At last we can actually define submanifolds. If and are both manifolds with as topological spaces — the points of form a subset of the points of and the topology of agrees with the subspace topology from — then we say that is a submanifold of if the inclusion map is an embedding. If the inclusion is only an immersion, we say that is an “immersed submanifold” of .
Now, if is any embedding of one manifold into another, then the image is a submanifold, as defined above. Similarly, the image of an injective immersion is an immersed submanifold. The tricky bit here is that if we have a situation like the second of our pathological immersions, we have to consider the topology on the image that does not consider the endpoints to be “close” to the middle point on the curve that they approach.
This motivates us to define an equivalence relation on injective immersions into : if and are two maps, we consider them equivalent if there is a diffeomorphism so that . Clearly, this is reflexive (we just let be the identity map), symmetric (a diffeomorphism is invertible), and transitive (the composition of two diffeomorphisms is another one).
The nice thing about this equivalence class is that every immersion is equivalent to a unique immersed submanifold, and so there is no real loss in speaking about an immersion as “being” an immersed submanifold. And of course the same goes for embeddings “being” submanifolds as well.