Show that the sphere, S, and $\mathbb{R}^2$ is not homeomorphic I am trying to show that the sphere $S^2$ and $\mathbb{R}^2$ are not homeomorphic.I understand that you can't 'compress' a 3D shape into a 2D plane but I don't know how I would express this formally.
$S^2 = \{(x, y, z) ∈ \mathbb{R}^3: x^2 + y^2 + z^2 = 1\}$
As always, any help is appreciated!

If $\;f:S^2\to\Bbb R^2\;$ is a homeomorphism then $\;f:S^2\setminus\{x_0\}\to\Bbb R\setminus\{f(x_0)\}\;$ also is, yet the sphere minus any point is simply connected whereas the plane minus any point isn't.
Or as above but with "contractible" instead of simply connected.
