Is there any difference between Immersion and embedding?

Question

Definition as below , I think they are same ,is right ?

Answer 1

As noted in the comments, immersions needn't be injective. Even if they are injective, they needn't be embeddings: consider the injective immersion of $(0,2\pi)$ in the plane as a figure eight, so that the point $(0,0)$ is the center of the cross. Explicitly, send $t\mapsto (2 \sin t,\sin(2t))$. This is an immersion that cannot be a homeomorphism onto its image, since the image has noncut points while $(0,2\pi)$ has none.

It is true, however, that every immersion is locally an embedding. This is deduced from the so called "constant rank theorems", for example.