TITLE: Is $\mathbb{Q}^2$ homeomorphic to $\mathbb{Q}^2\setminus \{0\}$?
QUESTION [24 upvotes]: I know that $\mathbb{R}^2$ and $\mathbb{R}^2\setminus\{(0,0)\}$ are not homeomorphic. (For examle $\pi_1(\mathbb{R}^2)=\{e\}$, but $\pi_1(\mathbb{R}^2\setminus\{(0,0)\})=\mathbb{Z}$). 
But what can be said about $\mathbb{Q}^2$ and $\mathbb{Q}^2\setminus\{(0,0)\}$? Is there a homeomorphism?

REPLY [20 votes]: The rational numbers are the unique (up to isomorphism) metric space which is both countable and have no isolated points.
$\Bbb Q^2\setminus\{(0,0)\}$ is countable and without isolated points. Therefore the answer is yes. There is a homeomorphism.