The wave model of metric spaces

Let $\Omega$ be a metric space, $A^t$ denote the metric neighborhood of the set $A\subset\Omega$ of the radius $t$; ${\mathfrak O}$ be the lattice of open sets in $\Omega$ with the partial order $\subseteq$ and the order convergence. The lattice of $\mathfrak O$-valued functions of $t\in(0,\infty)$ with the point-wise partial order and convergence contains the family ${I\mathfrak O}=\{A(\cdot)\,|\,\,A(t)=A^t,\,\,A\in{\mathfrak O}\}$. Let $\widetilde\Omega$ be the set of atoms of the order closure $\overline{I\mathfrak O}$. We describe a class of spaces for which the set $\widetilde\Omega$, equipped with an appropriate metric, is isometric to the original space $\Omega$. The space $\widetilde\Omega$ is the key element of the construction of the wave spectrum of a symmetric operator semi-bounded from below, which was introduced in a work of one of the authors. In that work, a program of constructing a functional model of operators of the aforementioned class was devised. The present paper is a step in realization of this program.