Lattices and Their Continuum Limits

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 $\mathcal{C}(M)$ of continuous functions on $M$. In a companion paper we shall extend this analysis to systems of noncommutative lattices (non Hausdorff lattices).

Comments: 11 pages, 1 Figure included in the LaTeX Source New version, minor modifications (typos corrected) and a correction in the list of authors