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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
518:
713:
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
1355:, 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.
76:
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."
1452:
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
724:
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.
1602:
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
1264:
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
1782:
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.
392:
1363:
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.
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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.
1258:
95:(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
31:
3093:
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".
1474:
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.
1802:. 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 (
1850:
1118:
1353:
1321:
1292:
887:
851:
822:
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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
1470:
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:
1493:, 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
1405:
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
1391:
1722:
1653:
1633:
1444:. 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
1076:
931:
34:
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).
710:
has at least one winner", from which we obtain a logical implication: Hex theorem implies
Brouwer fixed point theorem, which implies Jordan curve theorem.
2475:
Adler, Aviv; Daskalakis, Constantinos; Demaine, Erik D. (2016). Chatzigiannakis, Ioannis; Mitzenmacher, Michael; Rabani, Yuval; Sangiorgi, Davide (eds.).
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350:
consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The set
630:, completely lying in the interior region. An alternative and equivalent formulation of the Jordan–Schönflies theorem asserts that any Jordan curve
1941:
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
2973:
513:{\displaystyle {\tilde {H}}_{q}(Y)={\begin{cases}\mathbb {Z} ,&q=n-k{\text{ or }}q=n,\\\{0\},&{\text{otherwise}}.\end{cases}}}
2886:
Sakamoto, Nobuyuki; Yokoyama, Keita (2007), "The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic",
1532:
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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2535:
2502:
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1937:
that does not pass through any vertex of the polygon (all rays but a finite number are convenient). Then, compute the number
1479:
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have criticized Jordan, and each case the author has admitted to having no direct knowledge of an error in Jordan’s proof.
2854:
721:
The
Brouwer fixed point theorem, by being sandwiched between the two equivalent theorems, is also equivalent to both.
2930:
936:
1735:
1658:
1558:
A proof using constructive mathematics by Gordon O. Berg, W. Julian, and R. Mines et al. (
1401:
The statement of the Jordan curve theorem may seem obvious at first, but it is a rather difficult theorem to prove.
157:
in the plane is the image of an injective continuous map of a closed and bounded interval into the plane. It is a
595:
662:
of the plane. Unlike
Lebesgue's and Brouwer's generalization of the Jordan curve theorem, this statement becomes
3122:
2766:
1965:. This complements the earlier result by Maehara, that Brouwer's fixed point theorem implies Jordan's theorem.
1123:
234:
1815:
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is positive. Using a sequence of Jordan polygons (that converge to the given Jordan curve) we have a sequence
3117:
3064:
2716:
2630:
854:
2606:
2322:
Johnson, Dale M. (1977). "Prelude to dimension theory: the geometrical investigations of
Bernard Bolzano".
3059:
2602:
1962:
1917:
1818:
1566:
703:
528:
2147:
Nguyen, Phuong; Cook, Stephen A. (2007). "The
Complexity of Proving the Discrete Jordan Curve Theorem".
72:
connecting a point of one region to a point of the other intersects with the curve somewhere. While the
2533:
Berg, Gordon O.; Julian, W.; Mines, R.; Richman, Fred (1975), "The constructive Jordan curve theorem",
1494:
1203:
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17:
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1974:
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1811:
706:(in 2 dimensions), and the Brouwer fixed point theorem can be proved from the Hex theorem: "every
679:
598:, which states that the interior and the exterior planar regions determined by a Jordan curve in
222:
2982:
2674:
2625:
2063:
1923:
1458:
1410:
1370:
100:
1409:, but the problem came in generalizing it to all kinds of badly behaved curves, which include
737:, a binary picture is a discrete square grid of 0 and 1, or equivalently, a compact subset of
2575:
2451:
1707:
1638:
1618:
1548:
1081:
1926:, the Jordan curve theorem can be used for testing whether a point lies inside or outside a
1861:
3039:
2915:
2797:
2753:
2659:
2566:
2343:
2308:
2049:
1985:
1977:, a description of certain sets of points in the plane that can be subsets of Jordan curves
1946:
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715:
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8:
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1807:
675:
254:
124:
65:
61:
57:
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1957:
Adler, Daskalakis and Demaine prove that a computational version of Jordan's theorem is
3094:
3027:
2950:
2919:
2871:
2841:
2741:
2714:
Maehara, Ryuji (1984), "The Jordan Curve Theorem Via the Brouwer Fixed Point Theorem",
2663:
2215:
2190:
Hales, Thomas C. (December 2007). "The Jordan Curve Theorem, Formally and Informally".
