I am trying to prove that a submersion is an open map.
There is a hint that locally there are coordinates in which the submersion can be shown as the projection onto the first few coordinates and I know that projections are open maps, but I don't know how to show that the submersion is a projection.
Thanks!
Suppose $f:X^{n+k}\to Y^n$ is a submersion and $U\subset X$ is open. If $x\in U$, then there exist diffeomorphisms $\phi:\Bbb R^{n+k}\to X$ and $\psi:\Bbb R^n\to Y$ parametrizing $x$ and $f(x)$, respectively, such that $$ f = \psi\circ \pi\circ\phi^{-1}. $$ Where $\pi:\Bbb R^{n+k}\to\Bbb R^{n}$ is the canonical projection. If we take $\tilde U\subset U$ sufficiently small and open, then since diffeomorphisms are open, and the projection is open, then $f(\tilde U) \subset f(U)$ is a neighborhood of $f(x)$, so that $f$ is an open map.