The Non-Hausdorff Number of a Topological Space
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Implementation of Bourbaki's Elements of Mathematics in Coq
Implementation of Bourbaki’s Elements of Mathematics in Coq: Part Two, From Natural Numbers to Real Numbers JOSÉ GRIMM Inria Sophia-Antipolis, Marelle Team Email: [email protected] This paper describes a formalization of the rst book of the series Elements of Mathematics by Nicolas Bourbaki, using the Coq proof assistant. In a rst paper published in this journal, we presented the axioms and basic constructions (corresponding to a part of the rst two chapters of book I, Theory of sets). We discuss here the set of integers (third chapter of book I, theory of set), the sets Z and Q (rst chapter of book II, Algebra) and the set of real numbers (Chapter 4 of book III, General topology). We start with a comparison of the Bourbaki approach, the Coq standard library, and the Ssreect library, then present our implementation. 1. INTRODUCTION 1.1 The Number Sets This paper is the second in a series that explains our work on a project (named Gaia), whose aim is to formalize, on a computer, the fundamental notions of math- ematics. We have chosen the “Elements of Mathematics” of Bourbaki as our guide- line, the Coq proof assistant as tool, and ssreflect as proof language. As ex- plained in the first paper [Gri10], our implementation relies on the theory of sets, rather than a theory of types. In this paper we focus on the construction of some sets of numbers. Starting with the set of integers N (which is an additive monoid), one can define Z as its group of differences (this is a ring), then Q as its field of fractions (this is naturally a topological space), then R, the topological completion (that happens to be an Archimedean field); next comes the algebraic closure C of R. -
1.1.5 Hausdorff Spaces
1. Preliminaries Proof. Let xn n N be a sequence of points in X convergent to a point x X { } 2 2 and let U (f(x)) in Y . It is clear from Definition 1.1.32 and Definition 1.1.5 1 2F that f − (U) (x). Since xn n N converges to x,thereexistsN N s.t. 1 2F { } 2 2 xn f − (U) for all n N.Thenf(xn) U for all n N. Hence, f(xn) n N 2 ≥ 2 ≥ { } 2 converges to f(x). 1.1.5 Hausdor↵spaces Definition 1.1.40. A topological space X is said to be Hausdor↵ (or sepa- rated) if any two distinct points of X have neighbourhoods without common points; or equivalently if: (T2) two distinct points always lie in disjoint open sets. In literature, the Hausdor↵space are often called T2-spaces and the axiom (T2) is said to be the separation axiom. Proposition 1.1.41. In a Hausdor↵space the intersection of all closed neigh- bourhoods of a point contains the point alone. Hence, the singletons are closed. Proof. Let us fix a point x X,whereX is a Hausdor↵space. Denote 2 by C the intersection of all closed neighbourhoods of x. Suppose that there exists y C with y = x. By definition of Hausdor↵space, there exist a 2 6 neighbourhood U(x) of x and a neighbourhood V (y) of y s.t. U(x) V (y)= . \ ; Therefore, y/U(x) because otherwise any neighbourhood of y (in particular 2 V (y)) should have non-empty intersection with U(x). -
Set-Theoretic and Algebraic Properties of Certain Families of Real Functions
Set-theoretic and Algebraic Properties of Certain Families of Real Functions Krzysztof PÃlotka Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Krzysztof Chris Ciesielski, Ph.D., Chair Dening Li, Ph.D. Magdalena Niewiadomska-Bugaj, Ph.D. Jerzy Wojciechowski, Ph.D. Cun-Quan Zhang, Ph.D. Department of Mathematics Morgantown, West Virginia 2001 Keywords: Darboux-like, Hamel, Sierpi´nski-Zygmund functions; Martin’s Axiom Copyright °c 2001 Krzysztof PÃlotka ABSTRACT Set-theoretic and Algebraic Properties of Certain Families of Real Functions Krzysztof PÃlotka Given two families of real functions F1 and F2 we consider the following question: can every real function f be represented as f = f1 +f2, where f1 and f2 belong to F1 and F2, respectively? This question leads to the definition of the cardinal function Add: Add(F1; F2) is the smallest cardinality of a family F of functions for which there is no function g in F1 such that g + F is contained in F2. This work is devoted entirely to the study of the function Add for different pairs of families of real functions. We focus on the classes that are related to the additive properties and generalized continuity. Chapter 2 deals with the classes related to the generalized continuity. In particu- lar, we show that Martin’s Axiom (MA) implies Add(D,SZ) is infinite and Add(SZ,D) equals to the cardinality of the set of all real numbers. SZ and D denote the families of Sierpi´nski-Zygmund and Darboux functions, respectively. -
Six Strategies for Defeating the Runge Phenomenon in Gaussian Radial Basis Functions on a Finite Interval✩ John P
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 60 (2010) 3108–3122 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite intervalI John P. Boyd Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor MI 48109, United States article info a b s t r a c t Article history: Radial basis function (RBF) interpolation is a ``meshless'' strategy with great promise for Received 19 August 2010 adaptive approximation. Because it is meshless, there is no canonical grid to act as a starting Accepted 5 October 2010 point for function-adaptive grid modification. Uniform grids are therefore common with RBFs. Like polynomial interpolation, which is the limit of RBF interpolation when the RBF Keywords: functions are very wide (the ``flat limit''), RBF interpolants are vulnerable to the Runge Radial basis function Phenomenon, whether the grid is uniform or irregular: the N-point interpolant may diverge Runge Phenomenon as N ! 1 on the interval spanned by the interpolation points even for a function f .x/ that is analytic everywhere on and near this interval. In this work, we discuss six strategies for defeating the Runge Phenomenon, specializing to a uniform grid on a finite interval and one particular species of RBFs in one space dimension: φ.x/ D exp.−[α2=h2Ux2/ where h is the grid spacing. -