2152:
2129:
2037:
1612:
305:
82:
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is not simply connected, and hence not homeomorphic to the exterior of the unit ball.
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2903:
2825:
2785:
2733:
2647:
2590:
2580:
2579:. Herbert Robbins ( ed.). United Kingdom: Oxford University Press. p. 267.
2554:
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2432:
2399:(1924). "Bemerkungen zu den Beweisen von C. Jordan und Ch. J. de la Vallée Poussin".
2277:
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1522:
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1502:
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2923:
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2104:
Gale, David (December 1979). "The Game of Hex and the Brouwer Fixed-Point Theorem".
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2487:. Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik: 24:1–24:14.
2331:
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278:
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2481:
43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
2219:
3035:
2911:
2867:
2852:
Ross, Fiona; Ross, William T. (2011), "The Jordan curve theorem is non-trivial",
2793:
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1402:
359:
314:
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69:
162:
81:, Introduction)). More transparent proofs rely on the mathematical machinery of
2961:
2697:
2235:"First Principles of Computer Vision: Segmenting Binary Images | Binary Images"
1934:
1927:
1877:
1433:
1414:
548:
273:
The Jordan curve theorem was independently generalized to higher dimensions by
134:
92:
47:
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2255:
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2737:
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1980:
1958:
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does not converge to zero, using only the given Jordan curve, not the region
1725:
1510:
1498:
1445:
1437:
1418:
564:
96:
89:
2933:(1992), "The Jordan–Schönflies theorem and the classification of surfaces",
2780:
690:, but so twisted in space that the unbounded component of its complement in
3074:
2996:
2594:
1799:
1787:
1577:
1422:
603:
203:
With these definitions, the Jordan curve theorem can be stated as follows:
52:
1482:
had already been critically analyzed and completed by Schoenflies (1924).
68:" region containing all of the nearby and far away exterior points. Every
2166:
1887:
647:
551:), and with a bit of extra work, one shows that their common boundary is
274:
238:
158:
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3031:
2954:
2845:
2745:
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What is mathematics? : an elementary approach to ideas and methods
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30:
106:
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2117:
2073:. From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
2033:
1367:
The theorem states that: suppose you put bombs on some squares on a
2149:
22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
1608:
325:
39:
3099:
2157:
795:, such as number of components, might fail to be well-defined for
324:> 0), i.e. the image of an injective continuous mapping of the
85:, and these lead to generalizations to higher-dimensional spaces.
3006:(1905), "Theory on Plane Curves in Non-Metrical Analysis Situs",
2239:
1406:
892:
594:
There is a strengthening of the Jordan curve theorem, called the
196:
is a continuous loop, whereas the last condition stipulates that
73:
1635:
of the largest disk contained in the closed polygon. Evidently,
687:
591:) without boundary, its complement has 2 connected components.
172:
Alternatively, a Jordan curve is the image of a continuous map
138:
2804:
Osgood, William F. (1903), "A Jordan Curve of Positive Area",
2628:(2007a), "The Jordan curve theorem, formally and informally",
2452:"PNPOLY - Point Inclusion in Polygon Test - WR Franklin (WRF)"
1961:. As a corollary, they show that Jordan's theorem implies the
2483:. Leibniz International Proceedings in Informatics (LIPIcs).
1886:), the number of intersections of the ray and the polygon is
666:
in higher dimensions: while the exterior of the unit ball in
2573:
Courant, Richard (1978). "V. Topology". Written at Oxford.
506:
3071:
The full 6,500 line formal proof of Jordan's curve theorem
2020:
Kline, J. R. (1942). "What is the Jordan curve theorem?".
618:
on the Jordan curve, there exists a Jordan arc connecting
1294:, such as the Jordan curve theorem, do not generalize to
192:
to [0,1) is injective. The first two conditions say that
56:(a plane simple closed curve) divides the plane into an "
1326:
If the "6-neighbor square grid" structure is imposed on
2440:. Edinburg: University of Edinburgh. 1978. p. 267.
