I. Juh\'asz, L. Soukup, Z. Szentmikl\'ossy
Forcing countable networks for spaces satisfying $R(X^{\omega })={\omega }$

Comment.Math.Univ.Carolinae 37,1 (1996) 157-168.

Abstract: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 cannot get rid of using generic extensions, (ii) we have to consider all finite powers of $X$.

Keywords: net weight, weakly separated, Martin's Axiom, forcing
AMS Subject Classification: 54A25, 03E35