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Assume we have two closed subsets $F$ and $G$ of $\mathbb{C}^*$ which are proper for the multiplication, i.e. $$KF^{-1}\cap G$$ is a compact of $\mathbb{C}^*$ when $K$ is a compact of $\mathbb{C}^*$. The aim here is to find a closed neighbourhood $N_F$ of $F$ and $N_G$ of $G$ such that $N_F$ and $N_G$ remain proper for the multiplication.

Here is my try. For each connected component $i$ of $\mathbb{C}^*\backslash F$ we choose a $z_i$ and $r_i$ such that $\overline{D(z_i,r_i)}$ is included in that component. We set $$N_F = \bigcap_{i\in CC(\mathbb{C}^*\backslash F)} \mathbb{C}^*\backslash D(z_i,r_i).$$ This is clearly a closed neighbourhood of $F$ and we can do the same for $G$. But with that construction I can't proove that $N_F$ and $N_G$ are proper for the multiplication. I guess I have some lattitude in the choice of $z_i$ and $r_i$ (I can take the disks "big" as possible) but even with that, it seems complicated to me.

Edit : With further reflexion, I think the only problem are the non-bounded components of $\mathbb{C}^*\backslash F$. I start thinking that the property is false, so if someone can help to find a counterexample.

Any help will be much appreciated.

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