3092:
2532:
2474:
1559:
2085:
1821:
1738:
1710:
1661:
1641:
1621:
1485:
Due to the importance of the Jordan curve theorem in
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1300:
1271:
1206:
1126:
1084:
1052:
939:
907:
866:
830:
801:
772:
743:
395:
2477:"The Complexity of Hex and the Jordan Curve Theorem"
2003:
2001:
1988:, a mathematical group that preserves a Jordan curve
3052:
2511:
1478:Earlier, Jordan's proof and another early proof by
362:. It is first established that, more generally, if
107:
Definitions and the statement of the Jordan theorem
1844:
1770:
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1693:
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881:
845:
816:
787:
758:
728:
512:
3009:Transactions of the American Mathematical Society
2807:Transactions of the American Mathematical Society
2762:"A nonstandard proof of the Jordan curve theorem"
1998:
1728:bounded by the Jordan curve. However, we have to
650:in the plane, can be extended to a homeomorphism
575:and the reduced cohomology of its complement. If
547:has 2 connected components (which are, moreover,
27:A closed curve divides the plane into two regions
3109:
3081:Collection of proofs of the Jordan curve theorem
2885:
2256:"A digital analogue of the Jordan curve theorem"
1880:(in red) lies outside a simple polygon (region
1803:
1046:the "8-neighbor square grid", where each vertex
901:the "4-neighbor square grid", where each vertex
702:The Jordan curve theorem can be proved from the
1448:, who said the following about Jordan's proof:
1358:
1036:{\displaystyle (x+1,y),(x-1,y),(x,y+1),(x,y-1)}
2395:
1771:{\displaystyle \delta _{1},\delta _{2},\dots }
1704:converging to a positive number, the diameter
1694:{\displaystyle \delta _{1},\delta _{2},\dots }
583:-dimensional compact connected submanifold of
99:. However, this notion has been overturned by
1918:Point in polygon § Ray casting algorithm
1432:The first proof of this theorem was given by
1396:
2679:"Jordan's proof of the Jordan Curve theorem"
2068:"Jordan's proof of the Jordan curve theorem"
1615:, the closed polygon. Consider the diameter
487:
481:
268:
88:The Jordan curve theorem is named after the
2974:Bulletin of the London Mathematical Society
1903:) of a ray lies inside the polygon (region
1790:of the Jordan curve theorem was created by
1598:The root of the difficulty is explained in
2146:
1810:the Jordan curve theorem is equivalent to
860:There are two obvious graph structures on
3098:
3021:
2986:
2929:
2819:
2779:
2607:"An elementary proof of Jordan's theorem"
2548:
2492:
2297:"A discrete form of Jordan curve theorem"
2271:
2156:
1591:
1335:
1303:
1274:
1193:{\displaystyle |x-x'|\leq 1,|y-y'|\leq 1}
869:
833:
804:
775:
746:
436:
2960:
2851:
2601:
1952:
1860:
1599:
1541:
1537:
1442:Cours d'analyse de l'École Polytechnique
891:
626:and, with the exception of the endpoint
555:. A further generalization was found by
261:In contrast, the complement of a Jordan
78:
29:
2713:
2572:
2517:
2418:
2321:
2294:
2091:
1570:
896:8-neighbor and 4-neighbor square grids.
853:does not have an appropriately defined
14:
3110:
3087:A simple proof of Jordan curve theorem
3002:
2803:
2759:
2696:
2686:Studies in Logic, Grammar and Rhetoric
2673:
2624:
2384:
2373:
2361:
2253:
2075:Studies in Logic, Grammar and Rhetoric
2007:
1831:
1828:
1825:
1791:
1607:, the boundary of a bounded connected
1552:
1426:
2966:"A proof of the Jordan curve theorem"
2536:Rocky Mountain Journal of Mathematics
2324:Archive for History of Exact Sciences
2232:
2189:
2062:
2019:
1724:of the largest disk contained in the
2103:
1611:, call it the open polygon, and its
1536:Elementary proofs were presented by
606:to the interior and exterior of the
2855:Journal of Mathematics and the Arts
1845:{\displaystyle {\mathsf {RCA}}_{0}}
697:
614:in the interior region and a point
24:
1909:), the number of intersections is
563:between the reduced homology of a
245:) and the other is unbounded (the
25:
3134:
3046:
2717:The American Mathematical Monthly
2631:The American Mathematical Monthly
2192:The American Mathematical Monthly
2106:The American Mathematical Monthly
1480:Charles Jean de la Vallée Poussin
1253:{\displaystyle (x,y)\neq (x',y')}
539:, the zeroth reduced homology of
283:Jordan–Brouwer separation theorem
200:has no self-intersection points.
