Functions. Limits and Continuity. Arcs and Curves 169zo E A, by the convexity of A. It follows that A is contained in the region
RL = C - L, so is any cycle C with graph in A. Now, if the angles B; inDefinition 3.30 are measured from -L, we see that !lc(zo) = 0 since C
cannot wind about z 0 •
Definition 3.33 A region R that is not simply connected is said to be
multiply connected.
It may happen that there exists in Ra set of m cycles Ci, C2, ... , Cm
with the following properties:- The cycles are linearly independent, that is, any homology
ki C1 + k2C2 + ... +km Cm ,..., 0
in R implies that ki = kz = · · · = km = 0.
- Every cycle C in R is homologous to a linear combination with integral
coefficients of the m cycles in the set, i.e.,
(3.17-1)In such a case we say that there is an m-dimensional base in R. If follows,
as in linear algebra, that the coefficients kj (j = 1, ... ,m) in (3.17-1) are
uniquely determined, and that any two bases have the same number of
cycles.Definition 3.34 If a multiply connected region has a base consisting of
m cycles, the number m + 1 is called the connectivity of the region, and the
region is said to be of finite connectivity. If no base with a finite number
of cycles exists, the connectivity is said to be infinite.Examples
- The connectivity of the region A in Fig. 3.26 is 2, that of the region
A in Fig. 3.27 is 3.
2. The region R = C - {z: z = ki, k = 0, ±1, ±2, ... } is infinitely
connected.Exercises 3.4
l. Define a homotopy from 'Yi: z = 2 cost + 3i sin t, 0 ~ t ~ 67r into
')' 2 : z = eit, 0 ~ t ~ 67r.2. Find the winding number of the curve r with respect to a point in each
of the regions A, B, and C in Fig. 3.28 (a) intuitively, and (b) by usingDefinition 3.30. Hint: To apply the definition, divide r into a suitable
number of subarcs and measure the angles B; (-^1 / 2 7r < B; < %7r) with
a protractor.