Is there a simple bijective function between the surface of the unit sphere in $\mathbb{R}^3$ and the real plane? I am aware of stereographic projection and Riemann sphere but these seem to map the north pole to infinity.
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Start from the stereographic bijection $B$ between the sphere minus its North pole $p$ and the plane. One needs a point of the plane to be the image of $p$ hence we will make some room for it. Choose any injective sequence $(x_k)_{k\ge0}$ of points in the plane and define a shift $S$ on the plane by $S(x_k)=x_{k+1}$ for every $k\ge0$ and $S(x)=x$ for every other point $x$ of the plane. Then $S\circ B$ is a bijection between the sphere minus $p$ and the plane minus $x_0$. Extend it to a bijection between the sphere and the plane by sending $p$ to $x_0$. You are done. |
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It is not possible if you want a continuous bijection: Assume there is such a bijection h. Then $h^-1$ is also a bijection between $S^2$ and the plane. But $S^2$ is compact, and the plane is Hausdorff; a continuous bijection between compact and Hausdorff is a homeomorphism. But $S^2$ is compact, and the plane is not, and compactness is a topological property, i.e., it is preserved by homeomorphisms (basically since compactness is defined in terms of open sets.) |
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The unit sphere is compact. Any image of it under a continuous function is compact. The plane is not bounded; hence it is not compact. Therefore no such homeomorphism exists. |
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