  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).
