Let Omega be a metric space and D be the lattice of its open sets with the partial order subset of, equipped with the order topology. For a set A subset of Omega, let A(t) denote its metric t-neighborhood. In the lattice of D-valued functions on (0, + infinity), consider the family ID = {A (.) vertical bar A(t) = A(t), A is an element of D}. Let (Omega) over tilde be the set of atoms of its closure, (ID) over bar, in the topology of pointwise convergence. Under certain conditions one can define a metric on (Omega) over tilde. We describe a class of metric spaces Omega such that (Omega) over tilde is isometric to Omega.