A stratification of a topological space is a finite filtration by closed subsets Fi , such that the difference between successive members Fi and F(i − 1) of the filtration is either empty or a smooth submanifold of dimension i.
Let X and Y be two disjoint (locally closed) submanifolds of Rn, of dimensions i and j. John Mather first pointed out that Whitney's condition B implies Whitney's condition A in the notes of his lectures at Harvard in 1970, which have been widely distributed.
David Trotman showed in his 1977 Warwick thesis that a stratification of a closed subset in a smooth manifold M satisfies Whitney's condition A if and only if the subspace of the space of smooth mappings from a smooth manifold N into M consisting of all those maps which are transverse to all of the strata of the stratification, is open (using the Whitney, or strong, topology).
More general singular spaces can be given Whitney stratifications, such as semialgebraic sets (due to René Thom) and subanalytic sets (due to Heisuke Hironaka).
In a thesis under the direction of Wieslaw Pawlucki at the Jagellonian University in Kraków, Poland, the Vietnamese mathematician Ta Lê Loi proved further that every definable set in an o-minimal structure can be given a Whitney stratification.