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BASES AND BOREL SELECTORS FOR TALL FAMILIES

Published online by Cambridge University Press:  30 January 2019

JAN GREBÍK  and
CARLOS UZCÁTEGUI
JAN GREBÍK
Affiliation:
INSTITUTE OF MATHEMATICS ACADEMY OF SCIENCES OF THE CZECH REPUBLIC ŽITNÁ 609/25, 110 00PRAHA 1-NOVÉ MĚSTO CZECH REPUBLIC E-mail: grebik@math.cas.cz
CARLOS UZCÁTEGUI
Affiliation:
ESCUELA DE MATEMÁTICAS UNIVERSIDAD INDUSTRIAL DE SANTANDER CRA. 27 CALLE 9 UIS EDIFICIO 45 BUCARAMANGA, COLOMBIAE-mail: cuzcatea@saber.uis.edu.co
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Abstract

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Given a family ${\cal C}$ of infinite subsets of ${\Bbb N}$, we study when there is a Borel function $S:2^{\Bbb N} \to 2^{\Bbb N} $ such that for every infinite $x \in 2^{\Bbb N} $, $S\left( x \right) \in {\Cal C}$ and $S\left( x \right) \subseteq x$. We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an $F_\sigma $ tall ideal on ${\Bbb N}$ without a Borel selector. The proof is not constructive since it is based on complexity considerations. We construct a ${\bf{\Pi }}_2^1 $ tall ideal on ${\Bbb N}$ without a tall closed subset.

Keywords

Primary 03E1503E05Borel idealstall familiesBorel selectorGalvin’s Lemma
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 359 - 375
Copyright
Copyright © The Association for Symbolic Logic 2019 

References

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