Topological invariants in the Cohen model
One of the main results of this paper is a Parovičenko style structure theorem for \({\mathcal P}(N)/{\mathcal F}inite\) (or \(\beta N\setminus N\)) which is valid in the model obtained by adding \(\omega_ 2\) Cohen reals to a model of CH. The key property is that if \(X\subset N\) is added by Cohen forcing over \(V\), then the ideal \(\{I\in V\): \(I\subset X\}\) is countably generated. A consequence is that in this model, all \(P\)-points of \(\beta N\setminus N\) of the same topological type (as W. Rudin showed under CH). This in turn is applied to a question of Fremlin and Nyikos. An ultrafilter \(p\) is \((\kappa,\omega)\)-saturating if the ultrapower by \(p\) of each countable structure is \(\kappa^ +\)-saturated. Two \(P\)- points are constructed in the Cohen model exactly one of which is \((\omega_ 1,\omega)\)-saturating, whereas Fremlin and Nyikos had shown that \(\text{MA}(\omega_ 1)\) implies autohomeomorphisms of \(\beta N\setminus N\) preserve the property of being \((\omega_ 1,\omega)\)- saturating.
scientific article; zbMATH DE number 62943
scientific article