In this paper we show that it is relatively consistent with ZFC that every gamma-set is countable while not every strong measure zero set is countable. This answers a question of Paul Szeptycki. A set is a gamma-set iff every omega-cover contains a gamma-subcover. An open cover is an omega-cover iff every finite set is covered by some element of the cover. An open cover is a gamma-cover iff every element of the space is in all but finitely many elements of the cover. Gerlits and Nagy proved that every gamma-set has strong measure zero. We also show that is consistent that every strong gamma-set is countable while there exists an uncountable gamma-set. On the other hand every strong measure zero set is countable iff every set with the Rothberger property is countable.
In this paper we prove that if there is a Borel-cover gamma-set of cardinality the continuum, then there is one which is not hereditary. A set of reals X is a Borel-cover gamma-set iff for every countable family of Borel sets which is an omega-cover contains a gamma-cover. This is also denoted S_1(Borel_omega, Borel_gamma). This result partially answers a question of Bartoszynski and Tsaban. Tsaban points out that it also gives an example of set which is both a gamma-set and sigma-set but is not hereditarily gamma, which answers a question of Bukovsky, Reclaw, and Repicky.
There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations. In this paper, we shall give an overview of some work that has been done on Martin's conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and…
We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (\eg the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Add…
We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszyński and Judah.
We study groups which satisfy Gardner's equidecomposition conjecture for uniformly distributed sets. We prove that an amenable group has this property if and only if it does not admit $(\mathbb{Z}/2\mathbb{Z}) *(\mathbb{Z}/2\mathbb{Z})$ as a quotient by a finite subgroup. Our technical contribution is an algorithm for rounding Borel flows for actions of amenable groups.
We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.
We show that a {\it Borel} action of a Polish group on a standard Borel space is Borel isomorphic to a {\it continuous} action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures, and the Topological Vaught Conjecture. We establish the existence of universal actions for any given Polish group, extending a result of Mackey and Varadarajan for the locally compact case. We prove an analog of Tarski's theorem on paradoxical decompositions by showing that the existence of an inva…
We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $κ$, let {\sf BC}$_κ$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_κ$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (…
A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence relations imposed by the notion of Borel reducibility. We will survey results and conjectures concerning the problem of identifying the place in this hierarchy of these equivalence relations from…
Let $Γ$ and $Δ$ be sufficiently distinct countable groups. We show that there is an orbit equivalence relation $E$, induced by an action of the Polish wreath product group $Γ\wrΓ$, so that $E$ is generically $F$-ergodic for any orbit equivalence relation $F$ induced by an action of $Δ\wrΔ$. More generally, we establish generic ergodicity between $Γ$-jumps and the iterated $Δ$-jumps, answering a question of Clemens and Coskey. The proofs follow a translation between Borel homomorphisms and definable pins.
We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is $\aleph_0$-categorical, has finitely many countable models (at least three), is Borel equivalent to equality on the reals, is Borel equivalent to "countable sets of reals," or is Borel complete. Furthermore, each of these cases corresponds in a natural way to a count of models of all sizes, up to back-and-forth equivalence. All these cases are possible and we compute precise…
We show that the Dual Borel Conjecture implies that ${\mathfrak d}> \aleph_1$ and find some topological characterizations of perfectly meager and universally meager sets.
We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning …
Let $F_{ω_1}$ be the countable admissible ordinal equivalence relation defined on ${}^ω2$ by $x \ F_{ω_1} \ y$ if and only if $ω_1^x = ω_1^y$. It will be shown that $F_{ω_1}$ is classifiable by countable structures and must be classified by structures of high Scott rank. If $E$ and $F$ are equivalence relations, then $E$ is almost Borel reducible to $F$ if and only if there is a Borel reduction of $E$ to $F$, except possibly on countably many $E$-classes. Let $E_{ω_1}$ denote the equivalence of order types of reals coding well-orderings. It will be shown that in the constructible universe $L$ …
We consider the question, which of the major classes defined by topological diagonalizations of open or Borel covers is hereditary. Many of the classes in the open case are not hereditary already in ZFC, and none of them is provably hereditary. This is contrasted with the Borel case, where some of the classes are provably hereditary. Two of the examples are counter-examples of sizes d$ and b, respectively, to the Menger and Hurewicz Conjectures, and one of them answers a question of Steprans on perfectly meager sets.
We establish the Borel computability of various C$^*$-algebra invariants, including the Elliott invariant and the Cuntz semigroup. As applications we deduce that AF algebras are classifiable by countable structures, and that a conjecture of Winter and the second author for nuclear separable simple C*-algebras cannot be disproved by appealing to known standard Borel structures on these algebras.
Jayne and Rogers proved that every function from an analytic space into a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_σ$ set under it is again $F_σ$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals $α$ …
Let S1(Gamma,Gamma) be the statement: For each sequence of point-cofinite open covers, one can pick one element from each cover and obtain a point-cofinite cover. b is the minimal cardinality of a set of reals not satisfying S1(Gamma,Gamma). We prove the following assertions: (1) If there is an unbounded tower, then there are sets of reals of cardinality b, satisfying S1(Gamma,Gamma). (2) It is consistent that all sets of reals satisfying S1(Gamma,Gamma) have cardinality smaller than b. These results can also be formulated as dealing with Arhangel'skii's property alpha_2 for spaces of continuo…
We describe a simple machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are: 1. The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measure zero/is strongly meager, respectively. 2. Using Scheepers' notation for selection principles: Sfin(Omega,Omega^gp)\cap S1(O,O)=S1(Omega,Omega^gp), and Borel's Conjecture for S1(Omega,Omega) (or just S1(Omega,Omega^gp)) implies Borel's Conjecture. These results extend results of Scheepers and Mill…
We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of the real line R (thus strictly o-bounded) which have the Hurewicz property but are not sigma-compact, and show that the product of two o-bounded subgroups of R^N may fail to be o-bounded, even when they satisfy the stronger property S1(Borel_Omega,Borel_Omega). This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups G of …
We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of the Hjorth's conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube hom…
