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I have to be honest that I am very lost on the same kind of problem about proving existence of smooth function. I have not done much topology, hence unfamiliar with the "covering" business.

In the proof of problem 4 of the provided solution. The solution is brief and I am having trouble in writing the mathematics in detail. Could anyone help in providing a more complete/detailed solution. I am very unclear (in term of writing the mathematics formally) in the sentence "For each $i$, we apply problem 2 and find a $w_i\equiv 1$ on $W_i$ and $0$ near the boundary $\partial V_i$." I know this is the $C^{\infty}$ version of Usysohn's lemma. But I am having difficulty in linking things well in writing proper sentences due to my lack of training in topology.

enter link description here Appreciate for any helps.

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  • $\begingroup$ What does $\subset\subset$ mean in this text? $\endgroup$ – DisintegratingByParts Jul 7 '15 at 5:08
  • $\begingroup$ compactly embedded $\endgroup$ – math101 Jul 7 '15 at 5:11
  • $\begingroup$ Because $\{W_i\}_{i=1}^{N}$ a covering of U. Every point $x\in U$ is in $W_i$ for some $i$. So $\sum_i w_i(x)\neq 0$ $\endgroup$ – Euler88 ... Jul 7 '15 at 5:21
  • $\begingroup$ $w_{i} \equiv 1$ on $W_{i}$ and $U\subset\subset\cup W_{i}$. So $\sum_{i} w_{i}$ does not vanish on $U$. Was that your question? $\endgroup$ – DisintegratingByParts Jul 7 '15 at 5:22
  • $\begingroup$ Thank you both of you! What bothered me most is as edited... $\endgroup$ – math101 Jul 7 '15 at 10:05

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