Forcings constructed along morasses

In a previous paper [11], we introduced a way of constructing a forcing along a simplified (κ, 1)-morass such that the forcing satisfies a chain condition. The basic idea is to generalize iterated forcing with finite support as introduced by Solovay and Tennenbaum, which works with continuous, commutative systems of complete embeddings. However, instead of considering a linear system of embeddings, we take a two-dimensional system. These two-dimensional systems behave in some ways like forcing iterations, in other respects they do not. In the present paper, the theory of these higher-dimensional systems is further developed. We generalize the approach of [11] to three-dimensional systems constructed along simplified (κ, 2)-morasses. Moreover, we observe that the forcings obtained in these twoor three-dimensional constructions are usually equivalent to the respective forcing constructed along a subsystem of size κ. An immediate consequence of this and [11] is: If there is a simplified (ω1, 1)-morass, then there exists a ccc forcing of size ω1 that adds an ω2-Suslin tree. The main result is: 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. By a theorem of Hajnal and Juhasz, card(X) ≤ 2 s(X) holds for all Hausdorff spaces X. Since a forcing as ours preserves GCH , our result answers the open question if card(X) = 2 s(X) is consistent for a regular space X with s(X) = ω1. However, there are many statements about ω2 whose consistency one would naturally prove with a ccc forcing but which contradict GCH . Therefore, we will also discuss how to change our approach so that this becomes possible. As an example, we construct a ccc forcing which adds a chain 〈Xα | α < ω2〉 such thatXα ⊆ ω1, Xβ−Xα is finite andXα−Xβ has size ω1 for all β < α < ω2. Such a forcing was first found by Koszmider using only 2.