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Tests à la Hurewicz dans le plan

Dominique Lecomte (1998)

Fundamenta Mathematicae

Nous donnons, pour une certaine catégorie de boréliens d'un produit de deux espaces polonais, comprenant les boréliens à coupes dénombrables, une caractérisation du type "test d'Hurewicz" de ceux ne pouvant pas être rendus différence transfinie d'ouverts par changement des deux topologies polonaises.

The covering property for σ-ideals of compact, sets

Carlos Uzcátegui (1992)

Fundamenta Mathematicae

The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of 2 ω .

The effective Borel hierarchy

M. Vanden Boom (2007)

Fundamenta Mathematicae

Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is Σ α (respectively, Π α ) in the Borel hierarchy iff K is axiomatized by an infinitary Σ α (respectively, Π α ) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective Σ α or effective Π α Borel set with a computable infinitary sentence of the same complexity. This result...

The generic isometry and measure preserving homeomorphism are conjugate to their powers

Christian Rosendal (2009)

Fundamenta Mathematicae

It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of (ℚ, +) by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring 𝔄 of finite adèles. ...

The ideal (a) is not G δ generated

Marta Frankowska, Andrzej Nowik (2011)

Colloquium Mathematicae

We prove that the ideal (a) defined by the density topology is not G δ generated. This answers a question of Z. Grande and E. Strońska.

The isomorphism relation between tree-automatic Structures

Olivier Finkel, Stevo Todorčević (2010)

Open Mathematics

An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that...

The Point of Continuity Property: Descriptive Complexity and Ordinal Index

Bossard, Benoit, López, Ginés (1998)

Serdica Mathematical Journal

∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142Let X be a separable Banach space without the Point of Continuity Property. When the set of closed subsets of its closed unit ball is equipped with the standard Effros-Borel structure, the set of those which have the Point of Continuity Property is non-Borel. We also prove that, for any separable Banach space X, the oscillation rank of the identity on X (an ordinal index which quantifies the Point of Continuity Property) is determined by the subspaces...

The point of continuity property, neighbourhood assignments and filter convergences

Ahmed Bouziad (2012)

Fundamenta Mathematicae

We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment ( V x ) x ∈ X of X such that d(f(x),f(y)) < ε whenever ( x , y ) ∈ V y × V x . We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.

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