Local transversals to a regular foliation

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the standing choice assumption of the smooth-distribution setting (Smooth distributions on a manifold). Let $F$ be a regular foliation of codimension $q$ on a smooth $n$-manifold $M$ (Regular foliation atlases); by Regular foliations and integrable distributions correspond its tangent distribution $D=TF$ is the integrable rank-$(n-q)$ smooth subbundle of $TM$ whose maximal connected integral manifolds are the leaves.

A subset $T\subseteq M$ is a local transversal to $F$ at $x\in T$ when $T$ is an embedded submanifold of $M$ of dimension $q$ containing $x$ (Embedded submanifolds and slice charts, Codimension and hypersurfaces) with

$T_{x}M=D_x\oplus T_{x}T\mathrm{.}$

Equivalently, $T_{x}T$ is a linear complement of $D_x$ in $T_{x}M$. Since $\dim T_{x}T=q$ and $\dim D_x=n-q$, the sum condition alone already forces the sum to be direct and to fill $T_{x}M$; writing it as a direct sum records both clauses at once. The condition is imposed at the single point $x$: it is a local transversality condition at $x$, and it says exactly that $T$ meets the leaf through $x$ transversely at $x$. In particular a codimension-one submanifold meeting a leaf tangentially at $x$ is not a local transversal to $F$ at $x$. The definition specialises the general transversality of a submanifold to the leaf distribution $D$; the atlas convention and the dimension $q$ come from the foliation, so no new structure is introduced.