Isometric model of metric spaces

Let Ω be a metric space and $\mathfrak{D}$ be the lattice of its open sets with the partial order ⊆, equipped with the order topology. For a set A ⊆ Ω, let A^t denote its metric t-neighborhood. In the lattice of O-valued functions on (0, +∞), consider the family $I\mathfrak{D} = \left\{ {A\left( \cdot \right)\left| {A\left( t \right)} \right. = {A^t} \in \mathfrak{D}} \right\}$. Let $\tilde \Omega $ be the set of atoms of its closure, $\overline {I\mathfrak{D}} $, in the topology of point-wise convergence. Under certain conditions one can define a metric on $\tilde \Omega $. We describe a class of metric spaces Ω such that $\tilde \Omega $ is isometric to Ω.