Only the usual axioms of set theory are needed to prove the existence of a Baire space whose square is not a Baire space. Assuming the continuum hypothesis (CH), Oxtoby [9] constructed a Baire space whose square is not Baire. We will show in this paper that the assumption of CH is unnecessary. Such results are greatly enhanced by Krom [5], who showed that if there is such an example, then there...

We establish some results on the Wadge degrees and on the Boolean hierarchy of k-partitions of some spaces, where k is a natural number. The main attention is paid to the Baire space, Baire domain and their close relatives. For the case of Δ2-measurable k-partitions the structures of Wadge degrees are characterized completely. For many degree structures, undecidability of the first-order theori...

We give a new characterization of the Baire class 1 functions (defined on an ultrametric space) by proving that they are exactly the pointwise limits of sequences of full functions, which are particularly simple Lipschitz functions. Moreover we highlight the link between the two classical stratifications of the Borel functions by showing that the Baire class functions of some level are exactly ...

Throughout, by a graph we mean a simple undirected graph, where the degree of a vertex is its number of neighbors, and a d-coloring is a function assigning each vertex one of d colors so that adjacent vertices are mapped to different colors. This paper examines measurable analogues of Brooks’s Theorem. While a straightforward compactness argument extends Brooks’s Theorem to infinite graphs, suc...

Laczkovich proved that if bounded subsets A and B of R have the same non-zero Lebesgue measure and the upper box dimension of the boundary of each set is less than k, then there is a partition of A into finitely many parts that can be translated to form a partition of B. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, thi...

We investigate the extension of Monadic Second Order logic, interpreted over infinite words and trees, with generalized “for almost all” quantifiers interpreted using the notions of Baire category and Lebesgue measure. All three authors were supported by the Polish National Science Centre grant no. 2014-13/B/ST6/03595. The work of the second author has also been supported by the project ANR-16-...

In a recent paper E. J. McShane [3]2 has given a theorem which is the common core of a variety of results about Baire sets, Baire functions, and convex sets in topological spaces including groups and linear spaces. In general terms his theorem states that if J is a family of open maps defined in one topological space Xi into another, X2, the total image JiS) of a second category Baire set S in ...

We obtain game–theoretic characterizations for meagerness and rareness of filters on ω. One of the classical methods for obtaining a set of real numbers which does not have the property of Baire, is to interpret appropriate filters on N = {1,2,3, . . . } as subsets of [0, 1]. Filters which result in a set having the property of Baire have nice combinatorial characterizations, due to Talagrand [...