Immersions, submersions, embeddings

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

  1. ,
  2. is injective,
  3. is a homeomorphism onto with subspace topology.

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