On two topological cardinal invariants of an order-theoretic flavour (Q714720)

Following [\textit{S. A. Peregudov}, Commentat. Math. Univ. Carol. 38, No. 3, 581--586 (1997; Zbl 0937.54003)], the author defines for a topological space \(X\) the \textit{Noetherian type of} \(X\), denoted \(\operatorname{Nt}(X)\), as the minimum cardinal \(\kappa\) such that every \(\subseteq\)-bounded subfamily of a base for the space has cardinality strictly less than \(\kappa\). He also defines \(\pi \operatorname{Nt}(X)\) in the obvious way using a \(\pi\)-base instead of a base. The author improves a theorem by D. Milovich showing that the Noetherian \(\pi\)-type of a \(\kappa\)-Souslin line is \(\kappa^+\) whenever \(\kappa<\aleph_\omega\); he remarks that the general case remains open. He also studies the effect of Chang's conjecture for \(\aleph_\omega\) to compute the Noetherian \(\pi\)-type of countable supported box products of some regular spaces. For instance, assuming Chang's conjecture for \(\aleph_\omega\), the \(\sigma\)-product of \(2^{\aleph_\omega}\) with the topology induced by the countable supported boxes has Noetherian \(\pi\)-type equal to \(\aleph_2\). He ends the paper studying the Noetherian type of Pixley-Roy hyperspaces. A nice result he presents, is that the gap between the Noetherian type of \(X\) and that of its Pixley-Roy hyperspace can be as large as the gap between \(\aleph_4\) and \(\aleph_\alpha\), for any \(\alpha<\omega_1\).