The notation "CP^1" refers to the complex projective line, which is a one-dimensional complex manifold. It can be thought of as the set of all lines through the origin in C^2, where C represents the complex numbers. This space is often visualized as a sphere, specifically the Riemann sphere, which connects complex analysis and geometry. In mathematical terms, CP^1 can be represented using homogeneous coordinates, typically denoted as [z_0 : z_1], where not both z_0 and z_1 are zero. This representation allows for a compact and elegant way to study properties of complex numbers and their geometric interpretations, making it a fundamental concept in areas like algebraic geometry and topology.