Perfect pre-images of \(\omega _ 1\) and the PFA
type: Item
instance of: scholarly article
title: Perfect pre-images of \(\omega _ 1\) and the PFA
zbMATH Open document ID: 0656.54001
DOI: 10.1016/0166-8641(88)90072-7
author: David H. Fremlin
published in: Topology and its Applications
publication date: 1988-01-01T00:00:00Z
review text: The main result of the paper states that assuming the Proper Forcing Axiom (PFA), spaces of character \(\leq \omega_ 1\) being a closed preimage of an uncountable subspace of \(\omega_ 1\) and satisfying (rather a technical) condition \((+)\) contain a copy of \(\omega_ 1\). The following condition (\(\beta)\) is stronger than \((+)\) for any such space X: \((\beta)2^{\omega}=\omega_ 2\) and every subset A of X of size \(2^{\omega}\) contains a subset B of size \(\leq \omega\), such that \(A\subset U\{c1C:\) \(C\subset B\) and \(| C| \leq \omega \}\). Numerous consequences are derived from the main theorem. For example, under PFA, a manifold is metrizable iff it is hereditarily collectionwise Hausdorff and does not contain a copy of \(\omega_ 1\); or under \(PFA+2^{\omega}=\omega_ 2\), a compact Hausdorff space of size \(\omega_ 1\) has countable tightness iff it does not contain a copy of \(\omega_ 1\).
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 57N99
zbMATH DE Number: 4071873
zbMATH Keywords: Proper Forcing Axiom
zbMATH Keywords: character
zbMATH Keywords: hereditarily collectionwise Hausdorff
zbMATH Keywords: countable tightness
reviewed by: A. Szymański
MaRDI profile type: Publication
full work available at URL: 0166-8641(88)90072-7
OpenAlex ID: W2026196056
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description: scientific article; zbMATH DE number 4071873