Is there a pair of continuous surjective functions say $f\_1$ and $f\_2$ from $\mathbb{Q}$ to itself such that for every $x, y \in \mathbb{Q}$, $f\_1^{-1}(x) \cap f\_2^{-1}(y)$ is non-empty? 



Continuous Functions
Take the two projections from ${\mathbb Q}\times{\mathbb Q}$ to ${\mathbb Q}$ and use the [fact](https://mathoverflow.net/a/148680/10503) that ${\mathbb Q}\times{\mathbb Q}$ is homeomorphic to ${\mathbb Q}$.


