Super properties and net weight (Q2307617)

In what follows, all topological spaces are \(T_3\) (i.e., Hausdorff and regular) and we assume the reader is familiar with \textit{Martin's Axiom}, \(\mathbf{MA}(\kappa)\), and with the \textit{Continuum Hypothesis}, \(\mathbf{CH}\). The cardinality of a set \(X\) is denoted by \(|X|\). A \textit{network} for a topological space \(X\) is a family \(\mathcal{N}\) of (non-necessarily open) subsets of \(X\) such that every open set of \(X\) is the union of a subfamily of \(\mathcal{N}\). The \textit{net weight} of \(X\), denoted by \(nw(X)\), is the least cardinality of a network for \(X\). Every base is a network and \(\{\{x\}: x \in X\}\) is a network, so clearly \(nw(X) \leqslant \min\{|X|,w(X)\}\) -- where \(w(X)\) (the \textit{weight} of \(X\)) is the least cardinality of a base of \(X\). Given a topological space \(X\) and a cardinal \(\kappa\), a \textit{\(\kappa\)-assignment} for \(X\) is a sequence \(\mathcal{U} = \{(x_\alpha,U_\alpha): \alpha < \kappa\}\), where each \(U_\alpha\) is open in \(X\) and \(x_\alpha \in U_\alpha\). An \textit{assignment} for \(X\) is an \(\omega_1\)-assignment. A topological space \(X\) is said to be HG (\textit{Hereditarily Good}) if for all assignments \(\mathcal{U}\) of \(X\) there are \(\alpha \neq \beta\) such that \([x_\beta \in U_\alpha\,\, \&\,\, x_\alpha \in U_\beta]\). The property HG was introduced in [\textit{J. E. Hart} and \textit{K. Kunen}, Topol. Proc. 55, 147--174 (2020; Zbl 1444.54016)] and it is a natural strengthening of the properties \(HS\) (i.e. Hereditarily Separable) and \(HL\) (i.e. Hereditarily Lindelöf). A topological space \(X\) is \textit{strongly} HG, or \(\text{st}\)HG, if all finite powers of \(X\) are HG (equivalently, \(X^\omega\) is HG), and it is said to be \textit{super} HG, or \(\text{su}\)HG, if for all assignments for \(X\) there is some \(S\subseteq\omega_1\) with \(|S| = \aleph_1\) such that for all \(\alpha, \beta \in S\) one has \([x_\beta \in U_\alpha\,\, \&\,\, x_\alpha \in U_\beta]\). \(\text{su}\)HG trivially implies HG and, as finite products of \(\text{su}\)HG spaces are \(\text{su}\)HG spaces, one also has that \(\text{su}\)HG implies \(\text{st}\)HG. It was previously shown by the authors in [loc. cit.] that \(\mathbf{CH}\) produces an example of a \(\text{st}\)HG space that is not \(\text{su}\)HG; however, in the very same paper it was established that \textbf{MA}\((\aleph_1)\) implies that every \(\text{st}\)HG space is \(\text{su}\)HG. In the paper under review, the authors proceed with the investigation of super properties. The main results of the paper are the following: (i) \(\mathbf{MA}(\kappa)\) implies that every \(\text{st}\)HG space with \(|X| \leqslant \kappa\) and \(w(X) \leqslant \kappa\) has countable net weight. The paper includes a brief proof of the above result and remarks that it is essentially a consequence of Theorem 2.1 of [\textit{I. Juhász} et al., Commentat. Math. Univ. Carol. 37, No. 1, 159--170 (1996; Zbl 0862.54003)]. (ii) If \(nw(X) \leqslant \aleph_0\) then \(X\) is \(\text{su}\)HG. The above result is an easy consequence of a pigeonhole argument. It follows that spaces with countable net weight are trivial examples of \(\text{su}\)HG spaces. Nevertheless, in the paper also the following is proved: (iii) It is consistent with \(\mathbf{ZFC}\) to have \textbf{MA} and \(\mathfrak{c}=\aleph_2\) and a first countable \(\text{su}\)HG space \(X\) with \(|X| = w(X) = nw(X) = \aleph_2\).