1.6 The Topology of Complex Numbers

    In this section we investigate some basic ideas concerning sets of points in the plane.  The first concept is that of a curve. Intuitively, we think of a curve as a piece of string placed on a flat surface in some type of meandering pattern.  More formally, we define a curve to be the range of a continuous complex-valued function defined on the interval .  That is, a curve C is the range of a function given by  ,  for ,  where both and are continuous real-valued functions.  If both and are differentiable, we say that the curve is smooth.  A curve for which and are differentiable except for a finite number of points is called piecewise smooth.  

Definition  (Curve).  

    We specify a curve C as  

            ,  

and say that is a parametrization for the curve C.  Notice that with this parametrization, we are specifying a direction the curve C, and we say that C is a curve that goes from the initial point    to the terminal point  .  If we had another function whose range was the same set of points as z(t) but whose initial and final points were reversed, we would indicate the curve this function defines by -C.

Example 1.22.  Find parametrizations for C and -C, where C is the straight line segment beginning at and ending at .  

Solution.  Refer to Figure 1.21. The vector form of a line shows that the direction of C is .  As is a point on C, its vector equation is

  

Clearly one parametrization for    is

            

Remark.  Note that  ,  which illustrates a general principle:  If C is a curve parametrized by    for  ,  then one parametrization for    will be  ,  for  .  

Explore Solution 1.22.

            Figure 1.21  The straight-line segment C joining .

Extra Example 1.22  Find the equation of the line segment with the initial point    and the terminal point  .  

Explore Extra Solution 1.22

    A curve C having the property that    is said to be a closed curve.  The line segment (1-48) is not a closed curve.  The range of  ,  where    for    is a closed curve because  .  The range of is the four-leaved rose shown in Figure 1.22.  Note that, as t goes from , the point is on leaf  1 ;   from to , it is on leaf  2 ;  between , it is on leaf 3;  and finally, for t between , it is on leaf 4.  

        Figure 1.22  The curve for , which forms a four-leaved rose.

    Note further, in Figure 1.22, that at the point , the curve has crossed over itself (at points other than those corresponding with   );  we want to be able to distinguish when a curve does not cross over itself in this way.  The curve C is called simple if it does not cross over itself, except possibly at its initial and terminal points.  In other words, the curve  ,  for  ,  is simple provided that    whenever  ,  except possibly when  .

Extra Example 2.  The curve    for  .  
Remark. The curve looks like a "four leafed rose".

Explore Extra Solution 2.

Example 1.23.  Show that the circle C with center and radius can be parameterized to form a simple closed curve.

Solution.  Note that the required parametrization is

              

    Figure 1.23 shows that, as t varies from , the circle is traversed counterclockwise.  If you were traveling around the circle in this manner, its interior would be on your left.  When a simple closed curve is parametrized in this fashion, we say that the curve has a positive orientation.  We will have more to say about this idea shortly.

Explore Solution 1.23.

        Figure 1.23  The simple closed curve  ,  for  .  

    We need to develop some vocabulary that will help describe sets of points in the plane.  One fundamental idea is that of an neighborhood of the point .  It is the open disk of radius about shown in Figure 1.24.  Formally, it is the set of all points satisfying the inequality   and is denoted .  That is,  

(1-49)            .

            Figure 1.24   An neighborhood of the point .

Example 1.24.  The solution sets of the inequalities  ,  ,  and    are neighborhoods of the points  ,  with radii  ,  respectively.  They can also be expressed as  ,  ,  and  .  

    We also define  ,  the closed disk of radius centered at , and  ,  the punctured disk of radius centered at as   

(1-50)            ,   and  

(1-51)            .  

Definitions (Interior Point, Exterior Point, Boundary Point).  

    The point is said to be an interior point of the set S provided that there exists an neighborhood of that contains only points of S;   is called an exterior point of the set S if there exists an neighborhood of that contains no points of S.  If is neither an interior point nor an exterior point of S, then it is called a boundary point of S and has the property that each neighborhood of contains both points in S and points not in S.  Figure 1.25 illustrates this situation.

            Figure 1.25  The interior, exterior, and boundary of a set S.

    The boundary of    is the circle depicted in Figure 1.23.  We denote this circle    and refer to it as the circle of radius R centered at .  Thus  

(1-52)            .

    We use the notation to indicate that the parametrization we chose for this simple closed curve resulted in a positive orientation;   denotes the same circle, but with a negative orientation.  (In both cases, counterclockwise denotes the positive direction.)  Using notation that we have already introduced, we get  .

        Figure 1.C  The positively oriented curve and its opposite .

Example 1.25.  Let  .  (a) Find the interior of S.  (b) Find the exterior of S.  (c) Find boundary of  S.

