Generalized side-conditions and Moore-Mrówka
type: Item
instance of: scholarly article
title: Generalized side-conditions and Moore-Mrówka
zbMATH Open document ID: 1359.54004
author: Alan Dow
published in: Topology and its Applications
publication date: 2015-12-08T00:00:00Z
review text: 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 gen…
reviewed by: David J. Fernández-Bretón
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54G12
zbMATH DE Number: 6517543
zbMATH Keywords: countable tightness
zbMATH Keywords: sequential space
zbMATH Keywords: compact space
zbMATH Keywords: Martin's axiom
zbMATH Keywords: proper forcing
MaRDI profile type: Publication
full work available at URL: j.topol.2015.10.016
OpenAlex ID: W2126641060
cites work: On Compact Hausdorff Spaces of Countable Tightness
cites work: Q3472092
cites work: A first countable, initially \(\omega _1\)-compact but non-compact space
cites work: Minimally generated Boolean algebras
cites work: On families of Lindelöf and related subspaces of~\(2^{\omega_1}\)
cites work: Forcing minimal extensions of Boolean algebras
cites work: Necessary use of induction in a reversal
cites work: On Countably Compact, Perfectly Normal Spaces
cites work: An 𝜔₂-minimal Boolean algebra
DOI: 10.1016/J.TOPOL.2015.10.016
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description: scientific article; zbMATH DE number 6517543