Projective Space: CP1, Homeomorphism to 2-Sphere?

[Thorn]

I have a question about complex projective space... specifically CP1 which can be thought of as the action of C on C^2\{0} which gives rise to the equivalence classes of "lines" passing through the origin in C^2 (but not including the 0) Now, any vector in complex space, when multiplied by the set of all complex numbers of a given norm will give rise to a circle...and in a sense, when C (numbers of all norm!) acts of C^2 you simply get equivalence classes of planes... I think anyway..someone correct me if I am wrong...but the ultimate question is.. is this homeomorphic to a 2 sphere?

 

[OrderOfThings]

Yes, CP¹ is the 2-sphere. This construction of S² from C² is on way to do the celibrated Hopf fibration.