The intersection of a transversal section and a plaque of a foliation I have the following question about the transversal sections of a foliation. Let $M$ be a manifold and $F$ a foliation of codimension $q$. We know that if $\phi:U \to U_1 \times U_2 \in F$ then the sets $D= \phi^{-1}(\\{x\\} \times U_2)$ and $P = \phi^{-1}(U_1 \times \\{y\\})$ have exactly one intersection point and $D$ is transversal to the foliation $F$. Is the same afirmation valid for any transversal section of $F$? If $T$ is a transversal section of $F$ and $T$ intersects a plaque $P$ then $T \cap P$ have exactly one point? Thanks in advance.

It depends on what you mean by transverse section.

In general if $T$ is allowed to be an immersed submanifold the it does not work. If you think about the torus being foliated by $S^1 \times $ {$\theta$} with $\theta \in S^1 $ and $T:\mathbb{R} \to S^1\times S^1 $ with irrational slope then T is transverse. However, for each plaque $P$ the intersection $T(\mathbb{R})\cap P subset P$ is dense.

If you require $T$ to be embedded, you should always be able to choose the plaque small enough such that it holds.