Via identification with one-point compactification of : the interior of the disk can be identified with, and the boundary point is identified with the point at infinity. Quotient of closed unit disk in by the identification of all points in its boundary with each other, i.e., Homeomorphism arising from a straight line homotopy
Identification of antipodal points gives the double cover from to One-point compactification of the Euclidean planeĭouble cover (and hence also universal cover) of the real projective plane or Stereographic projection hence homeomorphic and diffeomorphic How it is equivalent to the 2-sphere viewed geometrically
Note that all 2-spheres are equivalent up to translations and dilations, and in particular, they are homeomorphic as topological spaces. In particular, the unit 2-sphere centered at the origin is defined as the following subset of : The 2-sphere in with center and radius is defined as the following subset of : The 2-sphere, denoted, is defined as the sphere of dimension 2.
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