Smooth maps transverse to a regular foliation

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the standing assumption of the smooth-distribution setting (Smooth distributions on a manifold). Let $F$ be a regular foliation of $M$ with tangent distribution $D=TF$, the integrable smooth subbundle of $TM$ supplied by Regular foliations and integrable distributions correspond (Vector subbundles). Let $N$ be a smooth manifold and $f:N\to M$ a smooth map with differential $d{f}_x:T_{x}N\to T_{f(x)}M$ (The differential of a smooth map).

Then $f$ is transverse to $F$ at $x\in N$ when

$d{f}_x(T_{x}N)+D_{f(x)}=T_{f(x)}M,$

and transverse to $F$, written $f⋔F$, when this holds at every $x\in N$. When $N\subseteq M$ and $f$ is the inclusion of an embedded submanifold, the condition reads $T_{x}N+D_x=T_{x}M$ at every $x\in N$: this is the pointwise sum condition of Transverse smooth maps applied with the leaf distribution $D$ in place of the tangent space of a second submanifold, so the definition specialises the published transversality of smooth maps to the leaf distribution. The condition forces $\dim N\ge \mathrm{codim}F$ at every point, since $\dim (d{f}_x(T_{x}N))\le \dim N$ and the sum with the $(n-q)$-dimensional space $D_{f(x)}$ fills the $n$-dimensional space $T_{f(x)}M$. When $\mathrm{codim}F=0$ the condition is automatic, because then $D_{f(x)}=T_{f(x)}M$.