Jordan's set of curves

The Jordanian curve set is a result in the mathematical sub-area of topology .

statement

closed Jordan curve

Every closed Jordan curve in the Euclidean plane divides this into two disjoint regions , the common edge of which is the Jordan curve and the union of which with the Jordan curve is the whole plane. Exactly one of the two areas is restricted .

history

This theorem seems so obvious that generations of mathematicians have used it without explicitly formulating it, much less proving it. However, the proof is extremely difficult and time-consuming. A first - still incorrect - attempt at proof was published in 1887 by Camille Jordan in the third volume of his work Cours d'Analyse de l'Ecole Polytechnique . The first correct proof of the Jordanian curve theorem was provided in 1905 by Oswald Veblen . The Jordanian set of curves is used today in geographic information systems in the point-in-polygon test according to Jordan .

generalization

Jordan-Brouwer decomposition theorem

The Jordan set of curves was generalized by Luitzen Brouwer to the so-called Jordan-Brouwer decomposition theorem. This theorem states that the complement of a compact coherent -dimensional submanifold of exactly two connected components has. One of the two has the property that its closure forms a compact bounded manifold , the boundary of which is precisely the named submanifold. The proof of this theorem is mostly done with the degree of mapping or with the help of the algebraic topology . ${\ displaystyle (n-1)}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

Schoenflies' theorem

Another generalization is Schoenflies' theorem, according to which every homeomorphism between the unit circle and a Jordan curve in the plane can be continued on the whole plane. However, the generalization to higher dimensions does not apply here.

literature

MC Jordan: Cours d'Analyse de l'Ecole Polytechnique , Volume 3, Paris (1887). The passage on the Jordanian curve set is also available as a PDF document .
Oswald Veblen: Theory on plane curves in non-metrical analysis situs. In: Transactions of the American Mathematical Society , Volume 6 (1905), pp. 83-98.

Web links

Jordanscher Kurvsatz - Explanatory video by Edmund Weitz on YouTube

Individual evidence

↑ That is the predominant view of mathematicians and mathematicians (in the series of Veblen), e.g. B. Morris Kline . But it was questioned by Thomas C. Hales . In particular, he considers one of the main points of criticism, the lack of evidence for polygons in Jordan, to be unsound, since this part is relatively simple. Hales The Jordan curve theorem, formally and informally , The American Mathematical Monthly, Volume 114, 2007, pp. 882-894, Jordan's proof of the Jordan Curve theorem , Studies in Logic, Grammar and Rhetoric, Volume 10, 2007, pdf

[1] That is the predominant view of mathematicians and mathematicians (in the series of Veblen), e.g. B. Morris Kline . But it was questioned by Thomas C. Hales . In particular, he considers one of the main points of criticism, the lack of evidence for polygons in Jordan, to be unsound, since this part is relatively simple. Hales The Jordan curve theorem, formally and informally , The American Mathematical Monthly, Volume 114, 2007, pp. 882-894, Jordan's proof of the Jordan Curve theorem , Studies in Logic, Grammar and Rhetoric, Volume 10, 2007, pdf