Let $M$ be a manifold that has a transversely orientable foliation with leaves of codimension one. I am trying to show that the union of all compact leaves is closed and open.
I got the following guidelines:
1) Show that there exists a vector field transversal to the foliation, whose flow $\{\Phi_{t}\}$ takes leaves to leaves.
2) Let $N$ be the union of all compact leaves. If $L$ is a compact leaf, then show that $\bigcup_{t\in(-\epsilon,\epsilon)}\Phi_{t}(L)$ is an open neighborhood of $L$ contained in $N$.
I managed to prove $1)$, but I don't see why the set $\bigcup_{t\in(-\epsilon,\epsilon)}\Phi_{t}(L)$ is open. I appreciate your help.
Let $L$ be a compact leaf and $x\in L$, there exists an open subset $U$ containing $x$ such that $U$ is a chart of $M$ (i.e $U$ is diffeomorphic to an open subset of $R^n$, $x=0$) and the restriction of ${\cal F}$ to $U$ is simple and defined by $x_n=c$ on $U$ where $(x_1,...,x_n)$ are coordinates. Let $X$ be the vector transverse, and $\phi_t$ its flow, $\bigcup_{t\in (-\epsilon,\epsilon)\phi_t(L_x)=\bigcup_{t\in (-\epsilon,\epsilon)}(y_n=c_t)}$ where $y_n=c_t$ is the equation of the leaf $\phi_t(L_x)$. Since $X$ is transverse, $\{c_t, t\in (-\epsilon,\epsilon)\}$ contains an open interval which contains $0$. This implies that $\bigcup_{t\in (-\epsilon,\epsilon)}\phi_t(L_x)$ is open since it contains a neighborhood of $x$ for every $x\in L$.