Introduction and Simply Connected Regions

Abstract

AbstractWe work with a metric space which we understand as a complex plane, unless otherwise specified. The letter \(\Omega \) will denote an open set in the metric space. We introduce the notion of connectedness in Sect. 1.2, and we show in Theorem 1.1 that an open set in a metric space is a disjoint union of open connected sets which we call regions. Further, we prove Theorem 1.3 in Sect. 1.3 that the extended complex plane is a metric space homeomorphic to the Riemann sphere. We introduce curves and paths, complex integral over paths, index of a point with respect to a closed path, homotopic paths, simply connected regions and give their basic properties in Sects. 1.4 and 1.5. Then we prove in Theorem 1.6 that the index of a point with respect to two \(\Omega \)-homotopic closed paths in \(\Omega \) are equal whenever the point lies outside \(\Omega \) and we apply it to prove in Theorem 1.9 that the complement of a simply connected region in the extended complex plane is connected. We shall prove in Sect. 3.5 the Riemann mapping theorem which implies the converse of the above statement, i.e a region is simply connected if its complement in the extended complex plane is connected. Hence a region is simply connected if and only if its complement in the extended complex plane is connected. Thus a region is simply connected if and only if it is without holes. This is a very transparent criterion to determine whether a region is simply connected or not. For example, it implies that the unbounded strip \(\{z\mid a<\text {Re} (z)< b\}\) for given real numbers a and b is simply connected.