Generalized projections of Borel and analytic sets

Marek Balcerzak

Colloquium Mathematicae (1996)

  • Volume: 71, Issue: 1, page 47-53
  • ISSN: 0010-1354

Abstract

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For a σ-ideal I of sets in a Polish space X and for A ⊆ X 2 , we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where A x =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a ∑ 2 0 -supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [ ∑ 1 1 ( X 2 ) ] = ∑ 1 1 ( X ) for a wide class of ∑ 2 0 -supported σ-ideals.

How to cite

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Balcerzak, Marek. "Generalized projections of Borel and analytic sets." Colloquium Mathematicae 71.1 (1996): 47-53. <http://eudml.org/doc/210426>.

@article{Balcerzak1996,
abstract = {For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^2$, we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where $A_x$ =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a $∑_\{2\}^\{0\}$-supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [$∑_\{1\}^\{1\}(X^2)]=∑_\{1\}^\{1\}(X)$ for a wide class of $∑_\{2\}^\{0\}$-supported σ-ideals.},
author = {Balcerzak, Marek},
journal = {Colloquium Mathematicae},
keywords = {meager set; Effros Borel structure; analytic set; σ-ideal; -ideal},
language = {eng},
number = {1},
pages = {47-53},
title = {Generalized projections of Borel and analytic sets},
url = {http://eudml.org/doc/210426},
volume = {71},
year = {1996},
}

TY - JOUR
AU - Balcerzak, Marek
TI - Generalized projections of Borel and analytic sets
JO - Colloquium Mathematicae
PY - 1996
VL - 71
IS - 1
SP - 47
EP - 53
AB - For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^2$, we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where $A_x$ =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a $∑_{2}^{0}$-supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [$∑_{1}^{1}(X^2)]=∑_{1}^{1}(X)$ for a wide class of $∑_{2}^{0}$-supported σ-ideals.
LA - eng
KW - meager set; Effros Borel structure; analytic set; σ-ideal; -ideal
UR - http://eudml.org/doc/210426
ER -

References

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  1. [B] M. Balcerzak, Can ideals without ccc be interesting? Topology Appl. 55 (1994), 251-260. Zbl0795.54052
  2. [BR] M. Balcerzak and A. Rosłanowski, On Mycielski ideals, Proc. Amer. Math. Soc. 110 (1990), 243-250. Zbl0708.04002
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  4. [Ke] A. S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1994. 
  5. [KLW] A. S. Kechris, A. Louveau and W. H. Woodin, The structure of σ-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 263-288. Zbl0633.03043
  6. [KS] A. S. Kechris and S. Solecki, Approximation of analytic by Borel sets and definable chain conditions, Israel J. Math. 89 (1995), 343-356. Zbl0827.54023
  7. [Ku] K. Kuratowski, Topology, Vols. 1, 2, PWN and Academic Press, Warszawa and New York, 1966, 1968. 
  8. [Mo] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980. 
  9. [My] J. Mycielski, Some new ideals of sets on the real line, Colloq. Math. 20 (1969), 71-76. Zbl0203.05701
  10. [P] Gy. Petruska, On Borel sets with small covers: a problem of M. Laczkovich, Real Anal. Exchange 18 (1992-93), 330-338. Zbl0783.28001
  11. [R] A. Rosłanowski, Mycielski ideals generated by uncountable systems, Colloq. Math. 66 (1994), 187-200. Zbl0833.04002
  12. [Sh] R. M. Shortt, Product sigma-ideals, Topology Appl. 23 (1986), 279-290. Zbl0594.28002
  13. [So] S. Solecki, Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), 1022-1031. Zbl0808.03031

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