A Geometric Proof of the Existence of Whitney Stratifications

A stratification of a set, e.g. an analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together “regularly”. Stratification theory was originated by Thom and Whitney for algebraic and analytic sets. It was one of the key ingredients in Mather’s proof of the topological stability theorem [Ma] (see [GM] and [PW] for the history and further applications of stratification theory). In this paper, given a partition of a singular set (which we know always exists), we prove that there is a “regular” partition. Our proof is based on a remark that if there are two parts of the partition V and W of different dimension and V ⊂ W , then irregularity of the partition at a point x in V corresponds to the existence of nonunique limits of tangent planes TyW as y approaches x. Consider either the category of (semi)analytic (or (semi)algebraic) sets. Call a subset V ⊂ R (or C) a semivariety if locally at each point x ∈ R (or C) it is a finite union of subsets defined by equations and inequalities