Paracompact Spaces
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[Math.DG] 1 Feb 2002
The BIC of a conical fibration. Martintxo Saralegi-Aranguren∗ Robert Wolak† Universit´ed’Artois Uniwersytet Jagiellonski November 14, 2018 Abstract In the paper we introduce the notions of a singular fibration and a singular Seifert fi- bration. These notions are natural generalizations of the notion of a locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations defined by such fibra- tions we prove a de Rham type theorem for the basic intersection cohomology introduced the authors in a recent paper. One of important examples of such a structure is the natural projection onto the leaf space for the singular Riemannian foliation defined by an action of a compact Lie group on a compact smooth manifold. The failure of the Poincar´eduality for the homology and cohomology of some singular spaces led Goresky and MacPherson to introduce a new homology theory called intersection homology which took into account the properties of the singularities of the considered space (cf. [10]). This homology is defined for stratified pseudomanifolds. The initial idea was generalized in several ways. The theories of simplicial and singular homologies were developed as well as weaker notions of the perversity were proposed (cf. [11, 12]). Several versions of the Poincar´eduality were proved taking into account the notion of dual perversities. Finally, the deRham intersection cohomology was defined by Goresky and MacPherson for Thom-Mather stratified spaces. The first written reference is the paper by J.L. Brylinski, cf. [6]. This led to the search for the ”de Rham - type” theorem for the intersection cohomology. The first author in his thesis and several subsequent publications, (cf. -
Topology and Data
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, April 2009, Pages 255–308 S 0273-0979(09)01249-X Article electronically published on January 29, 2009 TOPOLOGY AND DATA GUNNAR CARLSSON 1. Introduction An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data. -
Professor Smith Math 295 Lecture Notes
Professor Smith Math 295 Lecture Notes by John Holler Fall 2010 1 October 29: Compactness, Open Covers, and Subcovers. 1.1 Reexamining Wednesday's proof There seemed to be a bit of confusion about our proof of the generalized Intermediate Value Theorem which is: Theorem 1: If f : X ! Y is continuous, and S ⊆ X is a connected subset (con- nected under the subspace topology), then f(S) is connected. We'll approach this problem by first proving a special case of it: Theorem 1: If f : X ! Y is continuous and surjective and X is connected, then Y is connected. Proof of 2: Suppose not. Then Y is not connected. This by definition means 9 non-empty open sets A; B ⊂ Y such that A S B = Y and A T B = ;. Since f is continuous, both f −1(A) and f −1(B) are open. Since f is surjective, both f −1(A) and f −1(B) are non-empty because A and B are non-empty. But the surjectivity of f also means f −1(A) S f −1(B) = X, since A S B = Y . Additionally we have f −1(A) T f −1(B) = ;. For if not, then we would have that 9x 2 f −1(A) T f −1(B) which implies f(x) 2 A T B = ; which is a contradiction. But then these two non-empty open sets f −1(A) and f −1(B) disconnect X. QED Now we want to show that Theorem 2 implies Theorem 1. This will require the help from a few lemmas, whose proofs are on Homework Set 7: Lemma 1: If f : X ! Y is continuous, and S ⊆ X is any subset of X then the restriction fjS : S ! Y; x 7! f(x) is also continuous. -
The Long Line
The Long Line Richard Koch November 24, 2005 1 Introduction Before this class began, I asked several topologists what I should cover in the first term. Everyone told me the same thing: go as far as the classification of compact surfaces. Having done my duty, I feel free to talk about more general results and give details about one of my favorite examples. We classified all compact connected 2-dimensional manifolds. You may wonder about the corresponding classification in other dimensions. It is fairly easy to prove that the only compact connected 1-dimensional manifold is the circle S1; the book sketches a proof of this and I have nothing to add. In dimension greater than or equal to four, it has been proved that a complete classification is impossible (although there are many interesting theorems about such manifolds). The idea of the proof is interesting: for each finite presentation of a group by generators and re- lations, one can construct a compact connected 4-manifold with that group as fundamental group. Logicians have proved that the word problem for finitely presented groups cannot be solved. That is, if I describe a group G by giving a finite number of generators and a finite number of relations, and I describe a second group H similarly, it is not possible to find an algorithm which will determine in all cases whether G and H are isomorphic. A complete classification of 4-manifolds, however, would give such an algorithm. As for the theory of compact connected three dimensional manifolds, this is a very ex- citing time to be alive if you are interested in that theory. -
Paracompactness and C2 -Paracompactness
