Topology in the Complex Plane

Abstract

function theory. (In later editions, he did include a proof of part of this theorem, to illustrate the power of the concept of winding number for closed curves. It is worth mentioning that the notion of winding number is never used in this article.) It turns out that the basic facts about topology in R2 can be explained efficiently by identifying the real plane with the complex plane (also known as the complex line). The key advantages in this approach, as we shall see, come not only from the direct presence of multiplication, but most of all from the availability of the exponential function. The big disadvantage, of course, is that some of the proofs do not transfer to the study of topology in R' for n > 2. It is the purpose of this article to provide an exposition of this method, which was introduced by Eilenberg [4], and in particular to provide a fairly short proof of the Alexander duality theorem in the plane, and thus an easy proof of the Jordan curve theorem. The Alexander duality theorem in the plane is proved in Eilenberg's paper, which however is rather long, including as it does his doctoral thesis. It is also to be found proved by this method in Dieudonne's book [3]. I believe the argument presented here to be simpler. Since first submitting this article to the MONTHLY, I have discovered that the proof given here of Theorem 23, which I had proudly believed to be my discovery, had in fact been found many years ago by Graham Allan, though never published. The reader of this article is assumed to have a basic knowledge of point set topology, to know what a group is, and to have a slight acquaintance with complex numbers, in particular to know that ez = ex(cosy + i sin y) when z = x + iy with x and y real numbers. The brief final section assumes an acquaintance with sheaf theory and Cech cohomology.