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 (ω 1 , 1)-morass, then there exists a ccc forcing of size ω 1 that adds an ω 2 -Suslin tree. (2) If there is a simplified (ω 1 , 2)-morass, then there exists a ccc forcing of size ω 1 that adds a 0-dimensional Hausdorff topology τ on ω 3 which has spread s(τ) = ω 1 . 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 < ω 2 〉 such that X α ⊆ ω 1 , X β — X α is finite and X α — X β has size ω 1 for all β < α < ω 2