2401:Jahresber. Deutsch. Math.-Verein
1440:, and was published in his book
1348:{\displaystyle \mathbb {Z} ^{2}}
1316:{\displaystyle \mathbb {Z} ^{2}}
1287:{\displaystyle \mathbb {R} ^{2}}
882:{\displaystyle \mathbb {Z} ^{2}}
846:{\displaystyle \mathbb {Z} ^{2}}
817:{\displaystyle \mathbb {Z} ^{2}}
788:{\displaystyle \mathbb {R} ^{2}}
759:{\displaystyle \mathbb {Z} ^{2}}
2468:
2444:
2424:
2411:
2389:
2378:
2367:
2351:
2315:
2301:Annales Mathematicae Silesianae
2288:
1423:a Jordan curve of positive area
729:Application to image processing
610:. In particular, for any point
523:This is proved by induction in
217:be a Jordan curve in the plane
2888:Archive for Mathematical Logic
2767:Pacific Journal of Mathematics
2644:10.1080/00029890.2007.11920481
2247:
2226:
2204:10.1080/00029890.2007.11920481
2183:
2140:
2097:
2081:(23). University of Białystok.
2056:
2013:
1856:
1323:under either graph structure.
1247:
1225:
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1207:
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908:
421:
415:
403:
13:
1:
3053:M.I. Voitsekhovskii (2001) ,
2935:American Mathematical Monthly
2526:
2022:American Mathematical Monthly
543:has rank 1, which means that
237:. One of these components is
3083:at Andrew Ranicki's homepage
2868:10.1080/17513472.2011.634320
2494:10.4230/LIPIcs.ICALP.2016.24
2260:Discrete Applied Mathematics
2233:Nayar, Shree (Mar 1, 2021).
1933:From a given point, trace a
1359:Steinhaus chessboard theorem
766:. Topological invariants on
50:in 1887, asserts that every
7:
3060:Encyclopedia of Mathematics
1968:
1963:Brouwer fixed-point theorem
1567:Brouwer fixed point theorem
704:Brouwer fixed point theorem
265:in the plane is connected.
188:(1) and the restriction of
10:
3139:
2417:Richard Courant (
2295:SurĂłwka, Wojciech (1993).
2151:. IEEE. pp. 245–256.
1915:
1397:History and further proofs
678:onto the unit sphere, the
354:is their common boundary.
281:in 1911, resulting in the
279:L. E. J. Brouwer
233:, consists of exactly two
2900:10.1007/s00153-007-0050-6
2273:10.1016/j.dam.2002.11.003
1527:Arthur Moritz Schoenflies
1386:{\displaystyle n\times n}
596:Jordan–Schönflies theorem
372:reduced integral homology
269:Proof and generalizations
2550:10.1216/RMJ-1975-5-2-225
2254:Ĺ lapal, J (April 2004).
1992:
1582:complete bipartite graph
1487:low-dimensional topology
161:that is not necessarily
3089:(PDF) by David B. Gauld
2781:10.2140/pjm.1971.36.219
2434:1. Jordan curve theorem
1717:{\displaystyle \delta }
1648:{\displaystyle \delta }
1628:{\displaystyle \delta }
1113:{\displaystyle (x',y')}
680:Alexander horned sphere
529:Mayer–Vietoris sequence
366:is homeomorphic to the
338:. Then the complement
2760:Narens, Louis (1971),
1924:computational geometry
1913:
1892:If the initial point (
1865:If the initial point (
1846:
1772:
1718:
1695:
1649:
1629:
1476:
1468:
1455:
1411:nowhere differentiable
1387:
1349:
1317:
1288:
1254:
1194:
1114:
1072:
1037:
927:
897:
883:
847:
818:
789:
760:
559:, who established the
514:
35:
3123:Theorems about curves
1953:Computational aspects
1864:
1847:
1773:
1719:
1696:
1650:
1630:
1549:non-standard analysis
1472:
1463:
1450:
1388:
1350:
1318:
1289:
1255:
1195:
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1073:
1071:{\displaystyle (x,y)}
1038:
928:
926:{\displaystyle (x,y)}
895:
884:
848:
819:
790:
761:
515:
64:by the curve and an "
33:
3118:Theorems in topology
2997:10.1112/blms/12.1.34
2167:10.1109/lics.2007.48
1986:Quasi-Fuchsian group
1975:Denjoy–Riesz theorem
1819:
1736:
1708:
1659:
1639:
1619:
1413:curves, such as the
1371:
1330:
1298:
1269:
1204:
1124:
1082:
1050:
937:
905:
864:
828:
799:
770:
741:
393:
235:connected components
44:Jordan curve theorem
1808:reverse mathematics
1436:in his lectures on
293: —
257:of each component.
