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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 Q is composed of lines which are parallel to either the x-axis or y-axis

Solution Preview

In the following I assume that the metric under which G is open is the standard one: d(z_1, z_2) = |z_1-z_2| and that the polygon is compact.

Take any point z_p on polygon P. Since G is open there is some e(z_p) > 0 such that all the points in the e-vicinity of z_p are in G.

Consider an "e-ball" - an open ball of radius e and a "half-e-ball" - an open ball of radius e/2, both centered on z_p.

Next consider all the points z_p ...