We show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [N]N, although its Borel chromatic number is א0. We also show that if add(null) = c, then the universally measurable chromatic number of every treei...

We call a subset S of a topological vector space V linearly Borel, if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V . It will be shown that a Hamel base of an infinite dimensional Banach space can never be linearly Borel. This answers a question of Anatolij Plichko. In the sequel, let X be any infinite dimensional Banach space. A subset S ...

Let A be a commutative unital C-algebra and let  denote its Gelfand spectrum. We find some necessary and sufficient conditions for a nondegenerate representation of A to be unitarily equivalent to a multiplicative representation on a space L(Â, μ), where μ is a positive measure on the Baire sets of Â. We also compare these conditions with the multiplicity-free property of a representation.

We prove several dichotomy theorems which extend some known results on σ -bounded and σ -compactpointsets. In particular we show that, given a finite number of Δ1 equivalence relations F1 , . . . , Fn , any Σ1 set A of the Baire space either is covered by compact Δ1 sets and lightface Δ1 equivalence classes of the relations Fi , or A contains a superperfect subset which is pairwise Fi -inequiva...

It is consistent (relative to ZFC) that each union of max{b, g} many families in the Baire space ω which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter U , the cofinality of the reduced ultrapower ω/U is greater than max{b, g}. The model is constructed by oracle chain condition forcing, to which we give a self-contained intr...

Strong variants of the Operator Speed-up Theorem, Operator Gap Theorem and Compression Theorem are obtained using an effective version of Baire Category Theorem. It is also shown that all complexity classes of recursive predicates have effective measure zero in the space of recursive predicates and, on the other hand, the class of predicates with almost everywhere complexity above an arbitrary ...

Abstract We prove that the lower density operator associated with Baire category points in real line has Borel values of class $$\pmb \Pi ^0_3$$ Π 3 0 which is analogous to measure case. also introduce notion point a subset ...

For a free ultrafilter U on ω we study several cardinal characteristics which describe part of the combinatorial structure of U . We provide various consistency results; e.g. we show how to force simultaneously many characters and many π–characters. We also investigate two ideals on the Baire space ω naturally related to U and calculate cardinal coefficients of these ideals in terms of cardinal...

We show that many so called discrete weak semilattices considered earlier in a series of author’s publications have hereditary undecidable first-order theories. Since such structures appear naturally in some parts of computability theory, we obtain several new undecidability results. This applies e.g. to the structures of complete numberings, of m-degrees of index sets and of the Wadge degrees ...

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rati...