A differentiable mapping between two Manifolds such that its Differential is injective (i.e. the linear operator doesn't "collapse" any direction, and so its invertible)

If an immersion is a Homeomorphism, then it is an Embedding

For local properties it is enough to work with an immersion, because an immersion is locally (in a certain sense) an embedding. (Proposition 3.7 in DoCarmo: given an immersion there is always a neighbourhood around each point such that the restriction to that nhbd is an embedding. A consequence of the Inverse function theorem, which ensures that the immersion is a local Diffeomorphism, and thus a homeomorphism)