Brouwer fixed-point theorem

3441: 890: 1053:, 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". 1217: 5379:. 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 2988: 5069:, 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. 872:

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

1095:

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

876:

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

955:

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

852:

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

1178:. 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 857:

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.

841:

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

1182:

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

1080:. 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 6014:

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

3935: 853:

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

4183: 3247: 900:

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

6654:"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 2847: 1191:

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."

5743: 877:

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

1865: 683:

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

1195:

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

1132: 245:, 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 1250:

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

873:

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.

5491: 3755: 2375: 1306:

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

2256: 6399:

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

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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.

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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

6163:

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.

5291: 846:= 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. 1142:

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.

1187:, 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. 5925: 2983:{\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} }).} 5092:

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.

6298:

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

5879: 5211: 5133: 1150:

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.

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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

2128: 5751:

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

4454: 4267: 1400: 4782: 4573: 2379:

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

842:

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

799:-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 5637: 4871: 1993: 4410: 4222: 5812: 2542:.) 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 1429: 5756: 3525:

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

1272: 149: 2025: 1764: 1056:

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

6598: 4690: 4630: 4042: 986: 614: 592: 5967: 912:

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

2475: 2436: 1692: 1520: 567: 5542: 5347:, 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 1461: 509: 4986: 785: 7233:

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

2072: 732: 100: 6339: 6126: 4945: 1617: 1034: 1026: 6042: 5032: 4894: 1283:, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the 1231:

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

326:

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:

5780: 5009: 4916: 4822: 4730: 4710: 4650: 4596: 4514: 4494: 4474: 2397: 2151: 2045: 1927: 1907: 1756: 1736: 1716: 1657: 1637: 1588: 1564: 1544: 1485: 232: 212: 189: 169: 66: 6659: 5403:

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

5084:, 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 3654:. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group. 703:

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

6083: 3930:{\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,} 6512:

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

3697:

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

6808:[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]. 5409: 8145: 2266: 1204:. This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became 8104: 2159: 7951: 7863: 7836: 7785: 6749: 6494: 6469: 4178:{\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\}.} 8155: 6420: 6262: 3242:{\displaystyle {\mathbf {X} }({\mathbf {x} },t)=(-t\,{\mathbf {w} }({\mathbf {x} }),{\mathbf {x} }\cdot {\mathbf {w} }({\mathbf {x} })).} 6772: 4275: 6359: 6059: 5969:. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers. 7879: 4804:

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

6617: 5601:

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

5359:, which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area of 305: 7970: 7726: 7695: 7627: 7596: 7503: 7276: 7054: 7033: 6716: 6685: 6311: 6286: 6150: 5054: 3722: 2721: 6806:"Первое доказательство теоремы о неподвижной точке при непрерывном отображении шара в себя, данное латышским математиком П.Г.Болем" 5748:

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.

5065:. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by 1096:

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

8048: 7150: 6093: 7939: 5219: 1523: 7184: 2811:

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

7809: 5987:(functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set. 1344: 1201: 336: 749:

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

261:. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about 6577: 8150: 8140: 8070: 7272: 7204: 6708: 5991: 1288: 1264: 1042: 998:

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

810: 1108: 7998: 6212: 5973: 1319: 1010: 991: 411: 274: 42: 6770:

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

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

1314:

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

1119:. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a 1092: 6334: 6119: 5884: 3949:

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

7962: 7801: 7655: 6073: 1163: 1112: 1100:

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

941: 6656: 3943:

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

1235:. Brouwer's theorem is probably the most important. It is also among the foundational theorems on the topology of 6099: 1267:

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

214:

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

5998:

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

5053:

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

2484:

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

707:, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball 6528: 6404:

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

1567: 6540: 6049: 5579: 5565: 1252: 258: 7045:"... Brouwer's fixed point theorem, perhaps the most important fixed point theorem." p xiii V. I. Istratescu 5832: 5138: 5103: 7234: 6908:... cette dernière propriété, bien que sous des hypothèses plus grossières, ait été démontré par H. Poincaré 6765: 6078: 3440: 3397:),0). The advantage of this proof is that it uses only elementary techniques; more general results like the 1280: 1104: 1033: 1025: 630: 301: 296:

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

254: 5738:{\displaystyle y_{0}={\sqrt {1-\|x\|_{2}^{2}}}\quad {\text{ and}}\quad y_{n}=x_{n-1}{\text{ for }}n\geq 1.} 2081: 278: 4415: 4227: 3619: 3618:> 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of 1361: 5504:

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

4735: 4526: 809:

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

8124: 8092: 6221: 6185: 4827: 3541: 1932: 1348: 1284: 889: 266: 262: 6992: 6828: 4379: 4191: 734:. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also 6441:

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

5785: 1860:{\displaystyle \operatorname {deg} _{p}(f)=\sum _{x\in f^{-1}(p)}\operatorname {sign} \,\det(df_{x}).} 1405: 1263:

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

1087:

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

1045:

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

8060: 7888: 7182:

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

6929: 6888: 6850: 6698: 2510:

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

1695: 1323: 1322:. He became the originator and zealous defender of a way of formalising mathematics that is known as 1240: 1236: 1197: 1014: 246: 8101: 7524: 6791: 6741: 6735: 5597:

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

1878:, with sheets counted oppositely if they are oppositely oriented. This is thus a generalization of 105: 8023: 7550: 5984: 5591: 5572: 5561: 5505: 5383:-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the 3414: 3398: 3006: 2627: 1998: 1276: 1247: 1120: 973: 969: 854: 282: 242: 46: 34: 4665: 4605: 4017: 597: 575: 8032: 7761: 7567: 7520: 7420: 7088: 6532: 6238: 6011: 5930: 3987: 3954: 3658: 3588: 3402: 2501: 2492:

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

1315: 1232: 1205: 1179: 1155: 1084:. If the area is a circular band, or if it is not closed, then this is not necessarily the case. 1057: 1050: 999: 965: 949: 343: 250: 192: 6037: 3999: 3746: 1308: 1037:

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

6165: 5100:

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

2441: 2402: 1662: 1490: 528: 7966: 7947: 7859: 7832: 7805: 7781: 7722: 7691: 7623: 7592: 7050: 7029: 6977:, then the open set is never homeomorphic to an open subset of a Euclidean space of dimension 6745: 6731: 6712: 6681: 6569: 6553: 6524: 6490: 6465: 6401: 6307: 6282: 6257: 6146: 6054: 5995: 5814:

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

5511: 5066: 3584: 1434: 1167: 1097: 1046: 473: 297: 7877:(1976). "A constructive proof of the Brouwer fixed point theorem and computational results". 7792:(see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction) 6216: 4953: 752: 443:(functions that have the same set as the domain and codomain) and for nonempty sets that are 7896: 7753: 7690:. Cahiers Scientifiques (in French). Vol. IX. Paris: Gauthier-Villars. pp. 44–47. 7664: 7559: 7512: 7473: 7412: 7216: 7167: 7126: 7097: 6938: 6920: 6897: 6879: 6859: 6841: 6769: 6356: 6301: 6230: 6180: 6088: 5994:

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

5376: 1303: 1299: 1279:. One also meets the theorem and its variants outside topology. It can be used to prove the 1220: 1188: 1171: 1135: 995: 743: 315: 290: 8044: 8006: 7980: 7908: 7846: 7819: 7736: 7705: 7678: 7637: 7606: 7579: 7532: 6585:, on the website of l'Association roumaine des chercheurs francophones en sciences humaines 2050: 1318:, and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of 1029:

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

710: 78: 17: 8108: 8040: 8002: 7976: 7904: 7842: 7815: 7732: 7701: 7674: 7633: 7619: 7602: 7575: 7528: 7263: 7188: 6776: 6663: 6581: 6363: 6343: 6266: 6130: 5981: 5822: 4921: 2489: 1593: 739: 594:

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

366: 235: 7516: 7892: 7068: 5014: 4947:

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

4876: 1183:

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

8015: 7992: 7874: 7646: 5765: 5073: 5046: 4994: 4901: 4807: 4715: 4695: 4635: 4581: 4499: 4479: 4459: 2382: 2136: 2030: 1912: 1892: 1879: 1741: 1721: 1701: 1642: 1622: 1573: 1549: 1529: 1470: 1216: 1175: 1002: 217: 197: 174: 154: 51: 8118: 7669: 7650: 7498: 7478: 7461: 896:

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

8134: 7773: 7718: 7651:"Finding zeroes of maps: Homotopy methods that are constructive with probability one" 7259: 7145: 6962: 6943: 6924: 6902: 6883: 6864: 6845: 6242: 6190: 5819: 5602: 5598: 5590:

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

5077: 5062: 5042: 3730: 1464: 1116: 1065: 878: 704: 452: 388: 286: 4786:

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

2736:

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

1009:

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

8096: 7181: 6596:

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

6536: 6010:

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

5815: 5363:

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

3651: 3608: 1268: 1006: 421: 7831:. Mathematics and its Applications. Vol. 7. Dordrecht–Boston, MA: D. Reidel. 7130: 5976:

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

4662:

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

8016:"Analytic proofs of the 'hairy ball theorem' and the Brouwer fixed-point theorem" 7591:. Pure and Applied Mathematics. Vol. 120 (Second ed.). Academic Press. 5782:

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

1287:. The theorem can also be found in existence proofs for the solutions of certain 7988: 6197:(Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477 (French) 6124: 5762:

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

5384: 5058: 3742: 3726: 1295: 1224: 1077: 1073: 362: 270: 69: 7548:

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

6574: 6303:

General Equilibrium Analysis: Existence and Optimality Properties of Equilibria

3607:= 2 can also be proven by contradiction based on a theorem about non-vanishing 1351:. Several modern accounts of the proof can be found in the literature, notably 285:

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

8112: 7494: 5826: 5752: 5486:{\displaystyle \displaystyle \sum _{n}(-1)^{n}\operatorname {Tr} (f|H_{n}(B))} 5372: 3583:= 2 is less obvious, but can be proven by using basic arguments involving the 1327: 1143: 1069: 981: 826: 735: 455:

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

385: 340: 72: 5355:

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

5296:

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

3342:) are both non-zero). This contradiction proves the fixed point theorem when 514:

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

6805: 5582:, so this gives a precise description of the strength of Brouwer's theorem. 3734: 2370:{\displaystyle H(t,x)={\frac {x-tf(x)}{\sup _{y\in K}\left|y-tf(y)\right|}}} 837:

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

822: 7102: 7083: 1111:. Picard's approach is based on a result that would later be formalised by 1060:? Poincaré discovered that the answer can be found in what we now call the 806:

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

7959:

From calculus to cohomology: de Rham cohomology and characteristic classes

7220: 7117:

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

7015:

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

6559:

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

6378: 5088:

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

6958: 5606: 3545: 1200:

to arbitrary dimension and established the properties connected with the

1184: 1159: 1061: 38: 2251:{\displaystyle g(x)={\frac {x-f(x)}{\sup _{y\in K}\left|y-f(y)\right|}}} 1870:

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

8036: 7765: 7571: 7424: 7154: 6234: 4790:-dimensional simplex whose vertices are colored with the entire set of 4011: 787:, which is a continuous function from the unit circle to itself. Since 151:. The simplest forms of Brouwer's theorem are for continuous functions 7403:

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

1131: 6357:

Théorèmes du Point Fixe et Applications aux Equations Différentielles

3434: 1147: 964:

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

273:. In economics, Brouwer's fixed-point theorem and its extension, the 7934: 7900: 7757: 7744:

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

7563: 7416: 4002:. We now give an outline of the proof for the special case in which 3540:

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

1330:. Brouwer disavowed his original proof of the fixed-point theorem. 1076:, then the trajectory either becomes stationary, or it approaches a 7589:

An introduction to differentiable manifolds and Riemannian geometry

7011: 6667:

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

3685:

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

3587:

of the respective spaces: the retraction would induce a surjective

3354:

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

2399:

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

7918:"An integral theorem and its applications to coincidence theorems" 7009:

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

6511: 5045:, based on the impossibility of a differentiable retraction. The 4366:{\displaystyle \sum _{i=0}^{n}{P_{i}}=1=\sum _{i=0}^{n}{f(P)_{i}}} 3603:

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

3559:

Intuitively it seems unlikely that there could be a retraction of

3439: 2817:

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

1215: 1130: 1064:

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

1032: 1024: 405:

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

5609:

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

2589:

sufficiently small, a routine computation shows that the mapping

318:

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

7499:"A Borsuk–Ulam equivalent that directly implies Sperner's lemma" 6531:'s mathematical competition in 1889 for his work on the related 6425: 5395:

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

3485:) are distinct. Because they are distinct, for every point x in 416:

Every continuous function from a nonempty convex compact subset

6925:"The cradle of modern topology, according to Brouwer's inedita" 6884:"The cradle of modern topology, according to Brouwer's inedita" 6846:"The cradle of modern topology, according to Brouwer's inedita" 6515:

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

2488:, and then to continuous functions. The more modern advent of 1239:

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

5343:) (that is, the Lebesgue measure of the image of the ball via 3657:

The impossibility of a retraction can also be shown using the

821:

The continuous function in this theorem is not required to be

7917: 5135:

is a smooth retraction, one considers the smooth deformation

6418:

This citation comes originally from a television broadcast:

1017:

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

888: 7715:

A history of algebraic and differential topology, 1900–1960

6737:

A History of Algebraic and Differential Topology, 1900–1960

4824:

which satisfies the labeling condition in all coordinates:

3681:- 1, and vanishes otherwise. If a retraction existed, then 7069:

Brouwer's Fixed Point Theorem and the Jordan Curve Theorem

6379:"Why is convexity a requirement for Brouwer fixed points?" 5496:

and in particular if the Lefschetz number is nonzero then

1343:

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

1227:

to prove the existence of an equilibrium strategy profile.

