Teorema de Jordan–Schönflies

El teorema de Jordan-Schönflies és un resultat en l'àmbit de la topologia. Rep el seu nom dels matemàtics Camille Jordan, qui l'enuncià, i Arthur Moritz Schönflies, qui el demostrà l'any 1908.[1] És una generalització del teorema de la corba de Jordan.

Enunciat

Sigui una corba simple tancada en un pla que separa en dues regions. Aleshores existeix un homeomorfisme pel qual la imatge de , és un cercle unitari.[2]

Aquest enunciat planteja que una corba simple tancada en el pla euclidià parteix el pla en exactament dues parts: la interior i l'exterior a la corba. Tot i que aquest resultat sembla evident per la intuïció, és fascinantment difícil de demostrar.[3]

Un enunciat equivalent és el següent: Sigui una corba simple tancada en , aleshores l'adherència d'una de les components de és homeomorfa a la 2-bola.[4]

Demostracions

Es coneixen diverses demostracions del teorema de Jordan-Schönflies, entre elles una fent servir el teorema de la representació conforme de Riemann (o teorema del mapeig de Riemann), una del noruec Helge Tverberg que es val només d'aproximacions per polígons[3][5] i una basada en el teorema de Kuratowski sobre grafs planars, basada en que no és planar.[3] Ara bé, una de les demostracions més senzilles és la que es val de la teoria de Morse.[4]

Vegeu també

Referències

  1. ↑ Alcalde Cuesta, Fernando. «Teorema de separación de Jordan-Brouwer» (en castellà). Universidade de Santiago de Compostela. Arxivat de l'original el 2012-06-21. [Consulta: 11 març 2016].
  2. ↑ Cairns, Stewart S. «An elementary proof of the Jordan-Schoenflies Theorem» (en anglès). Proceedings of the American Mathematical Society, 2, 1951, p. 860-867. ISSN: 0002-9939 [Consulta: 11 març 2016].
  3. ↑ 3,0 3,1 3,2 Thomassen, Carsten. «The Jordan-Schönflies Theorem and the Classification of Surfaces» (en anglès). Lyngby, Dinamarca: Insitut de Matemàtiques de la Universitat Tècnica de Dinamarca. [Consulta: 11 març 2016].
  4. ↑ 4,0 4,1 Weisstein, Eric W., Theorem.html «Teorema de Jordan–Schönflies» a MathWorld (en anglès).
  5. ↑ Tveberg, Helge Arnulf «A proof of the Jordan Curve Theorem» (en anglès). Bulletin of the London Mathematical Society, 12, 1980, p. 34-38.

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