Weight, net weight, and elementary submodels

In this note we prove several theorems that are related to some results and problems from [6]. We answer two of the main problems that were raised in [6]. First we give a ZFC example of a Hausdorff space in $C(\omega_1)$ that has uncountable net weight. Then we prove that after adding any number of Cohen reals to a model of CH, in the extension every regular space in $C(\omega_1)$ has countable net weight. We prove in ZFC that for any regular topology of uncountable weight on $\omega_1$ there is a non-stationary subset that has uncountable weight as well. Moreover, if all final segments of $\omega_1$ have uncountable weight then the assumption of regularity can be dropped. By [6], the analogous statements for the net weight are independent from ZFC. Our proofs of all these results make essential use of elementary submodels.