956:

respect to its original position on the unfolded string."

352:

This can be generalized to an arbitrary finite dimension:

5315:

is a constant function, which is a contradiction because

5286:{\displaystyle \varphi (t):=\int _{B}\det Dg^{t}(x)\,dx.} 2571:. It can be extended radially to a small spherical shell 2689:. This gives a contradiction, because, if the dimension 2565:

is a continuously differentiable unit tangent vector on

2477:

are precisely the fixed points of the original function

795:

has no fixed point. The analogous example works for the

7686:

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

6186:

Note sur quelques applications de l'indice de Kronecker

3505:(see illustration). By calling this intersection point 3451:

Suppose, for contradiction, that a continuous function

2724:, it can be uniformly approximated by a polynomial map 1041:

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

7798:

Homology theory: An introduction to algebraic topology

7205:"L. J. E. Brouwer : Topologie et constructivisme" 6195:

Introduction à la théorie des fonctions d'une variable

7379: 6680:

Kluwer Academic Publishers (réédition de 2001) p 113

6161:

More exactly, according to Encyclopédie Universalis:

5933: 5887: 5835: 5788: 5768: 5640: 5514: 5413: 5412: 5222: 5141: 5106: 5017: 4997: 4956: 4924: 4904: 4879: 4830: 4810: 4738: 4718: 4698: 4668: 4638: 4608: 4584: 4529: 4502: 4482: 4462: 4418: 4382: 4278: 4230: 4194: 4053: 4020: 3758: 3733:

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

3140: 2850: 2444: 2405: 2385: 2269: 2162: 2139: 2084: 2053: 2033: 2001: 1935: 1915: 1895: 1767: 1744: 1724: 1704: 1665: 1645: 1625: 1596: 1576: 1552: 1532: 1493: 1473: 1437: 1431:

centered at the origin. Suppose for simplicity that

1408: 1364: 755: 713: 633: 600: 578: 531: 476: 314:-dimensional closed ball was first proved in 1910 by 220: 200: 177: 157: 108: 81: 54: 7922:

Acta Universitatis Carolinae. Mathematica et Physica

6965:

to an open subset of a Euclidean space of dimension

6489:. Dordrecht-Boston, Mass.: D. Reidel Publishing Co. 3501:

and follow the ray until it intersects the boundary

2636:

and that the volume of its image is a polynomial in

691:

have a fixed point for the closed interval , namely

6258:

Applications du lemme de Sperner pour les triangles

5757:

