Embeddings of manifolds need not be injective.

Question

By the embedding of a smooth manifold I mean a smooth $f : M \to N$ such that $f(M)$ is a submanifold of $N$. The usual criterion that one sees in books requires $f$ to be an immersion and a homeomorphism onto its image. It seems to me that one needs only $f$ to be an immersion that is open on its image, i.e. the global injectivity of $f$ is useless (the problematic self intersections are taken care of by the openness condition). Am I correct? If I am, why don't the books mention this? The proof is exactly the same.