An exact Ramsey principle for block sequences.
Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem
We prove the following theorem: Given a⊆ω and , if for some and all u ∈ WO of length η, a is , then a is . We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: -Turing-determinacy implies the existence of .
Analytic gaps
We investigate when two orthogonal families of sets of integers can be separated if one of them is analytic.
Analytic partial orders and oriented graphs
We prove that there is no maximum element, under Borel reducibility, in the class of analytic partial orders and in the class of analytic oriented graphs. We also provide a natural jump operator for these two classes.
Analytic sets and Borel isomorphisms
Application of covering sets
Applicazioni della teoria descrittiva degli insiemi a problemi di classificazione per classi di strutture e relazioni d’equivalenza
Assertion Q distinguishes topologically ω* and when regular and
Asymmetric tie-points and almost clopen subsets of
A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of . One especially important application, due to Veličković, was to the existence of nontrivial involutions on . A tie-point of has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set...
Automata, Borel functions and real numbers in Pisot base
This note is about functions ƒ : Aω → Bω whose graph is recognized by a Büchi finite automaton on the product alphabet A x B. These functions are Baire class 2 in the Baire hierarchy of Borel functions and it is decidable whether such function are continuous or not. In 1920 W. Sierpinski showed that a function is Baire class 1 if and only if both the overgraph and the undergraph of f are Fσ. We show that such characterization is also true for functions on infinite words if we replace the real...
Bad Wadge-like reducibilities on the Baire space
We consider various collections of functions from the Baire space into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of...
Banach spaces without minimal subspaces – Examples
We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in [11],by classifying them according to which side of the dichotomies they fall.
Base-base paracompactness and subsets of the Sorgenfrey line
A topological space is called base-base paracompact (John E. Porter) if it has an open base such that every base has a locally finite subcover . It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample.
Bernoulli convolutions-an application of set theory in analysis
Beyond Lebesgue and Baire II: Bitopology and measure-category duality
We re-examine measure-category duality by a bitopological approach, using both the Euclidean and the density topologies of the line. We give a topological result (on convergence of homeomorphisms to the identity) obtaining as a corollary results on infinitary combinatorics due to Kestelman and to Borwein and Ditor. We hence give a unified proof of the measure and category cases of the Uniform Convergence Theorem for slowly varying functions. We also extend results on very slowly varying functions...
Borel and Baire reducibility
We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures.
Borel chromatic number of closed graphs
We construct, for each countable ordinal ξ, a closed graph with Borel chromatic number 2 and Baire class ξ chromatic number ℵ₀.
Borel classes in AST. Measurability, cuts and equivalence
Borel completeness of some ℵ₀-stable theories
We study ℵ₀-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of λ-Borel completeness and prove that such theories are λ-Borel complete. Using this, we conclude that an ℵ₀-stable theory satisfies for all cardinals λ if and only if T either has eni-DOP or is eni-deep.