fixed-point theorems in infinite-dimensional spaces

3469:fixed point. This means that, for every point x in 3258:is a continuous vector field on the unit sphere of 1107:, a contemporary mathematician who generalized the 7265:Topics in Linear and Nonlinear Functional Analysis 6676:Poincaré's theorem is stated in: V. I. Istratescu 5961: 5919: 5873: 5806: 5774: 5737: 5622:) from the closed unit ball of ℓ to the sequence ( 5536: 5485: 5285: 5205: 5127: 5026: 5003: 4980: 4939: 4910: 4888: 4865: 4816: 4776: 4724: 4704: 4684: 4644: 4624: 4590: 4567: 4508: 4488: 4468: 4448: 4404: 4365: 4261: 4216: 4177: 4036: 3929: 3721:is smooth, since it can be approximated using the 3241: 2982: 2469: 2430: 2391: 2369: 2250: 2145: 2122: 2066: 2039: 2019: 1987: 1929:be two continuously differentiable functions, and 1921: 1901: 1859: 1750: 1730: 1710: 1686: 1651: 1631: 1611: 1582: 1558: 1546:is non-singular at every point of the preimage of 1538: 1514: 1479: 1455: 1423: 1394: 1246:Besides the fixed-point theorems for more or less 1162:. His initial interest lay in an attempt to solve 779: 726: 672: 608: 586: 561: 503: 226: 206: 183: 163: 143: 94: 60: 4659:coordinates which are not zero on this sub-face. 1275:, a result generalized further by S. Kakutani to 7935:A First Course in Sobolev Spaces: Second Edition 7618:. Graduate Texts in Mathematics. Vol. 139. 7466:Proceedings of the American Mathematical Society 5564:, Brouwer's theorem can be proved in the system 5248: 2318: 2202: 1832: 1311:), financial equilibria and incomplete markets. 7946:. American Mathematical Society. pp. 734. 7311: 6176: 6174: 6096:– equivalent to the Brouwer fixed-point theorem 5391:A proof using the Lefschetz fixed-point theorem 2642:. On the other hand, as a contraction mapping, 987:Journal für die reine und angewandte Mathematik 375:A slightly more general version is as follows: 265:and is covered in most introductory courses on 191:in the real numbers to itself or from a closed 7367: 7049:Kluwer Academic Publishers (new edition 2001) 7028:Kluwer Academic Publishers (new edition 2001) 6557:Les méthodes nouvelles de la mécanique céleste 6207: 6205: 6203: 5920:{\displaystyle U_{i}\cap U_{j}\neq \emptyset } 4376:Hence, by the pigeonhole principle, for every 3571:= 1, the impossibility is more basic, because 2133:If there is no fixed point of the boundary of 1302:used the theorem to prove that in the game of 447:(thus, in particular, bounded and closed) and 439:The theorem holds only for functions that are 4269:Hence the sum of their coordinates is equal: 8: 6599:Journal de Mathématiques Pures et Appliquées 5868: 5836: 5669: 5662: 4443: 4425: 3968:) which is isomorphic to the homology group 3575:(i.e., the endpoints of the closed interval 2802:|| is a smooth unit tangent vector field on 1166:. In 1909, during a voyage to Paris, he met 8093:Brouwer's Fixed Point Theorem for Triangles 7437: 7166:For context and references see the article 6436: 6434: 6084:Infinite compositions of analytic functions 5578:Brouwer's theorem for a square implies the 3689:- 1 would have to vanish, a contradiction. 1103:Poincaré's method was analogous to that of 7994:Topology from the differentiable viewpoint 7262:(2019). "10. The Brouwer mapping degree". 6544:Site du Ministère Culture et Communication 3076:) is strictly positive. From the original 2999:has no fixed points, it follows that, for 1694:is defined as the sum of the signs of the 384:Every continuous function from a nonempty 7856:Fixed Points: Algorithms and Applications 7668: 7477: 7345: 7101: 6942: 6901: 6863: 6462:Fixed points: algorithms and applications 6261:Bulletin AMQ, V. XLVI N° 4, (2006) p 17. 6217:"Über Abbildungen von Mannigfaltigkeiten" 5948: 5934: 5932: 5905: 5892: 5886: 5862: 5843: 5834: 5787: 5767: 5721: 5709: 5696: 5686: 5677: 5672: 5654: 5645: 5639: 5519: 5513: 5464: 5455: 5437: 5418: 5411: 5327:(1) is zero. The geometric idea is that 5273: 5258: 5242: 5221: 5146: 5140: 5105: 5016: 4996: 4955: 4923: 4903: 4878: 4857: 4844: 4829: 4809: 4765: 4752: 4737: 4717: 4697: 4673: 4667: 4637: 4613: 4607: 4583: 4556: 4534: 4528: 4501: 4481: 4461: 4417: 4393: 4381: 4356: 4342: 4336: 4325: 4305: 4300: 4294: 4283: 4277: 4250: 4229: 4205: 4193: 4159: 4147: 4138: 4125: 4120: 4114: 4103: 4084: 4080: 4079: 4058: 4052: 4025: 4019: 3903: 3893: 3868: 3858: 3836: 3823: 3801: 3788: 3769: 3757: 3677:– (0) is one-dimensional in degree 0 and 3224: 3223: 3214: 3213: 3204: 3203: 3191: 3190: 3181: 3180: 3179: 3152: 3151: 3142: 3141: 3139: 2968: 2967: 2958: 2957: 2956: 2947: 2946: 2937: 2936: 2918: 2917: 2916: 2904: 2903: 2894: 2893: 2884: 2883: 2862: 2861: 2852: 2851: 2849: 2449: 2443: 2410: 2404: 2384: 2321: 2291: 2268: 2205: 2178: 2161: 2138: 2108: 2089: 2083: 2058: 2052: 2032: 2000: 1940: 1934: 1914: 1894: 1845: 1831: 1808: 1797: 1772: 1766: 1743: 1723: 1703: 1664: 1644: 1624: 1595: 1575: 1551: 1531: 1492: 1472: 1436: 1415: 1411: 1410: 1407: 1371: 1363: 1326:, which at the time made a stand against 754: 718: 712: 649: 632: 602: 601: 599: 580: 579: 577: 530: 475: 219: 199: 176: 156: 135: 119: 107: 86: 80: 53: 7149:CMI Université Paul Cézanne (2008–2009) 7143:These examples are taken from: F. Boyer 6804:Myskis, A. D.; Rabinovic, I. M. (1955). 6017: 3599:, but the latter group is isomorphic to 3413:The proof uses the observation that the 791:holds for any point of the unit circle, 269:. It appears in unlikely fields such as 8081:. New York-Toronto-London: McGraw-Hill. 7448: 7334: 7323: 7146:Théorèmes de point fixe et applications 6414: 6412: 6410: 6335:Point fixe, et théorèmes du point fixe 6110: 5874:{\displaystyle \{U_{1},\ldots ,U_{m}\}} 5548:acts as the identity on this group, so 5323:-dimensional volume of the ball, while 5206:{\displaystyle g^{t}(x):=tr(x)+(1-t)x,} 5128:{\displaystyle r\colon B\to \partial B} 2784:is polynomial and nowhere vanishing on 616:is convex and closed, but not bounded. 346:to itself has at least one fixed point. 7957:Madsen, Ib; Tornehave, Jørgen (1997). 7796:Hilton, Peter J.; Wylie, Sean (1960). 7356: 7300: 7246: 5613: : ℓ → ℓ which sends a sequence ( 5571:, and conversely over the base system 4898:Because the sum of the coordinates of 3715:. In that case it can be assumed that 2543: 1352: 1138:helped Brouwer to formalize his ideas. 673:{\displaystyle f(x)={\frac {x+1}{2}},} 7391: 7192:Institut Henri Poincaré, Paris (2007) 7084:"Der Fixpunktsatz in Funktionsräumen" 6780:Journal de Mathématiques p 217 (1893) 6043:Knaster–Kuratowski–Mazurkiewicz lemma 4632:then by the same argument, the index 3489:, we can construct a unique ray from 2559:. By scaling, it can be assumed that 2522:. (The tangency condition means that 2123:{\displaystyle \deg _{p}f=\deg _{p}g} 1885:The degree satisfies the property of 7: 7645:Chow, Shui Nee; Mallet-Paret, John; 7525:10.4169/amer.math.monthly.120.04.346 7517:10.4169/amer.math.monthly.120.04.346 7072:University of Auckland, New Zealand. 6367:Université de Nice-Sophia Antipolis. 5759:for a discussion of these theorems. 5605:. For example, in the Hilbert space 5072:R. Bruce Kellogg, Tien-Yien Li, and 4449:{\displaystyle j\in \{0,\ldots ,n\}} 4262:{\displaystyle f(P)\in \Delta ^{n}.} 3409:A proof using homology or cohomology 3324:has norm strictly less than 1, then 3294:) is nowhere vanishing (because, if 3264:, satisfying the tangency condition 2695:of the Euclidean space is odd, (1 + 2653:must restrict to a homeomorphism of 2496:A proof using the hairy ball theorem 1395:{\displaystyle K={\overline {B(0)}}} 1255:says that a continuous map from the 572:which is a continuous function from 7380:Chow, Mallet-Paret & Yorke 1978 7271:. Graduate Studies in Mathematics. 7026:Fixed Point Theory. An Introduction 4777:{\displaystyle f(P)_{j}\leq P_{j}.} 4568:{\displaystyle P_{j}\geq f(P)_{j}.} 3661:of open subsets of Euclidean space 3082:-dimensional space Euclidean space 1874:lying over a small open set around 1463:is continuously differentiable. A 968:, and is the basis of more general 7880:SIAM Journal on Numerical Analysis 7873:Kellogg, R. Bruce; Li, Tien-Yien; 7047:Fixed Point Theory an Introduction 6678:Fixed Point Theory an Introduction 6645:on ]0, 1[ has no fixed point. 5914: 5119: 4866:{\displaystyle f(P)_{j}\leq P_{j}} 4670: 4610: 4390: 4247: 4202: 4055: 4022: 3789: 3770: 3579:) is not even connected. The case 1988:{\displaystyle H_{t}(x)=tf+(1-t)g} 1271:. This generalization is known as 868:Explanations attributed to Brouwer 25: 7746:The American Mathematical Monthly 7670:10.1090/S0025-5718-1978-0492046-9 7504:The American Mathematical Monthly 7479:10.1090/S0002-9939-1956-0078693-4 7405:The American Mathematical Monthly 6973:is a positive integer other than 5399:from a finite simplicial complex 5371:A quite different proof given by 5055:Weierstrass approximation theorem 4516:th coordinate of its image under 4405:{\displaystyle P\in \Delta ^{n},} 4217:{\displaystyle P\in \Delta ^{n},} 3723:Weierstrass approximation theorem 2722:Weierstrass approximation theorem 1570:, every point of the preimage of 1294:Other areas are also touched. In 1005:, they ran contrary to Brouwer's 952:in ; this zero is a fixed point. 361:Every continuous function from a 8054:from the original on 2022-10-09. 6279:Calcul différentiel et géométrie 6143:Calcul différentiel et géométrie 5807:{\displaystyle f:X\rightarrow X} 5556:A proof in a weak logical system 5041:There is also a quick proof, by 4496:is greater than or equal to the 4006:is a function from the standard 3225: 3215: 3205: 3192: 3182: 3153: 3143: 2969: 2959: 2948: 2938: 2919: 2905: 2895: 2885: 2863: 2853: 1424:{\displaystyle \mathbb {R} ^{n}} 435:Importance of the pre-conditions 7940:Graduate Studies in Mathematics 7282:from the original on 2022-10-09 6990:J. J. O'Connor E. F. Robertson 6826:J. J. O'Connor E. F. Robertson 6789:J. J. O'Connor E. F. Robertson 6740:. Boston: Birkhäuser. pp. 5691: 5685: 4652:can be selected from among the 3669:≥ 2, the de Rham cohomology of 3444:Illustration of the retraction 2504:states that on the unit sphere 2438:is not empty. The elements of 1402:denote the closed unit ball in 881:'s, that guarantee uniqueness. 8146:Theory of continuous functions 8059:Sobolev, Vladimir I. (2001) , 7827:Istrăţescu, Vasile I. (1981). 