Geodesic
Forcing countable networks for spaces satisfying $\operatorname{R}(X^\omega)=\omega$
Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 1, pp. 159-170

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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 $X$.
Classification : 03E35, 54A25, 54A35
Keywords: net weight; weakly separated; Martin's Axiom; forcing 
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     title = {Forcing countable networks for spaces satisfying $\operatorname{R}(X^\omega)=\omega$},
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     pages = {159--170},
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Juhász, I.; Soukup, L.; Szentmiklóssy, Z. Forcing countable networks for spaces satisfying $\operatorname{R}(X^\omega)=\omega$. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 1, pp. 159-170. http://geodesic.mathdoc.fr/item/CMUC_1996__37_1_a10/