DOCSLIB.ORG

Mycielski Ideal and the Perfect Set Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mycielski Ideal and the Perfect Set Theorem PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 7, Pages 2141{2150 S 0002-9939(04)07360-5 Article electronically published on January 23, 2004 MYCIELSKI IDEAL AND THE PERFECT SET THEOREM MIROSLAV REPICKY´ (Communicated by Alan Dow) Abstract. We make several observations on the Mycielski ideal and prove a version of the perfect set theorem concerning this ideal for analytic sets: If A ⊆ !2 is an analytic set all projections of which are uncountable, then there is a perfect set B ⊆ A a projection of which is the whole space. We also prove that (a modification of) an infinite game of Mycielski is determined for analytic sets. 1. Introduction For a 2 [!]! and A ⊆ !2letΓ(A; a) be the infinite game for players I and II in which both players choose consecutive elements of a sequence x 2 !2. Player II chooses x(n)forn 2 a, and the other moves are done by player I. Player I wins if x 2 A, and otherwise player II wins. This game was used by J. Mycielski [6] for introducing new invariant ideals on !2 that are orthogonal to the ideals of sets of Lebesgue measure zero and of meager sets. A. Ros lanowski [8] introduced the next two ideals related to this game: ! ! C2 = fA ⊆ 2:(8a 2 [!] ) player II has a winning strategy in Γ(A; a)g; ! ! a P2 = fA ⊆ 2:(8a 2 [!] ) Aa =6 2g; where Aa = fxa : x 2 Ag. The ideals C2, P2 are translation-invariant σ-ideals, P2 ⊆ C2,everysetinC2 having the Baire property is meager, and every measurable set in C2 has measure zero. In this paper we make several observations on these two ideals and prove variants of the perfect set theorem for them (Theorems 2.2 and 3.2). In the case of the ideal C2, this perfect set theorem is the determinacy of a modified game for analytic sets. Namely, let Γ(A) be the following modification of the above game: At the beginning of the game, player I chooses a 2 [!]!, and then the players follow the rules of the game Γ(A; a). Clearly, ! C2 = fA ⊆ 2 : player II has a winning strategy in Γ(A)g: Theorem 3.2 says that the game Γ(A) is determined for analytic sets A ⊆ !2. We use the standard notation. If p ⊆ <!2(orp ⊆ <!!)isatreeands 2 p,then <! we denote [p]=fx :(8n) xn 2 pg and ps = ft 2 p : s ⊆ t _ t ⊆ sg.Fors 2 2or s 2 <!!,[s] denotes the corresponding clopen basic set fx : s ⊂ xg in !2or!!. Received by the editors August 21, 2002 and, in revised form, March 27, 2003. 2000 Mathematics Subject Classification. Primary 03E15; Secondary 03E17, 91A44. Key words and phrases. Mycielski ideal, analytic sets, perfect set theorem. This work was supported by a grant of Slovak Grant Agency VEGA 2/7555/20. c 2004 American Mathematical Society 2141 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2142 MIROSLAV REPICKY´ Let us recall that s is the minimal cardinality of a set A ⊆ [!]! such that for every a 2 [!]! there is b 2 A such that ja n bj = ja \ bj = !; h is the least cardinal κ such that P(!)=fin is not κ-distributive (see [5]). 2. The ideal P2 The ideal P2 is called the Mycielski ideal (see [8] and [3]). We consider the following subideal I0 of P2, defined by ! ! ! I0 = fA ⊆ 2:(8a 2 [!] )(9b 2 [a] ) jAb|≤!g: This is the σ-ideal IX defined in [7] for the sequence of open sets X = hUn : n 2 !i with U = fx 2 !2:x(n)=0g where n S T S T ! ! ! ! IX = fA ⊆ 2:(8a 2 [!] )(9b 2 [a] ) A ⊆ Un [ ( 2 n Un)g: m n2bnm m n2bnm Theorem 2.1. (i) For every cardinal number κ with !<κ≤ s, ! ! ! I0 = fA ⊆ 2:(8a 2 [!] )(9b 2 [a] ) jAbj <κg ( P2: ! (ii) s =minfjAj : A ⊆ 2 and A 2 P2 nI0g. (iii) add(I0) ≥ h. Proof. (i) The inclusions from left to right and the first equality are consequences of definitions. The only nontrivial part is the inequality (. Consider a base matrix B in [!]