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