Solution.  We show that every point of S is an interior point of S.  Let be a point of S.  Then  ,  and we can choose  .  We claim that  .  If  ,  then

            

Hence the neighborhood of is contained in S, which shows that is an interior point of S.  
It follows that the interior of S is the set S itself.

    Similarly, it can be shown that the exterior of S is the set .  The boundary of S is the unit circle  .  This condition is true because if    is any point on the circle, then any -neighborhood of will contain the point  ,  which belongs to S, and  ,  which does not belong to S.  We leave the details as an exercise.

Explore Solution 1.25 (a).  Find the interior of S.

Explore Solution 1.25 (b). Find the exterior off S.

Explore Solution 1.25 (c).  Find the boundary of S.

    The point is called an accumulation point of the set S if, for each , the punctured disk contains at least one point of S.  We ask you to show in the exercises that the set of accumulation points of is , and that there is only one accumulation point of , namely, the point 0.  
We also ask you to prove that a set is closed if and only if it contains all of its accumulation points.

    A set S is called an open set if every point of S is an interior point of  S.  Thus, Example 1.25 shows that is open.  A set S s called a closed set if it contains all its boundary points.  A set S is said to be a connected set if every pair of points contained in S can be joined by a curve that lies entirely in S.  Roughly speaking, a connected set consists of a "single piece."   The unit disk    is a connected open set.  We ask you to verify in the exercises that, if lie in , then the straight-line segment joining them lies entirely in .  The annulus    is a connected open set because any two points in A can be joined by a curve C that lies entirely in A, as shown in Figure 1.26.  The set    consists of two disjoint disks.  We leave it as an exercise for you to show that the set is not connected, as shown in Figure 1.27.

        Figure 1.26  The annulus    is a connected set.

        Figure 1.27  The set    is not a connected set.

    We call a connected open set a domain.  In the exercises we ask you to show that the open unit disk    is a domain and that the closed unit disk    is not a domain.  The term domain is a noun and is a type of set.  In Chapter 2 we note that it also refers to the set of points on which a function is defined.  In the latter context, it does not necessarily mean a connected open set.

Example 1.26.  Show that the right half-plane    is a domain.

Solution.  First we show that H is connected. Let be any two points in H.  We claim the obvious, that the straight-line segment ,  for    lies entirely within  H.  To prove this claim, we let  ,  for some  ,  be an arbitrary point on .  We must show that  .  Now,  
        
     

If , the last expression becomes , which is greater than zero because .  Likewise, if , then Equation (1-53) becomes , which also is positive.  Finally, if  ,  then each term in Equation (1-53) is positive, so in this case we also have  .
    To show that H is open, we suppose without loss of generality that  .  We claim that  ,  where .  We leave the proof of this claim as an exercise.

    A domain, together with some, none, or all its boundary points, is called a region.  For example, the horizontal strip    is a region.  A set formed by taking the union of a domain and its boundary is called a closed region; thus    is a closed region.  A set S is said to be a bounded set if it can be completely contained in some closed disk, that is, if there exists an    such that for each z in S we have .  The rectangle given by   is bounded because it is contained inside the disk .  A set that cannot be enclosed by any closed disk is called an unbounded set.  

    We mentioned earlier that a simple closed curve is positively oriented if its interior is on the left when the curve is traversed.  How do we know, though, that any given simple closed curve will have an interior and exterior?  Theorem 1.6 guarantees that this is indeed the case.  It is due in part to the work of the French mathematician Marie Ennemond Camille Jordan (1838--1922).

Theorem 1.6 (Jordan Curve Theorem).  The compliment of any simple closed curve C can be partitioned into two mutually exclusive domains I and E in such a way that I is bounded,  E is unbounded, and C is the boundary for both I and E.   In addition    is the entire complex plane.  The domain I is called the interior of C,  and  the domain E is called the exterior of C.

    The Jordan curve theorem is a classic example of a result in mathematics that seems obvious but is very hard to demonstrate, and its proof is beyond the scope of this book.  Jordan's original argument, in fact, was inadequate, and not until 1905 was a correct version finally given by the American topologist Oswald Veblen (1880-1960).  The difficulty lies in describing the interior and exterior of a simple closed curve analytically, and in showing that they are connected sets.  For example, in which domain (interior or exterior) do the two points depicted in Figure 1.28 lie?  If they are in the same domain, how, specifically, can they be connected with a curve?  If you appreciated the subtleties involved in showing that the right half-plane of Example 1.26 is connected, you can begin to appreciate the obstacles that Veblen had to navigate.

        Figure 1.28  Are the points in the interior or exterior of this simple closed curve?

Exploration

    Although an introductory treatment of complex analysis can be given without using this theorem, we think it is important for the well-informed student at least to be aware of it.

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