Topology Of The Complex Plane
Topology of the Complex Plane:
If P is a complex number and r > 0, then we set
The first of these is the open disc with center P and radius r; the second is the closed disc with center P and radius r (Figure 1.3). We often use the simpler symbols D and 
to denote, respectively, the discs D(0, 1) and (0, 1). We say that a subset U 
C is open if, for each P 
C, there is an r > 0 such that D(P, r) U. Thus an open set is one with the property that each point P of the set is surrounded by neighboring points that are still in the set (that is, the points of distance less than r from P)—see Figure 1.4. Of course the number r will depend on P. As examples, U = {z C : Re z > 1} is open, but F = {z C : Re z > 1} is not (Figure 1.5). A set E C is said to be closed if C \ E {z C : z 
E} (the complement of E in C) is open. The set F in the last paragraph is closed. It is not the case that any given set is either open or closed. For example, the set W = {z C : 1 < Re z ≤ 2} is neither open nor closed (Figure 1.6). We say that a set E C is connected if there do not exist non-empty disjoint open sets U and V such that E = (U
E) 
[(V E). Refer to Figure 1.7 for these ideas. It is a useful fact that if E C is an open set, then E is connected if and only if it is path-connected; this last means that any two points of E can be connected by a continuous path or curve.
