I was wondering if the following is true:
Let $M,N$ be two manifolds such that $\dim M\leq \dim N$ and $f:M\rightarrow N$ an smooth immersion.
Assume that for any open set $U\subset M$, $f(U)$ is open in $f(M)$, does it imply that $f(M)$ is a submanifold of $N$ ?
I know that if we also ask $f$ to be injective, then it is an embedding and $f(M)$ is automatically a submanifold of $N$. But without this assumption, I am not sure that the result holds.
Being an open map on its image somehow tells us that there is no bad self-intersection in $f(M)$ but I am not sure this is enough to have a submanifold.