! (constructed in [1]). Every projection Ba again is a base matrix in [a]! and has the size of the continuum. Therefore B 2 P2 nI0. (ii) Every set of size < s is in both P2 and I0.Ifs < c, then any splitting family of size s is in P2 nI0.Ifs = c, then the set of the size of the continuum (base matrix) used in the proof of (i) belongs to P2 nI0. (iii) The inequality add(I0) ≥ h is due to the fact that A 2I0 if and only if the set fa 2 [!]! :(8x 2 A)((81n 2 a) x(n)=0_ (81n 2 a) x(n)=1)g is open dense in P(!)=fin. Another example of a set belonging to P2 nI0 is constructed in Theorem 4.3. Now we prove a version of a perfect set theorem, which in a particular case says that if every projection of an analytic set in !2 is uncountable, then it contains a perfect set a projection of which is the whole space. ! Theorem 2.2. If an analytic subset of 2 is not in I0, then it contains a perfect set that is not in P2. Hence there is no analytic set in P2 nI0. ! Proof. Let A ⊆ 2 be analytic, and let A=2I0.Letf be a continuous mapping ! <! from ! onto A.Letfun : n 2 !g be an enumeration of !, and by induction on n 2 ! let us construct a sequence of infinite sets an ⊆ ! such that ! (i) (8b 2 [a0] ) jAbj >!, ! (ii) (8n 2 !)(jf([un])an|≤! _ (8b 2 [an] ) jf([un])bj >!), and (iii) (8n 2 !) an+1 ⊆ an. Condition (i) is due to the assumption that A=2I0, and condition (ii) is due to the an−1 ! fact that either f([un])an−1 belongs to the ideal I0 (in 2) or not. Let a 2 [!] <! be a pseudo-intersection of fan : n 2 !g and let T = fu 2 ! : jf([u])aj >!g. For every u 2 T and every b 2 [a]!, S <! f([Tu])b ⊇ f([u])b n ff([v])b : v 2 ! n T g; License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MYCIELSKI IDEAL AND THE PERFECT SET THEOREM 2143 and therefore ! (2.1) (8u 2 T )(8b 2 [a] ) jf([Tu])bj >!: (u) <! (u) For u 2 T let p = fs 2 2:[s] \ f([Tu]) =6 ;g, i.e., f([Tu]) = [p ]. Then 1 (u) n+1 (2.2) (8u 2 T )(8 n 2 a)(9s1;s2 2 p \ 2) s1(n) =6 s2(n); since otherwise there exists an infinite set b ⊆ a such that j[p(u)]bj =1,which contradicts (2.1). Condition (2.2) enables us to construct, by induction on n = jsj <! for s 2 2, an infinite set b = fkn : n 2 !}⊆a with kn+1 >kn,andsystems <! <! <! X = fus : s 2 2}⊆T , Y = fts : s 2 2}⊆ 2sothatu; = ; = t;, (us) kn+1 ts_0;ts_1 2 p \ 2aresuchthatts_i(kn)=i for i =0,1,andus_0;us_1 2 n ⊆ _ 2 ⊆ _ Tus are such that f([Tus_i ]) [ts i]fors 2andi = 0, 1. Clearly, ts ts i. The systems X and Y determine perfect trees in <!! and <!2, respectively, namely the trees <! T0 = fu 2 T :(9v 2 X) u ⊆ vg;p0 = fs 2 2:(9t 2 Y ) s ⊆ tg: Hence the sets B =[T0]andC =[p0] are perfect compact, and clearly C ⊆ A,since C = f(B). From the construction of Y it follows that Cb = b2. Corollary 2.3. The union of less than h analytic sets from P2 is in P2. Let P be the set of all perfect trees in <!2. For n 2 ! and p 2 P we define n n p ≤n q if p ⊆ q and p \ 2=q \ 2. A set D ⊆ P is said to be !-dense if for every tree p 2 P and every n 2 ! there is q 2 D such that q ≤n p. In [7] it was said that an ideal I on !2 is tall if it contains all singletons and every perfect set A ⊆ !2 has a perfect subset in I. Let us recall that the condition in this definition concerning perfect sets is the so-called property (P) introduced by M. Balcerzak and studied in a number of papers; see, e.g., [2] for references. In [7] the question arose whether the ideals IX defined for various sequences of open sets are tall. The referee has kindly informed us that this is not true, because W. D¸ebski, J. Kleszcz, and Sz. Plewik in [4] have proved that the ideal P2 (and hence also I0) does not have property (P). Anyway, we can make several observations in this direction. Let us note that if I is a tall ideal on !2, then for every p 2 P there exists q ≤n p such that [q] 2I, i.e., the family of perfect trees DI = fp 2 P :[p] 2Igis !-dense in P . Theorem 2.4. (i) The ideal I0 contains perfect sets. In fact, for every set B 2 P2 there is a perfect set A 2I0 disjoint from B. <! ! (ii) For every perfect tree p ⊆ 2 there are a perfect tree q ≤n p and c 2 [!] such that for every a 2 [c]! there is b 2 [a]! such that [q]b is countable c (hence [q]c 2I0 inside 2).
