Approximation of subanalytic sets by normal cones

In this paper, a notion of ‘approximation of order s ’ (called ‘ s ‐equivalence’) between two closed subanalytic subsets of ℝ n along a common submanifold is introduced. It is proved that the normal cone N X ( A ) to A along X is 1‐equivalent to A along X , assuming that X is a stratum of a stratification of A satisfying Verdier's condition ( w ). Furthermore, the normal cone is shown to be a complete invariant for the classes of 1‐equivalence of subanalytic sets along a common stratum.