6697:Voitsekhovskii, M.I. (2001) , 6594:This question was studied in: 6060:Lusternik–Schnirelmann theorem 5949: 5935: 5798: 5531: 5525: 5479: 5476: 5470: 5456: 5449: 5434: 5424: 5270: 5264: 5232: 5226: 5194: 5182: 5176: 5170: 5158: 5152: 5116: 5049:starts by noting that the map 4966: 4960: 4934: 4928: 4841: 4834: 4749: 4742: 4553: 4546: 4353: 4346: 4240: 4234: 3915: 3909: 3883: 3874: 3848: 3842: 3813: 3807: 3591:from the fundamental group of 3233: 3230: 3220: 3197: 3187: 3170: 3164: 3148: 2974: 2964: 2953: 2927: 2913: 2910: 2900: 2874: 2868: 2858: 2464: 2458: 2425: 2419: 2356: 2350: 2312: 2306: 2285: 2273: 2237: 2231: 2196: 2190: 2172: 2166: 1979: 1967: 1952: 1946: 1851: 1835: 1823: 1817: 1787: 1781: 1681: 1675: 1606: 1600: 1509: 1503: 1447: 1383: 1377: 1345:degree of a continuous mapping 1289:partial differential equations 1273:Schauder's fixed-point theorem 1202:degree of a continuous mapping 765: 759: 643: 637: 541: 535: 486: 480: 369:into itself has a fixed point. 304:. Proving results such as the 144:{\displaystyle f(x_{0})=x_{0}} 125: 112: 1: 7854:Karamardian, S., ed. (1977). 7273:American Mathematical Society 7209:Revue d'Histoire des Sciences 7201:For a long explanation, see: 7131:10.1215/S0012-7094-41-00838-4 6460:Karamardian, Stephan (1977). 5992:Lefschetz fixed-point theorem 5829:has a finite open refinement 5076:turned Hirsch's proof into a 3998:The BFPT can be proved using 3693:A proof using Stokes' theorem 2835:-dimensional Euclidean space 2020:{\displaystyle 0\leq t\leq 1} 1265:Lefschetz fixed-point theorem 1043:stability of the solar system 1011:constructivism in mathematics 811:Lefschetz fixed-point theorem 277:, play a central role in the 31:Brouwer's fixed-point theorem 7999:University Press of Virginia 7587:Boothby, William M. (1986). 7368:Kellogg, Li & Yorke 1976 6993:Luitzen Egbertus Jan Brouwer 6944:10.1016/0315-0860(75)90111-1 6903:10.1016/0315-0860(75)90111-1 6865:10.1016/0315-0860(75)90111-1 6829:Luitzen Egbertus Jan Brouwer 6541:Célébrations nationales 2004 6464:. New York: Academic Press. 6300:Florenzano, Monique (2003). 6141:See page 15 of: D. Leborgne 5974:Kakutani fixed point theorem 5500:must have a fixed point. If 4685:{\displaystyle \Delta ^{n},} 4625:{\displaystyle \Delta ^{n},} 4037:{\displaystyle \Delta ^{n},} 3367:dimensions and the mapping 3088:, construct a new auxiliary 2549:In fact, suppose first that 1387: 609:{\displaystyle \mathbb {R} } 587:{\displaystyle \mathbb {R} } 412:Schauder fixed point theorem 275:Kakutani fixed-point theorem 75:to itself, there is a point 8156:Theorems in convex geometry 8125:Brouwer Fixed Point Theorem 8066:Encyclopedia of Mathematics 7312:Madsen & Tornehave 1997 6704:Encyclopedia of Mathematics 6657:Poincaré, Henri (1854–1912) 6485:Istrăţescu, Vasile (1981). 6033:Brouwer fixed-point theorem 5962:{\displaystyle |i-j|\leq 1} 5351:passes from 0 to 1 the map 5096:A proof using oriented area 3282:) = 0. Moreover, 2557:continuously differentiable 1113:another fixed-point theorem 18:Brouwer fixed point theorem 8172: 8077:Spanier, Edwin H. (1966). 7963:Cambridge University Press 7802:Cambridge University Press 7656:Mathematics of Computation 6810:Успехи математических наук 6618:Poincaré–Bendixson theorem 6342:December 26, 2008, at the 6129:December 26, 2008, at the 6074:Banach fixed-point theorem 5367:A proof using the game Hex 5335:) is the oriented area of 4692:the color of every vertex 2536:= 0 for every unit vector 2027:. Suppose that the point 990:). It was later proved by 942:intermediate value theorem 306:Poincaré–Bendixson theorem 7916:Kulpa, Władysław (1989). 7119:Duke Mathematical Journal 7066:E.g.: S. Greenwood J. Cao 6572:taken from: P. A. Miquel 6100:Topological combinatorics 4602:-dimensional sub-face of 3729:with non-negative smooth 2470:{\displaystyle g^{-1}(0)} 2431:{\displaystyle g^{-1}(0)} 1687:{\displaystyle p\in B(0)} 1566:. In particular, by the 1515:{\displaystyle p\in B(0)} 1347:, stemming from ideas in 562:{\displaystyle f(x)=x+1,} 428:itself has a fixed point. 399:itself has a fixed point. 45:. It states that for any 43:L. E. J. (Bertus) Brouwer 8014:Milnor, John W. (1978). 7932:Leoni, Giovanni (2017). 7713:Dieudonné, Jean (1989). 7614:Bredon, Glen E. (1993). 6575:La catégorie de désordre 6277:Page 15 of: D. Leborgne 6094:Poincaré–Miranda theorem 5537:{\displaystyle H_{0}(B)} 5375:is based on the game of 5213:and the smooth function 3955:de Rham cohomology group 3940:giving a contradiction. 3745:on the boundary then by 3513:), we define a function 3318:) is non-zero; while if 2823:of the closed unit ball 2701:) is not a polynomial. 1568:inverse function theorem 1456:{\displaystyle f:K\to K} 1109:Cauchy–Lipschitz theorem 1093:Encyclopædia Universalis 980:= 3 first was proved by 504:{\displaystyle f(x)=x+1} 395:of a Euclidean space to 7462:"A fixed point theorem" 7438:Hilton & Wylie 1960 7235:constructive set theory 6957:If an open subset of a 6306:. Springer. p. 7. 6120:Théorèmes du point fixe 6079:Fixed-point computation 4981:{\displaystyle f(P)=P.} 4412:there must be an index 3039:, the scalar product 3027:) is non-zero; and for 2808:, a contradiction. 2714:unit tangent vector on 1281:Hartman-Grobman theorem 1259:-dimensional sphere to 1164:Hilbert's fifth problem 972:which are important in 780:{\displaystyle f(x)=-x} 255:invariance of dimension 171:from a closed interval 8119:Reconstructing Brouwer 7780:. New York: Springer. 7203:Dubucs, J. P. (1988). 7187:June 11, 2011, at the 7103:10.4064/sm-2-1-171-180 6616:This follows from the 6117:E.g. F & V Bayart 5980:, but considers upper 5963: 5921: 5875: 5808: 5776: 5739: 5538: 5487: 5287: 5207: 5129: 5028: 5005: 4982: 4941: 4912: 4890: 4867: 4818: 4778: 4726: 4706: 4686: 4646: 4626: 4592: 4569: 4510: 4490: 4470: 4450: 4406: 4367: 4341: 4299: 4263: 4218: 4179: 4119: 4038: 3931: 3692: 3552:) is a fixed point of 3448: 3243: 2984: 2471: 2432: 2393: 2371: 2252: 2147: 2124: 2068: 2047:is a regular value of 2041: 2021: 1989: 1923: 1903: 1882:to higher dimensions. 1861: 1752: 1732: 1718:over the preimages of 1712: 1688: 1653: 1633: 1613: 1584: 1560: 1540: 1516: 1481: 1457: 1425: 1396: 1228: 1139: 1038: 1030: 984:in 1904 (published in 893: 781: 728: 674: 624:Consider the function 610: 588: 563: 522:Consider the function 505: 467:Consider the function 263:differential equations 228: 208: 185: 165: 145: 96: 62: 7778:Differential Topology 7616:Topology and geometry 7221:10.3406/rhs.1988.4094 7082:Schauder, J. (1930). 6509:See F. Brechenmacher 6265:June 8, 2011, at the 6222:Mathematische Annalen 5964: 5922: 5876: 5809: 5777: 5740: 5539: 5488: 5288: 5208: 5130: 5029: 5006: 4983: 4942: 4913: 4891: 4868: 4819: 4779: 4727: 4707: 4687: 4647: 4627: 4593: 4570: 4511: 4491: 4471: 4451: 4407: 4368: 4321: 4279: 4264: 4219: 4180: 4099: 4039: 3994:A combinatorial proof 3932: 3544:: every point of the 3443: 3244: 2985: 2472: 2433: 2394: 2372: 2261:is well-defined, and 2253: 2153:, then the function 2148: 2125: 2069: 2067:{\displaystyle H_{t}} 2042: 2022: 1990: 1924: 1904: 1862: 1753: 1733: 1713: 1689: 1654: 1634: 1614: 1585: 1561: 1541: 1517: 1482: 1458: 1426: 1397: 1349:differential topology 1285:Central Limit Theorem 1237:topological manifolds 1219: 1134: 1036: 1028: 936:and ≤ 0 on 892: 782: 729: 727:{\displaystyle D^{n}} 675: 611: 589: 564: 506: 267:differential geometry 229: 209: 186: 166: 146: 97: 95:{\displaystyle x_{0}} 63: 8151:Theorems in topology 8141:Fixed-point theorems 8115:with attached proof. 7721:. pp. 166–203. 6930:Historia Mathematica 6889:Historia Mathematica 6851:Historia Mathematica 6443:J. Reine Angew. Math 6381:. Math StackExchange 6354:C. Minazzo K. Rider 5985:set-valued functions 5931: 5885: 5833: 5786: 5766: 5638: 5592:fixed-point theorems 5512: 5410: 5220: 5139: 5104: 5015: 5011:is a fixed point of 4995: 4954: 4940:{\displaystyle f(P)} 4922: 4902: 4877: 4828: 4808: 4736: 4716: 4696: 4666: 4636: 4606: 4582: 4527: 4500: 4480: 4460: 4416: 4380: 4276: 4228: 4192: 4051: 4018: 3756: 3636:) is trivial, while 3138: 3097:)-dimensional space 2848: 2442: 2403: 2383: 2267: 2160: 2137: 2082: 2051: 2031: 1999: 1933: 1913: 1893: 1765: 1742: 1722: 1702: 1696:Jacobian determinant 1663: 1643: 1623: 1612:{\displaystyle B(0)} 1594: 1574: 1550: 1530: 1491: 1471: 1435: 1406: 1362: 1339:A proof using degree 1277:set-valued functions 1241:Jordan curve theorem 1223:used the theorem in 1198:Jordan curve theorem 1049:, who worked on the 970:fixed point theorems 932:. It is ≥ 0 on 885:One-dimensional case 753: 711: 631: 598: 576: 529: 474: 302:Charles Émile Picard 247:Jordan curve theorem 243:fixed-point theorems 218: 198: 175: 155: 106: 79: 52: 8024:Amer. Math. Monthly 7997:. Charlottesville: 7893:1976SJNA...13..473K 7551:Amer. Math. Monthly 7493:Nyman, Kathryn L.; 7460:Eldon Dyer (1956). 7012:Topological algebra 6428:, 21 septembre 1999 6332:V. & F. Bayart 6050:Borsuk–Ulam theorem 5682: 5562:reverse mathematics 5552:has a fixed point. 5506:simplicial homology 4161: for all 3709:onto its boundary ∂ 3401:require tools from 3399:Borsuk-Ulam theorem 3113:, with coordinates 2628:contraction mapping 1887:homotopy invariance 1659:at a regular value 1253:Borsuk–Ulam theorem 974:functional analysis 855:shaken, not stirred 337:continuous function 283:general equilibrium 259:Borsuk–Ulam theorem 68:mapping a nonempty 47:continuous function 35:fixed-point theorem 27:Theorem in topology 8107:2007-03-19 at the 8079:Algebraic topology 7858:. Academic Press. 7829:Fixed Point Theory 7688:Éléments d'analyse 7495:Su, Francis Edward 7089:Studia Mathematica 7007:algebraic topology 6775:2011-07-16 at the 6662:2010-10-08 at the 6629:Multiplication by 6580:2016-03-03 at the 6533:three-body problem 6487:Fixed point theory 6362:2018-04-04 at the 6235:10.1007/BF01456931 6021:Algebraic topology 6012:algebraic topology 6006:Equivalent results 5959: 5917: 5871: 5804: 5772: 5735: 5668: 5580:weak Kőnig's lemma 5534: 5483: 5482: 5423: 5283: 5203: 5125: 5027:{\displaystyle f.} 5024: 5001: 4978: 4937: 4908: 4889:{\displaystyle j.} 4886: 4863: 4814: 4797:available colors. 