1851:
882:
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Nearly every modern citation that I have found agrees that the first correct proof is due to Veblen... In view of the heavy criticism of Jordan’s proof, I was surprised when I sat down to read his proof to find nothing objectionable about it. Since then, I have contacted a number of the authors who
507:
702:
It is clear that Jordan curve theorem implies the "strong Hex theorem": "every game of Hex ends with exactly one winner, with no possibility of both sides losing or both sides winning", thus the Jordan curve theorem is equivalent to the strong Hex theorem, which is a purely
1344:, then it is the hexagonal grid, and thus satisfies the strong Hex theorem, allowing the Jordan curve theorem to generalize. For this reason, when computing connected components in a binary image, the 6-neighbor square grid is generally used.
65:
seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."
1441:
His proof, however, is unsatisfactory to many mathematicians. It assumes the theorem without proof in the important special case of a simple polygon, and of the argument from that point on, one must admit at least that all details are not
713:
In reverse mathematics, and computer-formalized mathematics, the Jordan curve theorem is commonly proved by first converting it to an equivalent discrete version similar to the strong Hex theorem, then proving the discrete version.
1591:
as follows. It is relatively simple to prove that the Jordan curve theorem holds for every Jordan polygon (Lemma 1), and every Jordan curve can be approximated arbitrarily well by a Jordan polygon (Lemma 2). A Jordan polygon is a
1253:
Both graph structures fail to satisfy the strong Hex theorem. The 4-neighbor square grid allows a no-winner situation, and the 8-neighbor square grid allows a two-winner situation. Consequently, connectedness properties in
1771:
bounded by the curve. This is the point of
Tverberg's Lemma 3. Roughly, the closed polygons should not thin to zero everywhere. Moreover, they should not thin to zero somewhere, which is the point of Tverberg's Lemma 4.
381:
1352:
The
Steinhaus chessboard theorem in some sense shows that the 4-neighbor grid and the 8-neighbor grid "together" implies the Jordan curve theorem, and the 6-neighbor grid is a precise interpolation between them.
1187:
1030:
1765:
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1382:
chessboard, so that a king cannot move from the bottom side to the top side without stepping on a bomb, then a rook can move from the left side to the right side stepping only on bombs.
1247:
84:(1838–1922), who published its first claimed proof in 1887. For decades, mathematicians generally thought that this proof was flawed and that the first rigorous proof was carried out by
20:
3082:
Brown, R.; Antolino-Camarena, O. (2014). "Corrigendum to "Groupoids, the
Phragmen-Brouwer Property, and the Jordan Curve Theorem", J. Homotopy and Related Structures 1 (2006) 175-183".
1463:
Jordan’s proof is essentially correct... Jordan’s proof does not present the details in a satisfactory way. But the idea is right, and with some polishing the proof would be impeccable.
1791:. Both the Mizar and the HOL Light proof rely on libraries of previously proved theorems, so these two sizes are not comparable. Nobuyuki Sakamoto and Keita Yokoyama (
1839:
1107:
1342:
1310:
1281:
876:
840:
811:
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753:
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system, in
January 2005, and contained about 60,000 lines. Another rigorous 6,500-line formal proof was produced in 2005 by an international team of mathematicians using the
1459:
Hales also pointed out that the special case of simple polygons is not only an easy exercise, but was not really used by Jordan anyway, and quoted
Michael Reeken as saying:
1482:, it received much attention from prominent mathematicians of the first half of the 20th century. Various proofs of the theorem and its generalizations were constructed by
1394:
was the first to formulate a precise conjecture, observing that it was not a self-evident statement, but that it required a proof. It is easy to establish this result for
1380:
1711:
1642:
1622:
1433:. There is some controversy about whether Jordan's proof was complete: the majority of commenters on it have claimed that the first complete proof was given later by
1065:
920:
23:
Illustration of the Jordan curve theorem. The Jordan curve (drawn in black) divides the plane into an "inside" region (light blue) and an "outside" region (pink).
699:
has at least one winner", from which we obtain a logical implication: Hex theorem implies
Brouwer fixed point theorem, which implies Jordan curve theorem.
2464:
Adler, Aviv; Daskalakis, Constantinos; Demaine, Erik D. (2016). Chatzigiannakis, Ioannis; Mitzenmacher, Michael; Rabani, Yuval; Sangiorgi, Davide (eds.).
