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Real Analysis : Fold Lines

By an n-fold line subdivision of the plane P, we mean any collection of n-distinct (infinite) lines in P, together with the open regions in P that they determine. (We don't count the lines as part of the regions.) Let us say that two such regions are adjacent if their boundaries have a positive-length or infinite line segment in common. Prove the following theorem by induction on n:

Given any n-fold line subdivion of P, each of the regions in the subdivision may be colored either red or blue in such a way that no two adjacent regions have the same color.

(Hints: You may also use (without proof) the following facts: (1) Every infinite lines in P divides P into two regions of which it is the common boundary. (2) Given an open region R in a k-fold line subdivision of P, if one introduces a new line L distinct from the others, then either L does not meet R or L subdivides R into two adjacent regions with common boundary segment contained in L.)


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