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. Introduction. The Borel reducibility theory of Pol...

We investigate the Baire category of I-convergent subsequences and rearrangements of a divergent sequence s = (sn) of reals, if I is an ideal on N having the Baire property. We also discuss the measure of the set of I-convergent subsequences for some classes of ideals on N. Our results generalize theorems due to H. Miller and C. Orhan (2001).

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or compu...

Computable analysis provides ways of representing points in a topological space, and therefore defining notion computable the space. In this article, we investigate when two topologies on same space induce different sets points. We first study purely version problem, which is to understand are not -homeomorphic. obtain characterization leading an effective version, prove that satisfying conditi...

One way to formulate the Baire Category Theorem is that no compact space can be covered by countably many nowhere dense sets. Soon after Cohen’s discovery of forcing, it was realized that it was natural to consider strengthenings of this statement in which one replaces countably many with א1-many. Even taking the compact space to be the unit interval, this already implies the failure of the Con...

We prove a Hindman-type partition theorem for Baire partitions of [0, 1].

We introduce two resource-bounded Baire category notions on small complexity classes such as P, QUASIPOLY, SUBEXP and PSPACE and on probabilistic classes such as BPP, which differ on how the corresponding finite extension strategies are computed. We give an alternative characterization of small sets via resource-bounded Banach-Mazur games. As an application of the first notion, we show that for...

Given f : X → R ∪ {+∞} a convex and lower semi-continuous function defined on a reflexive Banach space X, and L, a closed linear manifold of X over which f takes at least a real value, the aim of this note is to prove the following Baire category result: in the Euclidean setting, the set of affine functions dominated by f on L for which there is no dominated extension to X is always of first Ba...

Considering two different metrics on the space of two-dimensional copulas C we prove some Baire category results for important subclasses of copulas, including the families of exchangeable, associative, and Archimedean copulas. From the point of view of Baire categories, with respect to the uniform metric d∞, a typical copula is not symmetric and a typical symmetric copula is not associative, w...