2997:
2795:
339:
consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The set
619:, completely lying in the interior region. An alternative and equivalent formulation of the Jordan–Schönflies theorem asserts that any Jordan curve
1930:
of intersections of the ray with an edge of the polygon. Jordan curve theorem proof implies that the point is inside the polygon if and only if
2962:
502:{\displaystyle {\tilde {H}}_{q}(Y)={\begin{cases}\mathbb {Z} ,&q=n-k{\text{ or }}q=n,\\\{0\},&{\text{otherwise}}.\end{cases}}}
2875:
Sakamoto, Nobuyuki; Yokoyama, Keita (2007), "The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic",
1521:
New elementary proofs of the Jordan curve theorem, as well as simplifications of the earlier proofs, continue to be carried out.
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2524:
2491:
2223:
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1926:
that does not pass through any vertex of the polygon (all rays but a finite number are convenient). Then, compute the number
1468:
1455:
have criticized Jordan, and each case the author has admitted to having no direct knowledge of an error in Jordan’s proof.
2843:
710:
The
Brouwer fixed point theorem, by being sandwiched between the two equivalent theorems, is also equivalent to both.
2919:
925:
1724:
1647:
1547:
A proof using constructive mathematics by Gordon O. Berg, W. Julian, and R. Mines et al. (
1390:
The statement of the Jordan curve theorem may seem obvious at first, but it is a rather difficult theorem to prove.
146:
in the plane is the image of an injective continuous map of a closed and bounded interval into the plane. It is a
584:
651:
of the plane. Unlike
Lebesgue's and Brouwer's generalization of the Jordan curve theorem, this statement becomes
3111:
2755:
1954:. This complements the earlier result by Maehara, that Brouwer's fixed point theorem implies Jordan's theorem.
1112:
223:
1804:
1644:
is positive. Using a sequence of Jordan polygons (that converge to the given Jordan curve) we have a sequence
3106:
3053:
2705:
2619:
843:
2595:
2311:
Johnson, Dale M. (1977). "Prelude to dimension theory: the geometrical investigations of
Bernard Bolzano".
3048:
2591:
1951:
1906:
1807:
1555:
692:
517:
2136:
Nguyen, Phuong; Cook, Stephen A. (2007). "The
Complexity of Proving the Discrete Jordan Curve Theorem".
61:
connecting a point of one region to a point of the other intersects with the curve somewhere. While the
2522:
Berg, Gordon O.; Julian, W.; Mines, R.; Richman, Fred (1975), "The constructive Jordan curve theorem",
1483:
1192:
545:
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852:
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1515:
1507:
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421:
1963:
1570:
1475:
723:
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1800:
695:(in 2 dimensions), and the Brouwer fixed point theorem can be proved from the Hex theorem: "every
668:
587:, which states that the interior and the exterior planar regions determined by a Jordan curve in
211:
2971:
2663:
2614:
2052:
1912:
1447:
1399:
1359:
89:
1398:, but the problem came in generalizing it to all kinds of badly behaved curves, which include
726:, a binary picture is a discrete square grid of 0 and 1, or equivalently, a compact subset of
2564:
2440:
1696:
1627:
1607:
1537:
1070:
1915:, the Jordan curve theorem can be used for testing whether a point lies inside or outside a
1850:
3028:
2904:
2786:
2742:
2648:
2555:
2332:
2297:
2038:
1974:
1966:, a description of certain sets of points in the plane that can be subsets of Jordan curves
1935:
1038:
893:
704:
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8:
2056:
1796:
664:
243:
113:
54:
50:
46:
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1946:
Adler, Daskalakis and Demaine prove that a computational version of Jordan's theorem is
3083:
3016:
2939:
2908:
2860:
2830:
2730:
2703:
Maehara, Ryuji (1984), "The Jordan Curve Theorem Via the Brouwer Fixed Point Theorem",
2652:
2204:
2179:
Hales, Thomas C. (December 2007). "The Jordan Curve Theorem, Formally and Informally".
2141:
2118:
2026:
1601:
294:
71:
683:
is not simply connected, and hence not homeomorphic to the exterior of the unit ball.
2954:
2892:
2814:
2774:
2722:
2636:
2579:
2569:
2568:. Herbert Robbins ( ed.). United Kingdom: Oxford University Press. p. 267.
2543:
2487:
2421:
2388:(1924). "Bemerkungen zu den Beweisen von C. Jordan und Ch. J. de la Vallée Poussin".