211: —
117:simple closed curve
2931:Thomassen, Carsten
2709:, pp. 587–594
2336:10.1007/BF00499625
1914:
1842:
1812:weak KĹ‘nig's lemma
1768:
1732:that the sequence
1714:
1691:
1645:
1625:
1565:A proof using the
1383:
1345:
1313:
1284:
1250:
1190:
1110:
1078:is connected with
1068:
1033:
933:is connected with
923:
898:
879:
843:
814:
785:
756:
686:homeomorphic to a
510:
505:
370:-sphere, then the
306:topological sphere
291:
209:
83:algebraic topology
36:
2614:Uspekhi Mat. Nauk
2586:978-0-19-502517-0
2504:978-3-95977-013-2
2358:Oswald Veblen
2176:978-0-7695-2908-0
1806:) showed that in
1523:Alfred Pringsheim
1515:Friedrich Hartogs
1503:Ludwig Bieberbach
646:is viewed as the
561:Alexander duality
498:
463:
406:
289:
249:), and the curve
207:
16:(Redirected from
3130:
3104:
3102:
3067:
3055:"Jordan theorem"
3042:
3025:
2999:
2990:
2970:
2957:
2926:
2878:
2848:
2823:
2800:
2783:
2756:
2710:
2708:
2693:
2683:
2670:
2626:Hales, Thomas C.
2621:
2611:
2598:
2569:
2552:
2521:
2515:
2509:
2508:
2496:
2472:
2466:
2465:
2463:
2462:
2456:wrf.ecse.rpi.edu
2448:
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2415:
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2393:
2387:
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2365:
2355:
2349:
2347:
2319:
2313:
2312:
2292:
2286:
2285:
2275:
2266:(1–3): 231–251.
2251:
2245:
2244:
2230:
2224:
2223:
2187:
2181:
2180:
2160:
2144:
2138:
2137:
2101:
2095:
2089:
2083:
2082:
2072:
2064:Hales, Thomas C.
2060:
2054:
2053:
2017:
2011:
2005:
1944:
1940:
1908:
1907:
1902:
1901:
1885:
1884:
1875:
1874:
1851:
1849:
1848:
1843:
1841:
1840:
1835:
1834:
1814:over the system
1777:
1775:
1774:
1769:
1761:
1760:
1748:
1747:
1723:
1721:
1720:
1715:
1700:
1698:
1697:
1692:
1684:
1683:
1671:
1670:
1654:
1652:
1651:
1646:
1634:
1632:
1631:
1626:
1592:Thomassen (1992)
1491:complex analysis
1392:
1390:
1389:
1384:
1354:
1352:
1351:
1346:
1344:
1343:
1338:
1322:
1320:
1319:
1314:
1312:
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1306:
1293:
1291:
1290:
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1283:
1282:
1277:
1259:
1257:
1256:
1251:
1246:
1235:
1199:
1197:
1196:
1191:
1183:
1178:
1164:
1150:
1145:
1131:
1119:
1117:
1116:
1111:
1106:
1095:
1077:
1075:
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1069:
1042:
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1034:
932:
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929:
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888:
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880:
878:
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852:
850:
849:
844:
842:
841:
836:
823:
821:
820:
815:
813:
812:
807:
794:
792:
791:
786:
784:
783:
778:
765:
763:
762:
757:
755:
754:
749:
735:image processing
698:Discrete version
672:simply connected
519:
517:
516:
511:
509:
508:
499:
496:
464:
461:
439:
414:
413:
408:
407:
399:
386:are as follows:
313:+1)-dimensional
294:
212:
141:into the plane,
46:, formulated by
21:
3138:
3137:
3133:
3132:
3131:
3129:
3128:
3127:
3108:
3107:
3049:
3023:10.2307/1986378
2988:10.1.1.374.2903
2968:
2962:Tverberg, Helge
2947:10.2307/2324180
2821:10.2307/1986455
2730:10.2307/2323369
2724:(10): 641–643,
2706:
2703:Cours d'analyse
2698:Jordan, Camille
2681:
2638:(10): 882–894,
2609:
2603:Filippov, A. F.
