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 $X$.
@article{118820,
author = {Istv\'an Juh\'asz and Lajos Soukup and Zolt\'an Szentmikl\'ossy},
title = {Forcing countable networks for spaces satisfying $\operatorname{R}(X^\omega)=\omega$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {37},
year = {1996},
pages = {159-170},
zbl = {0862.54003},
mrnumber = {1396168},
language = {en},
url = {http://dml.mathdoc.fr/item/118820}
}
Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán. Forcing countable networks for spaces satisfying $\operatorname{R}(X^\omega)=\omega$. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 159-170. http://gdmltest.u-ga.fr/item/118820/
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