Introduction and Simply Connected Regions

Abstract

We 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.