A Wave Model of Metric Spaces

Abstract

Let Ω be a metric space. By A^t we denote the metric neighborhood of radius t of a set A ⊂ Ω and by \(\mathfrak{D}\), the lattice of open sets in Ω with partial order ⊆ and order convergence. The lattice of \(\mathfrak{D}\)-valued functions of t ∈ (0, ∞) with pointwise partial order and convergence contains the family I\(\mathfrak{D}\) = {A(·)| A(t) = A^t, A ∈ \(\mathfrak{D}\)}. Let ̃Ω be the set of atoms of the order closure \(\overline{I\mathfrak{D}}\). We describe a class of spaces for which the set ̃Ω equipped with an appropriate metric is isometric to the original space Ω.

The space ̃Ω is the key element of the construction of the wave spectrum of a lower bounded symmetric operator, which was introduced in a work of one of the authors. In that work, a program for constructing a functional model of operators of the aforementioned class was laid down. The present paper is a step in the realization of this program.