Hart and Kunen and, independently, Ríos-Herrejón defined and studied the class $C({\omega }_1)$ of topological spaces $X$ having the property that for every neighborhood assignment $U(y):y\in Y$ with $Y\in [X{]}^{{\omega }_1}$ there is $Z\in [Y{]}^{{\omega }_1}$ such that $Z\subset \bigcap U(z):z\in Z.$ It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. We present several independence results concerning the relationships of these two and several other natural classes that are sandwiched between them.

In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in $C({\omega }_1)$ has countable net weight, answering a question that was raised by Hart and Kunen.

These results are joint with L. Soukup and Z. Szentmiklossy.