Complex Analysis Volume 2

Author: J.B. Conway
Publisher:
Publish Date: 1997-09-01
Features:
Fragment: Chapter 13 Return to Basics In this chapter a few results of somewhat elementary nature are collected. These will be used quite often in the remainder of this volume.
Regions and Curves In this first section a few definitions and facts about regions and curves in the plane are given. Some of these may be familiar to the reader. Indeed some will be recollections from the first volume. Begin by recalling that a region is an open connected set and a simply connected region is one for which every closed curve is contractible to a point (see 4.6.14). In Theorem 8.2.2 numerous statements equivalent to simple connectedness were given. We begin by recalling one of these equivalent statements and giving another. Do not forget that Coodenotes the extended complex nunibers and Gdenote S the boundary of the set G in . That is, when G is bounded and when G is unbounded. It is often convenient to give results about subsets of the extended plane rather than about C. If something was proved in the first volume for a subset of C, but it holds for subsets of C with little change in the proof, we will not hesitate to quote the appropriate reference from the first twelve chapters as though the result for C was proved there.
1.1 Proposition. If G is a region in C, the following statements are equivalent.
(a) G is simply connected.
(b) C\G is connected
(c) G is connected.
Proof. The equivalence of (a) and (b) has already been established in (8.2.2). In fact, the equivalence of (a) and (b) was established without assuming that G is connected. That is, it was only assumed that G was a simply connected open set; an open set with every component simply connected. The reader must also pay attention to the fact that the connectedness of G will not be used when it is shown that (c) implies (b). This will be used when it is shown that (b) implies (c).