Level sets are isosurfaces of an implicit function F : R3 \3 R, that is the set of points satisfying F(x, y, z) = O. In this paper we introduce an algorithm to move at interactive speed through the different level sets. Furthermore the meshes of the level sets are isotopic to the isosurface itself, as long as the surface stays away from singularities. When the surface moves close to singularities, the algorithm indicates arbitrarily small boxes where the topology is not certified. In this case, the user can decide to decrease the size of the boxes by further refinement. For special classes of functions, such as algebraic surfaces, other methods could be used to determine the topology inside the singular boxes.