Load more

Recommended publications
  • 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]
  • Set Ideal Topological Spaces
    Set Ideal Topological Spaces W. B. Vasantha Kandasamy Florentin Smarandache ZIP PUBLISHING Ohio 2012 This book can be ordered from: Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: (614) 485-0721 E-mail: [email protected] Website: www.zippublishing.com Copyright 2012 by Zip Publishing and the Authors Peer reviewers: Prof. Catalin Barbu, V. Alecsandri National College, Mathematics Department, Bacau, Romania. Prof. Valeri Kroumov, Okayama Univ. of Science, Japan. Dr. Sebastian Nicolaescu, 2 Terrace Ave., West Orange, NJ 07052, USA. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-13: 978-1-59973-193-3 EAN: 9781599731933 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two SET IDEALS IN RINGS 9 Chapter Three SET IDEAL TOPOLOGICAL SPACES 35 Chapter Four NEW CLASSES OF SET IDEAL TOPOLOGICAL SPACES AND APPLICATIONS 93 3 FURTHER READING 109 INDEX 111 ABOUT THE AUTHORS 114 4 PREFACE In this book the authors for the first time introduce a new type of topological spaces called the set ideal topological spaces using rings or semigroups, or used in the mutually exclusive sense. This type of topological spaces use the class of set ideals of a ring (semigroups). The rings or semigroups can be finite or infinite order. By this method we get complex modulo finite integer set ideal topological spaces using finite complex modulo integer rings or finite complex modulo integer semigroups. Also authors construct neutrosophic set ideal toplogical spaces of both finite and infinite order as well as complex neutrosophic set ideal topological spaces.
    [Show full text]
  • ON BERNSTEIN SETS Let Us Recall That a Subset of a Topological Space
    ON BERNSTEIN SETS JACEK CICHON´ ABSTRACT. In this note I show a construction of Bernstein subsets of the real line which gives much more information about the structure of the real line than the classical one. 1. BASIC DEFINITIONS Let us recall that a subset of a topological space is a perfect set if is closed set and contains no isolated points. Perfect subsets of real line R have cardinality continuum. In fact every perfect set contains a copy of a Cantor set. This can be proved rather easilly by constructing a binary tree of decreasing small closed sets. There are continnum many perfect sets. This follows from a more general result: there are continuum many Borel subsets of real. We denote by R the real line. We will work in the theory ZFC. Definition 1. A subset B of the real line R is a Bernstein set if for every perfect subset P of R we have (P \ B 6= ;) ^ (P n B 6= ;) : Bernstein sets are interesting object of study because they are Lebesgue nonmeasurable and they do not have the property of Baire. The classical construction of a Berstein set goes as follows: we fix an enumeration (Pα)α<c of all perfect sets, we define a transfinite sequences (pα), (qα) of pairwise different points such that fpα; qαg ⊆ Pα, we put B = fpα : α < cg and we show that B is a Bernstein set. This construction can be found in many classical books. The of this note is to give slightly different construction, which will generate simultanously a big family of Bermstein sets.We start with one simple but beautifull result: Lemma 1.
