DOCSLIB.ORG

THE VERY BASICS of DESCRIPTIVE SET THEORY 1. Standard Borel Spaces Definition 1.1. (A) a Metric Space Px, Dq Is Called Polish If

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
THE VERY BASICS of DESCRIPTIVE SET THEORY 1. Standard Borel Spaces Definition 1.1. (A) a Metric Space Px, Dq Is Called Polish If THE VERY BASICS OF DESCRIPTIVE SET THEORY PHILIPP SCHLICHT Abstract. This note presents elementary proofs that any two uncountable Polish metric spaces are Borel isomorphic, a function is Borel measurable if and only if its graph is a Borel set, and the auxiliary result that disjoint analytic sets can be separated by Borel sets. The proofs are much shorter than those in the literature. 1. Standard Borel spaces Definition 1.1. (a) A metric space pX; dq is called Polish if it is countably based and complete. (b) A topological space pX; τq is called Polish if there is a Polish metric d on X that induces τ. (c) A pair pX; σq is called a standard Borel space if σ is the Borel σ-algebra generated by a Polish topology on X. Our notation is pX; dq, pY; dq, pZ; dq for Polish metric spaces and the metric d is often omitted. We further write dpAq for the diameter supx;yPA dpx; yq of a subset A of X, and Bpxq for the open ball of radius around x. Example 1.2. Rn, Baire space NN, Cantor space 2N, compact metric spaces, countably based locally compact metric spaces. N N 1 Here N caries the product topology. The standard metric on N is defined by dpx; yq “ 2n for the least n with xpnq ‰ ypnq, where x ‰ y. The basic open subsets of NN (clopen balls) are ă denoted Nt “ tx P NN | t Ď xu for t P N N. The following combinatorial presentation of NN via trees is useful for constructions of continuous maps. A tree on N is a subset of NăN that is closed under initial segments. It is easy to see that the closed subsets of NN are exactly the sets of the form rT s “ tx P NN | @n xæn P T u for trees T on N. Let |t| denote the length of a sequence t P NăN. Let xæn denote the restriction of a sequence x P NN to t0; : : : ; n ´ 1u. A tree is called pruned if it has no end nodes. Let LevnpT q denote the n-th level of a tree T , so the elements of LevnpT q are sequences of length n. Note that Polish spaces are closed under countable products. To see this, take a sequence ~ X “ xXn | n P Ny of Polish spaces and assume that each metric d is bounded by replacing d with maxtd; 1u. Then form the weighted sum over all coordinates with factors 2´n. It is easy to show that this metric on the product is complete and induces the product topology. We next show that any two uncountable Polish spaces are Borel isomorphic in the sense of a bijection that preserves Borel sets both ways. Proposition 1.3. Any two uncountable Polish spaces are Borel isomorphic. This will follow from the next two lemmas. To state them, we fix some notation. An -cover ~ of a subset A of X is a sequence A “ xAi | i ă Ny with N P N Y tNu whose union equals A with dpAiq ă and Ai Ď A for all i P N. An Fσ-set is a countable union of closed sets. Note that all open sets are Fσ. Lemma 1.4. Let ¡ 0. (1) Any open set has an -cover consisting of open sets. (2) Any Fσ-set has a disjoint -cover consisting of Fσ-sets. Date: November 15, 2019. 1 2 PHILIPP SCHLICHT Proof. (1) Suppose that A is an open subset of X. Fix a countable dense subset Q of X and let ~ A “ xAi | i P Ny list all open balls Bqpxq with x P Q, q P p0; 2 q X Q and Bqpxq Ď A. It suffices to show that A “ iPN Ai. To see this, take any x P A and find some q P Q with B2qpxq Ď A. Since Q is dense, there is some y P Bqpxq X Q. We then also have x P Bqpyq and thus it suffices to show Ť that Bqpyq Ď A. This holds since Bqpyq Ď B2qpxq Ď A. (2) Suppose that A is an Fσ subset of X. Take a sequence A~ “ xAi | i P Ny whose union equals A with Ai closed. Then A is the disjoint union of the sets Bi “ AizpA0 Y ¨ ¨ ¨ Y Ai´1q. Since Bi is of the form C X D with C open and D closed, it suffices to prove the claim for sets of this form. To see this, note that by (a), there is an -cover C~ “ xCi | i P Ky of C consisting of open sets. Let ~ ~ D “ xDi | i P Ky with Di “ pCizpC0 Y ¨ ¨ ¨ Y Ci´1qq X D. Then D is a disjoint -cover of C X D consisting of Fσ-sets. The existence of Borel isomorphisms between uncountable Polish spaces follows from the second part of the next lemma by noting that countable sets can be discarded. We call a subset of X co-countable if its complement is countable. Lemma 1.5. 