Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets.
In particular we want compact sets $K_j$'s and open sets $V_j$'s such that $K_j\subset V_j$ for each $j$ and \begin{equation}
\Omega=\cup K_j.
\end{equation}
Moreover, we may ask for more property like each compact subset of $\Omega$ intersects only finitely many $V_j$'s.
I have met such kind of construction many times, in real variable as well in Riemannian geometry, and there seem to be many ways to do such a partition. But now I guess it may help to have a collection of all possible ways to accomplish this, together with some special property of each partition.
I hope you guys can help me with this.
Thanks!