I. Juhász, L. Soukup, Z. Szentmiklóssy:
Forcing countable networks for spaces satisfying R(X^omega)=omega
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 0dimensional subspaces of countable weight.
We also show that
this theorem is sharp in two different senses:

we can't get rid of
using generic extensions,
 we have to consider all finite powers of X.
Downloading the paper