30 (z,t)∊S2-(0,1)→w=z/(1-t)∊Cν from the point P on S2 excluding the north pole to the point w on Cν exists. The inverse mapping w∊Cν→(z=2w/(|w|2+1),t=(|w|2-1)/(|w|2+1))∊S2-(0,1) of this exists, and is continuous, so the sphere S2 -(0,1) excluding the north pole and the complex plane Cν are homeomorphic and can be seen as the same. This sort of mapping also exists between the sphere S2 -(0,-1) excluding the south pole and the complex plane Cσ. If we now make the intersection with S2 of the line that connects the south pole (0,-1) of the sphere S2, and the point (r,0) on the complex plane Cσ(≄Cν) that includes the equator of S2, Q, then the continuous mapping (z,t)∊S2-(0,-1)→r=z/(1+t)∊Cσ from the point Q on S2 excluding the south pole to the point r on Cσ exists. The inverse mapping r∊Cσ→(z=2r/(|r|2+1),t=-(|r|2-1)/(|r|2+1))∊S2-(0,-1) of this exists, and is continuous, so the sphere S2 -(0,-1) excluding the south pole and the complex plane Cσ are homeomorphic and can be seen as the same. However, the union of the sphere S2 -(0,1) excluding the north pole and the sphere S2 - (0,-1) excluding the south pole is the sphere S2 itself. Therefore the union of the complex plane Cν that can be seen as the same as the sphere S2 -(0,1) excluding the north pole and the complex plane Cσ that can be seen as the same as the sphere S2 -(0,-1) excluding the south pole can also
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