TITLE: Different p-adic topologies on $\mathbb{Q} $ are homeomorphic
QUESTION [10 upvotes]: Endow $\mathbb{Q}$ (the rationals) with the p-adic (resp. q-adic) topology, where p,q are primes. Are these topological spaces homeomorphic?
I know the norms are not equivalent, therefore do not induce the same topology.
What I am asking is if the two distinct topologies are homeomorphic to one another. The reason I am asking is that I know that $\mathbb{Q}_p$ and $\mathbb{Q}_q$ are homeomorphic to each other.

REPLY [6 votes]: Here's an alternative to Lee Mosher's argument, using a different general theorem from topology.  Any two countable metric spaces with no isolated points are homeomorphic.  In particular, this implies the $p$-adic and $q$-adic topologies on $\mathbb{Q}$ give homeomorphic spaces, and in fact these spaces are also homeomorphic to $\mathbb{Q}$ with the usual topology.
(Actually, one way to prove this theorem is by going through facts about the Cantor set similar to the argument in Lee Mosher's answer.  You can find a proof along these lines as well as some other proofs in this nice paper.)