An elementary proof of a theorem concerning the division of a region into two

摘要

Intuitively obvious theorems which are hard to prove are nothing new in topology. The most celebrated case is certainly the Jordan curve theorem. For pedagogical reasons elementary proofs of such theorems never become obsolete. During their work with students of mathematics the following problem has forced itself on the authors of the paper: to prove in a reasonably elementary fashion that an open Jordan curve with its endpoints on a closed Jordan curve K, but otherwise located in the bounded part, divides the closure of the bounded part into two parts. In this paper we take the Jordan curve theorem (JCT) for granted and then prove, in a careful, elementary way, the related fact. Unfortunately, it seems that, even given the JCT, there is still a whole lot of work to do. But there are shortcuts. For instance, we do not need to consider the problem of approximating a general curve by polygons, or the delicate limit questions arising when going back from the easy polygon case to the general case.