Forcings constructed along morasses

Abstract

AbstractWe further develop a previously introduced method of constructing forcing notions with the help of morasses. There are two new results: (1) If there is a simplified (ω1, 1)-morass, then there exists a ccc forcing of sizeω1that adds an ω2-Suslin tree. (2) If there is a simplified (ω1, 2)-morass, then there exists a ccc forcing of sizeω1that adds a 0-dimensional Hausdorff topologyτonω3which has spreads(τ) =ω1. While (2) is the main result of the paper, (1) is only an improvement of a previous result, which is based on a simple observation. Both forcings preserveGCH. To show that the method can be changed to produce models where CH fails, we give an alternative construction of Koszmider's model in which there is a chain 〈Xα∣α<ω2〉 such thatXα⊆ω1.Xβ–Xαis finite andXα–Xβhas sizeω1for allβ<α<ω2.