PFA and \(\omega_1\)-free compact spaces
type: Item
instance of: scholarly article
title: PFA and \(\omega_1\)-free compact spaces
zbMATH Open document ID: 1499.54116
author: Alan Dow
author: K. P. Hart
published in: Acta Mathematica Hungarica
publication date: 2022-03-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/2104.13698
review text: All spaces considered here are assumed to be Hausdorff, and all convergent sequences are assumed to be injective. In the late seventies, M.\ Hušek asked if every infinite compact space contains a convergent \(\omega\)- or \(\omega_1\)-sequence, and I.\ Juhász independently raised the stronger question as to whether every non-first-countable compact space contains a convergent \(\omega_1\)-sequence. \textit{I. Juhász} et al. [Topology Appl. 156, No. 10, 1863--1879 (2009; Zbl 1168.54003)] obtained a consistent counterexample to the second question by forcing a first-countable, normal, locally compact, initially \(\omega_1\)-compact, but non-compact, space whose one-point compactification is a non-first-countable compact space containing no convergent \(\omega_1\)-sequence. \textit{A. Dow} and \textit{K. P. Hart} [Fundam. Math. 224, No. 3, 205--218 (2014; Zbl 1297.54050)] showed that Juhász' question has a positive answer in a large class of ccc forcing extensions of models in which CH holds.\par In the article under review, the authors note that the preceding result does not include all ccc extensions, and so they begin by considering the status under MA of the dichotomy, every compact space is either first-countable or has a convergent \(\omega_1\)-sequence. They show, using a space constructed by \textit{A. Dow} [Topology Appl. 197, 75--101 (2016; Zbl 1359.54004)] that \(MA + \mathfrak{c}=\aleph_2\) is not strong enough to imply the dichotomy. They then proceed to state and present an impressive proof of their main result, Theorem 1: (PFA) Every compact space either contains a convergent \(\omega_1\)-sequence or is first-countable. Thus, they have proved the dichotomy is consistent with and independent of \(MA +\mathfrak{c} =\aleph_2\). The authors develop a proof of Theorem 1 by assuming PFA, letting \(X\) denote a compact space which is not first-countable and contains no convergent \(\omega_1\)-sequences, and then deriving a contradiction by proving the following…
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E57
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 7491522
zbMATH Keywords: PFA
zbMATH Keywords: compact
zbMATH Keywords: convergence
zbMATH Keywords: \(\omega_1\)-sequence
zbMATH Keywords: first-countable
reviewed by: Robert M. Stephenson Jr.
MaRDI profile type: Publication
OpenAlex ID: W4206996851
arXiv ID: 2104.13698
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DOI: 10.1007/S10474-021-01197-9
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description: scientific article