1 (1) There is a closed subset A of NN and a continuous Borel isomorphism f : A Ñ X. (2) If X is uncountable, then there is a continuous Borel isomorphism f : NN Ñ X onto a co-countable subset of X. ` Proof. (1) Let ~ “ xi | i P Ny be a sequence in R converging to 0. By iterative applications ă ~ of Lemma 1.4, we obtain a tree T Ď N N without end nodes and a family A “ xAt | t P T y of nonempty Fσ-subsets of X with A0 “ X that satisfies the following properties for all t P T : (a) At “ tiPT Ati is a disjoint union, (b) Ati Ď At for ti P T , and (c) dpAtq ă n if |t| “ n. We then define f : rT s Ñ X by choosing fpxq as the unique element of Ax n. The latter nPN æ is nonemptyŤ by (b), (c) and completeness of X, and has a unique element by (c). Ş Claim. f is bijective. Proof. It follows from (a) that f is injective. To see that f is surjective, take any x P X. By (a), there is for each n P N a unique tn P T of length n with x P Atn . By (b), ti Ď tj for i ¤ j. Hence fpyq “ x for y “ ti. iPN Claim. f is continuous.Ť Proof. If U is open and fpxq P U, then Axæn Ď U for sufficiently large n by (c). Claim. f is a Borel isomorphism. Proof. Since f is continuous, it suffices that the images of Borel sets are Borel. This holds since f maps each basic open set Nt onto At and f is bijective. Thus f is as required. (2) A slight modification of the construction in (a) works. We will ensure that At is uncountable for all t P T . The change is that we remove a set Ati if it is countable. Note that one can easily ensure that Ati is uncountable for infinitely many i P N by partitioning the sets further. Therefore one can assume that ti P T for all i P N and thus T “ NăN. During the construction only a countable set is removed in total, yielding the claim. Note that in (2), for any X with isolated points, one cannot replace the co-countable set by X. 2. Analytic sets Let p: X ˆ Y Ñ X denote the first projection. Definition 2.1. A subset A of X is called analytic if A “ ppBq for some Borel subset B of X ˆ Y for some standard Borel space Y . 1In fact, any continuous bijection is a Borel isomorphism, but we don't prove this here. BASICS OF DESCRIPTIVE SET THEORY 3 Clearly this class contains all Borel sets. By Proposition 1.3, one can take Y to be a fixed uncountable Polish space. Moreover, the analytic subsets of an uncountable standard Borel space X are precisely the images of analytic subsets of NN under Borel isomorphisms. Hence we can work in NN to study their properties. To do this, we will go through a slight detour. We call a subset A of NN ˚analytic if A “ prCs for some closed subset C of NN ˆ NN, equivalently A “ prT s “ tx | Dy px; yq P rT su for some subtree T of pN ˆ NqăN. Lemma 2.2. (1) The class of ˚analytic sets is closed under countable unions and intersections. (2) Every analytic subset of NN is ˚analytic. Proof. (1) Closure under countable unions is easily seen by forming unions of disjointified trees. Closure under countable intersections is proved by merging trees. We do only the case of two ă trees, since the general case is similar. So suppose that T0 and T1 are subtrees of pN ˆ Nq N. We identify elements of pN ˆ NqăN with pairs of sequences. For t P NăN, define t0; t1 P NăN by letting t0piq “ tp2iq if 2i ă k and t1piq “ tp2i ` 1q if 2i ` 1 ă k. Let U be the subtree of pN ˆ NqăN k 0 1 that consists of all ps; tq P pN ˆ Nq with ps; t q P S and ps; t q P T1. One can now check that prSs X prT s “ prUs. (2) The collection of subsets generated from basic open sets by countable unions and intersec- tions forms a σ-algebra and thus equals the Borel σ-algebra.
