  Hart and Kunen, and independently in the recent preprint arXiv:2304.13113,
R\'ios-Herrej\'on 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. In
this paper we present several independence results concerning the relationships
of these and several other classes that are sandwiched between them. These
clarify some of the main problems that were raised in the above preprint.
  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.
