Abstract:We prove an analogue to Dordal's result in P.L. Dordal, {A model in which the base-matrix tree cannot have cofinal branches}, J. Symbolic Logic {52} (1980), 651--664. He obtained a model of ZFC in which there is a tree $\pi $-base for $\Bbb N^{\ast }$ with no $\omega _{2}$ branches yet of height $\omega _{2}$. We establish that this is also possible for $\Bbb R^{\ast }$ using a natural modification of Mathias forcing.
Keywords: distributivity of Boolean algebras, cardinal invariants of the continuum, Stone-\v {C}ech compactification, tree $\pi $-base
AMS Subject Classification: Primary 54G05, 54A35, 03E17, 06E15