EDB — 2FX

[2FX]Let \(C=𝜑([0,1])\), let \(P\) be the region bounded by the closed polygonal curve, and \(E\) the exterior; recall that \(C,P,E\) is a partion of the plane. Choose \(A,B∉ C\) and suppose that the segment \(AB\) meets \(C\) in \(k\) points, none a vertex. Then: if \(k\) is odd, \(A,B\) are in different regions, \(A∈ P⇔ B∉ P\); if \(k\) is even, \(A,B\) are in the same region, \(A∈ P⇔ B∈ P\).