《A proof of the Jordan curve theorem》.pdfVIP

A PROOF OF THE JORDAN CURVE THEOREM HELGE TVERBERG 1. Introduction Let F be a Jordan curve in the plane, i.e. the image of the unit circle 2 2 2 C = {(x,y);x + y = 1} under an injective continuous mapping y into R . The Jordan curve theorem [1] says that / ? 2 \ F is disconnected and consists of two components. (We shall use the original definition whereby two points are in the same component if and only if they can be joined by a continuous path (image of [0,1]).) Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it. The present paper is intended to provide a reasonably short and selfcontained proof or at least, failing that , to point at the need for one . 2. Prerequisites and lemmata Some elementary concepts and facts from analysis are needed, for instance uniform continuity. One must know that F is compact, and so is any continuous path . Also, if A and B are disjoint compact sets, inf {|a-b| ; a e A, b e B], to be denoted by d{A, B), is l 0 . Sometimes it is useful to keep in mind that y~ is continuous . It would have been possible to avoid the use of these results, at the cost of an extra page, by replacing their applications by arguments ad hoc . The deepest result needed would then be Weierstrass theorem to the effect that any bounded sequence of real numbe