### On Monotone Presentations of Borel Sets

Tamás Mátrai and Miroslav Zelený
Source: Real Anal. Exchange Volume 34, Number 2 (2008), 311-318.

#### Abstract

If $A$ is a ${\bf \Sigma}^{0}_{\xi}$ set and $A_{n}$ $(n \omega)$ are Borel sets then we call $\{A_{n} \colon n \omega\}$ a presentation of $A$ if $A = \bigcup_{n \omega}A_{n}$ and $A_{n}$ $(n \omega)$ have lower Borel class than $A$ has. We show that for $2 \leq \xi \omega_{1}$ it is not possible to assign a presentation to ${\bf \Sigma}^{0}_{\xi}$ sets in a monotone way; i.e., it is not possible to define functions $f_{n} \colon {\bf \Sigma}^{0}_{\xi} \rightarrow {\bf \Pi}^{0}_{\xi}$ $(n \omega)$ such that for every $A \in {\bf \Sigma}^{0}_{\xi}$ we have $A = \bigcup_{n \omega}f_{n}(A)$ and $A, A' \in {\bf \Sigma}^{0}_{\xi}$, $A \subseteq A'$ implies $f_{n}(A) \subseteq f_{n}(A')$ $(n \omega)$. This answers a question of M\'arton Elekes in the negative. We also show the nonexistence of monotone presentation for Borel functions.

First Page:
Primary Subjects: 03E15
Secondary Subjects: 54H05, 28A05, 28A10, 26A21, 54C50
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Permanent link to this document: http://projecteuclid.org/euclid.rae/1256835189
Mathematical Reviews number (MathSciNet): MR2569189
Zentralblatt MATH identifier: 1187.03038

### References

A. Andretta, G. Hjorth, I. Neeman, Effective cardinals of boldface pointclasses, J. Math. Log., 7(1) (2007), 35–82.
Mathematical Reviews (MathSciNet): MR2348578
Zentralblatt MATH: 1149.03036
Digital Object Identifier: doi:10.1142/S0219061307000615
M. Elekes, Linearly ordered families of Baire 1 functions, Real Anal. Exchange, 27(1) (2001/02), 49–63.
Mathematical Reviews (MathSciNet): MR1887681
Zentralblatt MATH: 1014.26010
Project Euclid: euclid.rae/1212763950
M. Elekes, A. Máthé, Can we assign the Borel hulls in a monotone way?, Fund. Math., to appear.
Mathematical Reviews (MathSciNet): MR2545447
Zentralblatt MATH: 1189.28002
Digital Object Identifier: doi:10.4064/fm205-2-2
D. H. Fremlin, Consequences of Martin's Axiom, Cambridge Tracts in Mathematics, 84, Cambridge University Press, Cambridge, 1984.
Mathematical Reviews (MathSciNet): MR780933
S. Gao, S. Jackson, M. Laczkovich, R. D. Mauldin, On the unique representation of families of sets, Trans. Amer. Math. Soc., 360(2) (2008), 939–958.
Mathematical Reviews (MathSciNet): MR2346478
Zentralblatt MATH: 1144.54025
Digital Object Identifier: doi:10.1090/S0002-9947-07-04243-2
G. Hjorth, An Absoluteness Principle for Borel Sets, J. Symbolic Logic, 63(2) (1998), 663–693.
Mathematical Reviews (MathSciNet): MR1625891
Zentralblatt MATH: 0909.03042
Digital Object Identifier: doi:10.2307/2586857
G. Hjorth, Cardinalities in the projective hierarchy, J. Symbolic Logic, 67(4) (2002), 1351–1372.
Mathematical Reviews (MathSciNet): MR1955242
Zentralblatt MATH: 1052.03024
Digital Object Identifier: doi:10.2178/jsl/1190150289
Project Euclid: euclid.jsl/1190150289
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.
Mathematical Reviews (MathSciNet): MR1321597
Zentralblatt MATH: 0819.04002
P. Komjáth, Ordered families of Baire-2-functions, Real Anal. Exchange, 15 (1989/90), 442–444.
Mathematical Reviews (MathSciNet): MR1059415
R. Laver, Linear orders in $(\omega)\sp\omega$ under eventual dominance, Logic colloquium '78 (Mons, 1978), Stud. Logic Foundations Math., 97, North-Holland, Amsterdam-New York, 1979, 299–302.
Mathematical Reviews (MathSciNet): MR567675
Zentralblatt MATH: 0464.03046