Question

Consider a sphere S . Lines on the sphere are taken to be great circles. On the sphere, does the Plane Separation Postulate hold? Give a short explanation to back up your claim that considers the various parts of the Plane Separation Postulate.

Answer 1

What you call 'projective sphere' is just a realization of the projective plane. (I guess you rather wanted S={{(x,y,z),(−x,−y,−z)}∣x2+y2+z2=1}S={{(x,y,z),(−x,−y,−z)}∣x2+y2+z2=1}.)

As you correctly note, a line on SS is 'sort of' a great circle. Correctly, a line is the image of a great circle under the canonical map S2→SS2→S where S2S2 denotes the unit sphere.

And yes, a line on the sphere divides it into two parts, but these two parts already coincide in SS, so SS is not divided into two parts.

The same goes if you consider projective plane as the Euclidean plane equipped with points in the infinity: starting from one half plane, going through a point in the infinity we can arrive to the other half plane. So, in the projective plane these are connected to each other, so is one part only.