Betweenness on the sphere. In Exercise 3.2.6, betweenness and segments were defined for the sphere..

Betweenness on the sphere. In Exercise 3.2.6, betweenness and segments were defined for the sphere S2. Define ray in the usual way (Definition 3.2). A great circle divides the sphere into two hemispheres, which we could think of as half-planes determined by the great circle. (a) Does the Plane Separation Postulate hold in this setting? Explain. (b) Let A and B be distinct, nonantipodal points on S2. Find all points C such that A _ B _ C. Sketch the ray _'AB. (c) Does Theorem 3.3.9 hold in this setting? Explain.