The wave model of metric spaces

Abstract

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 $\tilde{\Omega }$ be the set of atoms of the order closure $\overline{I\mathfrak{O}}$. We describe a class of spaces for which the set $\tilde{\Omega }$, equipped with an appropriate metric, is isometric to the original space $\Omega$. The space $\tilde{\Omega }$ 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.