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

(c) 2006 John H. Mathews, Russell W. Howell