# The decomposition of open set

If $U$ is an open set in $\mathbb R^n$, then there exists a sequence of open sets $\{U_i\}$, such that a.$U_i\subset \subset U_{i+1}$ (that is, ${\overline U _i}$ is compact and ${\overline U _i} \subset {U_{i + 1}}$), b.$U = \bigcup\limits_i {{U_i}}$.

My question is , can we choose all $U_i$ to have smooth boundary?

I come out this question as I was reading my PDE book, since we can in this way "approximate" a solution to an equation when $\partial U$ is rough.

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## 1 Answer

For any smooth manifold $M$, according to Morse Theory, we can find a smooth function $f$ on $M$ such that $f$ contains no degenerate critical point and for any $a$, $\{x \in M : f(x) \leq a\}$ is a compact set in $M$.

Now since non-degenerate critical points are isolated, we can choose a sequence of numbers $\{c_{i}\}$ such that $c_i \rightarrow \infty$ and the sets $\{x\in M:f(x)=c_i\}$ contain no critical points.

Therefore we can see sets $U_i=\{x\in M:f(x)< c_i\}$ satisify all the assumptions and especially they have smooth boundary $\{x\in M:f(x)=c_i\}$.

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