2266:
2196:
2159:
2110:
1511:
1503:
1491:
549:
2912:
2465:
2285:
2093:
Gale, David (December 1979). "The Game of Hex and the Brouwer Fixed-Point Theorem".
3006:
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2477:
2476:. Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik: 24:1–24:14.
2320:
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2192:
2188:
2151:
2102:
2018:
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360:
267:
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2470:
43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
2208:
3024:
2900:
2856:
2841:
Ross, Fiona; Ross, William T. (2011), "The Jordan curve theorem is non-trivial",
2782:
2750:
2738:
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2293:
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1391:
348:
303:
155:
58:
151:
70:, Introduction)). More transparent proofs rely on the mathematical machinery of
2950:
2686:
2224:"First Principles of Computer Vision: Segmenting Binary Images | Binary Images"
1923:
1916:
1866:
1422:
1403:
537:
262:
The Jordan curve theorem was independently generalized to higher dimensions by
123:
81:
36:
2888:
2261:
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3100:
2992:
2896:
2818:
2778:
2726:
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2547:
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2346:
2270:
2200:
2114:
1969:
1947:
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does not converge to zero, using only the given Jordan curve, not the region
1714:
1499:
1487:
1434:
1426:
1407:
553:
85:
78:
2922:(1992), "The Jordan–Schönflies theorem and the classification of surfaces",
2769:
679:, but so twisted in space that the unbounded component of its complement in
3063:
2985:
2583:
1788:
1776:
1566:
1411:
592:
192:
With these definitions, the Jordan curve theorem can be stated as follows:
41:
1471:
had already been critically analyzed and completed by Schoenflies (1924).
57:" region containing all of the nearby and far away exterior points. Every
2155:
1876:
636:
540:), and with a bit of extra work, one shows that their common boundary is
263:
227:
147:
3069:
3020:
2943:
2834:
2734:
2565:
What is mathematics? : an elementary approach to ideas and methods
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19:
95:
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2718:
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2106:
2062:. From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
2022:
1356:
The theorem states that: suppose you put bombs on some squares on a
2138:
22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
1597:
314:
28:
3088:
2146:
784:, such as number of components, might fail to be well-defined for
313:> 0), i.e. the image of an injective continuous mapping of the
74:, and these lead to generalizations to higher-dimensional spaces.
2995:(1905), "Theory on Plane Curves in Non-Metrical Analysis Situs",
2228:
1395:
881:
583:
There is a strengthening of the Jordan curve theorem, called the
185:
is a continuous loop, whereas the last condition stipulates that
62:
1624:
of the largest disk contained in the closed polygon. Evidently,
676:
580:) without boundary, its complement has 2 connected components.
161:
Alternatively, a Jordan curve is the image of a continuous map
127:
2793:
Osgood, William F. (1903), "A Jordan Curve of Positive Area",
2617:(2007a), "The Jordan curve theorem, formally and informally",
2441:"PNPOLY - Point Inclusion in Polygon Test - WR Franklin (WRF)"
1950:. As a corollary, they show that Jordan's theorem implies the
2472:. Leibniz International Proceedings in Informatics (LIPIcs).
1875:), the number of intersections of the ray and the polygon is
655:
in higher dimensions: while the exterior of the unit ball in
2562:
Courant, Richard (1978). "V. Topology". Written at Oxford.
495:
3060:
The full 6,500 line formal proof of Jordan's curve theorem
2009:
Kline, J. R. (1942). "What is the Jordan curve theorem?".
607:
on the Jordan curve, there exists a Jordan arc connecting
1283:, such as the Jordan curve theorem, do not generalize to
181:
to [0,1) is injective. The first two conditions say that
45:(a plane simple closed curve) divides the plane into an "
1315:
If the "6-neighbor square grid" structure is imposed on
2429:. Edinburg: University of Edinburgh. 1978. p. 267.
