In this work one proves that under quite general assumptions one can deduce the topology of a bounded open set in ℝn from an approximation of it. For this, one introduces the weak feature size (wfs) that extends for nonsmooth objects the notion of local feature size. Our results apply to open sets with positive wfs. This class includes subanalytic open sets which cover many cases encountered in practical applications. The proofs are based upon the study of distance functions to closed sets and their critical points. The notion of critical point is the same as the one used in riemannian geometry [22], [9], [20] and nonsmooth analysis [10]. As an application, one gives a way to compute the homology groups of open sets from noisy samples of points on their boundary.