4774: 4722: 4702: 4682: 4642: 4622: 4588: 4565: 4506: 4486: 4466: 4446: 4402: 4363: 4259: 4214: 4175: 4034: 3927: 3659:de Rham cohomology 3589:group homomorphism 3585:fundamental groups 3567:, and in the case 3449: 3403:algebraic topology 3239: 2980: 2790:; by construction 2502:hairy ball theorem 2467: 2428: 2389: 2367: 2332: 2248: 2216: 2143: 2120: 2064: 2037: 2017: 1985: 1919: 1899: 1857: 1827: 1748: 1728: 1708: 1684: 1649: 1639:). The degree of 1629: 1609: 1580: 1556: 1536: 1512: 1477: 1453: 1421: 1392: 1233:fixed-point theory 1229: 1206:algebraic topology 1180:hairy ball theorem 1156:mathematical logic 1140: 1051:three-body problem 1039: 1031: 966:algebraic topology 894: 863:Intuitive approach 777: 724: 670: 606: 584: 559: 501: 463:as an endomorphism 381:Convex compact set 358:In Euclidean space 279:proof of existence 251:hairy ball theorem 241:Among hundreds of 224: 204: 181: 161: 141: 92: 58: 8061:"Brouwer theorem" 7952:978-1-4704-2921-8 7865:978-0-12-398050-2 7838:978-90-277-1224-0 7787:978-0-387-90148-0 7774:Hirsch, Morris W. 7157:(August 1, 2010). 7024:V. I. Istratescu 6921:Freudenthal, Hans 6880:Freudenthal, Hans 6842:Freudenthal, Hans 6764:See for example: 6751:978-0-8176-3388-2 6699:"Brouwer theorem" 6496:978-90-277-1224-0 6471:978-0-12-398050-2 6338:on Bibmath.net. 6213:Brouwer, L. E. J. 6065: 6064: 5996:singular homology 5775:{\displaystyle X} 5724: 5689: 5683: 5414: 5387:theorem for Hex. 5037:A proof by Hirsch 5004:{\displaystyle P} 4911:{\displaystyle P} 4817:{\displaystyle P} 4725:{\displaystyle j} 4705:{\displaystyle P} 4645:{\displaystyle j} 4591:{\displaystyle P} 4509:{\displaystyle j} 4489:{\displaystyle P} 4476:th coordinate of 4469:{\displaystyle j} 4162: 4141: 4044:to itself, where 3988:de Rham's theorem 3300:has norm 1, then 2392:{\displaystyle g} 2365: 2317: 2246: 2201: 2146:{\displaystyle K} 2040:{\displaystyle p} 1922:{\displaystyle g} 1902:{\displaystyle f} 1793: 1751:{\displaystyle f} 1731:{\displaystyle p} 1711:{\displaystyle f} 1652:{\displaystyle f} 1632:{\displaystyle K} 1619:(the interior of 1583:{\displaystyle f} 1559:{\displaystyle p} 1539:{\displaystyle f} 1480:{\displaystyle f} 1390: 1098:fundamental group 665: 227:{\displaystyle K} 207:{\displaystyle D} 184:{\displaystyle I} 164:{\displaystyle f} 61:{\displaystyle f} 16:(Redirected from 8163: 8082: 8073: 8055: 8053: 8020: 8010: 7984: 7929: 7912: 7869: 7850: 7823: 7791: 7769: 7740: 7709: 7682: 7672: 7663:(143): 887–899. 7641: 7610: 7583: 7536: 7535: 7490: 7484: 7483: 7481: 7457: 7451: 7446: 7440: 7435: 7429: 7428: 7400: 7394: 7389: 7383: 7377: 7371: 7365: 7359: 7354: 7348: 7343: 7337: 7332: 7326: 7321: 7315: 7314:, pp. 39–48 7309: 7303: 7298: 7292: 7291: 7289: 7287: 7281: 7270: 7256: 7250: 7244: 7238: 7231: 7225: 7224: 7199: 7193: 7177: 7171: 7168:Hex (board game) 7164: 7158: 7141: 7135: 7134: 7114: 7108: 7107: 7105: 7079: 7073: 7064: 7058: 7043: 7037: 7022: 7016: 7003: 6997: 6988: 6982: 6955: 6949: 6948: 6946: 6917: 6911: 6910: 6905: 6876: 6870: 6869: 6867: 6838: 6832: 6824: 6818: 6817: 6801: 6795: 6787: 6781: 6762: 6756: 6755: 6728: 6722: 6721: 6694: 6688: 6674: 6668: 6652: 6646: 6644: 6642: 6641: 6638: 6635: 6627: 6621: 6614: 6608: 6607: 6592: 6586: 6566: 6560: 6551: 6545: 6522: 6516: 6507: 6501: 6500: 6482: 6476: 6475: 6457: 6451: 6450: 6438: 6429: 6416: 6405: 6397: 6391: 6390: 6388: 6386: 6374: 6368: 6352: 6346: 6330: 6324: 6323: 6321: 6320: 6296: 6290: 6275: 6269: 6253: 6247: 6246: 6209: 6198: 6181:Jacques Hadamard 6178: 6169: 6159: 6153: 6139: 6133: 6115: 6089:Nash equilibrium 6018: 5968: 5966: 5965: 5960: 5952: 5938: 5926: 5924: 5923: 5918: 5910: 5909: 5897: 5896: 5880: 5878: 5877: 5872: 5867: 5866: 5848: 5847: 5813: 5811: 5810: 5805: 5781: 5779: 5778: 5773: 5744: 5742: 5741: 5736: 5725: 5722: 5720: 5719: 5701: 5700: 5690: 5687: 5684: 5681: 5676: 5655: 5650: 5649: 5543: 5541: 5540: 5535: 5524: 5523: 5492: 5490: 5489: 5484: 5469: 5468: 5459: 5442: 5441: 5422: 5301: 5292: 5290: 5289: 5284: 5263: 5262: 5247: 5246: 5212: 5210: 5209: 5204: 5151: 5150: 5134: 5132: 5131: 5126: 5033: 5031: 5030: 5025: 5010: 5008: 5007: 5002: 4987: 4985: 4984: 4979: 4946: 4944: 4943: 4938: 4917: 4915: 4914: 4909: 4895: 4893: 4892: 4887: 4872: 4870: 4869: 4864: 4862: 4861: 4849: 4848: 4823: 4821: 4820: 4815: 4796: 4783: 4781: 4780: 4775: 4770: 4769: 4757: 4756: 4731: 4729: 4728: 4723: 4711: 4709: 4708: 4703: 4691: 4689: 4688: 4683: 4678: 4677: 4658: 4651: 4649: 4648: 4643: 4631: 4629: 4628: 4623: 4618: 4617: 4597: 4595: 4594: 4589: 4574: 4572: 4571: 4566: 4561: 4560: 4539: 4538: 4515: 4513: 4512: 4507: 4495: 4493: 4492: 4487: 4475: 4473: 4472: 4467: 4455: 4453: 4452: 4447: 4411: 4409: 4408: 4403: 4398: 4397: 4372: 4370: 4369: 4364: 4362: 4361: 4360: 4340: 4335: 4311: 4310: 4309: 4298: 4293: 4268: 4266: 4265: 4260: 4255: 4254: 4223: 4221: 4220: 4215: 4210: 4209: 4188:For every point 4184: 4182: 4181: 4176: 4171: 4167: 4163: 4160: 4152: 4151: 4142: 4139: 4131: 4130: 4129: 4118: 4113: 4095: 4094: 4083: 4063: 4062: 4043: 4041: 4040: 4035: 4030: 4029: 3985: 3979: 3967: 3961: 3952: 3948: 3936: 3934: 3933: 3928: 3908: 3907: 3898: 3897: 3873: 3872: 3863: 3862: 3841: 3840: 3828: 3827: 3806: 3805: 3796: 3795: 3777: 3776: 3740: 3720: 3714: 3708: 3702: 3622:: the homology 3464: 3396: 3390: 3384: 3378: 3372: 3366: 3359: 3353: 3347: 3341: 3335: 3329: 3323: 3317: 3311: 3305: 3293: 3287: 3281: 3275: 3269: 3263: 3257: 3252:By construction 3248: 3246: 3245: 3240: 3229: 3228: 3219: 3218: 3209: 3208: 3196: 3195: 3186: 3185: 3157: 3156: 3147: 3146: 3130: 3124: 3118: 3108: 3102: 3096: 3087: 3081: 3075: 3069: 3063: 3057: 3051: 3045: 3038: 3032: 3026: 3020: 3014: 3004: 2998: 2989: 2987: 2986: 2981: 2973: 2972: 2963: 2962: 2952: 2951: 2942: 2941: 2923: 2922: 2909: 2908: 2899: 2898: 2889: 2888: 2867: 2866: 2857: 2856: 2840: 2834: 2828: 2822: 2816: 2807: 2801: 2795: 2789: 2783: 2777: 2771: 2765: 2759: 2753: 2747: 2741: 2735: 2729: 2719: 2709: 2700: 2694: 2688: 2682: 2676: 2670: 2664: 2658: 2652: 2641: 2635: 2625: 2611: 2605: 2599: 2588: 2582: 2576: 2570: 2564: 2554: 2541: 2535: 2521: 2515: 2509: 2476: 2474: 2473: 2468: 2457: 2456: 2437: 2435: 2434: 2429: 2418: 2417: 2398: 2396: 2395: 2390: 2376: 2374: 2373: 2368: 2366: 2364: 2363: 2359: 2331: 2315: 2292: 2257: 2255: 2254: 2249: 2247: 2245: 2244: 2240: 2215: 2199: 2179: 2152: 2150: 2149: 2144: 2129: 2127: 2126: 2121: 2113: 2112: 2094: 2093: 2073: 2071: 2070: 2065: 2063: 2062: 2046: 2044: 2043: 2038: 2026: 2024: 2023: 2018: 1994: 1992: 1991: 1986: 1945: 1944: 1928: 1926: 1925: 1920: 1908: 1906: 1905: 1900: 1866: 1864: 1863: 1858: 1850: 1849: 1826: 1816: 1815: 1777: 1776: 1757: 1755: 1754: 1749: 1737: 1735: 1734: 1729: 1717: 1715: 1714: 1709: 1693: 1691: 1690: 1685: 1658: 1656: 1655: 1650: 1638: 1636: 1635: 1630: 1618: 1616: 1615: 1610: 1589: 1587: 1586: 1581: 1565: 1563: 1562: 1557: 1545: 1543: 1542: 1537: 1521: 1519: 1518: 1513: 1486: 1484: 1483: 1478: 1462: 1460: 1459: 1454: 1430: 1428: 1427: 1422: 1420: 1419: 1414: 1401: 1399: 1398: 1393: 1391: 1386: 1372: 1316:not constructive 1189:Hans Freudenthal 1172:Jacques Hadamard 1136:Jacques Hadamard 1021:Before discovery 1000:non-constructive 996:Jacques Hadamard 992:L. E. J. Brouwer 786: 784: 783: 778: 733: 731: 730: 725: 723: 722: 679: 677: 676: 671: 666: 661: 650: 615: 613: 612: 607: 605: 593: 591: 590: 585: 583: 568: 566: 565: 560: 510: 508: 507: 502: 316:Jacques Hadamard 313: 233: 231: 230: 225: 213: 211: 210: 205: 190: 188: 187: 182: 170: 168: 167: 162: 150: 148: 147: 142: 140: 139: 124: 123: 101: 99: 98: 93: 91: 90: 67: 65: 64: 59: 21: 8171: 8170: 8166: 8165: 8164: 8162: 8161: 8160: 8131: 8130: 8127:at Math Images. 8109:Wayback Machine 8102:Brouwer theorem 8089: 8076: 8058: 8051: 8018: 8013: 7989:Milnor, John W. 7987: 7973: 7956: 7915: 7901:10.1137/0713041 7875:Yorke, James A. 7872: 7866: 7853: 7839: 7826: 7812: 7795: 7788: 7772: 7758:10.2307/2320146 7752:(10): 818–827. 7743: 7729: 7712: 7698: 7685: 7647:Yorke, James A. 7644: 7630: 7620:Springer-Verlag 7613: 7599: 7586: 7564:10.2307/2317520 7547: 7544: 7539: 7492: 7491: 7487: 7459: 7458: 7454: 7447: 7443: 7436: 7432: 7417:10.2307/2320146 7411:(10): 818–827. 7402: 7401: 7397: 7390: 7386: 7378: 7374: 7366: 7362: 7355: 7351: 7344: 7340: 7333: 7329: 7322: 7318: 7310: 7306: 7299: 7295: 7285: 7283: 7279: 7268: 7258: 7257: 7253: 7249:, pp. 1–19 7245: 7241: 7232: 7228: 7202: 7200: 7196: 7189:Wayback Machine 7178: 7174: 7165: 7161: 7142: 7138: 7116: 7115: 7111: 7081: 7080: 7076: 7065: 7061: 7044: 7040: 7023: 7019: 7004: 7000: 6989: 6985: 6956: 6952: 6937:(4): 495–502 . 6919: 6918: 6914: 6896:(4): 495–502 . 6878: 6877: 6873: 6858:(4): 495–502 . 6840: 6839: 6835: 6825: 6821: 6803: 6802: 6798: 6788: 6784: 6777:Wayback Machine 6763: 6759: 6752: 6732:Dieudonné, Jean 6730: 6729: 6725: 6719: 6696: 6695: 6691: 6675: 6671: 6664:Wayback Machine 6653: 6649: 6639: 6636: 6633: 6632: 6630: 6628: 6624: 6615: 6611: 6595: 6593: 6589: 6582:Wayback Machine 6568:Quotation from 6567: 6563: 6552: 6548: 6523: 6519: 6508: 6504: 6497: 6484: 6483: 6479: 6472: 6459: 6458: 6454: 6449:(3/4): 179–276. 6440: 6439: 6432: 6417: 6408: 6398: 6394: 6384: 6382: 6376: 6375: 6371: 6364:Wayback Machine 6353: 6349: 6344:Wayback Machine 6331: 6327: 6318: 6316: 6314: 6299: 6297: 6293: 6276: 6272: 6267:Wayback Machine 6254: 6250: 6211: 6210: 6201: 6179: 6172: 6160: 6156: 6140: 6136: 6131:Wayback Machine 6116: 6112: 6108: 6070: 6038:Sperner's lemma 6008: 5982:hemi-continuous 5929: 5928: 5927:if and only if 5901: 5888: 5883: 5882: 5858: 5839: 5831: 5830: 5825:of which every 5823:Hausdorff space 5784: 5783: 5764: 5763: 5723: for 5705: 5692: 5641: 5636: 5635: 5630: 5621: 5588: 5586:Generalizations 5576: 5569: 5558: 5515: 5510: 5509: 5460: 5433: 5408: 5407: 5393: 5369: 5299: 5254: 5238: 5218: 5217: 5142: 5137: 5136: 5102: 5101: 5098: 5039: 5013: 5012: 4993: 4992: 4952: 4951: 4920: 4919: 4900: 4899: 4875: 4874: 4853: 4840: 4826: 4825: 4806: 4805: 4791: 4761: 4748: 4734: 4733: 4714: 4713: 4694: 4693: 4669: 4664: 4663: 4653: 4634: 4633: 4609: 4604: 4603: 4580: 4579: 4552: 4530: 4525: 4524: 4498: 4497: 4478: 4477: 4458: 4457: 4414: 4413: 4389: 4378: 4377: 4352: 4301: 4274: 4273: 4246: 4226: 4225: 4201: 4190: 4189: 4143: 4140: and 4121: 4078: 4071: 4067: 4054: 4049: 4048: 4021: 4016: 4015: 4000:Sperner's lemma 3996: 3981: 3978: 3969: 3963: 3957: 3950: 3944: 3899: 3889: 3864: 3854: 3832: 3819: 3797: 3784: 3765: 3754: 3753: 3747:Stokes' theorem 3738: 3716: 3710: 3704: 3698: 3695: 3645: 3631: 3620:homology groups 3452: 3411: 3392: 3386: 3380: 3374: 3368: 3361: 3355: 3349: 3343: 3337: 3331: 3325: 3319: 3313: 3307: 3301: 3299: 3289: 3283: 3277: 3271: 3265: 3259: 3253: 3136: 3135: 3126: 3120: 3114: 3104: 3098: 3091: 3089: 3083: 3077: 3071: 3065: 3059: 3053: 3047: 3041: 3040: 3034: 3028: 3022: 3016: 3010: 3000: 2994: 2846: 2845: 2836: 2830: 2824: 2818: 2812: 2803: 2797: 2791: 2785: 2779: 2773: 2767: 2761: 2755: 2749: 2743: 2737: 2731: 2725: 2715: 2705: 2696: 2690: 2684: 2678: 2672: 2666: 2660: 2654: 2651: 2643: 2637: 2631: 2613: 2607: 2601: 2598: 2590: 2584: 2578: 2572: 2566: 2560: 2550: 2537: 2523: 2517: 2511: 2505: 2498: 2490:homology theory 2445: 2440: 2439: 2406: 2401: 2400: 2381: 2380: 2337: 2333: 2316: 2293: 2265: 2264: 2221: 2217: 2200: 2180: 2158: 2157: 2135: 2134: 2104: 2085: 2080: 2079: 2054: 2049: 2048: 2029: 2028: 1997: 1996: 1936: 1931: 1930: 1911: 1910: 1891: 1890: 1841: 1804: 1768: 1763: 1762: 1740: 1739: 1720: 1719: 1700: 1699: 1661: 1660: 1641: 1640: 1621: 1620: 1592: 1591: 1572: 1571: 1548: 1547: 1528: 1527: 1489: 1488: 1469: 1468: 1433: 1432: 1409: 1404: 1403: 1373: 1360: 1359: 1341: 1336: 1309:Hotelling's law 1214: 1129: 1023: 1003:indirect proofs 962: 909:(light green). 