The class $C({\omega}_1)$ and countable net weight

TL;DR

This paper investigates the class of topological spaces with a specific neighborhood intersection property, exploring their relationship with countable net weight and showing that certain set-theoretic principles imply all regular spaces in this class have countable net weight.

Contribution

It provides independence results clarifying the relationship between the class $C({oldsymbol{ omannumeral1}})$ and countable net weight, and shows that super stick implies all regular spaces in the class have countable net weight.

Findings

Super stick principle implies all regular spaces in $C({oldsymbol{ omannumeral1}})$ have countable net weight.

The relationship between $C({oldsymbol{ omannumeral1}})$ and countable net weight is independent of ZFC.

Several independence results clarify main problems raised in prior work.

Abstract

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.