Kenneth Kuttler

35.9. EXERCISES 719

C along with a point called ∞, a sequence, zn converges to ∞ if and only if θzn converges to(0,0,2) if and only if |zn| converges to ∞ in the usual manner from calculus and a sequence,zn converges to z ∈ C if and only if θ (zn)→ θ (z) . This is interesting because of this lastpart. It gives a meaning for a sequence of complex numbers to converge to something called∞. To do this properly, we should define a metric on Ĉ and word everything in terms ofthis metric. However, it amounts to the same thing as saying what it means for sequences toconverge. Then, with this definition of what it means for a sequence of complex numbersto converge to ∞, the usual definition of connected sets and separated sets is identical withwhat was given earlier.

Definition 35.8.1 Let S ⊆ Ĉ the extended complex plane in which this extra point ∞ hasbeen included as just described. Then S is separated if there exist A,B not both empty suchthat S = A∪B, A∩B = /0 and no point of A is a limit of any sequence of points of B whileno point of B is the limit of any sequence of points of A. If S is not separated, then it iscalled connected.

Example 35.8.2 Consider the open set S≡ {z ∈ C such that Im(z)> 0} . Then S∪{∞} ≡Ŝ is connected in Ĉ .

It is obvious that S is connected in C because it is arcwise connected. Suppose Ŝ= A∪B where these two new sets separate Ŝ in Ĉ . Then one of them, say B must contain∞. Therefore, A is bounded since otherwise there would be a sequence of points of Aconverging to ∞ which is assumed not to happen. Then S = A∪ (B\{∞}) and A,B \ {∞}would separate S unless one is empty. If B \ {∞} = /0, then S would be bounded which isnot the case. Hence A = /0. Thus Ŝ is connected.

Definition 35.8.3 Let S ⊆ C. It is said to be simply connected if the set is connected andC\S∪{∞} is connected in Ĉ . Written more compactly, S is simply connected means S isconnected and also Ĉ \S is connected in Ĉ.

When looking at a set S in C, how do you determine whether it is simply connected?You consider θ

(SC)

in S2 and ask whether it is connected with the convention that if SC isunbounded, you must include (0,0,2) in the image of θ .

Example 35.8.4 Consider the set S ≡ {z ∈ C such that |z|> 1} . This is a connected set,but it is not simply connected because Ĉ \S is not connected. On S2 it consists of a piecenear the bottom of the sphere and the point (0,0,2) at the top.

Example 35.8.5 Consider S≡{z ∈ C such that |z| ≤ 1} . This connected set is simply con-nected because Ĉ \S corresponds to a connected set on S2.

35.9 ExercisesIn the following exercises, the term “simple closed curve” will be used repeatedly. Assumethat such curves Γ have an inside Ui and an outside and that Green’s theorem applies forUi with its boundary Γ if the boundary is oriented appropriately. This can be proved, but isnot in this book. It is one of these things which is mainly of mathematical interest. In theexamples of interest, it is typically not an issue.

35.9. EXERCISES 719C along with a point called oo, a sequence, z, converges to ¢ if and only if 0z, converges to(0,0, 2) if and only if |z,| converges to oo in the usual manner from calculus and a sequence,Zn converges to z € C if and only if 0 (z,) — 0 (z). This is interesting because of this lastpart. It gives a meaning for a sequence of complex numbers to converge to something calledco. To do this properly, we should define a metric on C and word everything in terms ofthis metric. However, it amounts to the same thing as saying what it means for sequences toconverge. Then, with this definition of what it means for a sequence of complex numbersto converge to oo, the usual definition of connected sets and separated sets is identical withwhat was given earlier.Definition 35.8.1 Let S C C the extended complex plane in which this extra point hasbeen included as just described. Then S is separated if there exist A,B not both empty suchthat S= AUB, ANB =0 and no point of A is a limit of any sequence of points of B whileno point of B is the limit of any sequence of points of A. If S is not separated, then it iscalled connected.Example 35.8.2 Consider the open set S= {z € C such that Im(z) > 0}. Then SU {00} =S is connected in C .It is obvious that S is connected in C because it is arcwise connected. Suppose S= AUB where these two new sets separate Sin C. Then one of them, say B must containco, Therefore, A is bounded since otherwise there would be a sequence of points of Aconverging to co which is assumed not to happen. Then S = AU (B\ {e}) and A,B \ {co}would separate S unless one is empty. If B \ {co} = 0, then S would be bounded which isnot the case. Hence A = 0. Thus S is connected.Definition 35.8.3 Let SC C. It is said to be simply connected if the set is connected andC\ SU {co} is connected in C.. Written more compactly, S is simply connected means S isconnected and also C \S is connected in C.When looking at a set S in C, how do you determine whether it is simply connected?You consider @ (S©) in S* and ask whether it is connected with the convention that if S isunbounded, you must include (0,0,2) in the image of 0.Example 35.8.4 Consider the set S = {z € C such that |z| > 1}. This is a connected set,but it is not simply connected because C \S is not connected. On S° it consists of a piecenear the bottom of the sphere and the point (0,0,2) at the top.Example 35.8.5 Consider S = {z € C such that |z| < 1}. This connected set is simply con-nected because C \S corresponds to a connected set on S”.35.9 ExercisesIn the following exercises, the term “simple closed curve” will be used repeatedly. Assumethat such curves I have an inside U; and an outside and that Green’s theorem applies forU; with its boundary I if the boundary is oriented appropriately. This can be proved, but isnot in this book. It is one of these things which is mainly of mathematical interest. In theexamples of interest, it is typically not an issue.