TITLE: $\mathbb{Z}$ and $\mathbb{Q}$ are not homeomorphic
QUESTION [8 upvotes]: Can anyone explain me why $\mathbb{Z}$ and $\mathbb{Q}$ are not homeomorphic?
Thanks.

REPLY [4 votes]: $\Bbb Z$ is a complete metric space. while $\Bbb Q$ is not completely metrizable - that means that there is no metric which is both complete and induces the standard $<$ topology on $\Bbb Q$. 
Therefore $\Bbb Q$ cannot be homeomorphic to any complete metric space and in particular it is not homeomorphic to $\Bbb Z$.