FORCING COUNTABLE NETWORKS FOR SPACES SATISFYING R(X ! ) = ω

We show that all finite powers of a Hausdorff space X do not contain uncountable weakly separated subspaces iff there is a c.c.c poset P such that in V P X is a countable union of 0-dimensional subspaces of countable weight. We also show that this theorem is sharp in two different senses: (i) we can't get rid of using generic extensions, (ii) we have to consider all finite powers of X.