It's an easy excercise in set theory to exhibit a bijection $\Bbb Q \cong \Bbb Q\times \Bbb Q$. However, none of the bijections I'm aware of respect the topologies on $\Bbb Q$ and $\Bbb Q^2$, inherited from their respective embeddings into $\Bbb R$ and $\Bbb R^2$.
Therefore, I'm asking whether there exists a homeomorphism $\phi: \Bbb Q^2 \to \Bbb Q$.
I don't believe that there is any such map, but since the standard techniques of algebraic topology don't enable one to discern between $\Bbb Q$ and a discrete space, I wasn't able to prove it. Maybe Cech cohomology provides a means to attack this problem, but I haven't even got the slightest Iiea how to calculate $H^1(\Bbb Q,\Bbb Z)$.
The set of the rational numbers with its usual topology is the unique countable metrizable space without isolated points.
Can you use that property to conclude?