Perfect pre-images of \(\omega _ 1\) and the PFA
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 \(\omega_ 1\).
scientific article; zbMATH DE number 4071873
scientific article