Load more

Recommended publications
  • Descriptive Set Theory of Complete Quasi-Metric Spaces
    数理解析研究所講究録 第 1790 巻 2012 年 16-30 16 Descriptive set theory of complete quasi-metric spaces Matthew de Brecht National Institute of Information and Communications Technology Kyoto, Japan Abstract We give a summary of results from our investigation into extending classical descriptive set theory to the entire class of countably based $T_{0}$ -spaces. Polish spaces play a central role in the descriptive set theory of metrizable spaces, and we suggest that countably based completely quasi-metrizable spaces, which we refer to as quasi-Polish spaces, play the central role in the extended theory. The class of quasi-Polish spaces is general enough to include both Polish spaces and $\omega$ -continuous domains, which have many applications in theoretical computer science. We show that quasi-Polish spaces are a very natural generalization of Polish spaces in terms of their topological characterizations and their complete- ness properties. In particular, a metrizable space is quasi-Polish if and only if it is Polish, and many classical theorems concerning Polish spaces, such as the Hausdorff-Kuratowski theorem, generalize to all quasi-Polish spaces. 1. Introduction Descriptive set theory has proven to be an invaluable tool for the study of separable metrizable spaces, and the techniques and results have been applied to many fields such as functional analysis, topological group theory, and math- ematical logic. Separable completely metrizable spaces, called Polish spaces, play a central role in classical descriptive set theory. These spaces include the space of natural numbers with the discrete topology, the real numbers with the Euclidean topology, as well as the separable Hilbert and Banach spaces.
    [Show full text]
  • Topology and Descriptive Set Theory
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 58 (1994) 195-222 Topology and descriptive set theory Alexander S. Kechris ’ Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Received 28 March 1994 Abstract This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. Instead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Bore1 actions of Polish groups and Bore1 equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory). There are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Bore1 actions. The last part concentrates on Bore1 equivalence relations. The exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline. Keywords: Polish spaces; Bore1 sets; Analytic sets; Polish groups; Bore1 actions; Bore1 equivalence relations 1.
    [Show full text]
  • Borel Structure in Groups and Their Dualso
    BOREL STRUCTURE IN GROUPS AND THEIR DUALSO GEORGE W. MACKEY Introduction. In the past decade or so much work has been done toward extending the classical theory of finite dimensional representations of com- pact groups to a theory of (not necessarily finite dimensional) unitary repre- sentations of locally compact groups. Among the obstacles interfering with various aspects of this program is the lack of a suitable natural topology in the "dual object"; that is in the set of equivalence classes of irreducible representations. One can introduce natural topologies but none of them seem to have reasonable properties except in extremely special cases. When the group is abelian for example the dual object itself is a locally compact abelian group. This paper is based on the observation that for certain purposes one can dispense with a topology in the dual object in favor of a "weaker struc- ture" and that there is a wide class of groups for which this weaker structure has very regular properties. If .S is any topological space one defines a Borel (or Baire) subset of 5 to be a member of the smallest family of subsets of 5 which includes the open sets and is closed with respect to the formation of complements and countable unions. The structure defined in 5 by its Borel sets we may call the Borel structure of 5. It is weaker than the topological structure in the sense that any one-to-one transformation of S onto 5 which preserves the topological structure also preserves the Borel structure whereas the converse is usually false.
    [Show full text]
  • Polish Spaces and Baire Spaces
    Polish spaces and Baire spaces Jordan Bell [email protected] Department of Mathematics, University of Toronto June 27, 2014 1 Introduction These notes consist of me working through those parts of the first chapter of Alexander S. Kechris, Classical Descriptive Set Theory, that I think are impor- tant in analysis. Denote by N the set of positive integers. I do not talk about universal spaces like the Cantor space 2N, the Baire space NN, and the Hilbert cube [0; 1]N, or \localization", or about Polish groups. If (X; τ) is a topological space, the Borel σ-algebra of X, denoted by BX , is the smallest σ-algebra of subsets of X that contains τ. BX contains τ, and is closed under complements and countable unions, and rather than talking merely about Borel sets (elements of the Borel σ-algebra), we can be more specific by talking about open sets, closed sets, and sets that are obtained by taking countable unions and complements. Definition 1. An Fσ set is a countable union of closed sets. A Gδ set is a complement of an Fσ set. Equivalently, it is a countable intersection of open sets. If (X; d) is a metric space, the topology induced by the metric d is the topology generated by the collection of open balls. If (X; τ) is a topological space, a metric d on the set X is said to be compatible with τ if τ is the topology induced by d.A metrizable space is a topological space whose topology is induced by some metric, and a completely metrizable space is a topological space whose topology is induced by some complete metric.
