Generalized side-conditions and Moore-Mrówka
The Moore-Mrówka problem asks whether every compact Hausdorff countably tight space must be sequential. \textit{A. J. Ostaszewski} [J. Lond. Math. Soc., II. Ser. 14, 505--516 (1976; Zbl 0348.54014)] and \textit{V. V. Fedorchuk} [Sov. Math., Dokl. 16, 651--655 (1975); translation from Dokl. Akad. Nauk SSSR 222, 302--305 (1975; Zbl 0331.54029)] independently found counterexamples to Moore-Mrówka (that is, compact countably tight spaces that are not sequential) assuming \(\diamondsuit\), whereas \textit{Z. Balogh} [Proc. Am. Math. Soc. 105, No. 3, 755--764 (1989; Zbl 0687.54006)] proved that the Proper Forcing Axiom \(\mathsf{PFA}\) implies that there are no such counterexamples. In this paper, the author proves that it is consistent with \(\mathsf{ZFC}+\mathsf{MA}\) that there exists a first countable initially \(\omega_1\)-compact (a space is initially \(\omega_1\)-compact if every open cover of size at most \(\omega_1\) has a finite subcover) space of size strictly larger than \(2^\omega\). This statement is known to be inconsistent with \(\mathsf{PFA}\), and showing its consistency with \(\mathsf{ZFC}\) answers a question by Arhangelskii. Then, by taking the one-point compactification of such a space, a counterexample to Moore-Mrówka is obtained (this counterexample is also separable), thus also proving the consistency of (the existence of) such a counterexample with \(\mathsf{ZFC}+\mathsf{MA}\) (and \(2^\omega=\omega_2\)). A further important result obtained in this paper is that it is consistent with \(\mathsf{ZFC}+\mathsf{MA}\) that there exists a compact separable first-countable space which, after forcing with an \(\omega_2\)-Suslin tree, is not Lindelöf in the corresponding generic extension. The main technical tools used to establish these results are two: minimally generated systems of Boolean algebras indexed by a tree, and Neeman's machinery of forcing with (finite \(\in\)-chains of) elementary submodels as side conditions. A \(T\)-system of minimally generated Boolean algebras (where \(T\) is a subtree of \(2^{\omega_2}\)) is roughly a family \(\langle a_t\mid t\text{ is a successor node of }T\rangle\) of subsets of \(T\) that satisfies certain coherence conditions. Under these conditions, the \(T\)-system helps define a topology on the set \(X_T\) of branches through \(T\), and there are appropriate lemmas relating properties of the \(T\)-system with properties of the resulting space \(X_T\) to the effect that an appropriate \(T\)-system will yield an \(X_T\) with appropriate subspaces witnessing the existential statements of interest, even after further forcing to ensure that \(\mathsf{MA}\) is satisfied. Hence the strategy is to produce an appropriate \(T\)-system by forcing, and the forcing notion \(\mathbb P\) consists of approximations to the desired \(T\) carrying, as side conditions, finite \(\in\)-chains of elementary submodels of \(H(\omega_2)\), each of which can be of size \(\omega\) or \(\omega_1\), in the spirit of Neeman's technique. The core of the paper consists of the proofs of the necessary lemmas that ensure that this indeed works.
scientific article; zbMATH DE number 6517543
scientific article