    [Show full text]
  • Lecture 8: the Real Line, Part II February 11, 2009
    Lecture 8: The Real Line, Part II February 11, 2009 Definition 6.13. a ∈ X is isolated in X iff there is an open interval I for which X ∩ I = {a}. Otherwise, a is a limit point. Remark. Another way to state this is that a is isolated if it is not a limit point of X. Definition 6.14. X is a perfect set iff X is closed and has no isolated points. Remark. This definition sounds nice and tidy, but there are some very strange perfect sets. For example, the Cantor set is perfect, despite being nowhere dense! Our goal will be to prove the Cantor-Bendixson theorem, i.e. the perfect set theorem for closed sets, that every closed uncountable set has a perfect subset. Lemma 6.15. If P is a perfect set and I is an open interval on R such that I ∩ P 6= ∅, then there exist disjoint closed intervals J0,J1 ⊂ I such that int[J0] ∩ P 6= ∅ and int[J1] ∩ P 6= ∅. Moreover, we can pick J0 and J1 such that their lengths are both less than any ² > 0. Proof. Since P has no isolated points, there must be at least two points a0, a1 ∈ I ∩ P . Then just pick J0 3 a0 and J1 3 a1 to be small enough. SDG Lemma 6.16. If P is a nonempty perfect set, then P ∼ R. Proof. We exhibit a one-to-one mapping G : 2ω → P . Note that 2ω can be viewed as the set of all infinite paths in a full, infinite binary tree with each edge labeled by 0 or 1.
    [Show full text]
  • Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets
    Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets by Rebecca Stettin A capstone project submitted in partial fulfillment of graduating from the Academic Honors Program at Ashland University May 2017 Faculty Mentor: Dr. Darren D. Wick, Professor of Mathematics Additional Reader: Dr. Gordon Swain, Professor of Mathematics Abstract Georg Cantor was made famous by introducing the Cantor set in his works of mathemat- ics. This project focuses on different Cantor sets and their properties. The ternary Cantor set is the most well known of the Cantor sets, and can be best described by its construction. This set starts with the closed interval zero to one, and is constructed in iterations. The first iteration requires removing the middle third of this interval. The second iteration will remove the middle third of each of these two remaining intervals. These iterations continue in this fashion infinitely. Finally, the ternary Cantor set is described as the intersection of all of these intervals. This set is particularly interesting due to its unique properties being uncountable, closed, length of zero, and more. A more general Cantor set is created by tak- ing the intersection of iterations that remove any middle portion during each iteration. This project explores the ternary Cantor set, as well as variations in Cantor sets such as looking at different middle portions removed to create the sets. The project focuses on attempting to generalize the properties of these Cantor sets. i Contents Page 1 The Ternary Cantor Set 1 1 2 The n -ary Cantor Set 9 n−1 3 The n -ary Cantor Set 24 4 Conclusion 35 Bibliography 40 Biography 41 ii Chapter 1 The Ternary Cantor Set Georg Cantor, born in 1845, was best known for his discovery of the Cantor set.
    [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]
  • Arxiv:1603.05121V3 [Math.GM] 31 Oct 2016 Definition of Set Derived Efc (I.E
    SOME EXISTENCE RESULTS ON CANTOR SETS Borys Alvarez-Samaniego´ N´ucleo de Investigadores Cient´ıficos Facultad de Ingenier´ıa, Ciencias F´ısicas y Matem´atica Universidad Central del Ecuador (UCE) Quito, Ecuador Wilson P. Alvarez-Samaniego´ N´ucleo de Investigadores Cient´ıficos Facultad de Ingenier´ıa, Ciencias F´ısicas y Matem´atica Universidad Central del Ecuador (UCE) Quito, Ecuador Jonathan Ortiz-Castro Facultad de Ciencias Escuela Polit´ecnica Nacional (EPN) Quito, Ecuador Abstract. The existence of two different Cantor sets, one of them contained in the set of Liouville numbers and the other one inside the set of Diophan- tine numbers, is proved. Finally, a necessary and sufficient condition for the existence of a Cantor set contained in a subset of the real line is given. MSC: 54A05; 54B05 Keywords: Cantor set; Liouville numbers; Diophantine numbers. arXiv:1603.05121v3 [math.GM] 31 Oct 2016 1. Introduction First, we will introduce some basic topological concepts. Definition 1.1. A nowhere dense set X in a topological space is a set whose closure has empty interior, i.e. int(X)= ∅. Definition 1.2. A nonempty set C ⊂ R is a Cantor set if C is nowhere dense and perfect (i.e. C = C′, where C′ := {p ∈ R; p is an accumulation point of C} is the derived set of C). Definition 1.3. A condensation point t of a subset A of a topological space, is any point t, such that every open neighborhood of t contains uncountably many points of A. Date: October 31, 2016. 1 2 B. ALVAREZ-SAMANIEGO,´ W.
