Topological invariants in the Cohen model
type: Item
instance of: scholarly article
title: Topological invariants in the Cohen model
zbMATH Open document ID: 0774.54001
DOI: 10.1016/0166-8641(92)90050-A
author: Juris Steprāns
published in: Topology and its Applications
publication date: 1992-09-27T00:00:00Z
review text: 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.
reviewed by: Alan Dow
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 03C20
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 62943
MaRDI profile type: Publication
cites work: Q4150804
cites work: Q3994998
cites work: Homogeneity problems in the theory of Cech compactifications
cites work: Proper forcing
cites work: PFA Implies all Automorphisms are Trivial
cites work: Non-trivial homeomorphisms of βN\N without the continuum hypothesis
full work available at URL: 0166-8641(92)90050-a
OpenAlex ID: W2162304332
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description: scientific article; zbMATH DE number 62943