I'm reading a book and have come across something that I cannot verify or fix. The assumption is that $\Omega_1, \Omega_2, ...$ is a sequence of connected open sets in $\mathbb{R}^n$ that converge in Hausdorff distance to some open connected set $\Omega$. After this the text reads "For convenience, we let $\phi_j : \Omega \to \Omega_j$ be diffeomorphisms such that the $\phi_j$ converge to the identity in a suitable topology."
My question is why can we say that these diffeomorphisms exist or how can we tweak the assumption to make them exist.
To illustrate my concern, here's an example that shows that, as stated, the above cannot be true. Let $B(a, r)$ denote the ball in $\mathbb{R}^n$ with center $a$ and radius $r$. Let $S_n = B(0, 1) - \overline{B(0, \frac{1}{n})}$. Then $S_n$ converges to (or accumulates to) $B(0,1)$ in Hausdorff distance. But for no $n$ are $S_n$ and $B(0,1)$ diffeomorphic, in fact for no $n$ are they homeomorphic since one can compute the homology groups for $S_n$ and $B(0,1)$ and see they are different.
I suppose that you use this book. It actually gives reference to this paper. In that one the theorem has additional condition. Namely "Assume that $\Omega$ and $\Omega_j$ are topologically equivalent."
Additionaly in the book there is plenty of phrases "smoothly bounded." Maybe somewhere in the beginning there is next assumption on domains.
Lastly, the author gives a refrence to one more book. In there Appendix A is about Hausdorff metric and there is some stuff which looks helpful. See for yourself:
Unfortunetly it is not exactly what you wanted, but I would GUESS that with this addition conditions (topological equivalence and smooth boundary) we would be able to say:
"For convenience, we let $\phi_j : \Omega \to \Omega_j$ diffeomorphisms such that the $\phi_j$ converge to the identity in a suitable topology."