Pole in Stereographic Projection

[FallenApple]

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.

 

[fresh_42]

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.

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 $\mathbb{R}^1$.