The class C(ω1) and countable net weight

Hart and Kunen in [5] and, independently, Ríos-Herrejón in [14] defined and studied the class C ( ω 1 ) of topological spaces X having the property that for every neighborhood assignment { U ( y ) : y ∈ Y } with Y ∈ [ X ] ω 1 there is Z ∈ [ Y ] ω 1 such that Z ⊂ ⋂ { U ( z ) : z ∈ 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 two and several other natural classes that are sandwiched between them, thus clarifying some of the main problems that were raised in [14] . In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick , implies that every regular space in C ( ω 1 ) has countable net weight, answering a question that was raised by Hart and Kunen in both [5] and [6] .