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)\} $.

Here's a slightly different way of looking at it that avoids fundamental groups (although has its own messy details to check).  One of the spaces, upon removing a compact set, can be separated into two connected components with noncompact closure.  The other can't.
