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!

Homeomorphism will preserve any "topological" property of spaces - in particular, $S^2$ is compact and $\mathbb R^2$ is not, so they can't be homeomorphic.
In fact, the image of a compact space under a continuous map is compact, so there is not even a surjective continuous map $S^2 \to \mathbb R^2$.
