# Generalized projections of Borel and analytic sets

Colloquium Mathematicae (1996)

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

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topBalcerzak, 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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- [P] Gy. Petruska, On Borel sets with small covers: a problem of M. Laczkovich, Real Anal. Exchange 18 (1992-93), 330-338. Zbl0783.28001
- [R] A. Rosłanowski, Mycielski ideals generated by uncountable systems, Colloq. Math. 66 (1994), 187-200. Zbl0833.04002
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- [So] S. Solecki, Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), 1022-1031. Zbl0808.03031

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