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?

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


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