Complex projective spaces, denoted as CP^n, are mathematical structures that generalize the concept of projective spaces to complex numbers. They consist of lines through the origin in C^{n+1}, where each line represents a point in CP^n. This means that each point in complex projective space corresponds to a set of equivalent points in C^{n+1} that differ by a non-zero complex scalar. These spaces are important in various fields, including algebraic geometry and string theory. They provide a framework for studying complex manifolds and can be used to understand properties of complex varieties. The topology of CP^n is rich, featuring interesting characteristics like being compact and having a well-defined notion of dimension.