An Example of a Manifold Let’s be a little more explicit about
our example from last time. The two-dimensional sphere consists
of all the points in of unit length. If we pick an orthonormal
basis for and write the coordinates with respect to this basis
as , , and , then we’re considering all triples with . We want
to show that this set is a manifold. We know that we can’t hope
to map the whole sphere into a plane, so we have to take some
points out. Specifically, let’s remove those points with , just
leaving one open hemisphere. We will map this hemisphere
homeomorphically to an open region in . But this is easy: just
forget the -component! Sending the point down to the point is
clearly a continuous map from the open hemisphere to the open
disk with . Further, for any point in the open disk, there is a
unique with . Indeed, we can write down This inverse is also
continuous, and so our map is indeed a homeomorphism. Similarly
we can handle all the points in the lower hemisphere . Again we
send to , but this ti