Let (M, d) be a complete topological 2-manifold, possibly with boundary, with a geodesic metric d.Let X ∪ M be a compact set. We show then that for all but countably many ε each component of the set S(X, ε)of points ε-distant from X is either a point, a simple closed curve disjoint from dM or an arc A such that A ∩M consists of both endpoints of A and that arcs and simple closed curves are dense in S(X, ε). In particular, if the boundary dM of M is empty, then each component of the set S(X,ε) is either a point or a simple closed curve and the simple closed curves are dense in S(X, ε). © 2008 American Mathematical Society.