158 Chapter3The following proposition was first stated by C. Jordan in 1892 [10],
but his original proof was inadequate. Correct proofs have since been
given by Veblen, Hartogs, Kerekjarto, Brouwer, Vallee-Poussin, Alexander,
Schonflies, and others. This theorem is a classical example of a property
that is intuitively obvious, yet whose proof is rather deep.
Theorem 3.13 (Jordan Curve Theorem). If C is a graph of a simple
closed curve in the complex plane, the complement of C is the union of
two disjoint regions, C being the common boundary of the two regions.
One of the regions is bounded (called the interior of C), the other region
is unbounded (called the exterior of C).
For a proof, see [7], [13], [15], or [20]. We also refer the reader to the
papers [11], [14], [18], and [19].
The following converse of Jordan's theorem was given by Schonflies in
1902: If a compact set Chas two complementary regions on the complex
plane, from each of which it is at every point accessible, then C is a Jordan
curve.
If A is a region in the plane and b E 8A, we say that b is an accessible
boundary point of A (or, from A) iff A U { b} is path connected.
The following important theorem is also due to Schonflies: All points
of a Jordan curve are accessible boundary points from each of the regions
determined by the curve.
Let C be a Jordan curve and let 'Y be an arc with endpoints on C such
that the graph of the arc less its endpoints is contained in the interior of
C. Then the arc "( is called a crosscut of C. If 'Y is a simple arc, it is
said that C U7 is a B-configuration (or, a B-curve), and an application of
the Jordan curve theorem yields the following result: The complement of a
B-configuration (in the complex plane) consists of three disjoint regions R 1 ,
Rz, and R3, one of which is unbounded while the other two are bounded.
Another useful theorem, especially in the theory of complex integration
(Chapter 7), is the following: If C and C' are two Jordan curves in the
plane such that C' C Int C, then Int C' C Int C.
For proofs of the preceding results and related properties, see [20],
Chapter VI.
Exercises 3.3
- Show that the property of an arc to be piecewise smooth is invariant
under a change of parameter t = h(7), provided that h'('Y) is continuous
and different from zero on [a', ,8']. - Show that the formula for the length of a regular arc applies also for
the computation of the length of a piecewise regular arc. Prove that the