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 questions that were raised in [6] . First we give a ZFC example of a Hausdorff space in C ( ω 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 ( ω 1 ) has countable net weight. In the last section we prove in ZFC the following two statements: (i) If S ⊂ ω 1 is stationary then for any regular topology on S of uncountable weight S has a non-stationary subset that has uncountable weight as well. (ii) For any topology on ω 1 , if all final segments of ω 1 have uncountable weight then ω 1 has a non-stationary subset of uncountable weight. In contrast to this, it was shown in [6] that the analogous statements for net weight are not provable in ZFC. It is remarkable that all our proofs of the above results make essential use of elementary submodels.