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 ${\mathbb{R}}^{2n+1}$.
Title  embedding 

Canonical name  Embedding 
Date of creation  20130322 14:52:46 
Last modified on  20130322 14:52:46 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 57R40 
Synonym  differential embedding 
Defines  Whitney’s theorem 