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Like the title says, what is the easiest way to see that CP^1 is topologically just a 2-sphere? Wikipedia says that CP^1 can be identified with C u {oo} (the 1-point compactification of C) but I don't see it.
quasar987 said:You call that an easy visualization?
CP^1 is the complex projective line, also known as the Riemann sphere. It is a one-dimensional complex manifold that can be visualized as a sphere with each point representing a line in the complex plane. S^2 is the two-dimensional sphere, also known as the surface of a ball. It is a two-dimensional manifold that can be visualized as a round, three-dimensional object.
CP^1 and S^2 are topologically equivalent, meaning they have the same underlying structure. They are both compact, connected, and orientable manifolds. In fact, CP^1 can be obtained by gluing two copies of S^2 together along their boundaries.
Visualizing CP^1 = S^2 can help us understand the concept of complex projective spaces and their relation to spheres. It also has applications in physics, particularly in quantum mechanics, where CP^1 is used to represent the state space of a spin-1/2 particle.
CP^1 can be visualized as a sphere with each point representing a complex line passing through the origin in the complex plane. S^2 can be visualized as a sphere with each point representing a point on the surface of a ball. By identifying the points on the boundary of CP^1 with the points on the surface of S^2, we can visualize the two manifolds as one.
Yes, there are several other ways to visualize CP^1 = S^2. For example, you can think of CP^1 as a sphere with antipodal points identified, or as the set of all lines through the origin in three-dimensional space. You can also use stereographic projection to map CP^1 to the plane, which can make it easier to visualize certain properties of the manifold.