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 $V$?
> 
> 
> 



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


