This note is about functions f : Aω → Bω whose graph is recognized by a Büchi finite automaton on the product alphabet A×B. These functions are Baire class 2 in the Baire hierarchy of Borel functions and it is decidable whether such fonction are continuous or not. In 1920 W. Sierpinski showed that a function f : R → R is Baire class 1 if and only if both the overgraph and the undergraph of f ar...

We study reflection principles involving nonmeager sets and the Baire Property which are consequences of the generic supercompactness of ω2, such as the principle asserting that any point countable Baire space has a stationary set of closed subspaces of weight ω1 which are also Baire spaces. These principles entail the analogous principles of stationary reflection but are incompatible with forc...

The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of R which is Baire (= has the property, BP) or measurable shift-compact: it contains some subsequence any null sequence to within translation by an element set. Here effective proofs are recognized yield (i) analogous category and Haar-measure metrizable generalizations for groups locally compact respectively, (ii) permit...

We generalize the Baire Category Theorem to the Borel and difference hierarchies, i.e. if Γ is any of the classes Σξ , Π 0 ξ , Dη(Σ 0 ξ) or Ďη(Σ 0 ξ) we find a representative set PΓ ∈ Γ and a Polish topology τΓ such that for every A ∈ Γ̌ from some assumption on the size of A ∩ PΓ we can deduce that A \ PΓ is of second category in the topology τΓ. This allows us to distinguish the levels of the B...

Say that a cardinal number κ is small relative to the space X if κ < ∆(X), where ∆(X) is the least cardinality of a non-empty open set in X . We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small n...

For a compact metric space ∆ we consider the compact space S(∆) of shift-invariant subsystems of the full shift ∆, endowed with the Hausdorf metric.. We show that when ∆ is perfect the space S(∆) and the homeomorphism group Homeo(K) of the Cantor set (with the topology of uniform convergence) are topologically generically equivalent in the sense of Glasner and King [7], ie a dynamical property ...

Introduction. It is well known that a function which is a derivative is in the first Baire class, and it is also known that a function which is an approximate derivative is in the first Baire class. The analogous statement for partial derivatives of functions of more than one variable is of course not true. For if g is a nonmeasurable function of one variable, then the function f(x, y)=x-g(y) h...

The aim of this paper is to present a detailed explanation of three models of Shelah. We show the rule of the amalgamation in the construction of models in which all deenable sets of reals have Baire property or are Lebesgue measurable. Next we construct a model in which every projective set of reals has Baire property, a model with the Uniformization Property and a model of ZF + DC in which al...

We begin the fine analysis of non Borel pointclasses. Working in ZFC + ̃ 11), we describe the Wadge hierarchy of the class of increasing differences of coanalytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets. Introduction. Collections of substets of the Baire space, the "logician’s reals", that are closed under continuous preimages have always been...