[Math] $\mathbb{Z}$ and $\mathbb{Q}$ are not homeomorphic

Question

Can anyone explain me why $\mathbb{Z}$ and $\mathbb{Q}$ are not homeomorphic?

Thanks.

Answer 1

$\mathbb{Z}$ is discrete (meaning that every subset is open), while $\mathbb{Q}$ has no isolated points (meaning that no singleton set is open; in fact, no finite set is open).