2587:
2529:
2524:
2516:
2512:
2505:
2473:
2469:
2460:
2458:
2450:
2449:
2445:
2437:
2431:"V. Topology".
2430:
2429:
2425:
2416:
2412:
2394:
2390:
2383:
2379:
2372:
2368:
2356:
2352:
2320:
2316:
2293:
2289:
2252:
2248:
2231:
2227:
2198:(10): 882–894.
2188:
2184:
2177:
2145:
2141:
2118:10.2307/2320146
2112:(10): 818–827.
2102:
2098:
2090:
2086:
2070:
2061:
2057:
2034:10.2307/2303093
2018:
2014:
2006:
1999:
1995:
1971:
1955:
1942:
1938:
1920:
1905:
1904:
1899:
1894:
1893:
1891:
1882:
1881:
1872:
1867:
1866:
1859:
1836:
1824:
1823:
1822:
1820:
1817:
1816:
1756:
1752:
1743:
1739:
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1734:
1733:
1709:
1706:
1705:
1679:
1675:
1666:
1662:
1660:
1657:
1656:
1640:
1637:
1636:
1620:
1617:
1616:
1605:polygonal chain
1600:Tverberg (1980)
1589:
1542:Tverberg (1980)
1538:Filippov (1950)
1519:Béla Kerékjártó
1507:Luitzen Brouwer
1495:J. W. Alexander
1459:Thomas C. Hales
1425:constructed by
1403:Bernard Bolzano
1399:
1372:
1369:
1368:
1361:
1339:
1334:
1333:
1331:
1328:
1327:
1307:
1302:
1301:
1299:
1296:
1295:
1278:
1273:
1272:
1270:
1267:
1266:
1239:
1228:
1205:
1202:
1201:
1179:
1171:
1160:
1146:
1138:
1127:
1125:
1122:
1121:
1099:
1088:
1083:
1080:
1079:
1051:
1048:
1047:
938:
935:
934:
906:
903:
902:
873:
868:
867:
865:
862:
861:
855:graph structure
837:
832:
831:
829:
826:
825:
808:
803:
802:
800:
797:
796:
779:
774:
773:
771:
768:
767:
750:
745:
744:
742:
739:
738:
731:
700:
682:is a subset of
557:J. W. Alexander
504:
503:
495:
493:
478:
477:
460:
443:
435:
428:
427:
409:
398:
397:
396:
394:
391:
390:
360:homology theory
358:The proof uses
356:
315:Euclidean space
292:
271:
259:
210:
109:
101:Thomas C. Hales
70:continuous path
28:
23:
22:
15:
12:
11:
5:
3136:
3126:
3125:
3120:
3106:
3105:
3090:
3084:
3078:
3068:
3048:
3047:External links
3045:
3044:
3043:
3004:Veblen, Oswald
3000:
2958:
2941:(2): 116–130,
2927:
2894:(5): 465–480,
2883:
2862:(4): 213–219,
2849:
2814:(1): 107–112,
2801:
2757:
2711:
2694:
2671:
2622:
2616:(in Russian),
2599:
2585:
2570:
2543:(2): 225–236,
2528:
2525:
2523:
2522:
2518:Maehara (1984)
2510:
2503:
2467:
2443:
2423:
2410:
2397:A. Schoenflies
2388:
2377:
2366:
2350:
2330:(3): 262–295.
2314:
2287:
2246:
2225:
2182:
2175:
2139:
2096:
2094:, p. 641.
2092:Maehara (1984)
2084:
2055:
2028:(5): 281–286.
