Forcing countable networks for spaces satisfying $R(x^\omega)=\omega$

{From the paper:''(...) we use a result of Shelah (the proof
of which presented here with his kind permission) saying that
in any generic extension by Cohen reals the ideal of the first
category subspaces of a space from the ground model is generated
by the subspaces of first category from the ground model.},
}
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