transversality
When two submanifolds meet, they can meet cleanly or tangentially. Two lines in a plane crossing at a point is a clean, stable meeting — wiggle them and they still cross at one point. A line tangent to a circle is a fragile meeting — wiggle it and the single touch becomes two crossings or none. Transversality is the precise condition for the clean, stable kind of intersection, and it is the differential-topology substitute for 'general position'.
Two submanifolds A and B of M meet transversally at a point p of their intersection if their tangent spaces together span the whole ambient tangent space: T_p A + T_p B = T_p M (the sum need not be direct). More generally a smooth map f: M -> N is transverse to a submanifold S if at every p with f(p) in S, the image of df_p together with T_{f(p)} S spans T_{f(p)} N. The payoff is a theorem: if f is transverse to S, then the preimage f^{-1}(S) is a submanifold of M whose codimension equals the codimension of S — the regular-value theorem is exactly the case where S is a single point. When A + B does not fill T_p M the intersection can be any mess; transversality forces it to be a submanifold of the expected dimension dim A + dim B - dim M.
Two facts make transversality the workhorse of the subject. First, stability: transverse intersections persist under small perturbations, so the intersection number is a robust invariant — this is the basis of degree theory and intersection theory. Second, genericity (the Thom transversality theorem, powered by Sard): any smooth map can be perturbed by an arbitrarily small amount to become transverse to a given S, so one may always assume transversality after a generic perturbation. The honest caveat is dimensional: if dim A + dim B is less than dim M, transversality forces the intersection to be empty (negative expected dimension means no intersection), which is correct but easy to forget — two generic curves in 3-space do not meet at all.
In R^3 a generic plane and a generic line meet transversally in a single point: the plane's 2-dimensional tangent and the line's 1-dimensional tangent span all 3 dimensions, so 2 + 1 - 3 = 0, a 0-manifold. If the line lies in the plane they are non-transverse and the intersection is the whole line instead.
A plane and a line in R^3 generically meet transversally in a point.
Transversality with dim A + dim B < dim M forces an EMPTY intersection — non-meeting is the transverse, generic outcome, not a failure. Transversality is a condition only at points that actually lie in the intersection.