2012:
1996:
1994:
1991:
1990:
1989:
1983:
1978:
1970:
1967:
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1951:
1928:simple polygon
1916:Main article:
1897:
1870:
1858:
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1833:
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1687:
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1674:
1669:
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1624:
1596:
1595:
1587:
1576:A proof using
1574:
1571:Maehara (1984)
1563:
1556:
1547:A proof using
1545:
1434:Camille Jordan
1419:fractal curves
1415:Koch snowflake
1398:
1395:
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549:path connected
521:
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287:
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135:continuous map
108:
105:
93:Camille Jordan
79:Tverberg (1980
48:Camille Jordan
26:
9:
6:
4:
3:
2:
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2882:
2881:author's site
2877:
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2675:Hales, Thomas
2672:
2669:
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2653:
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2385:Hales (2007b)
2381:
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2374:Hales (2007b)
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2027:
2023:
2016:
2009:
2008:Jordan (1887)
2004:
2002:
1997:
1987:
1984:
1982:
1981:Lakes of Wada
1979:
1976:
1973:
1972:
1966:
1964:
1960:
1959:PPAD-complete
1950:
1948:
1936:
1931:
1929:
1925:
1919:
1912:
1900:
1889:
1879:
1873:
1863:
1854:
1852:
1837:
1813:
1809:
1805:
1801:
1797:
1793:
1792:Hales (2007a)
1789:
1784:
1781:
1765:
1762:
1757:
1753:
1749:
1744:
1740:
1731:
1727:
1726:closed region
1711:
1703:
1688:
1685:
1680:
1676:
1672:
1667:
1663:
1642:
1622:
1614:
1610:
1606:
1601:
1593:
1590:was given by
1586:
1583:
1579:
1578:non-planarity
1575:
1572:
1568:
1564:
1561:
1557:
1554:
1553:Narens (1971)
1550:
1546:
1543:
1539:
1535:
1534:
1533:
1530:
1528:
1524:
1520:
1516:
1512:
1511:Arnaud Denjoy
1508:
1504:
1500:
1499:Louis Antoine
1496:
1492:
1488:
1483:
1481:
1475:
1471:
1467:
1462:
1460:
1454:
1449:
1447:
1446:Oswald Veblen
1443:
1439:
1438:real analysis
1435:
1430:
1428:
1427:Osgood (1903)
1424:
1420:
1416:
1412:
1408:
1404:
1394:
1380:
1377:
1374:
1365:
1356:
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1308:
1279:
1243:
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1222:
1216:
1213:
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1187:
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1132:
1103:
1100:
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1021:
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914:
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751:
736:
726:
722:
719:
717:
711:
709:
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695:
693:
689:
685:
681:
677:
674:, because it
673:
669:
665:
661:
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641:
637:
633:
629:
625:
621:
617:
613:
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586:
582:
578:
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570:
566:
562:
558:
554:
550:
546:
542:
538:
534:
530:
526:
500:
490:
484:
474:
471:
468:
465:
457:
454:
451:
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445:
440:
429:
424:
418:
410:
400:
389:
388:
387:
385:
381:
377:
373:
369:
365:
361:
355:
353:
349:
345:
341:
337:
333:
330:
328:
323:
319:
316:
312:
308:
307:
303:-dimensional
302:
298:
286:
284:
280:
276:
266:
264:
258:
256:
252:
248:
244:
240:
236:
232:
229: \
228:
224:
220:
216:
204:
201:
199:
195:
191:
187:
183:
179:
175:
170:
168:
164:
160:
156:
152:
148:
144:
140:
136:
133:
129:
126:
122:
119:in the plane
118:
114:
104:
102:
98:
97:Oswald Veblen
94:
91:
90:mathematician
86:
84:
80:
75:
71:
67:
63:
59:
55:
54:
49:
45:
41:
32:
19:
3058:
3016:(1): 83–98,
3013:
3007:
2981:(1): 34–38,
2978:
2972:
2938:
2934:
2891:
2887:
2859:
2853:
2811:
2805:
2771:
2765:
2721:
2715:
2702:
2689:
2685:
2635:
2629:
2620:(5): 173–176
2617:
2613:
2574:
2540:
2534:
2513:
2484:
2480:
2470:
2459:. Retrieved
2455:
2446:
2433:
2426:
2413:
2404:
2400:
2391:
2380:
2369:
2353:
2327:
2323:
2317:
2303:(7): 57–61.
2300:
2290:
2263:
2259:
2249:
2238:
2228:
2195:
2191:
2185:
2148:
2142:
2109:
2105:
2099:
2087:
2078:
2074:
2058:
2025:
2021:
2015:
1956:
1932:
1921:
1895:
1868:
1800:Mizar system
1788:formal proof
1785:
1779:
1729:
1701:
1597:
1584:
1531:
1484:
1477:
1473:
1469:
1464:
1456:
1451:
1441:
1431:
1400:
1366:
1362:
1325:
1263:
859:
732:
723:
720:
712:
701:
691:
683:
667:
663:
659:
655:
651:
643:
639:
635:
631:
627:
623:
619:
615:
611:
604:homeomorphic
599:
593:
588:
584:
580:
576:
572:
568:
552:
544:
540:
536:
532:
524:
522:
383:
379:
375:
367:
363:
357:
351:
347:
343:
339:
335:
331:
326:
321:
317:
310:
304:
300:
296:
288:
282:
272:
262:
260:
250:
246:
242:
230:
226:
218:
214:
206:
202:
197:
193:
189:
185:
181:
177:
173:
171:
154:
150:
146:
142:
127:
120:
116:
113:Jordan curve
112:
110:
103:and others.
