  We consider finite approximations of a topological space $M$ by
noncommutative lattices of points. These lattices are structure spaces of
noncommutative $C^*$-algebras which in turn approximate the algebra $\cc(M)$ of
continuous functions on $M$. We show how to recover the space $M$ and the
algebra $\cc(M)$ from a projective system of noncommutative lattices and an
inductive system of noncommutative $C^*$-algebras, respectively.