Turkish Journal of Mathematics Turk J Math (2019) 43: 9 – 20 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-1804-54 C -Paracompactness and C2 -paracompactness Maha Mohammed SAEED∗,, Lutfi KALANTAN,, Hala ALZUMI, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia Received: 17.04.2018 • Accepted/Published Online: 29.08.2018 • Final Version: 18.01.2019 Abstract: A topological space X is called C -paracompact if there exist a paracompact space Y and a bijective function f : X −! Y such that the restriction fjA : A −! f(A) is a homeomorphism for each compact subspace A ⊆ X .A topological space X is called C2 -paracompact if there exist a Hausdorff paracompact space Y and a bijective function f : X −! Y such that the restriction fjA : A −! f(A) is a homeomorphism for each compact subspace A ⊆ X . We investigate these two properties and produce some examples to illustrate the relationship between them and C -normality, minimal Hausdorff, and other properties. Key words: Normal, paracompact, C -paracompact, C2 -paracompact, C -normal, epinormal, mildly normal, minimal Hausdorff, Fréchet, Urysohn 1. Introduction We introduce two new topological properties, C -paracompactness and C2 -paracompactness. They were defined by Arhangel’skiĭ. The purpose of this paper is to investigate these two properties. Throughout this paper, we denote an ordered pair by hx; yi, the set of positive integers by N, the rational numbers by Q, the irrational numbers by P, and the set of real numbers by R. T2 denotes the Hausdorff property. A T4 space is a T1 normal space and a Tychonoff space (T 1 ) is a T1 completely regular space. -
Base-Paracompact Spaces
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 128 (2003) 145–156 www.elsevier.com/locate/topol Base-paracompact spaces John E. Porter Department of Mathematics and Statistics, Faculty Hall Room 6C, Murray State University, Murray, KY 42071-3341, USA Received 14 June 2001; received in revised form 21 March 2002 Abstract A topological space is said to be totally paracompact if every open base of it has a locally finite subcover. It turns out this property is very restrictive. In fact, the irrationals are not totally paracompact. Here we give a generalization, base-paracompact, to the notion of totally paracompactness, and study which paracompact spaces satisfy this generalization. 2002 Elsevier Science B.V. All rights reserved. Keywords: Paracompact; Base-paracompact; Totally paracompact 1. Introduction A topological space X is paracompact if every open cover of X has a locally finite open refinement. A topological space is said to be totally paracompact if every open base of it has a locally finite subcover. Corson, McNinn, Michael, and Nagata announced [3] that the irrationals have a basis which does not have a locally finite subcover, and hence are not totally paracompact. Ford [6] was the first to give the definition of totally paracompact spaces. He used this property to show the equivalence of the small and large inductive dimension in metric spaces. Ford also showed paracompact, locally compact spaces are totally paracompact. Lelek [9] studied totally paracompact and totally metacompact separable metric spaces. He showed that these two properties are equivalent in separable metric spaces. -
Counterexamples in Topology
Counterexamples in Topology Lynn Arthur Steen Professor of Mathematics, Saint Olaf College and J. Arthur Seebach, Jr. Professor of Mathematics, Saint Olaf College DOVER PUBLICATIONS, INC. New York Contents Part I BASIC DEFINITIONS 1. General Introduction 3 Limit Points 5 Closures and Interiors 6 Countability Properties 7 Functions 7 Filters 9 2. Separation Axioms 11 Regular and Normal Spaces 12 Completely Hausdorff Spaces 13 Completely Regular Spaces 13 Functions, Products, and Subspaces 14 Additional Separation Properties 16 3. Compactness 18 Global Compactness Properties 18 Localized Compactness Properties 20 Countability Axioms and Separability 21 Paracompactness 22 Compactness Properties and Ts Axioms 24 Invariance Properties 26 4. Connectedness 28 Functions and Products 31 Disconnectedness 31 Biconnectedness and Continua 33 VII viii Contents 5. Metric Spaces 34 Complete Metric Spaces 36 Metrizability 37 Uniformities 37 Metric Uniformities 38 Part II COUNTEREXAMPLES 1. Finite Discrete Topology 41 2. Countable Discrete Topology 41 3. Uncountable Discrete Topology 41 4. Indiscrete Topology 42 5. Partition Topology 43 6. Odd-Even Topology 43 7. Deleted Integer Topology 43 8. Finite Particular Point Topology 44 9. Countable Particular Point Topology 44 10. Uncountable Particular Point Topology 44 11. Sierpinski Space 44 12. Closed Extension Topology 44 13. Finite Excluded Point Topology 47 14. Countable Excluded Point Topology 47 15. Uncountable Excluded Point Topology 47 16. Open Extension Topology 47 17. Either-Or Topology 48 18. Finite Complement Topology on a Countable Space 49 19. Finite Complement Topology on an Uncountable Space 49 20. Countable Complement Topology 50 21. Double Pointed Countable Complement Topology 50 22. Compact Complement Topology 51 23. -
Differential Geometry: Curvature and Holonomy Austin Christian
University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics. -
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). -
Lecture Notes on Foliation Theory
INDIAN INSTITUTE OF TECHNOLOGY BOMBAY Department of Mathematics Seminar Lectures on Foliation Theory 1 : FALL 2008 Lecture 1 Basic requirements for this Seminar Series: Familiarity with the notion of differential manifold, submersion, vector bundles. 1 Some Examples Let us begin with some examples: m d m−d (1) Write R = R × R . As we know this is one of the several cartesian product m decomposition of R . Via the second projection, this can also be thought of as a ‘trivial m−d vector bundle’ of rank d over R . This also gives the trivial example of a codim. d- n d foliation of R , as a decomposition into d-dimensional leaves R × {y} as y varies over m−d R . (2) A little more generally, we may consider any two manifolds M, N and a submersion f : M → N. Here M can be written as a disjoint union of fibres of f each one is a submanifold of dimension equal to dim M − dim N = d. We say f is a submersion of M of codimension d. The manifold structure for the fibres comes from an atlas for M via the surjective form of implicit function theorem since dfp : TpM → Tf(p)N is surjective at every point of M. We would like to consider this description also as a codim d foliation. However, this is also too simple minded one and hence we would call them simple foliations. If the fibres of the submersion are connected as well, then we call it strictly simple. (3) Kronecker Foliation of a Torus Let us now consider something non trivial. -
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 .