Metric structure and dimensionality over a Borel set via uniform spaces

TL;DR

This paper introduces a novel method to define metric and dimensionality structures over Borel sets using uniform spaces and group structures, bypassing probabilistic assumptions and enabling new insights into low-order set dimensions.

Contribution

It presents a new pregeometry framework that constructs metrics and dimensions over Borel sets via uniformity bases derived from group structures, without relying on probability amplitudes.

Findings

A non-trivial metric can be generated over Borel sets using uniformity bases.

The method can suggest dimensionality over low-order sets, unlike traditional statistical graph approaches.

An example demonstrates a 3D polyhedron embedded in 4D space from a Z2 x Z4 set.

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.