A geometric proof of the existence of Whitney stratifications

In this paper we give a simple geometric proof of existence of so-called Whitney stratification for (semi)analytic and (semi)algebraic sets. Roughly, stratification is a partition of a singular set into manifolds so that these manifolds fit together “regularly”. The proof presented here does not use analytic formulas only qualitative considerations. It is based on a remark that if there are two manifolds of the partition $V$ and $W$ of different dimension and $V\subset\overline W$, then irregularity of the partition at a point $x$ in $V$ corresponds to the existence of nonunique limits of tangent planes $T_yW$ as $y$ approaches $x$.