3081:
2521:
2463:
1548:
2074:
1810:
1727:
1699:
1650:
1630:
1610:
1474:
Due to the importance of the Jordan curve theorem in
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1321:
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1260:
1195:
1115:
1073:
1041:
928:
896:
855:
819:
790:
761:
732:
384:
2466:"The Complexity of Hex and the Jordan Curve Theorem"
1992:
1990:
1977:, a mathematical group that preserves a Jordan curve
3041:
2500:
1467:Earlier, Jordan's proof and another early proof by
351:. It is first established that, more generally, if
96:
Definitions and the statement of the Jordan theorem
1833:
1759:
1705:
1682:
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1241:
1181:
1101:
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1024:
914:
870:
834:
805:
776:
747:
717:
501:
2998:Transactions of the American Mathematical Society
2796:Transactions of the American Mathematical Society
2751:"A nonstandard proof of the Jordan curve theorem"
1987:
1717:bounded by the Jordan curve. However, we have to
639:in the plane, can be extended to a homeomorphism
564:and the reduced cohomology of its complement. If
536:has 2 connected components (which are, moreover,
16:A closed curve divides the plane into two regions
3098:
3070:Collection of proofs of the Jordan curve theorem
2874:
2245:"A digital analogue of the Jordan curve theorem"
1869:(in red) lies outside a simple polygon (region
1792:
1035:the "8-neighbor square grid", where each vertex
890:the "4-neighbor square grid", where each vertex
691:The Jordan curve theorem can be proved from the
1437:, who said the following about Jordan's proof:
1347:
1025:{\displaystyle (x+1,y),(x-1,y),(x,y+1),(x,y-1)}
2384:
1760:{\displaystyle \delta _{1},\delta _{2},\dots }
1693:converging to a positive number, the diameter
1683:{\displaystyle \delta _{1},\delta _{2},\dots }
572:-dimensional compact connected submanifold of
88:. However, this notion has been overturned by
1907:Point in polygon § Ray casting algorithm
1421:The first proof of this theorem was given by
1385:
2668:"Jordan's proof of the Jordan Curve theorem"
2057:"Jordan's proof of the Jordan curve theorem"
1604:, the closed polygon. Consider the diameter
476:
470:
257:
77:The Jordan curve theorem is named after the
2963:Bulletin of the London Mathematical Society
1892:) of a ray lies inside the polygon (region
1779:of the Jordan curve theorem was created by
1587:The root of the difficulty is explained in
2135:
1799:the Jordan curve theorem is equivalent to
849:There are two obvious graph structures on
3087:
3010:
2975:
2918:
2808:
2768:
2596:"An elementary proof of Jordan's theorem"
2537:
2481:
2286:"A discrete form of Jordan curve theorem"
2260:
2145:
1580:
1324:
1292:
1263:
1182:{\displaystyle |x-x'|\leq 1,|y-y'|\leq 1}
858:
822:
793:
764:
735:
425:
2949:
2840:
2590:
1941:
1849:
1588:
1530:
1526:
1431:Cours d'analyse de l'École Polytechnique
880:
615:and, with the exception of the endpoint
544:. A further generalization was found by
250:In contrast, the complement of a Jordan
67:
18:
2702:
2561:
2506:
2407:
2310:
2283:
2080:
1559:
885:8-neighbor and 4-neighbor square grids.
842:does not have an appropriately defined
3099:
3076:A simple proof of Jordan curve theorem
2991:
2792:
2748:
2685:
2675:Studies in Logic, Grammar and Rhetoric
2662:
2613:
2373:
2362:
2350:
2242:
2064:Studies in Logic, Grammar and Rhetoric
1996:
1820:
1817:
1814:
1780:
1596:, the boundary of a bounded connected
1541:
1415:
2955:"A proof of the Jordan curve theorem"
2525:Rocky Mountain Journal of Mathematics
2313:Archive for History of Exact Sciences
2221:
2178:
2051:
2008:
1713:of the largest disk contained in the
2092:
1600:, call it the open polygon, and its
1525:Elementary proofs were presented by
595:to the interior and exterior of the
2844:Journal of Mathematics and the Arts
1834:{\displaystyle {\mathsf {RCA}}_{0}}
686:
603:in the interior region and a point
13:
1898:), the number of intersections is
552:between the reduced homology of a
234:) and the other is unbounded (the
14:
3123:
3035:
2706:The American Mathematical Monthly
2620:The American Mathematical Monthly
2181:The American Mathematical Monthly
2095:The American Mathematical Monthly
1469:Charles Jean de la Vallée Poussin
1242:{\displaystyle (x,y)\neq (x',y')}
528:, the zeroth reduced homology of
272:Jordan–Brouwer separation theorem
189:has no self-intersection points.