    [Show full text]
  • Absoluteness Theorems for Arbitrary Polish Spaces
    Revista Colombiana de Matem´aticas Volumen 53(2019)2, p´aginas109-123 Absoluteness theorems for arbitrary Polish spaces Teoremas de absolutidad para espacios polacos arbitrarios Diego Alejandro Mej´ıa1;a;B , Ismael E. Rivera-Madrid2;a 1Shizuoka University, Shizuoka, Japan 2Instituci´onUniversitaria Pascual Bravo, Medell´ın,Colombia Abstract. By coding Polish metric spaces with metrics on countable sets, we propose an interpretation of Polish metric spaces in models of ZFC and extend Mostowski's classical theorem of absoluteness of analytic sets for any Polish metric space in general. In addition, we prove a general version of Shoenfield’s absoluteness theorem. Key words and phrases. Mostowski's Absoluteness Theorem, Shoenfield’s Abso- luteness Theorem, Polish metric spaces. 2010 Mathematics Subject Classification. 03E15, 54H05. Resumen. Mediante la codificaci´onde espacios polacos con m´etricasde con- juntos contables, proponemos una interpretaci´onde espacios m´etricospolacos en modelos de ZFC y extendemos el cl´asicoTeorema de Absolutidad (para conjuntos anal´ıticos)de Mostowski para cualquier espacio m´etrico polaco en general. Adicionalmente, probamos una versi´ongeneral del Teorema de Ab- solutidad de Shoenfield. Palabras y frases clave. Teorema de Absolutidad de Mostowski, Teorema de Absolutidad de Shoenfield, espacios m´etricos polacos. 1. Introduction 1 Mostowski's Absoluteness Theorem (also known as Σ1 absoluteness) states that any analytic subset of the Baire space !! is absolute for transitive mod- els of ZFC. To be more precise, for any tree T ⊆ (! × !)<! the statement 9y 2 !!((x; y) 2 [T ]) is absolute (recall that any closed subset of (! × !)! is 109 110 DIEGO ALEJANDRO MEJ´ıA& ISMAEL E. RIVERA-MADRID characterized by such a tree, so they code all the analytic subsets).
    [Show full text]
  • New Sets and Topologies in Ideal Topological Spaces J
    International Journal of Pure and Applied Mathematics Volume 118 No. 20 2018, 93-98 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu NEW SETS AND TOPOLOGIES IN IDEAL TOPOLOGICAL SPACES J.Sebastian Lawrence Department of Mathematics Sathyabama University,Chennai-600119 Email: [email protected] R.Manoharan, Department of Mathematics, Sathyabama University, Chennai-600119. Email: [email protected] . ABSTRACT In this paper, in an ideal space(X, , ), we introduce and study about a new topology called RIC - topology which is finer than .Also we analyze the relationship with the already existing topologies . AMS Subject classification: 54A05,54A10 Keywords- R -perfect sets, R -open sets,R -open sets, R –topology. I IC I IC 1. Introduction A nonempty collection of subsets of a set X is said to be an ideal on X, if it satisfies the following two conditions: (i) if A and B A , then B (heredity); (ii) if A and B ,thenA B (finite additivity) . An ideal topological space ( or ideal space) (X, , ) means a topological space (X, ) with an ideal defined on X. For a given point x in a space ( X, ), the system of open neighborhoods of x is denoted by N(x) ={U : x U}. Let (X, ) be a topological space with an ideal defined on X. then for any subset A of X, A* ( , ) ={x X/A U for every U N(x)} is called the local function of A with respect to and . If there is no ambiguity, we will write A*( ) or simply A* for A*( , ) .
