The Topology of Ultrafilters As Subspaces of the Cantor Set and Other Topics
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Journal of Pure and Applied Algebra Cofiniteness Conditions, Projective
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 213 (2009) 458–475 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa Cofiniteness conditions, projective covers and the logarithmic tensor product theory Yi-Zhi Huang ∗ Institut des Hautes Études Scientifiques, Le Bois-Marie, 35, Route De Chartres, F-91440 Bures-sur-Yvette, France article info a b s t r a c t Article history: We construct projective covers of irreducible V -modules in the category of grading- Received 11 March 2008 restricted generalized V -modules when V is a vertex operator algebra satisfying the Received in revised form 26 June 2008 following conditions: 1. V is C1-cofinite in the sense of Li. 2. There exists a positive integer Available online 29 August 2008 N such that the differences between the real parts of the lowest conformal weights of Communicated by C.A. Weibel irreducible V -modules are bounded by N and such that the associative algebra AN .V / is finite dimensional. This result shows that the category of grading-restricted generalized MSC: V -modules is a finite abelian category over . Using the existence of projective covers, we 17B69 C 81T40 prove that if such a vertex operator algebra V also satisfies Condition 3 that irreducible 18D10 V -modules are R-graded and C1-cofinite in the sense of the author, then the category of × grading-restricted generalized V -modules is closed under operations P.z/ for z 2 C . -
Endomorphisms of Symbolic Algebraic Varieties
J. Eur. Math. Soc. 1, 109–197 c Springer-Verlag & EMS 1999 M. Gromov Endomorphisms of symbolic algebraic varieties Received August 3, 1998 / final version received January 22, 1999 Abstract. The theorem of Ax says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and investi- gated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case of proalgebraic varieties over infinite graphs. The paper intends to bring out relations between model theory, algebraic geometry, and symbolic dynamics. 1. Ax’ surjunctivity theorem 1.A. Strict embeddings and surjunctivity. A map between sets is called a strict embedding, denoted f : X ⊂ Y, if it is one-to-one but not onto. 6= Then, following Gottschalk, (see [Gott, 1972]) a map f : X → Y is called surjunctive if it is not a strict embedding. In other words f is surjunctive iff it is either surjective or non-injective. 1.B. Theorem [Ax]1. Every regular selfmapping of a complex algebraic variety X is surjunctive. In other words no X admits a strict embedding X ⊂ X. Or, put it yet 6= another way, “one-to-one” implies “onto” for every regular map f : X → X. If X = Cn this specializes to the following earlier result by Bialynicki- Barula and Rosenlicht, (see [BB-R, 1962]). 1.B0. Every complex polynomial self-mapping of Cn is surjunctive. Repeat, this signifies that no strict polynomial embedding Cn ⊂ Cn is 6= possible, i.e. -
Pacific Journal of Mathematics Vol 230 Issue 2, Apr 2007
PACIFIC JOURNAL OF MATHEMATICS Pacific Journal of Mathematics Volume 230 No. 2 April 2007 Pacific Journal of Mathematics 2007 Vol. 230, No. 2 Volume 230 No. 2 April 2007 PACIFIC JOURNAL OF MATHEMATICS http://www.pjmath.org Founded in 1951 by E. F. Beckenbach (1906–1982) F. Wolf (1904–1989) EDITORS V. S. Varadarajan (Managing Editor) Department of Mathematics University of California Los Angeles, CA 90095-1555 pacifi[email protected] Vyjayanthi Chari Darren Long Sorin Popa Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Riverside, CA 92521-0135 Santa Barbara, CA 93106-3080 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] Robert Finn Jiang-Hua Lu Jie Qing Department of Mathematics Department of Mathematics Department of Mathematics Stanford University The University of Hong Kong University of California Stanford, CA 94305-2125 Pokfulam Rd., Hong Kong Santa Cruz, CA 95064 fi[email protected] [email protected] [email protected] Kefeng Liu Alexander Merkurjev Jonathan Rogawski Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 Los Angeles, CA 90095-1555 [email protected] [email protected] [email protected] PRODUCTION pacifi[email protected] Paulo Ney de Souza, Production Manager Silvio Levy, Senior Production Editor Alexandru Scorpan, Production Editor SUPPORTING INSTITUTIONS ACADEMIA SINICA, TAIPEI UNIVERSIDAD DE LOS ANDES UNIV. OF CALIF., SANTA CRUZ CALIFORNIA INST. OF TECHNOLOGY UNIV. -
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. -
On the Definition and K-Theory Realization of a Modular Functor
ON THE DEFINITION AND K-THEORY REALIZATION OF A MODULAR FUNCTOR IGOR KRIZ AND LUHANG LAI Abstract. We present a definition of a (super)-modular functor which includes certain interesting cases that previous definitions do not allow. We also introduce a notion of topological twisting of a modular functor, and construct formally a realization by a 2- dimensional topological field theory valued in twisted K-modules. We discuss, among other things, the N = 1-supersymmetric mini- mal models from the point of view of this formalism. 1. Introduction This paper is being written on the 10th anniversary of the publi- cation of the first author’s paper [32], and it grew out of a project of writing a sequel of [32], solving certain questions posed there, and also commenting on how the material relates to some topics of cur- rent interest. During the process of writing the paper, the authors received many comments asking for examples of their theory. Supple- mented with those examples, which required venturing into other fields of mathematics, the scope of the paper now exceeds by far what was originally intended. The original aim of [32] was to work out the modularity behavior arXiv:1310.5174v3 [math.KT] 7 Jan 2018 of certain series known in mathematical physics as partition functions of chiral conformal field theories with 1-dimensional conformal anom- aly. The author intended to use the outline of Segal [49], and work things out in more mathematical detail. However, unexpected compli- cations arose. In correspondence with P. Deligne [11] (see also [10]), the author learned that an important example of 1-dimensional confor- mal anomaly, known as the Quillen determinant of a Riemann surface, only satisfies the desired gluing axioms when considered as a super- line instead of just an ordinary complex line. -
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. -
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. -
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. -
SEQUENCES FREED from ORDER 1. on Definitions of Sequence And
SEQUENCES FREED FROM ORDER SZYMON DOLECKI AND FRED´ ERIC´ MYNARD Abstract. Traditional definition of convergence of a sequence uses the order on the set of its indices, but the only structure, needed on that set to characterize convergence, is the cofinite topology. The only as- pect of a sequence from convergence point of view is that of the filter it generates. Sequential filters, the filters generated by sequences, are pre- cisely the images of cofinite filters of countably infinite sets. One cannot totally substitute sequences by the corresponding filters, because se- quences serve to list, but often it is useful to replace them by quences, that is, maps from countably infinite sets. 1. On definitions of sequence and subsequence A sequence on a set X is usually defined as a map from the ordered set of natural numbers to X and denoted by (x ) : Of course, a sequence can N n n N 2 be, and typically is, specified as the ordered list of its terms x0; x1; x2;::: Example 1.1. Consider the following sequences 1 1 1 (1.1) 1; 2 ; 3 ;::: n ;::: 1 1 1 (1.2) 1; 2 ; 1; 2 ; 1; 2 ;::: 1 1 1 1 1 1 1 (1.3) 1; 2 ; 2 ; 3 ; 3 ; 3 ;:::; n ;:::; n ;::: | {z } n times 1 1 1 1 1 1 1 (1.4) 1; 1; 2 ; 1; 2 ; 3 ; 1; 2 ; 3 ; 4 :::; 1;:::; n ;::: | {z } n terms on the set R (of real numbers). The sequence (1.1) is one-to-one, while (1.3) is finite-to-one. -
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. -
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. -
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.