# Cofinal ${\Sigma}_{1}^{1}$ and ${\Pi}_{1}^{1}$ subsets of ${\omega}^{\omega}$

Gabriel Debs; Jean Saint Raymond

Fundamenta Mathematicae (1999)

- Volume: 159, Issue: 2, page 161-193
- ISSN: 0016-2736

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topDebs, Gabriel, and Saint Raymond, Jean. "Cofinal $Σ^1_1$ and $Π^1_1$ subsets of $ω^ω$." Fundamenta Mathematicae 159.2 (1999): 161-193. <http://eudml.org/doc/212327>.

@article{Debs1999,

abstract = {We study properties of $∑^1_1$ and $π^1_1$ subsets of $ω^ω$ that are cofinal relative to the orders ≤ (≤*) of full (eventual) domination. We apply these results to prove that the topological statement “Any compact covering mapping from a Borel space onto a Polish space is inductively perfect” is equivalent to the statement "$∀α ∈ω^ω, ω^ω ∩ L(α )$ is bounded for ≤*".},

author = {Debs, Gabriel, Saint Raymond, Jean},

journal = {Fundamenta Mathematicae},

language = {eng},

number = {2},

pages = {161-193},

title = {Cofinal $Σ^1_1$ and $Π^1_1$ subsets of $ω^ω$},

url = {http://eudml.org/doc/212327},

volume = {159},

year = {1999},

}

TY - JOUR

AU - Debs, Gabriel

AU - Saint Raymond, Jean

TI - Cofinal $Σ^1_1$ and $Π^1_1$ subsets of $ω^ω$

JO - Fundamenta Mathematicae

PY - 1999

VL - 159

IS - 2

SP - 161

EP - 193

AB - We study properties of $∑^1_1$ and $π^1_1$ subsets of $ω^ω$ that are cofinal relative to the orders ≤ (≤*) of full (eventual) domination. We apply these results to prove that the topological statement “Any compact covering mapping from a Borel space onto a Polish space is inductively perfect” is equivalent to the statement "$∀α ∈ω^ω, ω^ω ∩ L(α )$ is bounded for ≤*".

LA - eng

UR - http://eudml.org/doc/212327

ER -

## References

top- [1] J. Brendle, G. Hjorth and O. Spinas, Regularity properties for dominating projective sets, Ann. Pure Appl. Logic 72 (1995), 291-307. Zbl0824.03025
- [2] J. P. R. Christensen, Necessary and sufficient conditions for the measurability of certain sets of closed sets, Math. Ann. 200 (1973), 189-193. Zbl0233.28002
- [3] G. Debs and J. Saint Raymond, Compact covering and game determinacy, Topology Appl. 68 (1996), 153-185. Zbl0848.54024
- [4] W. Just and H. Wicke, Some conditions under which tri-quotient or compact-covering maps are inductively perfect, ibid. 55 (1994), 289-305. Zbl0794.54019
- [5] A. Louveau, A separation theorem for ${\sum}_{1}^{1}$ sets, Trans. Amer. Math. Soc. 260 (1980), 363-378.
- [6] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.
- [7] A. V. Ostrovskiĭ, On new classes of mappings associated with k-covering mappings, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1994, no. 4, 24-28 (in Russian); English transl.: Moscow Univ. Math. Bull. 49 (1994), no. 4, 29-23.
- [8] J. Saint Raymond, Caractérisation d’espaces polonais d’après des travaux récents de J. P. R. Christensen et D. Preiss, Sém. Choquet, 11${}^{e}$-12${}^{e}$ années, Initiation à l’Analyse, exp. no. 5, Secrétariat Mathématique, Paris, 1973, 10 pp.
- [9] J. Saint Raymond, La structure borélienne d'Effros est-elle standard?, Fund. Math. 100 (1979), 201-210. Zbl0434.54028
- [10] O. Spinas, Dominating projective sets in the Baire space, Ann. Pure Appl. Logic 68 (1995), 327-342. Zbl0821.03021

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