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 they were homeomorphic, then they have the same homotopy groups $\pi_k$ for every $k\ge0$. Notice that $\mathbb{R}^2$ is contractible whereas $S^2$ is not. In particular $\pi_2(\mathbb{R}^2)$ is trivial whereas $\pi_2(S^2)\cong\mathbb{Z}$. So they are not homeomorphic.
