Lattices and Their Continuum Limits

TL;DR

This paper investigates how Hausdorff lattices can approximate a topological space through projective limits, providing a framework to recover the space and its continuous functions, with extensions to noncommutative lattices in future work.

Contribution

It introduces a projective system framework for approximating topological spaces and their function algebras using Hausdorff lattices, and dualizes the construction for continuous functions.

Findings

Established a universal space via projective limits of lattices

Demonstrated how to recover the topological space from the lattice system

Outlined extension to noncommutative lattices in subsequent research

Abstract

We address the problem of the continuum limit for a system of Hausdorff lattices (namely lattices of isolated points) approximating a topological space $M$. The correct framework is that of projective systems. The projective limit is a universal space from which $M$ can be recovered as a quotient. We dualize the construction to approximate the algebra ${\cal C}(M)$ of continuous functions on $M$. In a companion paper we shall extend this analysis to systems of noncommutative lattices (non Hausdorff lattices).