    [Show full text]
  • Descriptive Set Theory
    Descriptive Set Theory David Marker Fall 2002 Contents I Classical Descriptive Set Theory 2 1 Polish Spaces 2 2 Borel Sets 14 3 E®ective Descriptive Set Theory: The Arithmetic Hierarchy 27 4 Analytic Sets 34 5 Coanalytic Sets 43 6 Determinacy 54 7 Hyperarithmetic Sets 62 II Borel Equivalence Relations 73 1 8 ¦1-Equivalence Relations 73 9 Tame Borel Equivalence Relations 82 10 Countable Borel Equivalence Relations 87 11 Hyper¯nite Equivalence Relations 92 1 These are informal notes for a course in Descriptive Set Theory given at the University of Illinois at Chicago in Fall 2002. While I hope to give a fairly broad survey of the subject we will be concentrating on problems about group actions, particularly those motivated by Vaught's conjecture. Kechris' Classical Descriptive Set Theory is the main reference for these notes. Notation: If A is a set, A<! is the set of all ¯nite sequences from A. Suppose <! σ = (a0; : : : ; am) 2 A and b 2 A. Then σ b is the sequence (a0; : : : ; am; b). We let ; denote the empty sequence. If σ 2 A<!, then jσj is the length of σ. If f : N ! A, then fjn is the sequence (f(0); : : :b; f(n ¡ 1)). If X is any set, P(X), the power set of X is the set of all subsets X. If X is a metric space, x 2 X and ² > 0, then B²(x) = fy 2 X : d(x; y) < ²g is the open ball of radius ² around x. Part I Classical Descriptive Set Theory 1 Polish Spaces De¯nition 1.1 Let X be a topological space.
    [Show full text]
  • COUNTABLE BOREL EQUIVALENCE RELATIONS Introduction. These
    COUNTABLE BOREL EQUIVALENCE RELATIONS SIMON THOMAS AND SCOTT SCHNEIDER Introduction. These notes are based upon a day-long lecture workshop presented by Simon Thomas at the University of Ohio at Athens on November 17, 2007. The workshop served as an intensive introduction to the emerging theory of countable Borel equivalence relations. These notes are an updated and slightly expanded version of an earlier draft which was compiled from the lecture slides by Scott Schneider. 1. First Session 1.1. Standard Borel Spaces and Borel Equivalence Relations. A topological space is said to be Polish if it admits a complete, separable metric. If B is a σ-algebra of subsets of a given set X, then the pair (X, B) is called a standard Borel space if there exists a Polish topology T on X that generates B as its Borel σ-algebra; in which case, we write B = B(T ). For example, each of the sets R, [0, 1], NN, and 2N = P(N) is Polish in its natural topology, and so may be viewed, equipped with its corresponding Borel structure, as a standard Borel space. The abstraction involved in passing from a topology to its associated Borel structure is analagous to that of passing from a metric to its induced topology. Just as distinct metrics on a space may induce the same topology, distinct topologies may very well generate the same Borel σ-algebra. In a standard Borel space, then, one “remembers” only the Borel sets, and forgets which of them were open; it is natural therefore to imagine that any of them might have been, and indeed this is the case: Theorem 1.1.1.
    [Show full text]
  • 4 from Random Borel Functions to Random Closed Sets
    Tel Aviv University, 2012 Measurability and continuity 63 4 From random Borel functions to random closed sets 4a Random measurable maps . 63 4b Random continuous functions revisited . 65 4c Random semicontinuous functions . 66 4d Randomclosedsets ................. 70 Hints to exercises ....................... 75 Index ............................. 75 A random Borel set is not a random element of the Borel σ-algebra, but a random closed set is a random element of a standard Borel space of closed sets. 4a Random measurable maps Indicators of random Borel sets are a special case of random Borel functions. The supremum of a random Borel function is measurable due to Sect. 3. In Sect. 2c random functions were treated as random elements of some spaces of functions (see 2c10–2c13), much more restricted than all functions or all Borel functions. And indeed, the indicator of a random Borel set cannot be treated in this way (since the random Borel set cannot, recall Sect. 2d). Now we proceed in the spirit of 2d3. A probability space (Ω, F,P ) is assumed to be given, but rarely mentioned.1 4a1 Definition. A random Borel function is an equivalence class of maps Ω → RR that contains the map ω → x → f(ω, x) for some (F × B(R))-measurable f : Ω × R → R. As before, “equivalent” means “differ on a negligible set only”. As before, it is convenient to denote both objects by f; just denote f(ω) = f(ω, ·) = x → f(ω, x). You may easily give an equivalent definition in the spirit of 2d2. Compare it with 1d26–1d27 and 2c10–2c13.
