Some consequences of MA + ¬wKH

Abstract

Let wKH denote the following statement: There exists a ω1-tree of power ω1 with >ω1 ω1-branches. First, using methods of [6] and [13], we shall prove theconsistency of MA + ¬wKH. Then we shall prove that MA + ¬wKH implies the following: (a) There is no θ-dense (splitting) poset of power ω1; (b) Every LOTS of density ω1 has a θ-disjoint π-base; (c) There is no Baire LOTS of power ω1 without isolated points; (d) Every perfectly normal non-Archimedian space of weight ω1 is metrizable. These results are connected to problems from [4], [10], [7] and [15], respectively. A part of these results was announced in [17].