embedding

Let $M$ and $N$ be manifolds and $f:M\to N$ a smooth map. Then $f$ is an embedding if

1. $f(M)$ is a submanifold of $N$, and

2. $f:M\to f(M)$ is a diffeomorphism. (There’s an abuse of notation here. This should really be restated as the map $g:M\to f(M)$ defined by $g(p)=f(p)$ is a diffeomorphism.)

The above characterization can be equivalently stated:$f:M\to N$ is an embedding if

1. $f$ is an immersion, and

2. by abuse of notation, $f:M\to f(M)$ is a homeomorphism.

Remark. A celebrated theorem of Whitney states that every $n$ dimensional manifold admits an embedding into ${ℝ}^{2n+1}$.