582:D 3570:n 3566:D 3562:S 3558:n 3554:S 3550:D 3543:F 3539:S 3524:x 3520:x 3518:( 3516:F 3512:S 3508:D 3504:F 3500:x 3498:( 3496:F 3492:S 3488:x 3484:x 3482:( 3480:f 3476:D 3472:x 3470:( 3468:f 3464:x 3460:D 3451:D 3447:D 3443:f 3435:F 3420:n 3416:S 3412:D 3408:n 3383:x 3380:( 3377:f 3371:y 3368:, 3365:x 3362:( 3359:F 3352:n 3346:B 3340:n 3334:n 3328:x 3325:( 3322:w 3316:t 3310:x 3304:x 3301:( 3298:w 3292:x 3286:x 3280:y 3277:( 3274:X 3268:y 3265:( 3262:X 3256:y 3250:W 3244:X 3226:. 3223:) 3220:) 3215:x 3210:( 3205:w 3195:x 3190:, 3187:) 3182:x 3177:( 3172:w 3166:t 3160:( 3157:= 3154:) 3151:t 3148:, 3143:x 3138:( 3133:X 3117:t 3111:x 3105:y 3100:R 3095:V 3089:W 3082:n 3079:( 3074:V 3068:n 3062:x 3059:( 3056:f 3050:x 3044:x 3041:( 3038:w 3032:x 3025:S 3019:x 3013:x 3010:( 3007:w 3001:B 2991:x 2985:f 2967:. 2964:) 2959:x 2954:( 2949:f 2943:) 2938:x 2928:x 2920:1 2917:( 2909:x 2903:) 2900:) 2895:x 2890:( 2885:f 2875:x 2867:1 2864:( 2861:= 2858:) 2853:x 2848:( 2843:w 2827:V 2821:n 2815:B 2809:f 2803:n 2794:S 2788:v 2782:v 2776:A 2770:v 2764:x 2758:x 2755:( 2752:u 2746:x 2743:( 2740:u 2734:x 2731:( 2728:v 2722:A 2716:u 2706:S 2696:w 2687:t 2681:n 2675:A 2669:t 2663:A 2657:S 2651:t 2645:S 2638:t 2634:f 2628:t 2622:A 2613:) 2611:x 2609:( 2607:w 2604:t 2598:x 2592:x 2589:( 2585:t 2581:f 2575:t 2569:S 2563:A 2557:S 2551:w 2541:w 2528:x 2522:x 2518:x 2516:( 2514:w 2508:S 2502:w 2496:S 2475:f 2468:f 2454:) 2451:0 2448:( 2443:1 2436:g 2415:) 2412:0 2409:( 2404:1 2397:g 2376:g 2350:| 2346:) 2343:y 2340:( 2337:f 2334:t 2328:y 2324:| 2318:K 2312:y 2302:) 2299:x 2296:( 2293:f 2290:t 2284:x 2278:= 2275:) 2272:x 2269:, 2266:t 2263:( 2260:H 2231:| 2227:) 2224:y 2221:( 2218:f 2212:y 2208:| 2202:K 2196:y 2186:) 2183:x 2180:( 2177:f 2171:x 2165:= 2162:) 2159:x 2156:( 2153:g 2130:K 2107:g 2099:p 2091:= 2088:f 2080:p 2065:t 2049:t 2045:H 2024:p 2004:1 1998:t 1992:0 1972:g 1969:) 1966:t 1960:1 1957:( 1954:+ 1951:f 1948:t 1945:= 1942:) 1939:x 1936:( 1931:t 1927:H 1906:g 1886:f 1865:p 1861:f 1844:. 1841:) 1836:x 1832:f 1828:d 1825:( 1813:) 1810:p 1807:( 1802:1 1795:f 1788:x 1780:= 1777:) 1774:f 1771:( 1763:p 1735:f 1715:p 1695:f 1671:) 1668:0 1665:( 1662:B 1656:p 1636:f 1616:K 1596:) 1593:0 1590:( 1587:B 1567:f 1543:p 1523:f 1499:) 1496:0 1493:( 1490:B 1484:p 1464:f 1440:K 1434:K 1431:: 1428:f 1406:n 1401:R 1373:) 1370:0 1367:( 1364:B 1358:= 1355:K 1296:( 1250:R 1246:n 1071:t 967:n 935:g 927:b 923:a 919:x 915:x 913:( 911:f 907:x 903:g 896:x 892:x 887:f 833:n 790:f 786:n 782:f 764:x 758:= 755:) 752:x 749:( 746:f 709:n 705:D 682:f 675:f 657:, 652:2 648:1 645:+ 642:x 636:= 633:) 630:x 627:( 624:f 592:R 570:R 546:, 543:1 540:+ 537:x 534:= 531:) 528:x 525:( 522:f 488:1 485:+ 482:x 479:= 476:) 473:x 470:( 467:f 450:f 415:K 407:K 386:K 382:K 300:n 211:K 191:D 168:I 148:f 126:0 122:x 118:= 115:) 110:0 106:x 102:( 99:f 77:0 73:x 45:f

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Knowledge (XXG)

Brouwer fixed-point theorem

Index