Weight, net weight, and elementary submodels

Abstract

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 $C\left({\omega }_1\right)$ 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 $C\left({\omega }_1\right)$ has countable net weight.

In the last section we prove in ZFC the following two statements:

(i) If $S\subset {\omega }_1$ 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 ${\omega }_1$, if all final segments of ${\omega }_1$ have uncountable weight then ${\omega }_1$ 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.