La détermination projective
Le problème de la mesure
Lelek fan from a projective Fraïssé limit
We show that a natural quotient of the projective Fraïssé limit of a family that consists of finite rooted trees is the Lelek fan. Using this construction, we study properties of the Lelek fan and of its homeomorphism group. We show that the Lelek fan is projectively universal and projectively ultrahomogeneous in the class of smooth fans. We further show that the homeomorphism group of the Lelek fan is totally disconnected, generated by every neighbourhood of the identity, has a dense conjugacy...
Les boules peuvent-elles engendrer la tribu borélienne d'un espace métrisable non séparable ?
Les dérivations analytiques
Les dérivations en théorie descriptive des ensembles et le théorème de la borne
Les propriétés de réduction et de norme pour les classes de Boréliens
Lipschitz-quotients and the Kunen-Martin Theorem
We show that there is a universal control on the Szlenk index of a Lipschitz-quotient of a Banach space with countable Szlenk index. It is in particular the case when two Banach spaces are Lipschitz-homeomorphic. This provides information on the Cantor index of scattered compact sets and such that is a Lipschitz-quotient of (that is the case in particular when these two spaces are Lipschitz-homeomorphic). The proof requires tools of descriptive set theory.