Does a diffeomorphism always map boundaries to boundaries? If $F:\overline{M} \to \overline{M'}$ is a diffeomorphism between two open bounded domains $M$ and $M'$ in $\mathbb{R}^n$, what conditions do I need on $F$ or $M$ or $M'$ to make sure that $F(\partial M) = \partial M'$?
Or is this simply something that I must specify? Thanks
 A: Every homeomorphism between two bounded open sets $M$ and $M'$ in $\mathbb R^n$ induces a  boundary correspondence in the following sense: 


*

*$x\in \partial M$ corresponds to the cluster set $$\{y\in \mathbb R^n: \exists (x_n) \to x \text{ such that } F(x_n)\to y\}$$


The following properties hold: 


*

*Each cluster set is a nonempty closed subset of $\partial M'$

*The union of cluster sets over $x\in\partial M$ is equal to $\partial M'$.


Proofs:  


*

*"Nonempty and closed" are easy to see. Suppose $y$ is an element of a cluster set and $y\notin \partial M'$. Then $y\in M'$. Let $U$ be a neighborhood of $y$ such that $\overline{U}\subset M'$. The set $F^{-1}(\overline{U})$ is a compact subset of $M$. Any sequence $x_n$ in $M$ that approaches the boundary of $M$  will eventually leave $F^{-1}(\overline{U})$; thus, $F(x_n)$ will eventually be outside of $U$. Contradiction. 

*Take $y\in\partial M'$, pick a sequence $y_n\to y$ in $M'$, then choose a convergent subsequence of $F^{-1}(y_n)$.
So far,  no assumptions on $F$ were needed besides it being a homeomorphism. If you want to have boundary correspondence as a map,   point-goes-to-a-point, then add the assumption that for every $x\in\partial M$ the limit $\lim_{z\to x}F(x)$ exists. This limit, which can also be denoted $F(x)$, satisfies $F(\partial M ) = \partial M'$ by the above.  