    [Show full text]
  • Math 482 Final Paper: Cantor Spaces in R This Paper Describes Some Basic Properties of Cantor Subspaces of the Real Line
    Paul Gustafson Texas A&M University - Math 482 Instructor: Dr. David Larson Math 482 Final Paper: Cantor Spaces in R This paper describes some basic properties of Cantor subspaces of the real line. It also includes an application of these Cantor subspaces to a characterization of the countability of closed subsets of R in terms of some simple exterior measures. Recall that a perfect set is a set for which every point is a limit point. A set S is called totally disconnected if for every x; y 2 S, there exist disjoint open sets U; V ⊂ S such that x 2 U, y 2 V , and U [ V = S. Definition 1. A Cantor space is a non-empty, totally disconnected, perfect, compact metric space. Example 1. Let C0 := [0; 1], C1 := [0; 1=3] [ [2=3; 1], and C2 = [0; 1=9] [ [2=9; 1=3] [ [2=3; 7=9] [ [8=9; 1]. Similarly, for i > 2, let Ci be the closed set given by removing the open middle third of each interval of Ci−1. The ternary Cantor set 1 \ ∆ := Ci i=0 is a Cantor space. Proof. Since 0 2 Ci for all i, ∆ is non-empty. Since each interval in Ci is of length 3−i, ∆ is totally disconnected. It is closed and bounded, so compact by the Heine-Borel theorem. To see that ∆ is perfect, first note that the endpoints of any interval in any Ci remain endpoints of intervals in Ci+1, and Ci+1 ⊂ Ci. Hence, every point that is an endpoint of an interval in some Ci is in ∆.
    [Show full text]
  • An Irrational Problem
    FUNDAMENTA MATHEMATICAE * (200*) An irrational problem by Franklin D. Tall (Toronto) Abstract. Given a top ological space hX; Ti 2 M ,anelementary submo del of set theory,we de ne X to b e X \ M with top ology generated by fU \ M : U 2T \M g. M Supp ose X is homeomorphic to the irrationals; must X = X ?Wehave partial results. M M We also answer a question of Gruenhage byshowing that if X is homeomorphic to the M \Long Cantor Set", then X = X . M In [JT], we considered the elementary submodel topology , de ned as fol- lows. Let hX; Ti b e a top ological space which is an elementofM , an elemen- tary submo del of the universe of sets. (Actually, an elementary submo del of H ( ) for a suciently large regular cardinal. See [KT] or Chapter 24 of [JW] for discussion of this technical p oint.) Let T b e the top ology on M X \ M generated by fU \ M : U 2T \M g. Then X is de ned to b e X \ M M with top ology T .In[T]we raised the question of recovering X from X M M and proved for example: Theorem 1. If X isalocal ly compact uncountable separable metriz- M able space , then X = X . M In particular: Corollary 2. If X is homeomorphic to R, then X = X . M M The pro of pro ceeded byshowing that [0; 1] M |any de nable set of @ 0 power 2 would do. It followed that ! M , whence by relativization we 1 deduced that X had no uncountable left- or right-separated subspaces.
    [Show full text]
  • Some Properties Related to the Cantor-Bendixson Derivative on A
    SOME PROPERTIES RELATED TO THE CANTOR-BENDIXSON DERIVATIVE ON A POLISH SPACE BORYS ALVAREZ-SAMANIEGO´ N´ucleo de Investigadores Cient´ıficos Facultad de Ciencias Universidad Central del Ecuador (UCE) Quito, Ecuador ANDRES´ MERINO Escuela de Ciencias F´ısicas y Matem´atica Facultad de Ciencias Exactas y Naturales Pontificia Universidad Cat´olica del Ecuador (PUCE) Quito, Ecuador Abstract. We show a necessary and sufficient condition for any ordinal number to be a Polish space. We also prove that for each countable Polish space, there exists a countable ordinal number that is an upper bound for the first component of the Cantor-Bendixson characteristic of every compact countable subset of the aforementioned space. In addition, for any uncountable Polish space, for every countable ordinal number and for all nonzero natural number, we show the exis- tence of a compact countable subset of this space such that its Cantor-Bendixson characteristic equals the previous pair of numbers. Finally, for each Polish space, we determine the cardinality of the partition, up to homeomorphisms, of the set of all compact countable subsets of the aforesaid space. arXiv:2003.01512v1 [math.GM] 28 Feb 2020 1. Introduction First, we give some notations, definitions and basic facts that will be useful throughout this paper. E-mail addresses: [email protected], borys [email protected], [email protected]. Date: February 28, 2020. 2010 Mathematics Subject Classification. 54E50; 54A25; 03E15. Key words and phrases. Polish space; Cantor-Bendixson’s derivative; cardinality. 1 2 B. ALVAREZ-SAMANIEGO´ AND A. MERINO Definition 1.1 (Metrizable space). A topological space (E, τ) is called metrizable if there exists a metric on E that generates τ.
    [Show full text]