887: 870: 865: 835: 819: 751: 750: 714: 709: 708: 701: 651: 629: 628: 622: 596: 595: 574: 573: 527: 526: 520: 472: 471: 465: 437: 367:Euclidean space 324: 309: 236:Euclidean space 216: 215: 196: 195: 173: 172: 153: 152: 131: 115: 104: 103: 82: 77: 76: 50: 49: 28: 23: 22: 15: 12: 11: 5: 8169: 8167: 8159: 8158: 8153: 8148: 8143: 8133: 8132: 8129: 8128: 8122: 8116: 8099: 8088: 8087:External links 8085: 8084: 8083: 8074: 8056: 8031:(7): 521–524. 8011: 7985: 7971: 7954: 7930: 7913: 7887:(4): 473–483. 7870: 7864: 7851: 7837: 7824: 7810: 7793: 7786: 7770: 7741: 7727: 7710: 7696: 7683: 7642: 7628: 7611: 7597: 7584: 7558:(3): 237–249. 7543: 7540: 7538: 7537: 7511:(4): 346–354, 7485: 7472:(4): 662–672. 7452: 7441: 7430: 7395: 7384: 7372: 7360: 7349: 7346:Dieudonné 1982 7338: 7327: 7316: 7304: 7293: 7260:Teschl, Gerald 7251: 7239: 7226: 7215:(2): 133–155. 7194: 7172: 7159: 7136: 7125:(3): 457–459. 7109: 7074: 7059: 7038: 7017: 6998: 6983: 6950: 6912: 6871: 6833: 6819: 6812:(in Russian). 6796: 6782: 6757: 6750: 6723: 6717: 6689: 6669: 6647: 6622: 6609: 6587: 6570:Henri Poincaré 6561: 6554:Henri Poincaré 6546: 6529:King of Sweden 6525:Henri Poincaré 6517: 6502: 6495: 6477: 6470: 6452: 6430: 6406: 6402:Henri Poincaré 6392: 6369: 6347: 6325: 6312: 6291: 6270: 6248: 6199: 6170: 6166:Luizen Brouwer 6154: 6134: 6109: 6107: 6104: 6103: 6102: 6097: 6091: 6086: 6081: 6076: 6069: 6066: 6063: 6062: 6057: 6055:Tucker's lemma 6052: 6046: 6045: 6040: 6035: 6029: 6028: 6025: 6022: 6007: 6004: 5958: 5955: 5951: 5947: 5944: 5941: 5937: 5916: 5913: 5908: 5904: 5900: 5895: 5891: 5870: 5865: 5861: 5857: 5854: 5851: 5846: 5842: 5838: 5803: 5800: 5797: 5794: 5791: 5771: 5746: 5745: 5734: 5731: 5728: 5718: 5715: 5712: 5708: 5704: 5699: 5695: 5680: 5675: 5671: 5667: 5664: 5661: 5658: 5653: 5648: 5644: 5626: 5617: 5587: 5584: 5574: 5567: 5557: 5554: 5533: 5530: 5527: 5522: 5518: 5494: 5493: 5481: 5478: 5475: 5472: 5467: 5463: 5458: 5454: 5451: 5448: 5445: 5440: 5436: 5432: 5429: 5426: 5421: 5417: 5392: 5389: 5368: 5365: 5307:) = 0 for all 5294: 5293: 5282: 5279: 5276: 5272: 5269: 5266: 5261: 5257: 5253: 5250: 5245: 5241: 5237: 5234: 5231: 5228: 5225: 5202: 5199: 5196: 5193: 5190: 5187: 5184: 5181: 5178: 5175: 5172: 5169: 5166: 5163: 5160: 5157: 5154: 5149: 5145: 5124: 5121: 5118: 5115: 5112: 5109: 5097: 5094: 5074:James A. Yorke 5067:Sard's theorem 5063:bump functions 5047:indirect proof 5038: 5035: 5023: 5020: 5000: 4989: 4988: 4977: 4974: 4971: 4968: 4965: 4962: 4959: 4936: 4933: 4930: 4927: 4907: 4885: 4882: 4860: 4856: 4852: 4847: 4843: 4839: 4836: 4833: 4813: 4773: 4768: 4764: 4760: 4755: 4751: 4747: 4744: 4741: 4721: 4701: 4681: 4676: 4672: 4641: 4621: 4616: 4612: 4587: 4576: 4575: 4564: 4559: 4555: 4551: 4548: 4545: 4542: 4537: 4533: 4505: 4485: 4465: 4456:such that the 4445: 4442: 4439: 4436: 4433: 4430: 4427: 4424: 4421: 4401: 4396: 4392: 4388: 4385: 4374: 4373: 4359: 4355: 4351: 4348: 4345: 4339: 4334: 4331: 4328: 4324: 4320: 4317: 4314: 4308: 4304: 4297: 4292: 4289: 4286: 4282: 4258: 4253: 4249: 4245: 4242: 4239: 4236: 4233: 4213: 4208: 4204: 4200: 4197: 4186: 4185: 4174: 4170: 4166: 4158: 4155: 4150: 4146: 4137: 4134: 4128: 4124: 4117: 4112: 4109: 4106: 4102: 4098: 4093: 4090: 4087: 4082: 4077: 4074: 4070: 4066: 4061: 4057: 4033: 4028: 4024: 3995: 3992: 3973: 3953:generates the 3938: 3937: 3926: 3923: 3920: 3917: 3914: 3911: 3906: 3902: 3896: 3892: 3888: 3885: 3882: 3879: 3876: 3871: 3867: 3861: 3857: 3853: 3850: 3847: 3844: 3839: 3835: 3831: 3826: 3822: 3818: 3815: 3812: 3809: 3804: 3800: 3794: 3791: 3787: 3783: 3780: 3775: 3772: 3768: 3764: 3761: 3731:bump functions 3703:from the ball 3694: 3691: 3650:) is infinite 3640: 3626: 3548:(in this case 3410: 3407: 3295: 3250: 3249: 3238: 3235: 3232: 3227: 3222: 3217: 3212: 3207: 3202: 3199: 3194: 3189: 3184: 3178: 3175: 3172: 3169: 3166: 3163: 3160: 3155: 3150: 3145: 2991: 2990: 2979: 2976: 2971: 2966: 2961: 2955: 2950: 2945: 2940: 2935: 2932: 2929: 2926: 2921: 2915: 2912: 2907: 2902: 2897: 2892: 2887: 2882: 2879: 2876: 2873: 2870: 2865: 2860: 2855: 2647: 2594: 2497: 2494: 2466: 2463: 2460: 2455: 2452: 2448: 2427: 2424: 2421: 2416: 2413: 2409: 2388: 2362: 2358: 2355: 2352: 2349: 2346: 2343: 2340: 2336: 2330: 2327: 2324: 2320: 2314: 2311: 2308: 2305: 2302: 2299: 2296: 2290: 2287: 2284: 2281: 2278: 2275: 2272: 2259: 2258: 2243: 2239: 2236: 2233: 2230: 2227: 2224: 2220: 2214: 2211: 2208: 2204: 2198: 2195: 2192: 2189: 2186: 2183: 2177: 2174: 2171: 2168: 2165: 2142: 2119: 2116: 2111: 2107: 2103: 2100: 2097: 2092: 2088: 2061: 2057: 2036: 2016: 2013: 2010: 2007: 2004: 1984: 1981: 1978: 1975: 1972: 1969: 1966: 1963: 1960: 1957: 1954: 1951: 1948: 1943: 1939: 1918: 1898: 1880:winding number 1868: 1867: 1856: 1853: 1848: 1844: 1840: 1837: 1834: 1830: 1825: 1822: 1819: 1814: 1811: 1807: 1803: 1800: 1796: 1792: 1789: 1786: 1783: 1780: 1775: 1771: 1747: 1727: 1707: 1683: 1680: 1677: 1674: 1671: 1668: 1648: 1628: 1608: 1605: 1602: 1599: 1579: 1555: 1535: 1522:such that the 1511: 1508: 1505: 1502: 1499: 1496: 1476: 1452: 1449: 1446: 1443: 1440: 1418: 1413: 1389: 1385: 1382: 1379: 1376: 1370: 1367: 1340: 1337: 1335: 1334:Proof outlines 1332: 1320:constructivity 1213: 1210: 1168:Henri Poincaré 1128: 1125: 1115:, named after 1089:analysis situs 1047:Henri Poincaré 1022: 1019: 961: 958: 928:) − 886: 883: 869: 866: 864: 861: 860: 859: 850: 847: 834: 831: 818: 815: 805: 776: 773: 770: 767: 764: 761: 758: 721: 717: 705:homeomorphisms 700: 697: 690: 681: 680: 669: 664: 660: 657: 654: 648: 645: 642: 639: 636: 621: 618: 604: 582: 570: 569: 558: 555: 552: 549: 546: 543: 540: 537: 534: 519: 516: 512: 511: 500: 497: 494: 491: 488: 485: 482: 479: 464: 457: 436: 433: 432: 431: 430: 429: 414: 403: 402: 401: 400: 382: 373: 372: 371: 370: 359: 350: 349: 348: 347: 333: 323: 320: 298:Henri Poincaré 223: 203: 180: 160: 138: 134: 130: 127: 122: 118: 114: 111: 89: 85: 57: 41:, named after 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 8168: 8157: 8154: 8152: 8149: 8147: 8144: 8142: 8139: 8138: 8136: 8126: 8123: 8120: 8117: 8114: 8110: 8106: 8103: 8100: 8098: 8094: 8091: 8090: 8086: 8080: 8075: 8072: 8068: 8067: 8062: 8057: 8050: 8046: 8042: 8038: 8034: 8030: 8026: 8025: 8017: 8012: 8008: 8004: 8000: 7996: 7995: 7990: 7986: 7982: 7978: 7974: 7972:0-521-58059-5 7968: 7964: 7960: 7955: 7953: 7949: 7945: 7941: 7937: 7936: 7931: 7927: 7923: 7919: 7914: 7910: 7906: 7902: 7898: 7894: 7890: 7886: 7882: 7881: 7876: 7871: 7867: 7861: 7857: 7852: 7848: 7844: 7840: 7834: 7830: 7825: 7821: 7817: 7813: 7807: 7803: 7799: 7794: 7789: 7783: 7779: 7775: 7771: 7767: 7763: 7759: 7755: 7751: 7747: 7742: 7738: 7734: 7730: 7728:0-8176-3388-X 7724: 7720: 7716: 7711: 7707: 7703: 7699: 7697:2-04-011499-8 7693: 7689: 7684: 7680: 7676: 7671: 7666: 7662: 7658: 7657: 7652: 7648: 7643: 7639: 7635: 7631: 7629:0-387-97926-3 7625: 7621: 7617: 7612: 7608: 7604: 7600: 7598:0-12-116052-1 7594: 7590: 7585: 7581: 7577: 7573: 7569: 7565: 7561: 7557: 7553: 7552: 7546: 7545: 7541: 7534: 7530: 7526: 7522: 7518: 7514: 7510: 7506: 7505: 7500: 7496: 7489: 7486: 7480: 7475: 7471: 7467: 7463: 7456: 7453: 7450: 7445: 7442: 7439: 7434: 7431: 7426: 7422: 7418: 7414: 7410: 7406: 7399: 7396: 7393: 7388: 7385: 7381: 7376: 7373: 7369: 7364: 7361: 7358: 7353: 7350: 7347: 7342: 7339: 7336: 7331: 7328: 7325: 7320: 7317: 7313: 7308: 7305: 7302: 7297: 7294: 7278: 7274: 7267: 7266: 7261: 7255: 7252: 7248: 7243: 7240: 7236: 7230: 7227: 7222: 7218: 7214: 7210: 7206: 7198: 7195: 7191: 7190: 7186: 7183: 7176: 7173: 7169: 7163: 7160: 7156: 7152: 7151:Archived copy 7148: 7147: 7140: 7137: 7132: 7128: 7124: 7120: 7113: 7110: 7104: 7099: 7095: 7091: 7090: 7085: 7078: 7075: 7071: 7070: 7063: 7060: 7056: 7055:1-4020-0301-3 7052: 7048: 7042: 7039: 7035: 7034:1-4020-0301-3 7031: 7027: 7021: 7018: 7014: 7013: 7008: 7002: 6999: 6995: 6994: 6987: 6984: 6980: 6976: 6972: 6968: 6964: 6960: 6954: 6951: 6945: 6940: 6936: 6932: 6931: 6926: 6922: 6916: 6913: 6909: 6904: 6899: 6895: 6891: 6890: 6885: 6881: 6875: 6872: 6866: 6861: 6857: 6853: 6852: 6847: 6843: 6837: 6834: 6831: 6830: 6823: 6820: 6816:(3): 188–192. 6815: 6811: 6807: 6800: 6797: 6794: 6793: 6786: 6783: 6779: 6778: 6774: 6771: 6767: 6761: 6758: 6753: 6747: 6743: 6739: 6738: 6733: 6727: 6724: 6720: 6718:1-4020-0609-8 6714: 6710: 6706: 6705: 6700: 6693: 6690: 6687: 6686:1-4020-0301-3 6683: 6679: 6673: 6670: 6666: 6665: 6661: 6658: 6651: 6648: 6626: 6623: 6619: 6613: 6610: 6606:(4): 167–244. 6605: 6601: 6600: 6591: 6588: 6584: 6583: 6579: 6576: 6571: 6565: 6562: 6558: 6555: 6550: 6547: 6543: 6542: 6538: 6534: 6530: 6526: 6521: 6518: 6514: 6513: 6506: 6503: 6498: 6492: 6488: 6481: 6478: 6473: 6467: 6463: 6456: 6453: 6448: 6444: 6437: 6435: 6431: 6427: 6423: 6422: 6415: 6413: 6411: 6407: 6403: 6396: 6393: 6380: 6373: 6370: 6366: 6365: 6361: 6358: 6351: 6348: 6345: 6341: 6337: 6336: 6329: 6326: 6315: 6313:9781402075124 6309: 6305: 6304: 6295: 6292: 6288: 6287:2-13-037495-6 6284: 6280: 6274: 6271: 6268: 6264: 6260: 6259: 6252: 6249: 6244: 6240: 6236: 6232: 6228: 6225:(in German). 