Forcings constructed along morasses

Abstract

We 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 size ω₁ that adds an ω₂-Suslin tree. (2) If there is a simplified (ω₁,2)-morass, then there exists a ccc forcing of size ω₁ that adds a 0-dimensional Hausdorff topology τ on ω₃ which has spread s(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 preserve GCH. 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 〈 X_α | α < ω₂〉 such that X_α ⊆ ω₁, X_β -X_α is finite and X_α-X_β has size ω₁ for all β < α <ω₂.