The discussion centers on the concept of stereographic projection, specifically how the circle minus the pole is homeomorphic to the real number line. The pole, which corresponds to the point at infinity, signifies a number that does not exist on the real line, yet it is a crucial element in this mapping. The relationship between the unit circle and the projective number line is established through the equivalence of various mathematical structures, including ##\mathbb{S}^1##, ##SO(2,\mathbb{R})##, and ##\mathbb{P}(1,\mathbb{R})##.
PREREQUISITESMathematicians, students of topology, and anyone interested in the geometric interpretation of complex numbers and projective spaces will benefit from this discussion.
The pole corresponds to the point at infinity, which results is the projective number line. You have to define some point as pole, since ##\mathbb{S}^1 \simeq SO(2,\mathbb{R}) \simeq U(1,\mathbb{C}) \simeq \mathbb{P}(1,\mathbb{R})## is the unit circle, which is not the same topological space as ##\mathbb{R}^1##.FallenApple said:So the circle minus the pole is homeomorphic to the number line. Does that mean that the pole itself represents a number that doesn't exist on the real line? After all, the pole certainly exists and yet is the only point that isn't mapped.