2390:Jahresber. Deutsch. Math.-Verein
1429:, and was published in his book
1337:{\displaystyle \mathbb {Z} ^{2}}
1305:{\displaystyle \mathbb {Z} ^{2}}
1276:{\displaystyle \mathbb {R} ^{2}}
871:{\displaystyle \mathbb {Z} ^{2}}
835:{\displaystyle \mathbb {Z} ^{2}}
806:{\displaystyle \mathbb {Z} ^{2}}
777:{\displaystyle \mathbb {R} ^{2}}
748:{\displaystyle \mathbb {Z} ^{2}}
2457:
2433:
2413:
2400:
2378:
2367:
2356:
2340:
2304:
2290:Annales Mathematicae Silesianae
2277:
1412:a Jordan curve of positive area
718:Application to image processing
599:. In particular, for any point
512:This is proved by induction in
206:be a Jordan curve in the plane
2877:Archive for Mathematical Logic
2756:Pacific Journal of Mathematics
2633:10.1080/00029890.2007.11920481
2236:
2215:
2193:10.1080/00029890.2007.11920481
2172:
2129:
2086:
2070:(23). University of Białystok.
2045:
2002:
1845:
1312:under either graph structure.
1236:
1214:
1208:
1196:
1169:
1150:
1136:
1117:
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1074:
1054:
1042:
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1001:
995:
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953:
947:
929:
909:
897:
410:
404:
392:
1:
3042:M.I. Voitsekhovskii (2001) ,
2924:American Mathematical Monthly
2515:
2011:American Mathematical Monthly
532:has rank 1, which means that
226:. One of these components is
3072:at Andrew Ranicki's homepage
2857:10.1080/17513472.2011.634320
2483:10.4230/LIPIcs.ICALP.2016.24
2249:Discrete Applied Mathematics
2222:Nayar, Shree (Mar 1, 2021).
1922:From a given point, trace a
1348:Steinhaus chessboard theorem
755:. Topological invariants on
39:in 1887, asserts that every
7:
3049:Encyclopedia of Mathematics
1957:
1952:Brouwer fixed-point theorem
1556:Brouwer fixed point theorem
693:Brouwer fixed point theorem
254:in the plane is connected.
177:(1) and the restriction of
10:
3128:
2406:Richard Courant (
2284:SurĂłwka, Wojciech (1993).
2140:. IEEE. pp. 245–256.
1904:
1386:History and further proofs
667:onto the unit sphere, the
343:is their common boundary.
270:in 1911, resulting in the
268:L. E. J. Brouwer
222:, consists of exactly two
2889:10.1007/s00153-007-0050-6
2262:10.1016/j.dam.2002.11.003
1516:Arthur Moritz Schoenflies
1375:{\displaystyle n\times n}
585:Jordan–Schönflies theorem
361:reduced integral homology
258:Proof and generalizations
2539:10.1216/RMJ-1975-5-2-225
2243:Ĺ lapal, J (April 2004).
1981:
1571:complete bipartite graph
1476:low-dimensional topology
150:that is not necessarily
3078:(PDF) by David B. Gauld
2770:10.2140/pjm.1971.36.219
2423:1. Jordan curve theorem
1706:{\displaystyle \delta }
1637:{\displaystyle \delta }
1617:{\displaystyle \delta }
1102:{\displaystyle (x',y')}
669:Alexander horned sphere
518:Mayer–Vietoris sequence
355:is homeomorphic to the
327:. Then the complement
2749:Narens, Louis (1971),
1913:computational geometry
1902:
1881:If the initial point (
1854:If the initial point (
1835:
1761:
1707:
1684:
1638:
1618:
1465:
1457:
1444:
1400:nowhere differentiable
1376:
1338:
1306:
1277:
1243:
1183:
1103:
1061:
1026:
916:
886:
872:
836:
807:
778:
749:
548:, who established the
503:
24:
3112:Theorems about curves
1942:Computational aspects
1853:
1836:
1762:
1708:
1685:
1639:
1619:
1538:non-standard analysis
1461:
1452:
1439:
1377:
1339:
1307:
1278:
1244:
1184:
1104:
1062:
1060:{\displaystyle (x,y)}
1027:
917:
915:{\displaystyle (x,y)}
884:
873:
837:
808:
779:
750:
504:
53:by the curve and an "
22:
3107:Theorems in topology
2986:10.1112/blms/12.1.34
2156:10.1109/lics.2007.48
1975:Quasi-Fuchsian group
1964:Denjoy–Riesz theorem
1808:
1725:
1697:
1648:
1628:
1608:
1402:curves, such as the
1360:
1319:
1287:
1258:
1193:
1113:
1071:
1039:
926:
894:
853:
817:
788:
759:
730:
382:
224:connected components
33:Jordan curve theorem
1797:reverse mathematics
1425:in his lectures on
282: —
246:of each component.
