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Super properties and net weight. (English) Zbl 1444.54017
In what follows, all topological spaces are \(T_3\) (i.e., Hausdorff and regular) and we assume the reader is familiar with Martin’s Axiom, \(\mathbf{MA}(\kappa)\), and with the Continuum Hypothesis, \(\mathbf{CH}\). The cardinality of a set \(X\) is denoted by \(|X|\).
A 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 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 weight of \(X\)) is the least cardinality of a base of \(X\).
Given a topological space \(X\) and a cardinal \(\kappa\), a \(\kappa\)-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 assignment for \(X\) is an \(\omega_1\)-assignment. A topological space \(X\) is said to be HG (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 [J. E. Hart and 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 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 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 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 [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 MA and \(\mathfrak{c}=\aleph_2\) and a first countable \(\text{su}\)HG space \(X\) with \(|X| = w(X) = nw(X) = \aleph_2\).
MSC:
54D70 Base properties of topological spaces
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A35 Consistency and independence results in general topology
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