A foliation is taut if and only if it has no dead-end component

If a dead-end region $N$ exists, choose one boundary leaf. A hypothetical closed transversal through it can be oriented positively; its transverse direction then points inward at every boundary crossing. It enters $N$ but cannot exit, by the boundary defining-coordinate argument in F1. Boundary crossings are isolated and finite on the compact parameter circle; an entry with no exit contradicts periodicity. Therefore this boundary leaf meets no closed transversal, and the foliation is not taut. This uses a transversal through that leaf only, not a presumed transversal through every leaf.

Conversely, if the foliation is not taut, F1 gives a leaf meeting no closed transversal. F2 constructs its strict accessible closure $W$ as a compact proper region with nonempty finite compact-leaf boundary and inward positive directions. Thus $W$ is a dead-end region in the precise sense of F1. This argument works inside the ambient component of the chosen leaf and so also on a disconnected manifold.

For an arbitrary dead-end region the same entry/exit argument of step 1.1 applies to every boundary leaf. Its boundary is a compact embedded manifold locally equal to one plaque, so a finite boundary-chart cover supplies finitely many compact leaves, as in F2. In dimension three with ambient orientation these are cooriented oriented surfaces, and F3 makes each one a torus. The proof therefore uses the finite local accessibility and Euler suppliers rather than invoking Goodman's argument or an unproved definition theorem.