Is $\mathbb{Q}^2$ homeomorphic to $\mathbb{Q}^2\setminus \{0\}$?

general-topology metric-spaces

Solution 1:

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.

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