## Fernando Hern\'andez-Hern\'andez

*A tree $\pi $-base for $\Bbb R^{\ast }$ without cofinal branches *

Comment.Math.Univ.Carolinae 46,4 (2005) 721-734. **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

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