A linearly fibered Souslinean space under MA (Q2701854)

The purpose of the paper is to present a~general method for constructing a~c.c.c.\ non-separable compact space under Martin's axiom which does not map onto \([0,1]^{\aleph_1}\). The author constructs Boolean algebras from gaps in a~quotient \(\mathcal P(\mathbb N)/\mathcal I\) and finds some restrictions which must be placed on the gaps to prove some assertions about the Boolean algebras and their Stone spaces. Then he constructs a~gap so that the Stone space of the associated Boolean algebra maps continuously onto \([0,1]\) with linear fibers. By a~well-known result of Shapirovskij this compact space cannot be mapped onto \([0,1]^{\aleph_1}\). A~different construction of a~compact space that does not map onto \([0,1]^{\aleph_1}\) was given by \textit{S. Todorčević} [Chain-condition methods in topology, Topology Appl. 101, 45-82 (2000)].