Publication: El teorema de Schoenflies
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2018-06
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Abstract
En este trabajo demostraremos el teorema de Schoenflies. Introduciremos en primer lugar el teorema de la curva de Jordan y algunas de sus variantes; este resultado es la base del teorema de Schoenflies. Primero lo probaremos en el caso poligonal, y después construiremos una aproximación de una curva de Jordan por poligonales, que dará paso al final de la prueba del teorema. Como aplicación estudiaremos el teorema de Schoenflies en el plano proyectivo, que ilustra cómo el teorema de la curva de Jordan falla en las superficies compactas distintas de la esfera, y ofrece un criterio para diferenciar curvas de Jordan entre sí.
In this work we will prove the Schoenflies Theorem. We will introduce first the Jordan curve theorem and some variations; this is at the basis of Schoenflies theorem. Then we will prove the Shoenflies theorem in the polygonal case, and produce a polygonal approximation of any Jordan curve, which will give way to the final step of the proof. As an application we will study the theorem in the (real) projective plane, that shows how Jordan-Schoenflies fails in compact surfaces other than the sphere, and provides a criterion to distinguish Jordan curves.
In this work we will prove the Schoenflies Theorem. We will introduce first the Jordan curve theorem and some variations; this is at the basis of Schoenflies theorem. Then we will prove the Shoenflies theorem in the polygonal case, and produce a polygonal approximation of any Jordan curve, which will give way to the final step of the proof. As an application we will study the theorem in the (real) projective plane, that shows how Jordan-Schoenflies fails in compact surfaces other than the sphere, and provides a criterion to distinguish Jordan curves.