suppose $M$ is a smooth manifold, $O\subset M$ is an open domain and $K\subset O$ is some compact set. I am wondering if there always exists an open domain $D$ in $M$ with smooth boundary and such that $K\subset D\subset O$? Unfortunately I do not know how to approach this problem. Does anyone have an idea?
Best wishes
The answer is yes. Here's one way to do it.
Let $f\colon M\to \mathbb R$ be a smooth bump function that is identically equal to $1$ on $K$ and supported in $O$. (The existence of such a function can be proved easily using a partition of unity; see, e.g., Proposition 2.25 in my Introduction to Smooth Manifolds, 2nd ed.) By Sard's Theorem, there is a number $c\in (0,1)$ that is a regular value of $f$, and then the set $f^{-1}([c,1])$ is a domain with smooth boundary containing $K$ and contained in $O$.