Forcing with a coherent Souslin tree and locally countable subspaces of countably tight compact spaces
The paper is devoted mainly to a proof, from \(\roman{PFA}(S)[S]\), of the following (technical) statement, denoted~\(\sum\): Assume \(X\)~is compact and countably tight and let \(Y\) be a subset of \(X\) of size~\(\aleph_1\). Also let two sequences, \(\langle V_\alpha:\alpha\in\omega_1\rangle\) and \(\langle W_\alpha:\alpha\in\omega_1\rangle\), of open sets be given such that \(\overline{W_\alpha}\subseteq V_\alpha\), \(V_\alpha\cap Y\) is countable, and \(Y\subseteq\bigcup_\alpha W_\alpha\). Then \(Y\)~is \(\sigma\)-closed-discrete in \(\bigcup_\alpha W_\alpha\). The principle \(\roman{PFA}(S)\) is a version of the Proper Forcing Axiom obtained by considering (only) proper partial orders that preserve one fixed Souslin tree \(S\) and \(\roman{PFA}(S)[S]\) refers to any model obtained by forcing with \(S\) over a model of~\(\roman{PFA}(S)\). The statement \(\sum\) underlies many proofs of results on the structure of countably tight spaces. The paper also neatly surveys these results and some further background on \(\roman{PFA}(S)[S]\).
scientific article; zbMATH DE number 6506288
scientific article