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)\} $.

conceptually its because your mapping a continuous space to one with a hole in it. you cant have a continuous map from $\mathbb{R}^2$ to $\mathbb{R}^2\setminus (0,0)$ since to do so you'd have to tear a hole in it.
