Foliation components as mutual positive transverse-accessibility classes

Let F be a C² cooriented codimension-one foliation on a smooth manifold M without boundary. Its foliation component containing a leaf A is the saturated subset ${\mathcal{C}}_F(A)=\bigcup \{B:B\text{ is a leaf},A{⪰}_{F}B\text{ and }B{⪰}_{F}A\}\mathrm{.}$ The equivalence classes of leaves under mutual positive transverse accessibility partition the leaf set; their unions partition M into saturated subsets. This is Novikov’s connected component of the foliation. It is not defined as an ordinary connected component of M or M minus a leaf, and it differs from one-direction positive reach alone. A boundary leaf L of a distinct component means $L\cap {\mathcal{C}}_F(A)=\mathrm{\varnothing }$ and $L\subseteq {\mathrm{\partial }}_M{\mathcal{C}}_F(A)$, with boundary in the ambient topology. This definition does not assert that a component’s closure is a manifold with boundary.

The partition assertion also has a finite, choice-free justification, so it does not require importing the standing ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis of the cited preorder lemma. Two points of a connected leaf are joined by a finite plaque path: the points reachable by finite plaque paths and their complement are open in the leaf. Smooth the finitely many corners. Given a genuine positive segment and such initial and terminal plaque paths, choose a positive transverse field and defining form near their compact images using finitely many chart bumps. For its flow, the transverse error along a displaced leafwise path is bounded by $A\mathrm{\mid }u\mathrm{\mid }$, whereas its offset contributes at least $b{u}^{\mathrm{\prime }}$ for some $b>0$. Choose offsets with $u^{\mathrm{\prime }}=B\mathrm{\mid }u\mathrm{\mid }+\delta$ and $bB>A$, vanishing at the desired outer endpoints, and interpolate their small endpoint values along the original positive segment. This adjusts both endpoints and allows two genuine positive segments to concatenate; chartwise smoothing keeps their transverse derivatives positive. Equality cases are immediate. Transitivity, formal reflexivity and symmetry of mutual accessibility now give an equivalence relation, whose classes partition the leaves and whose saturated unions partition $M$ without choosing class representatives.