Metric structure and dimensionality over a Borel set via uniform spaces

Abstract

We introduce a pregeometry that provides a metric and dimensionality over a Borel set (Wheeler's "bucket of dust") without assuming probability amplitudes for adjacency. Rather, a non-trivial metric is produced over a Borel set X per a uniformity base generated via the discrete topological group structures over X. We show that entourage multiplication in this uniformity base mirrors the underlying group structure. One may exploit this fact to create an entourage sequence of maximal length whence a fine metric structure. Unlike the statistical approaches of graph theory, this method can suggest dimensionality over low-order sets. An example over Z2 x Z4 produces 3-dimensional polyhedra embedded in E4.