The region R as described in exercise 12.17 is such that any point on a sphere is identified with its antipodal point so that a loop on the sphere cannot be deformed to a singlular point (which I assume brings about the proof of 12.02). I am not clear on why a point is being identified with its antipodal point and the off the wall possibility that this proof may have something to do with a Steradian measure of a sphere is 4*pi? If anyone can clear this up it is sincerely appreciated.
Thanks
Tim