Cardinal invariants of dually CCC spaces (Q2029617)

The notion of the class \( P^* \) dual to a topological property \( P \) was introduced by [\textit{O. T. Alas} et al., Topol. Proc. 30(30), 25--38 (2006; Zbl 1127.54009)]. A neighborhood assignment for a space \( (X,\tau) \) is a function \( \phi:X \rightarrow \tau \) with \( x \in \phi(x) \); a set \( Y \subset X \) is a kernel of \( \phi \) if \( \phi(Y)=\{\phi(y): y \in Y \} \) covers \( X \). The class \( P^* \) consists of those spaces \( X \) such that each neighborhood assignment admits a kernel \( Y \) with property \( P \). In case \( P^*=P \), \( P \) is self-dual. In this paper, the authors consider the dual of the countable chain condition (CCC), weakly Lindelöf and separable. They show that: \begin{enumerate} \item The cardinality of a dually CCC (resp., dually weakly Lindelöf) first countable Hausdorff (resp., normal) space is at most \( 2^\mathfrak{c} \). \item Assuming \( 2^{<\mathfrak{c}}=\mathfrak{c} \), the extent of a normally dually CCC space with \( \chi (X) \leq \mathfrak{c} \) is at most \( \mathfrak{c} \), where \( \chi (X) \) is a character of \( X \). \item The extent of a dually separable Hausdorff space with a \( G_\delta^* \)-diagonal is at most \( \mathfrak{c} \). \item The cardinality (resp., cellularity) of a dually separable regular (resp., dually CCC Hausdorff) space with a \( G_\delta \)-diagonal is at most \( \mathfrak{c} \).\end{enumerate}