# Immersions, submersions, embeddings

Some tldr-excerpts from Lee and Spivak formalizing embeddings and stuff.

Topological embedding -- an injective continuous map $f: A\to X$ that is also a homeomorphism onto its image $f(A)$. We can think of $f(A)$ "as a homeomorphic copy of $A$ in $X$" (Lee, 2011).

A smooth map $F: M\to N$ is said to have rank $k$ at $p \in M$ if the linear map ${F_*}_{T_p M}$ ( the pushforward) has rank $k$. $F$ is of constant rank $k$ if it is of rank $k$ at every point.

Immersion -- smooth map $F:M \to N$ whose pushforward $F_*$ is injective at every point, that is $\operatorname{rank} F = \operatorname{dim} M$.

Submersion -- smooth map $F:M \to N$ whose pushforward is surjective at every point, that is $\operatorname{rank} F = \operatorname{dim} N$.

(Smooth) Embedding (of a manifold) -- an injective immersion $F: M\to N$ that is also a topological embedding.

So, a map $F: M\to N$ is an embedding, if

1. $\operatorname{rank}F = \operatorname{dim} M$,
2. $F$ is injective,
3. $F$ is a homeomorphism onto $F(M)$ with subspace topology.