    [Show full text]
  • Stationary Probability Measures and Topological Realizations
    ISRAEL JOURNAL OF MATHEMATICS ?? (2013), 1{14 DOI: 10.1007/s??????-??????-?????? STATIONARY PROBABILITY MEASURES AND TOPOLOGICAL REALIZATIONS BY Clinton T. Conley∗ 584 Malott Hall Cornell University Ithaca, NY 14853 USA e-mail: [email protected] URL: http://www.math.cornell.edu/People/Faculty/conley.html AND Alexander S. Kechris∗∗ Mathematics 253-37 Caltech Pasadena, CA 91125 USA e-mail: [email protected] URL: http://www.math.caltech.edu/people/kechris.html AND Benjamin D. Miller∗ Institut f¨urmathematische Logik und Grundlagenforschung Fachbereich Mathematik und Informatik Universit¨atM¨unster Einsteinstraße 62 48149 M¨unster Germany e-mail: [email protected] URL: http://wwwmath.uni-muenster.de/u/ben.miller 1 2 C.T. CONLEY, A.S. KECHRIS AND B.D. MILLER Isr. J. Math. ABSTRACT We establish the generic inexistence of stationary Borel probability mea- sures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no σ-compact realization. 1. Introduction It is well known that every standard Borel space is Borel isomorphic to a σ- compact Polish space (see [Kec95, Theorem 15.6]), and that every Borel action of a countable discrete group on a standard Borel space is Borel isomorphic to a continuous action on a Polish space (see [Kec95, Theorem 13.1]). Here we consider obstacles to simultaneously achieving both results. In x2, we give elementary examples of continuous actions of countable groups on Polish spaces which are not Borel isomorphic to continuous actions on σ- compact Hausdorff spaces.
    [Show full text]
  • Weak*-Polish Banach Spaces HASKELL ROSENTHAL*
    JOURNAL OF FUNCTIONAL ANALYSIS 76, 267-316 (1988) Weak*-Polish Banach Spaces HASKELL ROSENTHAL* Department of Mathemattcs, The University of Texas at Austin, Austin, Texas 78712 Communicated by the Editors Received February 27, 1986 A Banach space X which is a subspace of the dual of a Banach space Y is said to be weak*-Polish proviced X is separable and the closed unit bail of X is Polish in the weak*-topology. X is said to be Polish provided it is weak*-Polish when regar- ded as a subspace of ,I’**. A variety of structural results arc obtained for Banach spaces with the Point of Continuity Property (PCP), using the unifying concept of weak*-Polish Banach spaces. It is proved that every weak*-Polish Banach space contains an (infinite-dimensional) weak*-closed subspace. This yields the result of Edgar-Wheeler that every Polish Banach space has a reflexive subspace. It also yields the result of Ghoussoub-Maurey that every Banach space with the PCP has a subspace isomorphic to a separable dual, in virtue of the following known result linking the PCP and weak*-Polish Banach spaces: A separable Banach space has the PCP if (Edgar-Wheeler) and only if (GhoussoubMaurey) it is isometric to a weak*-Polish subspace of the dual of some separable Banach space. (For com- pleteness, a proof of this result is also given.) It is proved that if X and Y are Banach spaces with Xc Y*, then X is weak*-Polish if and only if X has a weak*- continuous boundedly complete skipped-blocking decomposition.