200: —
106:simple closed curve
2920:Thomassen, Carsten
2698:, pp. 587–594
2325:10.1007/BF00499625
1903:
1831:
1801:weak KĹ‘nig's lemma
1757:
1721:that the sequence
1703:
1680:
1634:
1614:
1554:A proof using the
1372:
1334:
1302:
1273:
1239:
1179:
1099:
1067:is connected with
1057:
1022:
922:is connected with
912:
887:
868:
832:
803:
774:
745:
675:homeomorphic to a
499:
494:
359:-sphere, then the
295:topological sphere
280:
198:
72:algebraic topology
25:
2603:Uspekhi Mat. Nauk
2575:978-0-19-502517-0
2493:978-3-95977-013-2
2347:Oswald Veblen
2165:978-0-7695-2908-0
1795:) showed that in
1512:Alfred Pringsheim
1504:Friedrich Hartogs
1492:Ludwig Bieberbach
635:is viewed as the
550:Alexander duality
487:
452:
395:
278:
238:), and the curve
196:
3119:
3093:
3091:
3056:
3044:"Jordan theorem"
3031:
3014:
2988:
2979:
2959:
2946:
2915:
2867:
2837:
2812:
2789:
2772:
2745:
2699:
2697:
2682:
2672:
2659:
2615:Hales, Thomas C.
2610:
2600:
2587:
2558:
2541:
2510:
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2498:
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2445:wrf.ecse.rpi.edu
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2281:
2275:
2274:
2264:
2255:(1–3): 231–251.
2240:
2234:
2233:
2219:
2213:
2212:
2176:
2170:
2169:
2149:
2133:
2127:
2126:
2090:
2084:
2078:
2072:
2071:
2061:
2053:Hales, Thomas C.
2049:
2043:
2042:
2006:
2000:
1994:
1933:
1929:
1897:
1896:
1891:
1890:
1874:
1873:
1864:
1863:
1840:
1838:
1837:
1832:
1830:
1829:
1824:
1823:
1803:over the system
1766:
1764:
1763:
1758:
1750:
1749:
1737:
1736:
1712:
1710:
1709:
1704:
1689:
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1672:
1660:
1659:
1643:
1641:
1640:
1635:
1623:
1621:
1620:
1615:
1581:Thomassen (1992)
1480:complex analysis
1381:
1379:
1378:
1373:
1343:
1341:
1340:
1335:
1333:
1332:
1327:
1311:
1309:
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1303:
1301:
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1224:
1188:
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1153:
1139:
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1120:
1108:
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1100:
1095:
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1066:
1064:
1063:
1058:
1031:
1029:
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1023:
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913:
877:
875:
874:
869:
867:
866:
861:
841:
839:
838:
833:
831:
830:
825:
812:
810:
809:
804:
802:
801:
796:
783:
781:
780:
775:
773:
772:
767:
754:
752:
751:
746:
744:
743:
738:
724:image processing
687:Discrete version
661:simply connected
508:
506:
505:
500:
498:
497:
488:
485:
453:
450:
428:
403:
402:
397:
396:
388:
375:are as follows:
302:+1)-dimensional
283:
201:
130:into the plane,
35:, formulated by
3127:
3126:
3122:
3121:
3120:
3118:
3117:
3116:
3097:
3096:
3038:
3012:10.2307/1986378
2977:10.1.1.374.2903
2957:
2951:Tverberg, Helge
2936:10.2307/2324180
2810:10.2307/1986455
2719:10.2307/2323369
2713:(10): 641–643,
2695:
2692:Cours d'analyse
2687:Jordan, Camille
2670:
2627:(10): 882–894,
2598:
2592:Filippov, A. F.
2576:
2518:
2513:
2505:
2501:
2494:
2462:
2458:
2449:
2447:
2439:
2438:
2434:
2426:
2420:"V. Topology".
2419:
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2405:
2401:
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2361:
2357:
2345:
2341:
2309:
2305:
2282:
2278:
2241:
2237:
2220:
2216:
2187:(10): 882–894.
2177:
2173:
2166:
2134:
2130:
2107:10.2307/2320146
2101:(10): 818–827.
