# Continuous Functions

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?

• I guess that this question would better suited for math.stackexchange.com rather than MO. – crskhr Nov 12 '13 at 14:54
• what is the topology? – Marc Palm Nov 12 '13 at 14:55
• Marc, the topology on $\mathbb{Q}$ is derived from $\mathbb{R}$ and the one on $\mathbb{Q}^2$, is the topology derived from $\mathbb{R}^2$ – Uday Bhaskar Nov 12 '13 at 15:02
• @MarcPalm I presume its metric topology. – crskhr Nov 12 '13 at 15:19

Take the two projections from ${\mathbb Q}\times{\mathbb Q}$ to ${\mathbb Q}$ and use the fact that ${\mathbb Q}\times{\mathbb Q}$ is homeomorphic to ${\mathbb Q}$.