Submanifolds (outline)

A subset $N$ of a manifold $M$ which is itself a manifold is a submanifold.

It can be an embedded manifold, if the topology of $N$ is the subset topology of $M$. If not, it is an injective immersed manifold. Keep an eye: if we don't require to be injective, an immersed manifold is not a submanifold!!