Let $X$,$Y$ be compact, connected, smooth manifolds of the same dimension. Suppose you have a surjective smooth map $f : X \rightarrow Y$, such that $|f^{-1}(p) | \leq k$ for all $p \in Y$.


Let $U \subset X$ be an open dense subset. 


**Question**: Is it possible to find $p \in Y$ such that $f^{-1}(p) \subset U$?



Map between manifolds and open dense subsets
Yes. I think you do not even have to assume that the preimages are finite. $f$ can be any $C^1$ mapping from $X$ onto $Y$. Let $y\in Y$ be a regular value of the mapping $f$. Then for some $\epsilon>0$
$$
f^{-1}(B(y,\epsilon))=\bigcup\_{i=1}^m U\_i,
$$
and $f|\_{U\_i}$ is a diffeomorphism of $U\_i$ onto $B(y,\epsilon)$. 
$f(U\cap U\_i)$ is open and sense in $B(y,\epsilon)$. If
$$
p\in\bigcap\_{i=1}^mf(U\_i\cap U)\subset B(y,\epsilon),
$$
then $f^{-1}(p)\subset U$.


