today we learn that
1. balls are not spheres. balls are not spheres. once you consider the boundary of a ball, you drop down a dimension 2. the 2-sphere lies in 3 dimensions, but is diffeomorphic to the real plane (plus a point) 3. the complex projective space lies in 4 dimensions, but is diffeomorphic to the complex plane (minus a point) (== real plane minus a point, up to diffeomorphism) 4. antipodal identification on RP^1 (x,y gets identified with -x,-y -> ratio is the same) is equivalent to CIRCULAR identification on CP^1 (one way to see this: all great circles intersecting the north pole on the unit sphere map to the same line).
In conclusion: the complex projective space is a two-dimensional object: it is not equivalent to the set of all complex vectors with unit norm; it is equivalent to the set of all complex vectors, period (well, minus a point.)
here’s a pretty video of the hopf fibration, yay.