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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
##### Keywords:
weakly separated; forcing; network; net weight; butterfly
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