I have been told the following fact: Given a foliation $F$ of a smooth manifold $M$, then the projection $\pi : M \to M /F$ onto the space of leaves (points on the same leaf are identified) is open.
Is is easy to see that?
I tried this using a foliation chart, $\varphi : M\supset U \to (-\varepsilon,\varepsilon)^r \times (-\varepsilon,\varepsilon)^q$ ($r$ is the dimension of $F$, $q$ codimension, $\dim M = r+q$, I assume $F$ is regular) so I described the leaves locally by plaquettes $\varphi^{-1}(c)$ for $c\in \{0\} \times (-\varepsilon,\varepsilon)^q$, but that did not really help me because I do not know which plaquettes will belong to the same leave.