Some tldr-excerpts from Lee and Spivak formalizing embeddings and stuff.
Topological embedding -- an injective continuous map that is also a homeomorphism onto its image . We can think of "as a homeomorphic copy of in " (Lee, 2011).
A smooth map is said to have rank at if the linear map ( the pushforward) has rank . is of constant rank if it is of rank at every point.
Immersion -- smooth map whose pushforward is injective at every point, that is .
Submersion -- smooth map whose pushforward is surjective at every point, that is .
(Smooth) Embedding (of a manifold) -- an injective immersion that is also a topological embedding.
So, a map is an embedding, if
- is injective,
- is a homeomorphism onto with subspace topology.