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1 Set Theoretic Notation.
If S is a set then the symbolic phrase s ∈ S means s is an element of S or s is in S. The symbolic phrase s /∈ S means that s is not an element of S. If S and T are sets then S ∪ T is the set whose elements are the elements of the set S or the elements of the set T. Symbolically S ∪ T = {x|x ∈ S or x ∈ T }. We call S ∪ T the union of S and T. If S and T are sets then S ∩ T is the set whose elements are the elements that are in both S and T. That is S ∩ T = {x|x ∈ S and x ∈ T }. We call S ∩ T the intersection of S and T. The symbol ∅ stands for the set without any elements and it is called the empty set. Two sets are equal if they have the same elements. That is x ∈ S implies that x ∈ T and x ∈ T implies that x ∈ S. We write S = T. The complement of T in S is the set {x ∈ S|x /∈ T } and is denoted S − T. We say that S and T are disjoint if they contain no elements in common. That is, if S ∩ T = ∅.
2 Connected sets in C.
A curve is a function from a closed interval in R to the complex plane. Thus a curve is a function f from an interval
I = [a, b] = {t ∈ R|a ≤ t ≤ b}
to C. This says that f (t) = x(t) + iy(t) with x(t) and y(t) real valued functions on I. If x(t)and y(t) are continuous then we will say that f is continuous and say that f defines a continuous curve. We write f : I → C.
Examples:
- A line segment is a straight line in the plane which can be parametrized as f (t) = ut + c + i(vt + d)
with u, v, c, d ∈ R and a ≤ t ≤ b.
- f(t) = eiπt^ for a ≤ t ≤ b.
If f : I → C is a continuous curve with I = [a, b] then we say that the curve connects f(a) to f (b). Thus if in the examples we take a = 0 and b = 1 then example 1 connects 0 to c + id and example 2 connects 1 to − 1.
A piecewise linear curve joining z to w is a continuous curve f : [a, b] → C such that there is a partition
a = t 0 < t 1 < ... < tn− 1 < tn = b
such that the restriction of f to each of the intervals [ti− 1 , ti] is a line segment for i = 1, ..., n and f (a) = z, f (b) = w. Example 3. Define f(t) = t + it for 0 ≤ t ≤ 12 and f(t) = t + i(1 − t) for 1 2 < t^ ≤^1. In this case^ [a, b] = [0,^ 1]^ and^ t^0 = 0, t^1 =^
1 2 and^ t^2 = 1. Notice that both formulas given take the same value 12 + i^12 for t = 12. So this is a piecewise linear curve joining 0 to 1.
We will say that an open subset, U, of C is arcwise connected if whenever z, w ∈ U there is a continuous curve joining z and w with all of its values in U. An open set, U, will be said to be connected if whenever we have U = A ∪ B with A and B open with no points in common then either A = U or B = U.
Theorem 1 Let U be open in C. Then the following three conditions are equivalent a) U is connected. b) U is connected in the sense of Brown and Churchill. c) U is arcwise connected.
