Saturated neighbourhoods of a leaf

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

An open set $U\subseteq M$ is saturated, or invariant, for $F$ when it is a union of leaves of $F$; equivalently, when for every $x\in U$ the whole leaf of $F$ through $x$ is contained in $U$. The equivalence is immediate from the definitions: a union of leaves has the pointwise property, and conversely the set of leaves meeting $U$ covers $U$ by the pointwise property, so $U$ is their union.

A saturated neighbourhood of $L$ is a saturated open set $U\subseteq M$ with $L\subseteq U$. If $U$ is saturated then $F$ restricts to a regular foliation of $U$: restrict the foliation charts to their intersections with $U$ and take connected components of their slices as plaques. These restricted charts form an atlas of the open submanifold $U$ (Regular foliation atlases). Since $U$ contains every leaf meeting it, every plaque chain in such a leaf remains in $U$, so the restricted leaves are exactly the leaves of $F$ meeting $U$ (Leaves of a regular foliation).

The neighbourhood conclusion of Reeb stability below is stated as the existence, for every neighbourhood $W$ of $L$ in $M$, of a saturated neighbourhood $U\subseteq W$ of $L$; this property of $L$ is recorded separately below as stability of the leaf.