Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$ Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus \{(0,0)\} $.

$\mathbb{R}^2\backslash\{(0,0)\}\cong S^1\times\mathbb{R}$.  you can use co/homology or fundamental groups (if that's in your toolkit).  you can note that one is contractible and the other is homotopy equivalent to a circle (and tell those apart).  you can note that one has euler characteristic 0 and the other has euler characteristic 1 (if this is something you can calculate).
