Geodesic
A tree $\pi $-base for $\Bbb R^\ast$ without cofinal branches
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 721-734.

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We prove an analogue to Dordal's result in P.L. Dordal, {\it A model in which the base-matrix tree cannot have cofinal branches\/}, J. Symbolic Logic {\bf 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.
Classification : 03E17, 06E15, 54A35, 54G05
Keywords: distributivity of Boolean algebras; cardinal invariants of the continuum; Stone-Čech compactification; tree $\pi$-base 
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Hernández-Hernández, Fernando. A tree $\pi $-base for $\Bbb R^\ast$ without cofinal branches. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 721-734. http://geodesic.mathdoc.fr/item/CMUC_2005__46_4_a10/