A tree π -base for ℝ * without cofinal branches

Fernando Hernández-Hernández

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 4, page 721-734
  • ISSN: 0010-2628

Abstract

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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 π -base for ℕ * with no ω 2 branches yet of height ω 2 . We establish that this is also possible for ℝ * using a natural modification of Mathias forcing.

How to cite

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Hernández-Hernández, Fernando. "A tree $\pi $-base for $\mathbb {R}^\ast $ without cofinal branches." Commentationes Mathematicae Universitatis Carolinae 46.4 (2005): 721-734. <http://eudml.org/doc/249519>.

@article{Hernández2005,
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 $\mathbb \{N\}^\{\ast \}$ with no $\omega _\{2\}$ branches yet of height $\omega _\{2\}$. We establish that this is also possible for $\mathbb \{R\}^\{\ast \}$ using a natural modification of Mathias forcing.},
author = {Hernández-Hernández, Fernando},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {distributivity of Boolean algebras; cardinal invariants of the continuum; Stone-Čech compactification; tree $\pi $-base; distributivity of Boolean algebras; cardinal invariants of the continuum; Stone-Čech compactification; tree -base},
language = {eng},
number = {4},
pages = {721-734},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A tree $\pi $-base for $\mathbb \{R\}^\ast $ without cofinal branches},
url = {http://eudml.org/doc/249519},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Hernández-Hernández, Fernando
TI - A tree $\pi $-base for $\mathbb {R}^\ast $ without cofinal branches
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 4
SP - 721
EP - 734
AB - 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 $\mathbb {N}^{\ast }$ with no $\omega _{2}$ branches yet of height $\omega _{2}$. We establish that this is also possible for $\mathbb {R}^{\ast }$ using a natural modification of Mathias forcing.
LA - eng
KW - distributivity of Boolean algebras; cardinal invariants of the continuum; Stone-Čech compactification; tree $\pi $-base; distributivity of Boolean algebras; cardinal invariants of the continuum; Stone-Čech compactification; tree -base
UR - http://eudml.org/doc/249519
ER -

References

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  11. Shelah S., Spinas O., The distributivity numbers of 𝒫 ( ø m e g a ) / f i n and its square, Trans. Amer. Math. Soc. 352 5 (2000), 2023-2047 (electronic). (2000) Zbl0943.03036MR1751223
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