TITLE: Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic?
QUESTION [5 upvotes]: If $p$ and $q$ are distinct prime number, are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic as topological space?

REPLY [6 votes]: Yes. Let $X$ be the product of countably infinitely many copies of the discrete two-point spaces (i.e., a Cantor set), and let $x\in X$ be any point; then $\Bbb Q_p$ and $\Bbb Q_q$ are both homeomorphic to $X\setminus\{x\}$. 
If you prefer, you can start with the middle-thirds Cantor set $C$ and delete any one point: they are homeomorphic to $C\setminus\{1\}$, for instance.