2091:
2087:
2079:
2075:
2059:
2050:
2046:
2023:10.2307/2303093
2007:
2003:
1995:
1988:
1984:
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1944:
1931:
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1668:
1664:
1655:
1651:
1649:
1646:
1645:
1629:
1626:
1625:
1609:
1606:
1605:
1594:polygonal chain
1589:Tverberg (1980)
1578:
1531:Tverberg (1980)
1527:Filippov (1950)
1508:Béla Kerékjártó
1496:Luitzen Brouwer
1484:J. W. Alexander
1448:Thomas C. Hales
1414:constructed by
1392:Bernard Bolzano
1388:
1361:
1358:
1357:
1350:
1328:
1323:
1322:
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1168:
1160:
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1114:
1111:
1110:
1088:
1077:
1072:
1069:
1068:
1040:
1037:
1036:
927:
924:
923:
895:
892:
891:
862:
857:
856:
854:
851:
850:
844:graph structure
826:
821:
820:
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815:
814:
797:
792:
791:
789:
786:
785:
768:
763:
762:
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739:
734:
733:
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728:
727:
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689:
671:is a subset of
546:J. W. Alexander
493:
492:
484:
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467:
466:
449:
432:
424:
417:
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349:homology theory
347:The proof uses
345:
304:Euclidean space
281:
260:
248:
199:
98:
90:Thomas C. Hales
59:continuous path
17:
12:
11:
5:
3125:
3115:
3114:
3109:
3095:
3094:
3079:
3073:
3067:
3057:
3037:
3036:External links
3034:
3033:
3032:
2993:Veblen, Oswald
2989:
2947:
2930:(2): 116–130,
2916:
2883:(5): 465–480,
2872:
2851:(4): 213–219,
2838:
2803:(1): 107–112,
2790:
2746:
2700:
2683:
2660:
2611:
2605:(in Russian),
2588:
2574:
2559:
2532:(2): 225–236,
2517:
2514:
2512:
2511:
2507:Maehara (1984)
2499:
2492:
2456:
2432:
2412:
2399:
2386:A. Schoenflies
2377:
2366:
2355:
2339:
2319:(3): 262–295.
2303:
2276:
2235:
2214:
2171:
2164:
2128:
2085:
2083:, p. 641.
2081:Maehara (1984)
2073:
2044:
2017:(5): 281–286.
2001:
1985:
1983:
1980:
1979:
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1917:simple polygon
1905:Main article:
1886:
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1585:
1584:
1576:
1565:A proof using
1563:
1560:Maehara (1984)
1552:
1545:
1536:A proof using
1534:
1423:Camille Jordan
1408:fractal curves
1404:Koch snowflake
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538:path connected
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124:continuous map
97:
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82:Camille Jordan
68:Tverberg (1980
37:Camille Jordan
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2870:author's site
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2374:Hales (2007b)
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1997:Jordan (1887)
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1991:
1986:
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1971:
1970:Lakes of Wada
1968:
1965:
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1961:
1955:
1953:
1949:
1948:PPAD-complete
1939:
1937:
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1781:Hales (2007a)
1778:
1773:
1770:
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1716:
1715:closed region
1700:
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1611:
1603:
1599:
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1582:
1579:was given by
1575:
1572:
1568:
1567:non-planarity
1564:
1561:
1557:
1553:
1550:
1546:
1543:
1542:Narens (1971)
1539:
1535:
1532:
1528:
1524:
1523:
1522:
1519:
1517:
1513:
1509:
1505:
1501:
1500:Arnaud Denjoy
1497:
1493:
1489:
1488:Louis Antoine
1485:
1481:
1477:
1472:
1470:
1464:
1460:
1456:
1451:
1449:
1443:
1438:
1436:
1435:Oswald Veblen
1432:
1428:
1427:real analysis
1424:
1419:
1417:
1416:Osgood (1903)
1413:
1409:
1405:
1401:
1397:
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292:-dimensional
291:
287:
275:
273:
269:
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255:
253:
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241:
237:
233:
229:
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218: \
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193:
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164:
159:
157:
153:
149:
145:
141:
137:
133:
129:
125:
122:
118:
115:
111:
108:in the plane
107:
103:
93:
91:
87:
86:Oswald Veblen
83:
80:
79:mathematician
75:
73:
69:
64:
60:
56:
52:
48:
44:
43:
38:
34:
30:
21:
3047:
3005:(1): 83–98,
3002:
2996:
2970:(1): 34–38,
2967:
2961:
2927:
2923:
2880:
2876:
2848:
2842:
2800:
2794:
2760:
2754:
2710:
2704:
2691:
2678:
2674:
2624:
2618:
2609:(5): 173–176
2606:
2602:
2563:
2529:
2523:
2502:
2473:
2469:
2459:
2448:. Retrieved
2444:
2435:
2422:
2415:
2402:
2393:
2389:
2380:
2369:
2358:
2342:
2316:
2312:
2306:
2292:(7): 57–61.
