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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!

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