Cardinal Functions for ¿-Spaces 357
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 68, Number 3, March 1978 CARDINALFUNCTIONS FOR ¿-SPACES JAMES R. BOONE, SHELDON W. DAVIS AND GARY GRUENHAGE Abstract. In this paper, four cardinal functions are defined on the class of fc-spaces. Some of the relationships between these cardinal functions are studied. Characterizations of various ¿-spaces are presented in terms of the existence of these cardinal functions. A bound for the ordinal invariant k of Arhangelskii and Franklin is established in terms of the tightness of the space. Examples are presented which exhibit the interaction between these cardinal invariants and the ordinal invariants of Arhangelskii and Franklin. 1. Introduction. The class of ¿-spaces, introduced by Arens [1], has become a widely used class of topological spaces, both because of their intrinsic interest and because they provide a natural category of spaces in which to study a variety of other topological problems. The ¿-spaces are precisely those spaces which are the quotient spaces of locally compact spaces. In particular, X is a k-space provided: H is closed in X if and only if H n K is closed in K, for each compact set K. That is, X has the weak topology induced by the collection of compact subspaces. Franklin [5] initiated the study of a restricted class of ¿-spaces, called sequential spaces. This is the class of ¿-spaces in which the weak topology is induced by the collection of convergent sequences. If A c X, let k c\(A) be the set of all points/? such that p E c\K(A n K) for some compact set K, and let s cl(A) be the set of all points/» such that there exists a sequence in A that converges top. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C. -
A Cardinal Function on the Category of Metric Spaces 1
International Journal of Contemporary Mathematical Sciences Vol. 9, 2014, no. 15, 703 - 713 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.4442 A Cardinal Function on the Category of Metric Spaces Fatemah Ayatollah Zadeh Shirazi Faculty of Mathematics, Statistics and Computer Science, College of Science University of Tehran, Enghelab Ave., Tehran, Iran Zakieh Farabi Khanghahi Department of Mathematics, Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran Copyright c 2014 Fatemah Ayatollah Zadeh Shirazi and Zakieh Farabi Khanghahi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In the following text in the metric spaces category we introduce a topologically invariant cardinal function, Θ (clearly Θ is a tool to classify metric spaces). For this aim in metric space X we consider cone metrics (X; H; P; ρ) such that H is a Hilbert space, image of ρ has nonempty interior, and this cone metric induces the original metric topology on X. We prove that for all sufficiently large cardinal numbers α, there exists a metric space (X; d) with Θ(X; d) = α. Mathematics Subject Classification: 54E35, 46B99 Keywords: cone, cone metric order, Fp−cone metric order, Hilbert space, Hilbert-cone metric order, metric space, solid cone 1 Introduction As it has been mentioned in several texts, cone metric spaces in the following form has been introduced for the first time in [9] as a generalization of metric 704 F. Ayatollah Zadeh Shirazi and Z. -
Power Homogeneous Compacta and the Order Theory of Local Bases
POWER HOMOGENEOUS COMPACTA AND THE ORDER THEORY OF LOCAL BASES DAVID MILOVICH AND G. J. RIDDERBOS Abstract. We show that if a power homogeneous compactum X has character κ+ and density at most κ, then there is a nonempty open U ⊆ X such that every p in U is flat, “flat” meaning that p has a family F of χ(p; X)-many neighborhoods such that p is not in the interior of the intersection of any infinite subfamiliy of F. The binary notion of a point being flat or not flat is refined by a cardinal function, the local Noetherian type, which is in turn refined by the κ-wide splitting numbers, a new family of cardinal functions we introduce. We show that the flatness of p and the κ-wide splitting numbers of p are invariant λ with respect to passing from p in X to hpiα<λ in X , provided that λ < χ(p; X), or, respectively, that λ < cf κ. The above <χ(p; X)-power- invariance is not generally true for the local Noetherian type of p, as shown by a counterexample where χ(p; X) is singular. 1. Introduction Definition 1.1. A space X is homogeneous if for any p; q 2 X there is a homeomorphism h: X ! X such that h(p) = q. There are several known restrictions on the cardinalities of homogeneous compacta. First we mention a classical result, and then we very briefly survey some more recent progress. Theorem 1.2. • Arhangel 0ski˘ı’sTheorem: if X is compact, then jXj ≤ 2χ(X). -
On Approximate Cardinal Preconditioning Methods for Solving Pdes with Radial Basis Functions
On Approximate Cardinal Preconditioning Methods for Solving PDEs with Radial Basis Functions Damian Brown Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH UK. Leevan Ling Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong. Edward Kansa Department of Mechanical and Aeronautical Engineering, One Shields Ave., University of California, Davis, Davis, CA 95616 USA. Jeremy Levesley Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH UK. Abstract The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis func- tions (RBFs). In [31], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This pre- conditioning technique is sufficiently general that it can be easily applied to many differential operators. In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [31] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance. Key words: radial basis function, partial differential equation, preconditioner, cardinal basis function Preprint submitted to Engng. Anal. Bound. Elem. 17 November 2004 Hardy's multiquadric φ(r) = pr2 + c2; Inverse multiquadric φ(r) = 1=pr2 + c2; cr2 Gaussian spline φ(r) = e− ; Table 1 Examples of infinitely smooth RBFs.