Are codimension one foliations of $\mathbb{R}^{n}-\{0\}$ with compact leaves, stable at origin? Assume that we  have  a  codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves.

Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of $0$, there is  a neighborhood $W\subset V$ containing $0$ such that the saturation of $W$ is  contained in $V$?

 A: Yes. Let $\mathcal{F}$ be the foliation (of arbitrary dimension in $\{1,\ldots,n-1\}$), which I assume regular and transversely continuous. It is sufficient to consider the case of a ball  $V$  of radius $r>0$. I claim that the set $A:=\mathrm{Sat_\mathcal{F}}(\partial{V})$ is compact, therefore $V\setminus A$ is a non-empty open set $W$, invariant by $\mathcal{F}$, giving the sought neighbourhood.
The fact that $A$ is compact follows from an elementary flow-box argument. For every $p\in \partial V$ there exists a small compact neighbourhood $V_p\ni p$ for which $\mathrm{Sat}_\mathcal{F}(V_p)$ is compact (the foliation is locally a $C^0$-fibration). By compactness of the sphere $\partial V$, the set $A$ is the union of finitely many such $\mathrm{Sat}_\mathcal{F}(V_p)$, thus compact.
Remark: The argument works as soon as one removes from $\mathbb{R}^n$ any compact  set $X$ and consider neighbourhoods $V$ of $X$. This hypothesis is somehow optimal: as the Hopf fibration shows, this is false for $n=3$ and $X$ a line.
