Complex Projective Space - Topology

Topology

The topology of CPn is determined inductively by the following cell decomposition. Let H be a fixed hyperplane through the origin in Cn+1. Under the projection map Cn+1\{0} → CPn, H goes into a subspace that is homeomorphic to CPn−1. The complement of H in CPn is homeomorphic to Cn. Thus CPn arises by attaching a 2n-cell to CPn−1:

Alternatively, if the 2n-cell is regarded instead as the open unit ball in Cn, then the attaching map is the Hopf fibration of the boundary. An analogous inductive cell decomposition is true for all of the projective spaces; see (Besse 1978).