1.6 • THE TOPOLOGY OF COMPLEX NUMBERS 45Figure 1.28 Are z 1 and z2 in the interior or exterior of this simple closed curve?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.
Although an introductory treatment of complex analysis can be given with-
out using this theorem, we think it is important for the well-informed student
at least to be aware of it.
-------.. EXERCISES FOR SECTION 1.6- Find a parametrization of the line that
(a) joins the origin to the point 1 + i.
(b) joins the point 1 to the point 1 + i.
( c) joins the point i to the point 1 + i.
( d) joins the point 2 to the point 1 + i.