Q:

Reference on diffeomorphisms between subspaces of ${\mathbb R}^n$

I am looking for a basic reference on Diffeomorphisms.
I am mainly interested on checking conditions to tell whether or not given two spaces $M,N \subseteq {\mathbb R}^n$, there exists at least one diffeomorphism between them. For example, I don't know if there exists a diffeomorphism between 
$[0,1]$ and $(0,1)$. As well as some other classical results on diffeomorphisms.

A:

Only general result I can think of is that of the inverse function theorem :http://en.wikipedia.org/wiki/Inverse_function_theorem , which tells you about the existence of a local diffeomorphism if the differential of the map is invertible as a matrix. And, of course, you first need for the two subspaces to be homeomorphic.

