LATTICES AND THEIR CONTINUUM LIMITS

We address the problem of the continuum limit for a system of Hausdorf f lattices (namely lattices of isolated points) approximating a topolo gical space M. The correct framework is that of projective systems. Th e projective limit is a universal space from which M can be recovered as a quotient. We dualize the construction to approximate the algebra C(M) of continuous functions on M. In a companion paper we shall exten d this analysis to systems of noncommutative lattices (non-Hausdorff l attices).