87:
53:Jordan curve
51:
43:
37:
2774:: 219–229,
2348:See p. 285.
1857:Application
708:game of Hex
648:unit circle
275:H. Lebesgue
221:. Then its
159:plane curve
3112:Categories
2838:34.0533.02
2527:References
2461:2021-07-18
2407:: 157–160.
1786:The first
1780:presumably
1702:presumably
1421:, or even
1417:and other
527:using the
374:groups of
223:complement
180:such that
155:Jordan arc
18:Jordan arc
3100:1404.0556
3065:EMS Press
2983:CiteSeerX
2908:0933-5846
2830:0002-9947
2790:0030-8730
2738:0002-9890
2677:(2007b),
2652:0002-9890
2559:0035-7596
2282:0166-218X
2212:0002-9890
2158:1002.2954
2126:0002-9890
1796:HOL Light
1766:…
1754:δ
1741:δ
1712:δ
1689:…
1677:δ
1664:δ
1643:δ
1623:δ
1457:However,
1378:×
1223:≠
1185:≤
1169:−
1152:≤
1136:−
1025:−
971:−
718:theorem.
608:unit disk
497:otherwise
455:−
404:~
167:algebraic
132:injective
60:" region
2964:(1980),
2924:33627222
2700:(1887),
2605:(1950),
2066:(2007).
1969:See also
1609:open set
1407:polygons
1244:′
1233:′
1176:′
1143:′
1104:′
1093:′
716:discrete
676:retracts
642:, where
309:in the (
255:boundary
247:exterior
243:interior
66:exterior
58:interior
40:topology
3040:1500697
3032:1986378
2955:2324180
2916:2321588
2876:3257011
2846:1986455
2798:0276940
2754:0769530
2746:2323369
2660:2363054
2595:6450129
2567:0410701
2360: (
2344:0446838
2309:1271184
2240:YouTube
2134:2320146
2050:0006516
2042:2303093
1876:) of a
1794:in the
1613:closure
1580:of the
1461:wrote:
567:subset
565:compact
531:. When
329:-sphere
290:Theorem
253:is the
239:bounded
208:Theorem
123:is the
74:theorem
62:bounded
3038:
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2906:
2874:
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2218:
2210:
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2048:
2040:
1525:, and
1453:given.
1200:, and
688:sphere
652:ψ
632:φ
579:is an
299:be an
184:(0) =
163:smooth
139:circle
130:of an
42:, the
3095:arXiv
3075:Mizar
3028:JSTOR
2969:(PDF)
2951:JSTOR
2920:S2CID
2872:S2CID
2842:JSTOR
2742:JSTOR
2707:(PDF)
2682:(PDF)
2664:S2CID
2610:(PDF)
2438:(PDF)
2216:S2CID
2153:arXiv
2130:JSTOR
2071:(PDF)
2038:JSTOR
1993:Notes
1730:prove
664:false
622:with
334:into
241:(the
176:: →
137:of a
125:image
115:or a
2904:ISSN
2826:ISSN
2786:ISSN
2734:ISSN
2692:(23)
2648:ISSN
2591:OCLC
2581:ISBN
2555:ISSN
2499:ISBN
2419:1978
2362:1905
2278:ISSN
2208:ISSN
2171:ISBN
2122:ISSN
1911:odd.
1888:even
1804:2007
1560:1975
1540:and
1489:and
1120:iff
602:are
587:(or
295:Let
277:and
213:Let
165:nor
153:. A
3073:in
3018:doi
2993:doi
2943:doi
2896:doi
2864:doi
2834:JFM
2816:doi
2776:doi
2726:doi
2640:doi
2636:114
2545:doi
2489:doi
2332:doi
2268:doi
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2200:doi
2196:114
2163:doi
2114:doi
2030:doi
1947:odd
1945:is
1935:ray
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1878:ray
1588:3,3
1569:by
1551:by
889::
824:if
733:In
670:is
571:of
346:in
342:of
263:arc
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3057:,
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3012:,
2991:,
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2024:.
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1949:.
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1943:n
1939:n
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