# 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

- ,
- is injective,
- is a homeomorphism onto with subspace topology.

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