ABSTRACT. We first go over constructions which induce topologies on subsets of a fixed domain or hold-all D in ¥Ln by using set parametrized functions in an appropriate function space. Secondly we concentrate on the family of oriented distance functions (algebraic or signed distance functions) which play an important role in the introduc tion of topologies which retain the classical geometric properties associated with sets: convexity, exterior normals, mean curvature, C k boundaries, etc . . .