In differential topology, transversality formalizes the idea of a generic intersection between two manifolds. It consists in asking an infinitesimal condition, namely on the tangent spaces, to be satisfied everywhere.
Definition. Let $f\colon M\rightarrow N$ be a $C^1$ map between two manifolds and let $A$ be a submanifold of $N$. One says that $f$ is transversal over $A$ if and only if for all $x\in f^{-1}(A)$, $\mathrm{d}_xf(T_xM)+T_{f(x)}A=T_{f(x)}N$.
For example, a submersion between two manifolds is transversal over any submanifold of its goal.
References.
M. Hirsch, Differential topology, Springer, 1976.
V. Guillemin, A. Pollack, Differential topology, American Mathematical Society, 1974
