Among the several ideals, Grothendieck's Esquisse d’un programme considers (or proposes) a stratified space with what he calls the tame topology.
A prestratification on a topological space X is a partition of X into subsets (called strata) such that (a) each stratum is locally closed, (b) it is locally finite and (c) (axiom of frontier) if two strata A, B are such that the closure of A intersects B, then B lies in the closure of A.
[citation needed] In the MacPherson's stratified pseudomanifolds; the strata are the differences Xi+i-Xi between sets in the filtration.
There is also a local conical condition; there must be an almost smooth atlas where locally each little open set looks like the product of two factors Rnx c(L); a euclidean factor and the topological cone of a space L. Classically, here is the point where the definitions turns to be obscure, since L is asked to be a stratified pseudomanifold.
The logical problem is avoided by an inductive trick which makes different the objects L and X.