We prove that the category $\mathsf{SBor}$ of standard Borel spaces is the (bi-)initial object in the 2-category of countably complete Boolean (countably) extensive categories. This means that $\mathsf{SBor}$ is the universal category admitting some familiar algebraic operations of countable arity (e.g., countable products, unions) obeying some simple compatibility conditions (e.g., products distribute over disjoint unions). More generally, for any infinite regular cardinal $κ$, the dual of the category $κ\mathsf{Bool}_κ$ of $κ$-presented $κ$-complete Boolean algebras is (bi-)initial in the 2-category of $κ$-complete Boolean ($κ$-)extensive categories.
In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel quasi-orders. In this paper we are concerned with universal countable Borel quasi-orders, i.e. countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We…
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…
In this paper we study the Borel structure of the space of left-orderings $\mathrm{LO}(G)$ of a group $G$ modulo the natural conjugacy action, and by using tools from descriptive set theory we find many examples of countable left-orderable groups such that the quotient space $\mathrm{LO}(G)/G$ is not standard. This answers a question of Deroin, Navas, and Rivas. We also prove that the countable Borel equivalence relation induced from the conjugacy action of $\mathbb{F}_{2}$ on $\mathrm{LO}(\mathbb{F}_{2})$ is universal, and leverage this result to provide many other examples of countable left-…
We consider countable Borel equivalence relations on quotient Borel spaces. We prove a generalization of the Feldman-Moore representation theorem, but provide some examples showing that other very simple properties of countable equivalence relations on standard Borel spaces may fail in the context of nonsmooth quotients.
We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property, and show that the analogous result for digraphs fails.
A function of two variables F(x,y)is universal iff for every other function G(x,y) there exists functions h(x) and k(y) with G(x,y) = F(h(x),k(y)) Sierpinski showed that assuming the continuum hypothesis there exists a Borel function F(x,y) which is universal. Assuming Martin's Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire class 1. We show that it is consistent that for each countable ordinal alpha>2 there is a universal function of class alpha but none of smaller class. We show that it is consistent with ZFC that there is no universal function (…
We introduce the notion of an invariantly universal pair (S,E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in…
We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic $\mathcal L_{ω_1ω}$: every countable $\mathcal L_{ω_1ω}$-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories $(\mathcal L, \mathcal T)$ and $(\mathcal L', \mathcal T')$ (in possibly different languages $\mathcal L, \mathcal L'$), every Borel functor $\mathsf{Mod}(\mathcal L', \mathcal T') \to \mathsf{Mod}(\mathcal L, \mathcal T)$ between the respective groupoids of count…
We prove that the homeomorphism problem for connected compact metric spaces is Borel bireducible with a universal orbit equivalence relation induced by a Borel action of a Polish group.
We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in *M.
In a recent work, the authors studied various Borel equivalence relations defined on the Polish space ${\rm{SA}}(H)$ of all (not necessarily bounded) self-adjoint operators on a separable infinite-dimensional Hilbert space $H$. In this paper we study the domain equivalence relation $E_{\rm{dom}}^{\rm{SA}(H)}$ given by $AE_{\rm{dom}}^{\rm{SA}(H)}B\Leftrightarrow {\rm{dom}}{A}={\rm{dom}}{B}$ and determine its exact Borel complexity: $E_{\rm{dom}}^{\rm{SA}(H)}$ is an $F_σ$ (but not $K_σ$) equivalence relation which is continuously bireducible with the orbit equivalence relation $E_{\ell^{\infty}}…
We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $τ$ on $X_G$ such that $(X_G, τ)$ is Polish and the quotient Borel structure on $X_G$ is equal to $Borel(X_G,τ)$. To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught's transform are valid for partial actions of groups.
Every beginning real analysis student learns the classic Heine-Borel theorem, that the interval [0,1] is compact. In this article, we present a proof of this result that doesn't involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The Heine-Borel theorem is an immediate corollary.
We study the complexity of the space $C^*_p(X)$ of bounded continuous functions with the topology of pointwise convergence. We are allowed to use descriptive set theoretical methods, since for a separable metrizable space $X$, the measurable space of Borel sets in $C^*_p(X)$ (and also in the space $C_p(X)$ of all continuous functions) is known to be isomorphic to a subspace of a standard Borel space. It was proved by A. Andretta and A. Marcone that if $X$ is a $σ$-compact metrizable space, then the measurable spaces $C_p(X)$ and $C^*_p(X)$ are standard Borel and if $X$ is a metrizable analytic…
CONTENTS OF THE ISSUE: Hurewicz-like tests for Borel subsets of the plane; Ordered Spaces, Metric Preimages, and Function Algebras; On the independence of a generalized statement of Egoroff's theorem from ZFC, after T. Weiss; Forty annotated questions about large topological groups; Strong compactness and a partition property; Countable Borel equivalence relations and quotient Borel spaces; Decisive creatures and large continuum; Models of real-valued measurability; Hausdorff ultrafilters; Block combinatorics; Maharam's problem; Universality of uniform Eberlein compacta; Linearly ordered compa…
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…
We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is infinitesimal. As an application to standard probability theory, we show that every finitely-additive Borel probability measure $P$ on a separable metric space is a limit of a sequence of countably-additive Borel probability measures if and only if the space is totally bounded.
