Stable leaves

Let $F$ be a regular foliation of a smooth manifold $M$ (Regular foliation atlases) and let $L$ be a leaf of $F$.

The leaf $L$ is stable when every open neighbourhood $W$ of $L$ in $M$ contains a saturated neighbourhood of $L$ (Saturated neighbourhoods of a leaf); equivalently, when the saturated neighbourhoods of $L$ form a fundamental system of neighbourhoods of $L$.

More generally, a subset $B\subseteq M$ is stable in the sense of Reeb when for every open neighbourhood $W$ of $B$ there is an open neighbourhood $W^{\mathrm{\prime }}\subseteq W$ of $B$ such that every leaf of $F$ meeting $W^{\mathrm{\prime }}$ is contained in $W$. For a leaf $B=L$ this is equivalent to stability of $L$: if such a $W^{\mathrm{\prime }}$ exists, then its saturation $U=\bigcup \{L^{\mathrm{\prime }}:L^{\mathrm{\prime }}\text{ a leaf and }{L}^{\mathrm{\prime }}\cap W^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }\}$ is open and saturated (each box carries an open set to its open plaque saturation, and finite plaque transport carries this property along every leaf), contains $L$, and satisfies $U\subseteq W$ by the property of $W^{\mathrm{\prime }}$, so $U$ is a saturated neighbourhood of $L$ inside $W$; conversely a saturated neighbourhood $U\subseteq W$ of $L$ is itself a neighbourhood $W^{\mathrm{\prime }}\subseteq W$ of $L$ every leaf meeting which is contained in $U\subseteq W$.

Stability of a leaf is a neighbourhood property: it depends only on the germ of the foliation along $L$, since both quantifiers involve neighbourhoods of $L$. This is the property that the local and global Reeb stability theorems below establish under their finiteness hypotheses.