6224: 6223: 6218: 6214: 6208: 6206: 6204: 6200: 6196: 6192: 6191:Jules Tannery 6188: 6187: 6182: 6177: 6175: 6171: 6168:by G. Sabbagh 6167: 6164: 6158: 6155: 6152: 6151:2-13-037495-6 6148: 6144: 6138: 6135: 6132: 6128: 6125: 6122: 6121: 6114: 6111: 6105: 6101: 6098: 6095: 6092: 6090: 6087: 6085: 6082: 6080: 6077: 6075: 6072: 6071: 6067: 6061: 6058: 6056: 6053: 6051: 6048: 6047: 6044: 6041: 6039: 6036: 6034: 6031: 6030: 6027:Set covering 6026: 6024:Combinatorics 6023: 6020: 6019: 6016: 6013: 6005: 6003: 6001: 5997: 5993: 5988: 5986: 5983: 5979: 5975: 5970: 5956: 5953: 5945: 5942: 5939: 5911: 5906: 5902: 5898: 5893: 5889: 5863: 5859: 5855: 5852: 5849: 5844: 5840: 5828: 5824: 5821: 5817: 5801: 5795: 5792: 5789: 5769: 5760: 5758: 5754: 5749: 5732: 5729: 5726: 5716: 5713: 5710: 5706: 5702: 5697: 5693: 5678: 5673: 5665: 5659: 5656: 5651: 5646: 5642: 5634: 5633: 5632: 5631:) defined by 5629: 5625: 5620: 5616: 5612: 5608: 5604: 5600: 5599:Hilbert space 5595: 5593: 5585: 5583: 5581: 5577: 5570: 5563: 5555: 5553: 5551: 5547: 5528: 5520: 5516: 5507: 5503: 5499: 5473: 5465: 5461: 5452: 5446: 5443: 5438: 5430: 5427: 5419: 5415: 5406: 5405: 5404: 5402: 5398: 5390: 5388: 5386: 5382: 5378: 5374: 5366: 5364: 5362: 5358: 5354: 5350: 5346: 5342: 5338: 5334: 5330: 5326: 5322: 5318: 5314: 5310: 5306: 5302: 5280: 5277: 5274: 5267: 5259: 5255: 5251: 5243: 5239: 5235: 5229: 5223: 5216: 5215: 5214: 5200: 5197: 5191: 5188: 5185: 5179: 5173: 5167: 5164: 5161: 5155: 5147: 5143: 5122: 5113: 5110: 5107: 5095: 5093: 5091: 5087: 5083: 5079: 5075: 5070: 5068: 5064: 5060: 5056: 5052: 5048: 5044: 5043:Morris Hirsch 5036: 5034: 5021: 5018: 4998: 4975: 4972: 4969: 4963: 4957: 4950: 4949: 4948: 4931: 4925: 4905: 4896: 4883: 4880: 4858: 4854: 4850: 4845: 4837: 4831: 4811: 4803: 4798: 4794: 4789: 4784: 4771: 4766: 4762: 4758: 4753: 4745: 4739: 4719: 4699: 4679: 4674: 4660: 4656: 4639: 4619: 4614: 4601: 4585: 4578:Moreover, if 4562: 4557: 4549: 4543: 4540: 4535: 4531: 4523: 4522: 4521: 4519: 4503: 4483: 4463: 4440: 4437: 4434: 4431: 4428: 4422: 4419: 4399: 4394: 4386: 4383: 4357: 4349: 4343: 4337: 4332: 4329: 4326: 4322: 4318: 4315: 4312: 4306: 4302: 4295: 4290: 4287: 4284: 4280: 4272: 4271: 4270: 4256: 4251: 4243: 4237: 4231: 4211: 4206: 4198: 4195: 4172: 4168: 4164: 4156: 4153: 4148: 4144: 4135: 4132: 4126: 4122: 4115: 4110: 4107: 4104: 4100: 4096: 4091: 4088: 4085: 4075: 4072: 4068: 4064: 4059: 4047: 4046: 4045: 4031: 4026: 4013: 4009: 4005: 4001: 3993: 3991: 3989: 3984: 3976: 3972: 3966: 3960: 3956: 3947: 3941: 3924: 3921: 3918: 3912: 3904: 3900: 3894: 3890: 3886: 3880: 3877: 3869: 3865: 3859: 3855: 3851: 3845: 3837: 3833: 3829: 3824: 3820: 3816: 3810: 3802: 3798: 3792: 3785: 3781: 3778: 3773: 3766: 3762: 3759: 3752: 3751: 3750: 3748: 3744: 3736: 3732: 3728: 3724: 3719: 3713: 3707: 3701: 3690: 3688: 3684: 3680: 3676: 3672: 3668: 3664: 3660: 3655: 3653: 3649: 3643: 3639: 3635: 3629: 3625: 3621: 3617: 3612: 3610: 3609:vector fields 3606: 3602: 3598: 3594: 3590: 3586: 3582: 3578: 3574: 3570: 3566: 3562: 3557: 3555: 3551: 3547: 3543: 3538: 3536: 3532: 3528: 3524: 3521: → 3520: 3517: : 3516: 3512: 3508: 3504: 3500: 3496: 3492: 3488: 3484: 3480: 3476: 3473:, the points 3472: 3468: 3463: 3459: 3455: 3447: 3442: 3438: 3436: 3432: 3428: 3424: 3420: 3416: 3408: 3406: 3404: 3400: 3395: 3389: 3383: 3377: 3371: 3364: 3358: 3352: 3348:is even. For 3346: 3340: 3334: 3328: 3322: 3316: 3310: 3304: 3298: 3292: 3286: 3280: 3274: 3268: 3262: 3256: 3236: 3210: 3200: 3176: 3173: 3167: 3161: 3158: 3134: 3133: 3132: 3129: 3123: 3117: 3112: 3107: 3101: 3094: 3086: 3080: 3074: 3068: 3062: 3056: 3050: 3044: 3037: 3031: 3025: 3019: 3015:, the vector 3013: 3008: 3003: 2997: 2977: 2943: 2933: 2930: 2924: 2890: 2880: 2877: 2871: 2844: 2843: 2842: 2839: 2833: 2827: 2821: 2815: 2809: 2806: 2800: 2794: 2788: 2782: 2776: 2770: 2764: 2758: 2752: 2746: 2740: 2734: 2728: 2723: 2718: 2713: 2708: 2702: 2699: 2693: 2687: 2681: 2675: 2669: 2663: 2657: 2650: 2646: 2640: 2634: 2629: 2623: 2619: 2616: 2610: 2604: 2597: 2593: 2587: 2581: 2575: 2569: 2563: 2558: 2553: 2547: 2545: 2544:Milnor (1978) 2540: 2534: 2530: 2526: 2520: 2514: 2508: 2503: 2495: 2493: 2491: 2487: 2482: 2480: 2461: 2453: 2450: 2446: 2422: 2414: 2411: 2407: 2386: 2377: 2360: 2353: 2347: 2344: 2341: 2338: 2334: 2328: 2325: 2322: 2309: 2303: 2300: 2297: 2294: 2288: 2282: 2279: 2276: 2270: 2262: 2241: 2234: 2228: 2225: 2222: 2218: 2212: 2209: 2206: 2193: 2187: 2184: 2181: 2175: 2169: 2163: 2156: 2155: 2154: 2140: 2131: 2117: 2114: 2109: 2105: 2101: 2098: 2095: 2090: 2086: 2077: 2059: 2055: 2034: 2014: 2011: 2008: 2005: 2002: 1982: 1976: 1973: 1970: 1964: 1961: 1958: 1955: 1949: 1941: 1937: 1916: 1896: 1888: 1883: 1881: 1877: 1873: 1854: 1846: 1842: 1838: 1828: 1820: 1812: 1809: 1805: 1801: 1798: 1794: 1790: 1784: 1778: 1773: 1769: 1761: 1760: 1759: 1745: 1725: 1705: 1697: 1678: 1672: 1669: 1666: 1646: 1626: 1603: 1597: 1577: 1569: 1553: 1533: 1525: 1506: 1500: 1497: 1494: 1474: 1466: 1465:regular value 1450: 1444: 1441: 1438: 1416: 1380: 1374: 1368: 1365: 1356: 1354: 1353:Milnor (1965) 1350: 1346: 1338: 1333: 1331: 1329: 1325: 1321: 1317: 1312: 1310: 1305: 1301: 1297: 1292: 1290: 1286: 1282: 1278: 1274: 1270: 1269:Banach spaces 1266: 1262: 1258: 1254: 1249: 1244: 1242: 1238: 1234: 1226: 1222: 1218: 1211: 1209: 1207: 1203: 1199: 1193: 1190: 1186: 1181: 1177: 1173: 1169: 1165: 1161: 1157: 1151: 1149: 1145: 1137: 1133: 1126: 1124: 1122: 1118: 1114: 1110: 1106: 1101: 1099: 1094: 1091:. The French 1090: 1085: 1083: 1079: 1075: 1071: 1067: 1063: 1059: 1054: 1052: 1048: 1044: 1035: 1027: 1020: 1018: 1016: 1013:, methods to 1012: 1008: 1004: 1001: 997: 993: 989: 988: 983: 979: 975: 971: 967: 959: 957: 953: 951: 947: 943: 939: 935: 931: 927: 923: 919: 915: 910: 908: 904: 899: 891: 884: 882: 880: 879:Stefan Banach 874: 867: 862: 856: 851: 848: 845: 840: 839: 838: 833:Illustrations 832: 830: 828: 824: 816: 814: 812: 807: 803: 802: 798: 794: 790: 774: 771: 768: 762: 756: 747: 745: 744:without holes 741: 737: 719: 715: 706: 698: 696: 694: 688: 687: 667: 662: 658: 655: 652: 646: 640: 634: 627: 626: 625: 619: 617: 556: 553: 550: 547: 544: 538: 532: 525: 524: 523: 517: 515: 498: 495: 492: 489: 483: 477: 470: 469: 468: 462: 459:The function 458: 456: 454: 450: 446: 442: 441:endomorphisms 434: 427: 423: 419: 415: 413: 410: 409: 408: 407: 406: 398: 394: 390: 387: 383: 380: 379: 378: 377: 376: 368: 364: 360: 357: 356: 355: 354: 353: 345: 342: 338: 334: 331: 330: 329: 328: 327: 321: 319: 317: 312: 307: 303: 299: 294: 292: 291:Gérard Debreu 288: 287:Kenneth Arrow 284: 280: 276: 272: 268: 264: 260: 256: 252: 248: 244: 239: 237: 221: 201: 194: 178: 158: 136: 132: 128: 120: 116: 109: 87: 83: 74: 71: 55: 48: 44: 40: 36: 32: 19: 8121:at MathPages 8097:cut-the-knot 8078: 8064: 8028: 8022: 7993: 7958: 7943: 7933: 7925: 7921: 7884: 7878: 7855: 7828: 7800:. New York: 7797: 7777: 7749: 7745: 7714: 7687: 7660: 7654: 7615: 7588: 7555: 7549: 7508: 7502: 7488: 7469: 7465: 7455: 7449:Spanier 1966 7444: 7433: 7408: 7404: 7398: 7387: 7375: 7363: 7352: 7341: 7335:Boothby 1986 7330: 7324:Boothby 1971 7319: 7307: 7296: 7284:. Retrieved 7264: 7254: 7242: 7229: 7212: 7208: 7197: 7180: 7175: 7162: 7144: 7139: 7122: 7118: 7112: 7093: 7087: 7077: 7067: 7062: 7046: 7041: 7025: 7020: 7010: 7006: 7001: 6991: 6986: 6978: 6974: 6970: 6966: 6963:homeomorphic 6953: 6934: 6928: 6915: 6907: 6893: 6887: 6874: 6855: 6849: 6836: 6827: 6822: 6813: 6809: 6799: 6790: 6785: 6768: 6766:Émile Picard 6760: 6736: 6726: 6702: 6692: 6677: 6672: 6655: 6650: 6625: 6612: 6603: 6597: 6590: 6573: 6564: 6556: 6549: 6539: 6537:Jacques Tits 6520: 6510: 6505: 6486: 6480: 6461: 6455: 6446: 6442: 6419: 6395: 6383:. Retrieved 6372: 6355: 6350: 6333: 6328: 6317:. Retrieved 6302: 6294: 6278: 6273: 6256: 6255:D. Violette 6251: 6226: 6220: 6194: 6184: 6162: 6157: 6142: 6137: 6118: 6113: 6032: 6009: 5999: 5989: 5977: 5971: 5881:, such that 5761: 5750: 5747: 5627: 5623: 5618: 5614: 5610: 5596: 5589: 5559: 5549: 5545: 5501: 5497: 5495: 5400: 5396: 5394: 5380: 5370: 5360: 5356: 5352: 5348: 5344: 5340: 5336: 5332: 5328: 5324: 5320: 5316: 5312: 5308: 5304: 5297: 5295: 5099: 5089: 5085: 5081: 5071: 5061:with smooth 5050: 5040: 4990: 4897: 4801: 4799: 4792: 4787: 4785: 4712:is an index 4661: 4654: 4599: 4577: 4517: 4375: 4187: 4007: 4003: 3997: 3982: 3974: 3970: 3964: 3958: 3945: 3942: 3939: 3717: 3711: 3705: 3699: 3696: 3686: 3682: 3678: 3674: 3670: 3666: 3662: 3656: 3647: 3641: 3637: 3633: 3627: 3623: 3615: 3613: 3604: 3600: 3596: 3592: 3580: 3576: 3572: 3568: 3564: 3560: 3558: 3553: 3549: 3539: 3534: 3530: 3526: 3522: 3518: 3514: 3510: 3506: 3502: 3498: 3494: 3490: 3486: 3482: 3478: 3474: 3470: 3466: 3461: 3457: 3453: 3450: 3445: 3430: 3426: 3422: 3418: 3412: 3393: 3387: 3381: 3375: 3369: 3362: 3356: 3350: 3344: 3338: 3332: 3326: 3320: 3314: 3308: 3302: 3296: 3290: 3284: 3278: 3272: 3266: 3260: 3254: 3251: 3127: 3121: 3115: 3110: 3105: 3099: 3092: 3084: 3078: 3072: 3066: 3060: 3054: 3048: 3042: 3035: 3029: 3023: 3017: 3011: 3001: 2995: 2992: 2837: 2831: 2825: 2819: 2813: 2810: 2804: 2798: 2792: 2786: 2780: 2774: 2768: 2762: 2756: 2750: 2744: 2738: 2732: 2726: 2716: 2711: 2706: 2703: 2697: 2691: 2685: 2679: 2673: 2667: 2661: 2655: 2648: 2644: 2638: 2632: 2621: 2617: 2614: 2608: 2602: 2595: 2591: 2585: 2579: 2573: 2567: 2561: 2556: 2551: 2548: 2538: 2532: 2528: 2524: 2518: 2512: 2506: 2499: 2485: 2483: 2478: 2378: 2263: 2260: 2132: 2075: 1886: 1884: 1875: 1871: 1869: 1357: 1342: 1324:intuitionism 1313: 1293: 1260: 1256: 1245: 1230: 1194: 1152: 1141: 1127:First proofs 1105:Émile Picard 1102: 1088: 1086: 1081: 1068:, i.e. both 1055: 1040: 1007:intuitionist 985: 977: 963: 954: 945: 937: 933: 929: 925: 921: 917: 913: 911: 906: 902: 897: 895: 875: 871: 843: 836: 820: 808: 800: 796: 792: 788: 748: 702: 692: 685: 682: 623: 571: 521: 513: 466: 460: 453:homeomorphic 448: 444: 440: 438: 425: 422:Banach space 417: 404: 396: 392: 374: 351: 332:In the plane 325: 310: 295: 240: 30: 29: 7928:(2): 83–90. 7357:Hirsch 1988 7301:Milnor 1978 7247:Milnor 1965 7096:: 171–180. 6377:Belk, Jim. 6281:Puf (1982) 6145:Puf (1982) 5385:determinacy 5319:(0) is the 3743:volume form 3595:to that of 1487:is a point 1296:game theory 1248:contracting 1225:game theory 1176:Émile Borel 1121:contraction 1078:limit cycle 1062:topological 1015:approximate 976:. The case 916:which maps 738:, bounded, 518:Boundedness 363:closed ball 271:game theory 238:to itself. 8135:Categories 8113:PlanetMath 7811:0521094224 7719:Birkhäuser 7542:References 7392:Kulpa 1989 7286:1 February 6792:Piers Bohl 6319:2016-03-08 6229:: 97–115. 