2289:
2279:
2252:
2248:
2238:
2227:
2217:
2184:
2180:
2174:
2137:
2131:
2098:
2094:
2088:
2076:
2067:
2063:
2047:
2014:
2010:
2004:
1945:
1921:
1910:
1884:
1857:
1789:Mizar system
1777:formal proof
1774:
1768:
1718:
1690:
1586:
1573:
1520:
1473:
1466:
1462:
1458:
1453:
1445:
1440:
1430:
1420:
1389:
1355:
1351:
1314:
1252:
848:
721:
712:
709:
701:
690:
680:
672:
656:
652:
648:
644:
640:
632:
628:
624:
620:
616:
612:
608:
604:
600:
593:homeomorphic
588:
582:
577:
573:
569:
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557:
541:
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529:
525:
521:
513:
511:
372:
368:
364:
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346:
340:
336:
332:
328:
324:
320:
315:
310:
306:
299:
293:
289:
285:
277:
271:
261:
251:
249:
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235:
231:
219:
215:
207:
203:
195:
191:
186:
182:
178:
174:
170:
166:
162:
160:
143:
139:
135:
131:
116:
109:
105:
102:Jordan curve
101:
99:
92:and others.
76:
42:Jordan curve
40:
32:
26:
2763:: 219–229,
2337:See p. 285.
1846:Application
697:game of Hex
637:unit circle
264:H. Lebesgue
210:. Then its
148:plane curve
3101:Categories
2827:34.0533.02
2516:References
2450:2021-07-18
2396:: 157–160.
1775:The first
1769:presumably
1691:presumably
1410:, or even
1406:and other
516:using the
363:groups of
212:complement
169:such that
144:Jordan arc
3089:1404.0556
3054:EMS Press
2972:CiteSeerX
2897:0933-5846
2819:0002-9947
2779:0030-8730
2727:0002-9890
2666:(2007b),
2641:0002-9890
2548:0035-7596
2271:0166-218X
2201:0002-9890
2147:1002.2954
2115:0002-9890
1785:HOL Light
1755:…
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1730:δ
1701:δ
1678:…
1666:δ
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1612:δ
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1367:×
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1174:≤
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1141:≤
1125:−
1014:−
960:−
707:theorem.
597:unit disk
486:otherwise
444:−
393:~
156:algebraic
121:injective
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2953:(1980),
2913:33627222
2689:(1887),
2594:(1950),
2055:(2007).
1958:See also
1598:open set
1396:polygons
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1222:′
1165:′
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705:discrete
665:retracts
631:, where
298:in the (
244:boundary
236:exterior
232:interior
55:exterior
47:interior
29:topology
3029:1500697
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1783:in the
1602:closure
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1450:wrote:
556:subset
554:compact
520:. When
318:-sphere
279:Theorem
242:is the
228:bounded
197:Theorem
112:is the
63:theorem
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1189:, and
677:sphere
641:ψ
621:φ
568:is an
288:be an
173:(0) =
152:smooth
128:circle
119:of an
31:, the
3084:arXiv
3064:Mizar
3017:JSTOR
2958:(PDF)
2940:JSTOR
2909:S2CID
2861:S2CID
2831:JSTOR
2731:JSTOR
2696:(PDF)
2671:(PDF)
2653:S2CID
2599:(PDF)
2427:(PDF)
2205:S2CID
2142:arXiv
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2060:(PDF)
2027:JSTOR
1982:Notes
1719:prove
653:false
611:with
323:into
230:(the
165:: →
126:of a
114:image
104:or a
2893:ISSN
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2723:ISSN
2681:(23)
2637:ISSN
2580:OCLC
2570:ISBN
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2160:ISBN
2111:ISSN
1900:odd.
1877:even
1793:2007
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576:(or
284:Let
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