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## Trichotomies for ideals of compact sets

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Venue: | J. SYMBOLIC LOGIC |

Citations: | 5 - 3 self |

### Citations

634 | Random sets and integral geometry - Matheron - 1975 |

597 | S.: Classical descriptive set theory - Kechris - 1995 |

391 | Classical descriptive set theory, Graduate Texts - Kechris - 1995 |

109 | Theory of capacities. Ann - Choquet - 1953 |

63 | The Infinite-Dimensional Topology of Function Spaces - Mill - 2001 |

57 | Capacités et Processus Stochastiques - Dellacherie - 1972 |

24 |
The structure of σ-ideals of compact sets
- Kechris, Louveau, et al.
- 1987
(Show Context)
Citation Context ... hereditary and stable under countable (compact) unions. It is well known that there exist connections between the descriptive complexity of an ideal and its structural properties (see e.g. [C], [E], =-=[KLW]-=-, [K2], [DSR]). For example, a Π 1 1 σ-ideal of compact sets is either Π 1 1-complete or Gδ; a Σ 1 1 σ-ideal is necessarily Gδ; and a Gδ ideal is necessarily a σ-ideal. In a similar vein, it was prove... |

18 | Analytic ideals and their applications, Ann - Solecki - 1999 |

15 | The partially ordered set of measure theory and Tukey’s ordering - Fremlin |

13 |
Descriptive Set Theory and the Structure of
- Kechris, Louveau
- 1989
(Show Context)
Citation Context ... For example, the above statement can be used to show the the family M p 0 := U perf 0 is a Π 0 3-complete subset of K(T), where U0 is the Π 1 1-complete σ-ideal of extended uniqueness sets in T (see =-=[KL]-=-). Since U0 is comeager in K(T) and all compact sets in U0 are nowhere dense, M p 0 is not Σ 0 3. Since U0 has a Σ 0 3 hereditary basis U ′ 0, M p 0 = (U ′ 0) perf is Π 0 3. Acknowledgement. We would ... |

9 |
Ensembles boréliens d’unicité et d’unicité au sens large
- Debs, Saint-Raymond
- 1987
(Show Context)
Citation Context ...nd stable under countable (compact) unions. It is well known that there exist connections between the descriptive complexity of an ideal and its structural properties (see e.g. [C], [E], [KLW], [K2], =-=[DSR]-=-). For example, a Π 1 1 σ-ideal of compact sets is either Π 1 1-complete or Gδ; a Σ 1 1 σ-ideal is necessarily Gδ; and a Gδ ideal is necessarily a σ-ideal. In a similar vein, it was proved in [M] that... |

7 | Cofinal types of topological directed orders - Solecki, Todorčević - 2005 |

6 | Analytic ideals and cofinal types - Louveau, Veličković - 1999 |

6 | Descriptive set theory of families of small sets - Matheron, Zelen´y |

5 |
Hereditary properties of the class of closed sets of uniqueness for trigonometric series
- Kechris
- 1991
(Show Context)
Citation Context ...tary and stable under countable (compact) unions. It is well known that there exist connections between the descriptive complexity of an ideal and its structural properties (see e.g. [C], [E], [KLW], =-=[K2]-=-, [DSR]). For example, a Π 1 1 σ-ideal of compact sets is either Π 1 1-complete or Gδ; a Σ 1 1 σ-ideal is necessarily Gδ; and a Gδ ideal is necessarily a σ-ideal. In a similar vein, it was proved in [... |

5 | Approximation of analytic by Borel sets and definable countable chain conditions - Kechris, Solecki - 1995 |

3 |
Engelen, On Borel Ideals
- van
- 1994
(Show Context)
Citation Context ...ch is hereditary and stable under countable (compact) unions. It is well known that there exist connections between the descriptive complexity of an ideal and its structural properties (see e.g. [C], =-=[E]-=-, [KLW], [K2], [DSR]). For example, a Π 1 1 σ-ideal of compact sets is either Π 1 1-complete or Gδ; a Σ 1 1 σ-ideal is necessarily Gδ; and a Gδ ideal is necessarily a σ-ideal. In a similar vein, it wa... |

3 |
Rudin-like sets and hereditary families of compact sets
- Matheron, Zelený
(Show Context)
Citation Context ...I ⊂ K(E) is an ideal with the property of Baire such that I \ {∅} is nonmeager in K(E), then I is rich in sequences at nonmeagerly many points of E. To derive the corollary, we will need a lemma from =-=[MZ]-=-. We reproduce its proof here for completeness. Lemma 1.7. Let A be a hereditary subset of K(E) with the Baire property. If A \ {∅} is nonmeager in K(E), then there exists a nonempty open set U ⊂ E su... |

3 | A Gδ Ideal of Compact Sets Strictly Above the Nowhere Dense - Moore, Solecki |

2 |
Some examples of true Fσδ sets
- Balcerzak, Darji
(Show Context)
Citation Context ... ∩ K(V ) is a Gδ σ-ideal. This result was used in [M] to show that many natural families of thin sets from Harmonic Analysis are Σ 0 3-complete. Some other aspects of these matters were considered in =-=[BD]-=-. In this note, we shall proceed further by proving some “trichotomy” results for ideals of compact sets. To formulate them, we need to introduce the following notion, which is crucial for our paper. ... |

2 | Polar σ-ideals of compact sets - Debs - 1995 |

2 | Calibrated thin Π 1 1 σ-ideals are Gδ - Zelen´y - 1997 |

1 |
Topology and Borel
- Christensen
- 1974
(Show Context)
Citation Context ...E which is hereditary and stable under countable (compact) unions. It is well known that there exist connections between the descriptive complexity of an ideal and its structural properties (see e.g. =-=[C]-=-, [E], [KLW], [K2], [DSR]). For example, a Π 1 1 σ-ideal of compact sets is either Π 1 1-complete or Gδ; a Σ 1 1 σ-ideal is necessarily Gδ; and a Gδ ideal is necessarily a σ-ideal. In a similar vein, ... |

1 |
How to recognize a true Σ
- Matheron
- 1998
(Show Context)
Citation Context ...], [DSR]). For example, a Π 1 1 σ-ideal of compact sets is either Π 1 1-complete or Gδ; a Σ 1 1 σ-ideal is necessarily Gδ; and a Gδ ideal is necessarily a σ-ideal. In a similar vein, it was proved in =-=[M]-=- that if an ideal I ⊂ K(E) is a countable union of Gδ hereditary sets and contains a dense Gδ hereditary subset of K(E), then either I is Σ 0 3-complete, or there exists a nonempty open set V ⊂ E such... |

1 |
FOR IDEALS OF COMPACT SETS 15 Université Bordeaux 1, 351 cours de la libération, 33405 Talence Cedex, France E-mail address: Etienne.Matheron@math.u-bordeaux1.fr Department of mathematics
- Todorcevic
- 1997
(Show Context)
Citation Context ...t follows that the family � Ĩ = b ⊂ ω; K ∪ � � Kn ∈ I is an ideal of P(ω) with the Baire property in P(ω). Moreover, ω /∈ Ĩ by assumption. By the so-called Jalali-Naini–Mathias–Talagrand Theorem (see =-=[T]-=-), it follows that there exists an increasing sequence of integers p0 < p1 < . . . such that no member of Ĩ contains infinitely many intervals an = [pn, pn+1[. So it is enough to put Fn � = j∈an Kj. �... |

1 | A note on Gδ ideals of compact sets, submitted - Saran |