    [Show full text]
  • Hyperfiniteness of Boundary Actions of Cubulated Hyperbolic Groups
    Hyperfiniteness of Boundary Actions of Cubulated Hyperbolic Groups Forte Shinko Department of Mathematics and Statistics McGill University, Montreal March 2017 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Master of Science in Mathematics ⃝c Forte Shinko 2017 Abstract A classical result due to Dougherty, Jackson and Kechris states that tail equivalence on Cantor space is a hyperfinite Borel equivalence relation, that is to say that it is the increasing union of finite Borel equivalence relations. This tail equivalence relation is Borel bireducible with the orbit equivalence relation induced by a free group on the boundary of its Cayley graph. We generalize this result to a wider class of hyperbolic groups. Namely, we prove that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced action on the Gromov boundary is hyperfinite. i Abrégé Un résultat classique dû à Dougherty, Jackson et Kechris affirme que l’équivalence de queue sur l’espace de Cantor est une relation d’équivalence borélienne hyperfinie, c’est-à-dire qu’elle est la réunion croissante de relations d’équivalence boréliennes finies. Cette relation d’équivalence de queue est Borel biréductible avec la relation d’équivalence d’orbite induite par l’action d’un groupe libre sur le bord de son graphe de Cayley. On généralise ce résultat à une classe de groupes hyperboliques plus large. À savoir, on prouve que si un groupe hyperbolique agit géométriquement sur un complexe cubique CAT(0), alors l’action induite sur le bord de Gromov est hyperfinie.
    [Show full text]
  • Souslin Quasi-Orders and Bi-Embeddability of Uncountable
    Souslin quasi-orders and bi-embeddability of uncountable structures Alessandro Andretta Luca Motto Ros Author address: Universita` di Torino, Dipartimento di Matematica, Via Carlo Alberto, 10, 10123 Torino, Italy E-mail address: [email protected] Universita` di Torino, Dipartimento di Matematica, Via Carlo Alberto, 10, 10123 Torino, Italy E-mail address: [email protected] arXiv:1609.09292v2 [math.LO] 18 Mar 2019 Contents 1. Introduction 1 2. Preliminaries and notation 14 3. The generalized Cantor space 22 4. Generalized Borel sets 30 5. Generalized Borel functions 37 6. The generalized Baire space and Baire category 41 7. Standard Borel κ-spaces, κ-analyticquasi-orders,andspacesofcodes 47 8. Infinitary logics and models 55 9. κ-Souslin sets 65 10. The main construction 76 11. Completeness 85 12. Invariant universality 91 13. An alternative approach 106 14. Definable cardinality and reducibility 115 15. Some applications 126 16. Further completeness results 132 Indexes 147 Concepts 147 Symbols 148 Bibliography 151 iii Abstract We provide analogues of the results from [FMR11, CMMR13] (which correspond to the case κ = ω) for arbitrary κ-Souslin quasi-orders on any Polish space, for κ an infinite cardinal smaller than the cardinality of R. These generalizations yield a variety of results concerning the complexity of the embeddability relation between graphs or lattices of size κ, the isometric embeddability relation between complete metric spaces of density character κ, and the linear isometric embeddability relation between (real or complex) Banach spaces of density κ. Received by the editor March 19, 2019. 2010 Mathematics Subject Classification. 03E15, 03E60, 03E45, 03E10, 03E47.
    [Show full text]
  • Arxiv:1703.05470V1 [Math.LO] 16 Mar 2017 Coding Polish Spaces
    Coding Polish spaces Diego Alejandro Mej´ıa Faculty of Science Shizuoka University 836 Ohya, Suruga-ku, 422-8529 Shizuoka, Japan [email protected] Abstract We use countable metric spaces to code Polish metric spaces and evaluate the complexity of some statements about these codes and of some relations that can be determined by the codes. Also, we propose a coding for continuous functions between Polish metric spaces. 1 Introduction A Polish metric space is a separable complete metric space hX,di and a Polish space is a topological space X which is homeomorphic to some Polish metric space (in the first notion the complete metric is required). As any Polish metric space is the completion of a countable metric space and the latter can be coded by reals in Rω×ω, we can use such reals to code Polish metric spaces. This coding was used by Clemens [Cle12] to formalize the isometry relation and to study other equivalence relations that can be reduced to that one. In this paper, we take a closer look to this coding and study the complexity of some statements about codes, some of them characterizing relations between Polish metric spaces. In particular, we provide a different proof of [Cle12, Lemma 4] that states that the isometry relation is analytic (Theorem 3.5(f)). We also code continuous functions between Polish metric spaces by Cauchy-continuous functions between the corresponding separable metric spaces and, like in the case of Polish metric spaces, we study the complexity of arXiv:1703.05470v1 [math.LO] 16 Mar 2017 some statements about this coding.
    [Show full text]