A Proof of the Jordan Curve Theorem

Xing Zhang

doi:10.4236/ns.2019.1112037

Abstract

The proof of the Jordan Curve Theorem (JCT) in this paper is focused on a graphic illustration and analysis ways so as to make the topological proof more understandable, and is based on the Tverberg’s method, which is acknowledged as being quite esoteric with no graphic explanations. The preliminary constructs a parametrisation model for Jordan Polygons. It takes quite a length to introduce four lemmas since the proof by Jordan Polygon is the approach we want to concern about. Lemmas show that JCT holds for Jordan polygon and Jordan curve could be approximated uniformly by a sequence of Jordan polygons. Also, lemmas provide a certain metric description of Jordan polygons to help evaluate the limit. The final part is the proof of the theorem on the premise of introduced preliminary and lemmas.

Highlights

INTRODUCTIONThough the definition of the Jordan Curve Theorem is not hermetic at all, the proof of the theorem is quite formidable and has experienced ups and downs throughout history
Though the definition of the Jordan Curve Theorem is not hermetic at all, the proof of the theorem is quite formidable and has experienced ups and downs throughout history.Bernard Bolzano was the first person who formulated a precise conjecture: it was not self-evident but required a hard proof
The Jordan Curve Theorem was named after Camille Jordan, a mathematician who came up with the first proof in his lectures on real analysis and published his findings in his book [1], yet critics doubted the completeness of his proof

Summary

INTRODUCTION

Though the definition of the Jordan Curve Theorem is not hermetic at all, the proof of the theorem is quite formidable and has experienced ups and downs throughout history. Veblen commented that “His (Jordon’s) proof, 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 given” [2]. He quoted Michael Reeken as saying “Jordan’s proof is essentially correct. The Jordan Curve Theorem had been successfully proved in the 20th century, mathematicians still sought more formal ways to prove it in the 21st century. Lemma 3 and Lemma 4 deal with the situation in limiting processes to prevent the cases from the polygons that may thin to zero somewhere

PRELIMINARIES

LEMMAS

PROOF OF JORDAN’S THEOREM

CONCLUSION