PFA and \(\omega_1\)-free compact spaces
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 lemmas: The space \(X\) has countable tightness (by a result of \textit{I. Juhász} and \textit{Z. Szentmiklóssy} [Proc. Am. Math. Soc. 116, No. 4, 1153--1160 (1992; Zbl 0767.54002)]. One may (and does) assume, wlog, \(X\) is separable. \(\vert X\vert \leq\mathfrak{c}\), using \textit{Z. Balogh} [Proc. Am. Math. Soc. 105, No. 3, 755--764 (1989; Zbl 0687.54006)] that PFA implies every compact space of countable tightness is sequential. \par Upon forcing with Fn\(({\omega_1},2,{\aleph_1})\), the space \(X\) remains compact. The space \(X\) is ultra-Fréchet, also upon forcing with Fn\((\omega_1,2,\aleph_1)\). The assumptions that \(X\) is compact, separable, not first-countable, and ultra-Fréchet, before and after forcing with Fn\((\omega_1,2,\aleph_1)\), imply that \(X\) has a convergent \(\omega_1\)-sequence. (The authors call a space \(Z\) \textit{ultra-Fréchet} ``if it has countable tightness and for each countable subset \(D\) of \(Z\) and each free ultrafilter \(\mathcal{U}\) on \(D\) that converges in \(Z\) there is a countable subfamily \(\mathcal{U}'\) of \(\mathcal{U}\) with the property that every infinite pseudointersection of \(\mathcal{U}'\) converges.'') Some portions of the proofs of the lemmas use prior results and joint results of the authors and others.
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