5827:open cover 5508:group is: 5373:David Gale 5078:computable 5059:convolving 4732:such that 4598:lies on a 3735:mollifying 3727:convolving 3542:retraction 3533:) must be 2712:continuous 2710:is only a 2677:onto (1 + 2659:onto (1 + 1328:set theory 1144:Piers Bohl 982:Piers Bohl 940:. By the 827:surjective 620:Closedness 102:such that 73:convex set 8071:EMS Press 7005:The term 6969:, and if 6709:EMS Press 6421:Archimède 6243:177796823 5954:≤ 5943:− 5915:∅ 5912:≠ 5899:∩ 5853:… 5799:→ 5753:convexity 5730:≥ 5714:− 5688: and 5670:‖ 5663:‖ 5660:− 5447:⁡ 5428:− 5416:∑ 5240:∫ 5224:φ 5189:− 5120:∂ 5117:→ 5111:: 4991:That is, 4851:≤ 4759:≤ 4671:Δ 4611:Δ 4541:≥ 4435:… 4423:∈ 4391:Δ 4387:∈ 4323:∑ 4281:∑ 4248:Δ 4244:∈ 4203:Δ 4199:∈ 4154:≥ 4101:∑ 4097:∣ 4076:∈ 4056:Δ 4023:Δ 3905:∗ 3891:∫ 3881:ω 3870:∗ 3856:∫ 3846:ω 3838:∗ 3821:∫ 3811:ω 3803:∗ 3790:∂ 3786:∫ 3779:ω 3771:∂ 3767:∫ 3211:⋅ 3174:− 2944:⋅ 2934:− 2925:− 2891:⋅ 2881:− 2720:, by the 2451:− 2412:− 2342:− 2326:∈ 2298:− 2226:− 2210:∈ 2185:− 2115:⁡ 2096:⁡ 2012:≤ 2006:≤ 1974:− 1810:− 1802:∈ 1795:∑ 1779:⁡ 1670:∈ 1498:∈ 1448:→ 1388:¯ 1300:John Nash 1221:John Nash 1212:Reception 994:in 1909. 823:bijective 772:− 746:, etc.). 740:connected 699:Convexity 695:(1) = 1. 322:Statement 8105:Archived 8049:Archived 7991:(1965). 7776:(1988). 7649:(1978). 7497:(2013), 7277:Archived 7185:Archived 7179:P. Bich 6959:manifold 6923:(1975). 6882:(1975). 6844:(1975). 6773:Archived 6734:(1989). 6660:Archived 6578:Archived 6527:won the 6360:Archived 6340:Archived 6263:Archived 6215:(1911). 6127:Archived 6068:See also 6015:column. 4873:for all 4800:Because 3546:codomain 3456: : 3415:boundary 3058:) = 1 – 3007:interior 2078:. Then 2074:for all 1590:lies in 1524:Jacobian 1185:homotopy 1160:topology 257:and the 39:topology 8111:, from 8045:0505523 8037:2320860 8007:0226651 7981:1454127 7909:0416010 7889:Bibcode 7847:0620639 7820:0115161 7766:2320146 7737:0995842 7706:0658305 7679:0492046 7638:1224675 7607:0861409 7580:0283792 7572:2317520 7533:3035127 7425:2320146 7155:WebCite 6643:⁠ 6631:⁠ 5820:compact 5603:compact 4012:simplex 3429:, the ( 3417:of the 3131:). Set 3005:in the 2829:of the 2778:. Thus 2583:. For 1148:Latvian 1074:bounded 1066:compact 960:History 445:compact 391:subset 389:compact 339:from a 70:compact 8043: 8035: 8005: 7979: 7969: 7950: 7907: 7862: 7845: 7835: 7818: 7808: 7784: 7764: 7735: 7725: 7704: 7694: 7677: 7636: 7626: 7605: 7595: 7578: 7570: 7531: 7523: 7423: 7053: 7032: 6748: 6715: 6684: 6493: 6468: 6385:22 May 6310: 6285: 6241: 6149: 5816:metric 5755:. See 5057:or by 3737:). If 3725:or by 3665:. For 3652:cyclic 3435:sphere 3421:-disk 2993:Since 2841:, set 1889:: let 1738:under 1174:, and 1117:Banach 1070:closed 948:has a 736:closed 449:convex 386:convex 341:closed 335:Every 253:, the 249:, the 8052:(PDF) 8033:JSTOR 8019:(PDF) 7762:JSTOR 7568:JSTOR 7521:JSTOR 7421:JSTOR 7280:(PDF) 7269:(PDF) 6742:17–24 6239:S2CID 6106:Notes 5311:, so 4224:also 3986:) by 3741:is a 3563:onto 3497:) to 3433:− 1)- 3385:) = ( 2671:and 2626:is a 817:Notes 420:of a 365:of a 33:is a 7967:ISBN 7948:ISBN 7860:ISBN 7833:ISBN 7806:ISBN 7782:ISBN 7723:ISBN 7692:ISBN 7624:ISBN 7593:ISBN 7288:2022 7051:ISBN 7030:ISBN 6746:ISBN 6713:ISBN 6682:ISBN 6491:ISBN 6466:ISBN 6426:Arte 6387:2015 6308:ISBN 6283:ISBN 6147:ISBN 5990:The 5972:The 5544:and 4918:and 3763:< 3614:For 3477:and 3465:has 3330:and 2772:) ⋅ 2760:) - 2748:) = 2606:) = 2531:) ⋅ 2500:The 1995:for 1909:and 1829:sign 1358:Let 1158:and 1146:, a 1072:and 1058:flow 950:zero 804:does 789:-x≠x 689:does 451:(or 344:disk 300:and 289:and 193:disk 8095:at 7944:181 7897:doi 7754:doi 7665:doi 7560:doi 7513:doi 7509:120 7474:doi 7413:doi 7217:doi 7153:at 7127:doi 7098:doi 6961:is 6939:doi 6898:doi 6860:doi 6447:127 6231:doi 6189:in 6123:on 5573:RCA 5566:WKL 5560:In 5377:Hex 5249:det 4795:+ 1 4657:+ 1 3425:is 3365:+ 1 3360:in 3119:= ( 3103:= 3095:+ 1 3033:in 3009:of 2796:/|| 2730:of 2704:If 2630:on 2577:of 2555:is 2546:. 2516:on 2319:sup 2203:sup 2106:deg 2087:deg 1833:det 1770:deg 1698:of 1526:of 1467:of 1304:Hex 920:to 905:to 825:or 424:to 281:of 234:of 37:in 8137:: 8069:, 8063:, 8047:. 8041:MR 8039:. 8029:85 8027:. 8021:. 8003:MR 8001:. 7977:MR 7975:. 7965:. 7961:. 7942:. 7938:. 7926:30 7924:. 7920:. 7905:MR 7903:. 7895:. 7885:13 7883:. 7843:MR 7841:. 7816:MR 7814:. 7804:. 7760:. 7750:86 7748:. 7733:MR 7731:. 7717:. 7702:MR 7700:. 7675:MR 7673:. 7661:32 7659:. 7653:. 7634:MR 7632:. 7622:. 7603:MR 7601:. 7576:MR 7574:. 7566:. 7556:78 7554:. 7529:MR 7527:, 7519:, 7507:, 7501:, 7468:. 7464:. 7419:. 7409:86 7407:. 7275:. 7213:41 7211:. 7207:. 7121:. 7092:. 7086:. 6933:. 6927:. 6906:. 6892:. 6886:. 6854:. 6848:. 6814:10 6744:. 6711:, 6707:, 6701:, 6602:. 6535:: 6445:. 6433:^ 6424:, 6409:^ 6237:. 6227:71 6219:. 6202:^ 6193:: 6183:: 6173:^ 6002:. 5818:) 5733:1. 5594:. 5444:Tr 5236::= 5162::= 4520:: 4014:, 3990:. 3980:(∂ 3977:-1 3962:(∂ 3749:, 3673:= 3644:−1 3630:−1 3611:. 3556:. 3537:. 3467:no 3460:→ 3437:. 3405:. 3306:⋅ 3270:⋅ 3125:, 3109:x 3064:⋅ 3046:⋅ 2683:) 2665:) 2612:+ 2481:. 2130:. 1758:: 1355:. 1298:, 1291:. 1243:. 1208:. 1170:, 1123:. 944:, 829:. 813:. 742:, 293:. 8009:. 7983:. 7911:. 7899:: 7891:: 7868:. 7849:. 7822:. 7790:. 7768:. 7756:: 7739:. 7708:. 7681:. 7667:: 7640:. 7609:. 7582:. 7562:: 7515:: 7482:. 7476:: 7470:7 7427:. 7415:: 7382:. 7370:. 7290:. 7237:. 7223:. 7219:: 7170:. 7133:. 7129:: 7123:8 7106:. 7100:: 7094:2 7057:. 7036:. 6996:. 6981:. 6979:p 6975:n 6971:p 6967:n 6947:. 6941:: 6935:2 6900:: 6894:2 6868:. 6862:: 6856:2 6754:. 6640:2 6637:/ 6634:1 6620:. 6604:2 6499:. 6474:. 6389:. 6322:. 6289:. 6245:. 6233:: 6000:D 5978:R 5957:1 5950:| 5946:j 5940:i 5936:| 5907:j 5903:U 5894:i 5890:U 5869:} 5864:m 5860:U 5856:, 5850:, 5845:1 5841:U 5837:{ 5802:X 5796:X 5793:: 5790:f 5770:X 5727:n 5717:1 5711:n 5707:x 5703:= 5698:n 5694:y 5679:2 5674:2 5666:x 5657:1 5652:= 5647:0 5643:y 5628:n 5624:y 5619:n 5615:x 5611:f 5607:ℓ 5575:0 5568:0 5550:f 5546:f 5532:) 5529:B 5526:( 5521:0 5517:H 5502:B 5498:f 5480:) 5477:) 5474:B 5471:( 5466:n 5462:H 5457:| 5453:f 5450:( 5439:n 5435:) 5431:1 5425:( 5420:n 5401:B 5397:f 5381:n 5361:r 5357:r 5353:g 5349:t 5345:g 5341:B 5339:( 5337:g 5333:t 5331:( 5329:φ 5325:φ 5321:n 5317:φ 5313:φ 5309:t 5305:t 5303:( 5300:′ 5298:φ 5281:. 5278:x 5275:d 5271:) 5268:x 5265:( 5260:t 5256:g 5252:D 5244:B 5233:) 5230:t 5227:( 5201:, 5198:x 5195:) 5192:t 5186:1 5183:( 5180:+ 5177:) 5174:x 5171:( 5168:r 5165:t 5159:) 5156:x 5153:( 5148:t 5144:g 5123:B 5114:B 5108:r 5090:q 5086:q 5082:q 5051:f 5022:. 5019:f 4999:P 4976:. 4973:P 4970:= 4967:) 4964:P 4961:( 4958:f 4935:) 4932:P 4929:( 4926:f 4906:P 4884:. 4881:j 4859:j 4855:P 4846:j 4842:) 4838:P 4835:( 4832:f 4812:P 4802:f 4793:n 4788:n 4772:. 4767:j 4763:P 4754:j 4750:) 4746:P 4743:( 4740:f 4720:j 4700:P 4680:, 4675:n 4655:k 4640:j 4620:, 4615:n 4600:k 4586:P 4563:. 4558:j 4554:) 4550:P 4547:( 4544:f 4536:j 4532:P 4518:f 4504:j 4484:P 4464:j 4444:} 4441:n 4438:, 4432:, 4429:0 4426:{ 4420:j 4400:, 4395:n 4384:P 4358:i 4354:) 4350:P 4347:( 4344:f 4338:n 4333:0 4330:= 4327:i 4319:= 4316:1 4313:= 4307:i 4303:P 4296:n 4291:0 4288:= 4285:i 4257:. 4252:n 4241:) 4238:P 4235:( 4232:f 4212:, 4207:n 4196:P 4173:. 4169:} 4165:i 4157:0 4149:i 4145:P 4136:1 4133:= 4127:i 4123:P 4116:n 4111:0 4108:= 4105:i 4092:1 4089:+ 4086:n 4081:R 4073:P 4069:{ 4065:= 4060:n 4032:, 4027:n 4010:- 4008:n 4004:f 3983:M 3975:n 3971:H 3965:M 3959:H 3951:ω 3946:M 3925:, 3922:0 3919:= 3916:) 3913:0 3910:( 3901:F 3895:B 3887:= 3884:) 3878:d 3875:( 3866:F 3860:B 3852:= 3849:) 3843:( 3834:F 3830:d 3825:B 3817:= 3814:) 3808:( 3799:F 3793:B 3782:= 3774:B 3760:0 3739:ω 3718:F 3712:B 3706:B 3700:F 3687:n 3683:U 3679:n 3675:E 3671:U 3667:n 3663:E 3648:S 3646:( 3642:n 3638:H 3634:D 3632:( 3628:n 3624:H 3616:n 3605:n 3601:Z 3597:S 3593:D 3581:n 3577:D 3573:S 3569:n 3565:S 3561:D 3554:F 3550:S 3535:x 3531:x 3529:( 3527:F 3523:S 3519:D 3515:F 3511:x 3509:( 3507:F 3503:S 3499:x 3495:x 3493:( 3491:f 3487:D 3483:x 3481:( 3479:f 3475:x 3471:D 3462:D 3458:D 3454:f 3446:F 3431:n 3427:S 3423:D 3419:n 3394:x 3391:( 3388:f 3382:y 3379:, 3376:x 3373:( 3370:F 3363:n 3357:B 3351:n 3345:n 3339:x 3336:( 3333:w 3327:t 3321:x 3315:x 3312:( 3309:w 3303:x 3297:x 3291:y 3288:( 3285:X 3279:y 3276:( 3273:X 3267:y 3261:W 3255:X 3237:. 3234:) 3231:) 3226:x 3221:( 3216:w 3206:x 3201:, 3198:) 3193:x 3188:( 3183:w 3177:t 3171:( 3168:= 3165:) 3162:t 3159:, 3154:x 3149:( 3144:X 3128:t 3122:x 3116:y 3111:R 3106:V 3100:W 3093:n 3090:( 3085:V 3079:n 3073:x 3070:( 3067:f 3061:x 3055:x 3052:( 3049:w 3043:x 3036:S 3030:x 3024:x 3021:( 3018:w 3012:B 3002:x 2996:f 2978:. 2975:) 2970:x 2965:( 2960:f 2954:) 2949:x 2939:x 2931:1 2928:( 2920:x 2914:) 2911:) 2906:x 2901:( 2896:f 2886:x 2878:1 2875:( 2872:= 2869:) 2864:x 2859:( 2854:w 2838:V 2832:n 2826:B 2820:f 2814:n 2805:S 2799:v 2793:v 2787:A 2781:v 2775:x 2769:x 2766:( 2763:u 2757:x 2754:( 2751:u 2745:x 2742:( 2739:v 2733:A 2727:u 2717:S 2707:w 2698:t 2692:n 2686:A 2680:t 2674:A 2668:S 2662:t 2656:S 2649:t 2645:f 2639:t 2633:A 2624:) 2622:x 2620:( 2618:w 2615:t 2609:x 2603:x 2600:( 2596:t 2592:f 2586:t 2580:S 2574:A 2568:S 2562:w 2552:w 2539:x 2533:x 2529:x 2527:( 2525:w 2519:S 2513:w 2507:S 2486:f 2479:f 2465:) 2462:0 2459:( 2454:1 2447:g 2426:) 2423:0 2420:( 2415:1 2408:g 2387:g 2361:| 2357:) 2354:y 2351:( 2348:f 2345:t 2339:y 2335:| 2329:K 2323:y 2313:) 2310:x 2307:( 2304:f 2301:t 2295:x 2289:= 2286:) 2283:x 2280:, 2277:t 2274:( 2271:H 2242:| 2238:) 2235:y 2232:( 2229:f 2223:y 2219:| 2213:K 2207:y 2197:) 2194:x 2191:( 2188:f 2182:x 2176:= 2173:) 2170:x 2167:( 2164:g 2141:K 2118:g 2110:p 2102:= 2099:f 2091:p 2076:t 2060:t 2056:H 2035:p 2015:1 2009:t 2003:0 1983:g 1980:) 1977:t 1971:1 1968:( 1965:+ 1962:f 1959:t 1956:= 1953:) 1950:x 1947:( 1942:t 1938:H 1917:g 1897:f 1876:p 1872:f 1855:. 1852:) 1847:x 1843:f 1839:d 1836:( 1824:) 1821:p 1818:( 1813:1 1806:f 1799:x 1791:= 1788:) 1785:f 1782:( 1774:p 1746:f 1726:p 1706:f 1682:) 1679:0 1676:( 1673:B 1667:p 1647:f 1627:K 1607:) 1604:0 1601:( 1598:B 1578:f 1554:p 1534:f 1510:) 1507:0 1504:( 1501:B 1495:p 1475:f 1451:K 1445:K 1442:: 1439:f 1417:n 1412:R 1384:) 1381:0 1378:( 1375:B 1369:= 1366:K 1307:( 1261:R 1257:n 1082:t 978:n 946:g 938:b 934:a 930:x 926:x 924:( 922:f 918:x 914:g 907:x 903:x 898:f 844:n 801:f 797:n 793:f 775:x 769:= 766:) 763:x 760:( 757:f 720:n 716:D 693:f 686:f 668:, 663:2 659:1 656:+ 653:x 647:= 644:) 641:x 638:( 635:f 603:R 581:R 557:, 554:1 551:+ 548:x 545:= 542:) 539:x 536:( 533:f 499:1 496:+ 493:x 490:= 487:) 484:x 481:( 478:f 461:f 426:K 418:K 397:K 393:K 311:n 222:K 202:D 179:I 159:f 137:0 133:x 129:= 126:) 121:0 117:x 113:( 110:f 88:0 84:x 56:f 20:)

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Knowledge (XXG)

Brouwer fixed-point theorem

Index