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Search Results: 1 - 10 of 2700 matches for " Dominique Lecomte "
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A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension
Dominique Lecomte
Mathematics , 2007,
Abstract: We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism.
Classes de Wadge potentielles des boréliens à coupes dénombrables
Dominique Lecomte
Mathematics , 2007,
Abstract: We give, for each non self-dual Wadge class C contained in the class of the Gdelta sets, a characterization of Borel sets which are not potentially in C, among Borel sets with countable vertical sections; to do this, we use results of partial uniformization.
On minimal non-potentially closed subsets of the plane
Dominique Lecomte
Mathematics , 2007,
Abstract: We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todorcevic dichotomy about analytic graphs.
Uniformisations partielles et critères à la Hurewicz dans le plan
Dominique Lecomte
Mathematics , 2007,
Abstract: We give characterizations of the Borel sets potentially in some Wadge class, among the Borel sets with countable vertical sections of a product of two Polish spaces. To do this, we use some partial uniformization results.
Omega-powers and descriptive set theory
Dominique Lecomte
Mathematics , 2007,
Abstract: We study the sets of the infinite sentences constructible with a dictionary over a finite alphabet, from the viewpoint of descriptive set theory. Among other things, this gives some true co-analytic sets. The case where the dictionary is finite is studied and gives a natural example of a set at the level omega of the Wadge hierarchy.
Classes de Wadge potentielles et théorèmes d'uniformisation partielle
Dominique Lecomte
Mathematics , 2007,
Abstract: We want to give a construction as simple as possible of a Borel subset of a product of two Polish spaces. This introduces the notion of potential Wadge class. Among other things, we study the non-potentially closed sets, by proving Hurewicz-like results. This leads to partial uniformization theorems, on big sets, in the sense of cardinality or Baire category.
Tests à la Hurewicz dans le plan
Dominique Lecomte
Mathematics , 2007,
Abstract: We give, for some Borel sets of a product of two Polish spaces, including the Borel sets with countable sections, a Hurewicz-like characterization of those which cannot become a transfinite difference of open sets by changing the two Polish topologies.
Hurewicz-like tests for Borel subsets of the plane
Dominique Lecomte
Mathematics , 2007,
Abstract: Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\bormxi$. We give a Hurewicz-like test to recognize potentially $\bormxi$ sets.
How can we recover Baire class one functions?
Dominique Lecomte
Mathematics , 2007,
Abstract: Let X and Y be separable metrizable spaces, and f:X-->Y be a function. We want to recover f from its values on a small set via a simple algorithm. We show that this is possible if f is Baire class one, and in fact we get a characterization. This leads us to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space.
Complexité des boréliens à coupes dénombrables
Dominique Lecomte
Mathematics , 2007,
Abstract: We give, for each level of complexity L, a Hurewicz-like characterization of the Borel subsets with countable sections of a product of two Polish spaces that cannot become in L by changing the two Polish topologies.
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