continuity complex plane
Let G be an open subset of C ( complex plane) and let P be a polygon in G from a to b. Use the following 2 theorems to show that there is a polygon Q in G from a to b which is composed of line segments which are parallel to either the real or imaginary axes.
The 2 theorems are:
1). Theorem: Suppose f: X –> omega is continuous and X is compact; then f is uniformly continuous. ( of course we are talking about complex plane remember that)
2).Theorem: If A and B are non-empty disjoint sets in X with B closed and A compact then
d(A,B) > 0.
Please I want a very detailed answer and justify every claim or statement in the solution. Please show where each theorem was used and why..I want to fully understand this problem. Thanks.
Polygon is a polygonal line..so Q is composed of lines which are parallel to either the x-axis or y-axis