Compact leaves with finite holonomy form an open saturated set

(Saturated and nonempty.) The set $S$ is a union of leaves, hence saturated by [F1], and it contains $L$ because $L$ is compact and diffeomorphic to itself; in particular $S$ is nonempty.

(Openness in the codimension-one case.) Let $x\in S$ and let $L_x$ be the leaf of $F$ through $x$; by definition of $S$, $L_x$ is compact and diffeomorphic to $L$, so ${\pi }_1(L_x)$ is finite; by [F2] the holonomy of $L_x$ is trivial and $L_x$ has a saturated product neighbourhood $U\cong L_x\times D$ all of whose leaves are compact and diffeomorphic to $L_x$, hence diffeomorphic to $L$. Therefore $U\subseteq S$, and since $x$ was arbitrary, $S$ is open [F1].

(General regular case.) Let $F$ be any regular foliation and let $S^{\mathrm{\prime }}\subseteq M$ be the union of the compact leaves with finite holonomy group. It is saturated as a union of leaves, and if $x\in S^{\mathrm{\prime }}$ then its leaf $L_x$ is compact with finite holonomy, so local Reeb stability supplies a saturated neighbourhood $U$ of $L_x$ all of whose leaves are compact with finite holonomy [F3]; hence $U\subseteq S^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}$ is open. In the transversely oriented codimension-one situation, $S$ is a subcollection of these leaves, and its openness follows specifically from the common diffeomorphism type argument in step 1.2. Trivial holonomy does not imply finite fundamental group.